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Forschungsseminar
Name  Datum  Uhrzeit  Ort  Titel 

Anton Dochtermann (Texas State University  20.07.22  16:30  MA 141  Root polytopes, tropical types, and toric edge ideals 
Abstract: We consider arrangements of tropical hyperplanes, where the apices of the hyperplanes can be `taken to infinity' in certain directions. Such an arrangement defines a decomposition of Euclidean space where a cell is defined by its `type' data, analogous to the covectors of an oriented matroid. By work of DevelinSturmfels and FinkRincon it is known that these `tropical complexes' are dual to (regular) subdivisions of root polytopes, which in turn are in bijection with mixed subdivisions of certain generalized permutohedra. Extending previous work with JoswigSanyal, we show how a natural monomial labeling of these complexes describes polynomial relations (syzygies) among `type ideals' which arise naturally from the combinatorial data of the arrangement. In particular, we show that the cotype ideal is Alexander dual to a squarefree initial ideal of the lattice ideal of a root polytope corresponding to the StanleyReisner ideal of a regular triangulation of the root polytope. This in turn leads to novel ways of studying algebraic properties of various monomial and lattice ideals. For instance, our methods of studying the dimension of a tropical complex provide new bounds on homological invariants of toric edge ideals of bipartite graphs, which have been extensively studied in the commutative algebra community. This is joint work with Ayah Almousa and Ben Smith. 

Büsra Sert  13.07.22  16:30  MA 141  The HalfPlane Property of Matroids, Related Concepts, and an Algorithm 
Abstract: Studies on homogeneous polynomials with the halfplane property were initially
motivated by the physical theory of electrical networks. They later became of interest
in convex optimization. In particular, a homogeneous polynomial with the halfplane
property is hyperbolic with respect to every point in the positive orthant, having an
associated hyperbolicity cone. Feasible sets of semidefinite programming, i.e.,
spectrahedral cones, are hyperbolicity cones for some polynomials. For the converse, the
generalized Lax conjecture states that every hyperbolicity cone is spectrahedral. 

Matthias Beck (SFSU)  22.06.22  16:30  MA 141  \(f^*\) and \(h^*\)vectors 
Abstract: If \(P\) is a lattice polytope (i.e., \(P\) is the convex hull of finitely many integer points in \({\bf R}^d\)), Ehrhart's theorem asserts that the integerpoint counting function \(L_P(m) = \#(mP \cap {\bf Z}^d)\) is a polynomial in the integer variable $m$. Our goal is to study structural properties of Ehrhart polynomialsessentiallly asking variants of the (way too hard) question which polynomials are Ehrhart polynomials? Similar to the situations with other combinatorial polynomials, it is useful to express \(L_P(m)\) in different bases. E.g., a theorem of Stanley (1976) says that \(L_P(m)\), expressed in the polynomial basis \(\binom m d, \binom{m+1} d, \dots, \binom{m+d} d\), has nonnegative coefficients; these coefficiencts form the \(h^*\)vector of \(P\). More recent work of Breuer (2012) suggests that one ought to also study \(L_P(m)\) as expressed in the polynomial basis \(\binom {m1} 0, \binom {m1} 1, \binom {m1} 2, \dots\); the coefficiencts in this basis form the \(f^*\)vector of \(P\). We will survey some old and new results (the latter joint work with Danai Deligeorgaki, Max Hlavaczek, and Jerónimo Valencia) about \(f^*\) and \(h^*\)vectors, including analogues and dissimiliarieties with \(f\) and \(h\)vectors of polytopes and polyhedral complexes. 

Julian Pfeifle (UPC)  15.06.22  16:30  Online  Coming soon to polymake: (sometimes) disproving convex realizability of spheres 
Abstract: Every so often, we find a family of simplicial spheres with interesting
combinatorial properties, and would like to know if they are realizable as
boundaries of convex polytopes. For instance, this happened recently to
Zheng; to Novik and Zheng; and to Criado and Santos. 

CG Week  08.06.22  FU Berlin  CG Week  
Abstract: 

Enis Kaya (MPI MiS Leipzig)  01.06.22  16:30  MA 141  Determining reduction types of Picard curves via tropical invariants 
Abstract: For a separable binary form of degree n over a complete nonarchimedean field, there is a canonical metric tree on n leaves associated to it. Furthermore, for every degree n, there are only finitely many tree types, which gives rise to a partition of the space of all nonarchimedean binary forms of degree n. 

Zoe Wellner (Carnegie Mellon University)  25.05.22  16:30  MA 141  Colorful BorsukUlam Results, Their Generalizations and Applications 
Abstract: In combinatorics and discrete geometry there are many colorful and rainbow results. Additionally topological tools have proven to be useful in generating interesting combinatorial results. In this line of approach we prove a colorful generalization of the Borsuk–Ulam theorem. We give a short proof of Ky Fan’s Lemma and generalize it from involutions to larger symmetry groups Z/p for prime p. We also present colorful generalizations of these nonexistence results for equivariant maps. As consequences we derive a colorful generalization of the Ham–Sandwich theorem and the necklace splitting theorem among other applications of BorsukUlam. This is joint work with Florian Frick. 

Niels Lindner (ZIB)  18.05.22  16:30  MA 141  On the tropical and zonotopal geometry of periodic timetabling 
Abstract: The timetable is the core of every public transportation system. The standard mathematical tool for optimizing periodic timetabling problems is the Periodic Event Scheduling Problem (PESP). A solution to PESP consists of three parts: a periodic timetable, a periodic tension, and integer periodic offset values. While the space of periodic tension has received much attention in the past, we explore geometric properties of the other two components, establishing novel connections between periodic timetabling and discrete geometry. Firstly, we study the space of feasible periodic timetables, and decompose it into polytropes, i.e., polytopes that are convex both classically and in the sense of tropical geometry. We then study this decomposition and use it to outline a new heuristic for PESP, based on the tropical neighbourhood of the polytropes. Secondly, we recognize that the space of fractional cycle offsets is in fact a zonotope. We relate its zonotopal tilings back to the hyperrectangle of fractional periodic tensions and to the tropical neighbourhood of the periodic timetable space. Finally, we also use this new understanding to give tight lower bounds on the minimum width of an integral cycle basis in terms of the number of spanning trees. 

Alex Elzenaar (MPI MiS Leipzig)  04.05.22  16:30  MA 141  Aspects of computation in hyperbolic geometry 
Abstract: Lowdimensional topology has the advantage that it is relatively easy to visualisethe key word being relatively. Some people have amazing powers of visualisation (famously, William Thurston); but most of us need to rely on visual aids to try to understand what is going on. There is a long history of computer experimentation in the area of hyperbolic geometry and Kleinian groups (groups which arise dually to hyperbolic 3manifolds), from the Thurston days (Robert Riley's work on knot groups in the 1970s and Jeff Weeks' study of triangulations of 3manifolds in the 1980s) to the modern era (including but not limited to work by David Mumford, Caroline Series, and David Wright; Masaaki Wada; and Sabetta Matsumoto and Henry Segerman). This talk will discuss some of the highlights of computation in hyperbolic geometry with an emphasis on visualisation, focusing on groups and manifolds related to twobridge knots (parameterised by the socalled Riley slice). 

Thomas Blomme (University of Geneva)  27.04.22  16:30  Online  Enumeration of tropical curves in abelian surfaces 
Abstract: Tropical geometry is a powerful tool that allows one to compute enumerative algebraic invariants through the use of some correspondence theorem, transforming an algebraic problem into a combinatorial problem. Moreover, the tropical approach also allows one to twist definitions to introduce mysterious refined invariants, obtained by counting tropical curves with polynomial multiplicities. So far, this correspondence has mainly been implemented in toric varieties. In this talk we will study enumeration of curves in abelian surfaces and use the tropical geometry approach to prove a multiple cover formula that enables an simple and elegant computation of enumerative invariants of abelian surfaces. 

Michael Joswig (TU Berlin)  20.04.22  16:30  Online  Lattice polygons and real roots 
Abstract: It is known from theorems of Bernstein, Kushnirenko and Khovanskii from the 1970s that the number of complex solutions of a system of multivariate polynomial equations can be expressed in terms of subdivisions of the Newton polytopes of the polynomials. For very special systems of polynomials Soprunova and Sottile (2006) found an analogue for the number of real solutions. In joint work with Ziegler we could give a simple combinatorial formula and an elementary proof for the signature of foldable triangulation of a lattice polygon. Via the SoprunovaSottile result this translates into lower bounds for the number of real roots of certain bivariate polynomial systems. 

PhD Students of DM/G (TU Berlin)  13.04.22  16:30  Online  Scientific Writing 
Abstract: 

Marie Brandenburg (MPI MiS Leipzig)  16.03.22  16:30  Online  Competitive Equilibrium and Lattice Polytopes 
Abstract: The question of existence of a competitive equilibrium is a game theoretic question in economics. It can be posed as follows: In a given auction, can we make an offer to all bidders, such that no bidder has an incentive to decline our offer? We consider a mathematical model of this question, in which an auction is modelled as weights on a simple graph. The existence of an equilibrium can then be translated to a condition on certain lattice points in a lattice polytope. In this talk, we discuss this translation to the polyhedral language. Using polyhedral methods, we show that in the case of the complete graph a competitive equilibrium is indeed guaranteed to exist. This is based on joint work with Christian Haase and Ngoc Mai Tran. 

Lorenzo Venturello (KTH)  09.03.22  16:30  Online  On the gammavector of symmetric edge polytopes 
Abstract: Symmetric edge polytopes (SEPs) are special lattice polytopes constructed from a simple graph. They play a role in physics, in the study of metric spaces and optimal transport. We study a numerical invariant of symmetric edge polytopes, called the gammavector, whose entries are linear combinations of the \(h^*\) numbers. In particular, we target a conjecture of Ohsugi and Tsuchiya which predicts nonnegativity of the gammavector of every SEP. In a joint work with Alessio D'Alì, Martina JuhnkeKubitzke and Daniel Köhne, we prove nonnegativity for one of the entries of the gammavector, and descriube the graphs whose SEP attains equality. Moreover, we consider random graphs in the Erdős–Rényi model, and we obtain asymptotic nonnegativity of the whole gammavector in certain regimes. 

Lei Xue (University of Washington)  16.02.22  16:30  Online  A Proof of Grünbaum’s Lower Bound Conjecture for polytopes, lattices, and strongly regular pseudomanifolds 
Abstract: In 1967, Grünbaum conjectured that any \(d\)dimensional polytope with \(d +s \leq 2d\) vertices has at least \( \phi_k (d + s, d) = {d +1 \choose k + 1} + { d \choose k + 1}  { d + 1  s \choose k + 1} \) \(k\)faces. In the talk, we will prove this conjecture and discuss equality cases. We will then extend our results to lattices with diamond property (the inequality part) and to strongly regular normal pseudomanifolds (the equality part). We will also talk about recent results on \(d\)dimensional polytopes with \(2d + 1\) or \(2d + 2\) vertices. 

Mara Belotti (TU Berlin)  09.02.22  16:30  Online  An enumerative problem for cubic (hyper)surfaces: point and line conditions 
Abstract: The moduli space of smooth cubic hypersurfaces in \(\mathbb{P}^n\) is an open subset of a projective space. We construct a compactification of the latter which allows us to count the number of smooth cubic hypersurfaces tangent to a prescribed number of lines and passing through a given number of points. We term it a \(1\)complete variety of cubic hypersurfaces in analogy to the space of complete quadrics. Paolo Aluffi explored the case of plane cubic curves. Starting from his work, we construct such a space in arbitrary dimensions by a sequence of five blowups. The counting problem is then reduced to the computation of five total Chern classes. Finally, we arrive at the desired numbers in the case of cubic surfaces. This is joint work with Alessandro Danelon, Claudia Fevola, Andreas Kretschmer. 

Sara Griffith (Brown University)  02.02.22  16:30  Online  The Torelli theorem, or how to determine a graph from its algebraic data 
Abstract: The Torelli theorem for graphs, due to CaporasoViviani and SuWagner, shows that with mild hypotheses a graph's matroid can be obtained from the integral lattice in its vector space of flows. New results of the speaker show that the graph itself can be recovered using additional data related to winnable chip firing games on the graph. 

Paco Santos (Universidad de Cantabria)  26.01.22  16:30  Online  Multitriangulations and tropical Pfaffians 
Abstract: Let \(V=\binom{[n]}{2}\) label the possible diagonals among the vertices of a convex \(n\)gon. A subset of size \(k+1\) is called a \((k+1)\)crossing if all elements mutually cross, and a general subset is called \((k+1)\)crossing free if it does not contain a \(k\)crossing. \((k+1)\)crossing free subsets form a simplicial complex that we call the \(k\)associahedron and denote \(Ass_k{n}\) since for \(k=1\) one (essentially) recovers the simplicial associahedron. The \(k\)associahedron on the \(n\)gon is known to be (essentially ) a shellable sphere of dimension \(k(n2k1)\) and conjectured to be polytopal (Jonsson 2003). It is also a subword complex in the root system of the A. 

Jonathan Boretsky (Harvard)  19.01.22  16:30  Online  The Totally NonNegative Complete Flag Variety and its Tropicalization 
Abstract: A complete flag in \(\mathbb{R}^n\) is a collection of linear subspaces \(\{0\}=V_0\subsetneq V_1\cdots \subsetneq V_n=\mathbb{R}^n\). The set of all complete flags in \(\mathbb{R}^n\) form a variety called the complete flag variety. We give a combinatorial parameterization of the totally nonnegative part of this variety. We also give a simple coordinatewise characterization of which flags lie in the totally nonnegative flag variety. We will also define the notion of a tropical variety and the nonnegative part of a tropical variety. There are two tropical varieties that can be described using tropical versions of the equations cutting out the complete flag variety. One, the flag Dressian, can be thought of as describing flags of tropical linear spaces and the second, the tropical flag variety, can be thought of as describing realizable flags of tropical linear space. While these tropical varieties are generally different, we will use our parameterization of the nonnegative complete flag variety to show that their nonnegative parts coincide. 

Dante Luber (TU Berlin)  12.01.21  16:30  Online  Generalized permutahedra and positive flag Dressians 
Abstract: A well known result relates matroid subdivisions of hypersimplices to tropical linear spaces. We continue the study of subdivisions of generalized permutahedra into cells that are generalized permutahedra. This talk will focus on subdivisions of the regular permutahedron into cells that correspond to flag matroids and intervals in a partial ordering on the symmetric group. This is joint work with Michael Joswig, Georg Loho, and Jorge Olarte 

Katharina Jochemko (KTH)  15.12.21  16:30  Online  The Eulerian transformation 
Abstract: Many polynomials arising in combinatorics are known or conjectured to have only real roots. One approach to these questions is to study transformations that preserve the realrootedness property. This talk is centered around the Eulerian transformation which is the linear transformation that sends the ith standard monomial to the ith Eulerian polynomial. Eulerian polynomials appear in various guises in enumerative and geometric combinatorics and have many favorable properties, in particular, they are realrooted and symmetric. We discuss how these properties carry over to the Eulerian transformation. In particular, we disprove a conjecture by Brenti (1989) concerning the preservation of real roots, extend recent results on binomial Eulerian polynomials and provide enumerative and geometric interpretations. This is joint work with Petter Brändén. 

Matthias Köppe (UC Davis)  08.12.21  16:30  Online  sagemathpolyhedra: A modularized Sage distribution 
Abstract: SageMath is an opensource generalpurpose mathematical system based
on Python that integrates computer algebra facilities and general
computational packages. Sage, initially developed by Stein and first
released in 2005, in over 1.5 decades of incubation in its
pseudodistribution comprising over 300 software packages, has grown
to provide over 500 Cython and over 2000 of its own Python modules,
ranging from sage.algebras.* over sage.geometry.* to sage.tensor.*,
with a total of over 2.2 million lines of code. 

Benjamin Schröter (KTH)  01.12.21  16:30  Online  Ehrhart Polynomials of RankTwo Matroids 
Abstract: There are many questions that are equivalent to the enumeration of lattice
points in convex sets. Ehrhart theory is the systematic study of lattice point counting
in dilations of polytopes. Over a decade ago De Loera, Haws and Köppe conjectured that
the lattice point enumerator of dilations of matroid basis polytopes is a polynomial with
positive coefficients. This intensively studied conjecture has recently been disproved in
all ranks greater than or equal to three. However, the questions of what characterizes
these polynomials remain wide open. 

Alexander Stoimenow (GIST College)  24.11.21  16:00  Online  Exchange moves and nonconjugate braid representatives of links 
Abstract: The braid groups $B_n$ were introduced in the 1930s
in the work of Artin. An element in the braid group $B_n$ is called an
$n$braid. Alexander related braids to knots and
links in 3dimensional space, by means of a closure
operation. In that realm, it became important to
understand the braid representatives of a given link.
Markov's theorem relates these
representatives by two moves, conjugacy in the braid group,
and (de)stabilization, which passes between braid groups. Markov's
moves and braid group algebra have become fundamental in Jones'
pioneering work and its later continuation towards
quantum invariants. Conjugacy
is, starting with Garside's, and later many others' work, now
relatively well grouptheoretically understood. In contrast, the effect
of (de)stabilization on conjugacy classes of braid representatives of
a given link is in general difficult to control. Only in very special
situations can these conjugacy classes be well described. 

Sinai Robins (Universidade de São Paulo)  17.11.21  16:30  Online  The null set of a polytope, and the Pompeiu property for polytopes 
Abstract: We study the null set $N(\P)$ of the Fourier transform of a polytope P in $\R^d$, and we find that this null set does not contain (almost all) circles in $\R^d$. As a consequence, the null set does not contain the algebraic varieties $\{ z \in \C^d \mid z_1^2 + \cdots + z_d^2 = \alpha \}$, for each fixed $\alpha \in \C$. In 1929, Pompeiu asked the following question. Suppose we have a convex subset P in R^d, and a function f, defined over R^d, such that the integral of f over P vanishes, and all of the integrals of f, taken over each rigid motion of P, also vanish. Does it necessarily follow that f = 0? If the answer is affirmative, then the convex body P is said to have the Pompeiu property. It is a conjecture that in every dimension, balls are the only convex bodies that do not have the Pompeiu property. Here we get an explicit proof that the Pompeiu property is true for all polytopes, by combining our work with the work of Brown, Schreiber, and Taylor from 1973. Our proof uses the BrionBarvinok theorem in combinatorial geometry, together with some properties of the Bessel functions. The original proof that polytopes (as well as other bodies) possess the Pompeiu property was given by Brown, Schreiber, and Taylor (1973) for dimension 2. In 1976, Williams observed that the same proof also works for $d>2$ and, using eigenvalues of the Laplacian, gave another proof valid for $d \geq 2$ that polytopes indeed have the Pompeiu property. The null set of the Fourier transform of a polytope has also been used in a different direction, by various researchers, to tackle problems in multitiling Euclidean space. Thus, the null set of a polytope is interesting for several applications, including discrete versions of this problem that we will mention, which are generally unsolved. This is joint work with Fabrício Caluza Machado. 

Austin Alderete (UT Austin)  10.11.21  16:30  Online  A polymake extension for computing the homology groups of minorclosed classes of matroids 
Abstract: We present a polymake extension for performing computations in the intersection ring of matroids. Addition in the ring is carried out on the indicator vectors of maximal chains of flats while the product is the usual matroid intersection when the intersection is loopfree and zero otherwise. The primary feature of this extension is the ability to compute the homology groups of the intersection ring induced by deletion and contraction as well as those of any subring generated by a minorclosed class of matroids. 

MaRDI  03.11.21  16:30  MPI Leipzig  MaRDI Kickoff Workshop 
Abstract: 

Davide Lofano (TU Berlin)  27.10.21  16:30  H 0107  Hadamard Matrix Torsion 
Abstract: After a brief overview about torsion in homology and examples of complexes with small torsion and few vertices we will focus our attention on a particular series of examples. In particular, we will construct a series HMT(n) of 2dimensional simplicial complexes with huge torsion, H_1(HMT(n))=det(H(n))=n^(n/2), where the construction is based on the Hadamard matrices H(n) and they have linearly many vertices in n. Our explicit series improves a previous construction by Speyer, narrowing the gap to the highest possible asymptotic torsion growth achieved by Newman via a randomized construction. 

Sylvain Spitz (TU Berlin)  20.10.21  16:30  H 0107  Generalized Permutahedra and Optimal Auctions 
Abstract: We study a family of convex polytopes, called SIMbodies, which were introduced by Giannakopoulos and Koutsoupias (2018) to analyze socalled StraightJacket Auctions. First, we show that the SIMbodies belong to the class of generalized permutahedra. Second, we prove an optimality result for the StraightJacket Auctions among certain deterministic auctions. Third, we employ computer algebra methods and mathematical software to explicitly determine optimal prices and revenues. 

Petter Brändén (KTH)  13.10.21  16:30  Online  Lorentzian polynomials on cones and the HeronRotaWelsh conjecture 
Abstract: We extend the theory of Lorentzian polynomials to convex cones. This is used to give a short proof of the logconcavity of the coefficients of the reduced characteristic polynomial of a matroid, and reprove the HodgeRiemann relations of degree one for the Chow ring of a matroid. This is joint work with Jonathan Leake. 

Georg Loho (University of Twente)  25.08.21  16:30  Online  Generalized permutahedra and complete classes of valuated matroids 
Abstract: Generalized permutahedra form an important class of polytopes occurring explicitly or implicitly in several branches of mathematics and beyond. I start with an overview of concepts related with generalized permutahedra from optimization and Discrete Convex Analysis. Valuated generalized matroids give rise to a particularly interesting subclass. I show some fundamental constructions for these objects. This leads to an answer to two open questions from auction theory and discrete convex analysis. 

Chiara Meroni (MPI MiS)  04.08.21  16:30  Online  Fiber convex bodies 
Abstract: Given a convex polytope P and a projection map it is possible to construct the associated fiber polytope, that intuitively is the average of the fibers of P under this projection and has interesting combinatorial properties. In this joint work with Léo Mathis we study this construction for all convex bodies. The aim is to relate aspects of the original convex body with those of its fiber body. I will discuss theoretical results as well as some concrete examples. 

Tobias Boege (Universität Magdeburg)  28.07.21  16:30  Online  The Gaussian conditional independence inference problem 
Abstract: Conditional independence is a ternary relation on subsets of a finite vector of random variables \xi. The CI statement \xi_i \CI \xi_j \mid \xi_K
asserts that ``whenever the outcome of all the variables \xi_k, k \in K,
is known, learning the outcome of \xi_i provides no further information
on \xi_j''. 

Claudia Yun (Brown University)  14.07.21  16:30  Online  The S_nequivariant rational homology of the tropical moduli spaces \Delta_{2,n} 
Abstract: Fix nonnegative integers g and n such that 2g2+n>0, the tropical moduli space \Delta_{g,n} is a topological space that parametrizes isomorphism classes of nmarked stable tropical curves of genus g with total volume 1. These spaces are important because their reduced rational homology is identified equivariantly with the top weight cohomology of the algebraic moduli spaces M_{g,n}, with respect to the S_naction of permuting marked points. In this talk, we focus on the genus 2 case and compute the characters of \tilde{H}_i(\Delta_{2,n};Q) as S_nrepresentations for n up to 8. To carry out the computations, we use a cellular chain complex for symmetric \Deltacomplexes, a technique developed by Chan, Galatius, and Payne. We will also briefly mention work in progress that extends the computations to n=10, in collaboration with Christin Bibby, Melody Chan, and Nir Gadish. All computations are done in SageMath. 

Ilia Itenberg (Sorbonne Université)  07.07.21  16:30  Online  Planes in cubic fourfolds 
Abstract: We discuss possible numbers of 2planes in a smooth cubic hypersurface in the 5dimensional projective space. We show that, in the complex case, the maximal number of planes is 405, the maximum being realized by the Fermat cubic. In the real case, the maximal number of planes is 357. The proofs deal with the period spaces of cubic hypersurfaces in the 5dimensional complex projective space and are based on the global Torelli theorem and the surjectivity of the period map for these hypersurfaces, as well as on Nikulin's theory of discriminant forms. Joint work with Alex Degtyarev and John Christian Ottem. 

Holger Eble (TU Berlin)  30.06.21  16:30  Online  Enumerating Chambers of Hyperplane Arrangements with Symmetry 
Abstract: We introduce a new algorithm for enumerating chambers of hyperplane arrangements which exploits their underlying symmetry groups. Our algorithm counts the chambers of an arrangement as a byproduct of computing its characteristic polynomial. We showcase our julia implementation, based on OSCAR, on examples coming from hyperplane arrangements with applications to physics and computer science. This is joint work with Taylor Brysiewicz and Lukas Kuehne. 

Manuel Radons (TU Berlin)  23.06.21  16:30  Online  Edgeunfolding nested prismatoids 
Abstract: A 3Prismatoid is the convex hull of two convex polygons A and B which lie in parallel planes H_A, H_B in R^3. Let A' be the orthogonal projection of A onto H_B. A prismatoid is called nested if A' is properly contained in B, or vice versa. We show that every nested prismatoid has an edgeunfolding to a nonoverlapping polygon in the plane. 

Christoph Hertrich (TU Berlin)  16.06.21  16:30  Online  Towards Lower Bounds on the Depth of ReLU Neural Networks 
Abstract: We contribute to a better understanding of the class of functions that is represented by a neural network with ReLU activations and a given architecture. Using techniques from mixedinteger optimization, polyhedral theory, and tropical geometry, we provide a mathematical counterbalance to the universal approximation theorems which suggest that a single hidden layer is sufficient for learning tasks. In particular, we investigate whether the class of exactly representable functions strictly increases by adding more layers (with no restrictions on size). We also present upper bounds on the sizes of neural networks required for exact function representation. 

Matteo Gallet (RICAM)  09.06.21  16:30  Online  Mobile Icosapods 
Abstract: Pods are mechanical devices constituted of two rigid bodies, the base and the platform, connected via spherical joints by a number of other rigid bodies, called legs. The maximal number, when finite, of legs of a mobile pod is 20. In 1904, Borel designed a technique to construct examples of such 20pods over the complex numbers. We show that recent results about spectrahedra provide a way to determine real solutions for Borel’s construction. This is a joint work with Georg Nawratil, Jon Selig, and Josef Schicho. 

Marieke van der Wegen (Utrecht University)  02.06.21  16:30  Online  Computing stable gonality is hard 
Abstract: Based on analogies between algebraic curves and graphs, there are several graph parameters defined. In this talk we will study the socalled stable gonality. The stable gonality is a measure for the complexity of a graph and is defined using morphisms from a graph to a tree. We show that computing the stable gonality of a graph is NPhard. This is joint work with Dion Gijswijt and Harry Smit. 

Guido Montúfar (MPI Leipzig and UCLA)  26.05.21  16:30  Online  Sharp bounds for the number of regions of maxout networks and vertices of Minkowski sums 
Abstract: We present results on the number of linear regions of the functions that can be represented by artificial feedforward neural networks with maxout units. A rankk maxout unit is a function computing the maximum of k linear functions. For networks with a single layer of maxout units, the linear regions correspond to the upper vertices of a Minkowski sum of polytopes. We obtain face counting formulas in terms of the intersection posets of tropical hypersurfaces or the number of upper faces of partial Minkowski sums, along with explicit sharp upper bounds for the number of regions for any input dimension, any number of units, and any ranks, in the cases with and without biases. Based on these results we also obtain asymptotically sharp upper bounds for networks with multiple layers. This is joint work with Yue Ren and Leon Zhang. 

Simon Telen (MPI MiS Leipzig)  19.05.21  16:30  Online  Likelihood equations and scattering amplitudes 
Abstract: We identify the scattering equations from particle physics as the likelihood equations for a particular statistical model. The scattering potential plays the role of the loglikelihood function. We employ recent methods from numerical nonlinear algebra to solve challenging instances of the scattering equations. We revisit the theory of stringy canonical forms proposed by ArkaniHamed, He and Lam, introducing positive statistical models and their amplitudes. This is joint work with Bernd Sturmfels. 

Sophie Rehberg (FU Berlin)  12.05.21  16:30  Online  Combinatorial reciprocity theorems for generalized permutahedra, hypergraphs, and pruned insideout polytopes 
Abstract: Generalized permutahedra are a class of polytopes with many interesting combinatorial subclasses. We introduce pruned insideout polytopes, a generalization of insideout polytopes introduced by Beck–Zaslavsky (2006), which have many applications such as recovering the famous reciprocity result for graph colorings by Stanley. We study the integer point count of pruned insideout polytopes by applying classical Ehrhart polynomials and Ehrhart–Macdonald reciprocity. This yields a geometric perspective on and a generalization of a combinatorial reciprocity theorem for generalized permutahedra by Aguiar–Ardila (2017) and Billera–Jia–Reiner (2009). Applying this reciprocity theorem to hypergraphic polytopes allows us to give an arguably simpler proof of a recent combinatorial reciprocity theorem for hypergraph colorings by Aval–Karaboghossian–Tanasa (2020). Our proof relies, aside from the reciprocity for generalized permutahedra, only on elementary geometric and combinatorial properties of hypergraphs and their associated polytopes. 

Sandra Di Rocco (KTH)  05.05.21  16:30  Online  Iterated discriminants 
Abstract: The Hyperdeterminant is a celebrated tool in tensor theory, geometrically it represents the discriminant of a Segre embedding. The Segre embedding is a toric map associated to a Cayley sum, probably the basic example of Cayley polytopes. Unlike other discriminants the hyperdeterminant is at times accessible to computation due to a decomposition method developed by Schäfli. We propose a generalisation of the Schäfli method to discriminants associated to general Cayley sums and show how this can be treated as a discriminant of a system of polynomials. This is joint work with A. Dickenstein and R. Morrison. 

Daniel Corey (University of WisconsinMadison)  28.04.21  16:30  Online  Initial degenerations of Grassmannians and spinor varieties 
Abstract: We construct closed immersions from initial degenerations of Gr_0(d,n)the open cell in the Grassmannian Gr(d,n) given by the nonvanishing of all Plücker coordinatesto limits of thin Schubert cells associated to diagrams induced by the face poset of the corresponding tropical linear space. These are isomorphisms in many cases, including (d,n) equal to (2,n), (3,6) and (3,7). As an application, Gr_0(3,7) is schön, and the Chow quotient of Gr(3,7) by the maximal torus in PGL(7) is the log canonical compactification of the moduli space of 7 points in P^2 in linear general position, making progress on a conjecture of Hacking, Keel, and Tevelev. Finally, I will discuss recent work on extending these results to the Lietype D setting. 

Mario Kummer (TU Dresden)  21.04.21  16:30  Online  Iso Edge Domains 
Abstract: A Conjecture by Conway and Sloane from the 90s can be phrased as ''Every tropical abelian variety is determined by its tropical theta constants''. It was proved by Conway and Sloane in dimension up to 3 and by Vallentin in dimension 4. We prove it in dimension 5. The prove relies on the classification of isoedge domains in dimension 5: These are a variant of the isoDelaunay decomposition introduced by Baranovskii and Ryshkov. We further characterize the socalled matroidal locus in terms of the tropical theta constants in dimension up to 5 and point out connections to the tropical Schottky problem. The talk is based on a joint work with Mathieu Dutour Sikirić. 

Antony Della Vecchia  14.04.21  16:30  Online  One Relator Groups and Regular Sectional Curvature 
Abstract: We present the notion of regular sectional curvature for angled 2complexes. It is known that fundamental groups of angled 2complexes with negative or nonpositive regular section curvature are coherent and, in certain cases, locally quasiconvex. We state some open problems for onerelator groups, describe a construction of angled 2complexes for onerelator groups and provide some computer generated results supporting the claims. 

Nicholas Williams (University of Leicester)  10.03.21  16:30  Online  New interpretations of the higher Stasheff–Tamari orders 
Abstract: The higher Stasheff–Tamari orders are two partial orders introduced in the 1990s by Kapranov, Voevodsky, Edelman and Reiner on the set of triangulations of a cyclic polytope. Edelman and Reiner conjectured these two orders to coincide, which remains an open problem today. In this talk we give new combinatorial interpretations of the orders. These interpretations differ depending on whether the cyclic polytope is 2ddimensional or (2d + 1)dimensional, but are expressed in each case in terms of the dskeleton of the triangulation. This naturally leads us to also characterise the dskeletons of triangulations of (2d + 1)dimensional cyclic polytopes, complementing the description of the dskeleton of a 2ddimensional triangulation given by Oppermann and Thomas. Our results make the conjectured equivalence of the orders a more tractable problem, and we use them to run computational experiments checking the conjecture, to which we find no counterexamples. Our results also have interpretations in the representation theory of algebras, but we will not discuss these connections. 

Kaie Kubjas (Aalto University)  03.03.21  16:30  Online  Identifying 3D Genome Organization in Diploid Organisms via Euclidean Distance Geometry 
Abstract: The 3D organization of the genome plays an important role for gene regulation. Chromosome conformation capture techniques allow one to measure the number of contacts between genomic loci that are nearby in the 3D space. In this talk, we study the problem of reconstructing the 3D organization of the genome from whole genome contact frequencies in diploid organisms, i.e. organisms that contain two indistinguishable copies of each genomic locus. In particular, we study the identifiability of the 3D organization of the genome and optimization methods for reconstructing it. This talk is based on joint work with Anastasiya Belyaeva, Lawrence Sun and Caroline Uhler. 

Malte Renken (TU Berlin)  24.02.21  16:30  Online  What does the Cyclic Polytope have to do with Visibility Graphs? 
Abstract: In the 90's the study of visibility graphs prompted the definition of
"persistent graphs", a class of vertexordered graphs defined by three
simple combinatorial properties.
We reveal a surprising connection of these persistent graphs to the
triangulations
of the 3dimensional cyclic polytope via a natural bijection. 

Jorge Olarte (TU Berlin)  10.02.2021  17:00  Online  The tropical symplectic Grassmannian 
Abstract: The symplectic Grassmannian SpGr(k,2n) is the space of a all linear subspaces of dimension k of a vector space of dimension 2n which are isotropic with respect to a symplectic form. We look at several equivalent characterizations of isotropic linear subspaces and formulate a tropical analog for each, such as, for example, being in the tropicalization of the symplectic Grassmannian. It turns out that in the tropical world these characterizations are no longer equivalent and we will see exactly for which k and n does one characterization imply another. This is joint work with George Balla. 

Oguzhan Yürük (TU Berlin)  10.02.2021  16:30  Online  Detecting the Regions of Multistaionarity in CRNT via Symbolic Nonnegativity Certificates 
Abstract: Parameterized ordinary differential equation systems are crucial for modeling in biochemical reaction networks under the assumption of massaction kinetics. Various questions concerning the signs of multivariate polynomials in positive orthant arise from studying the solutions' qualitative behavior with respect to parameter values, in particular the existence of multistationarity. In this work, we revisit a method of detecting multistationarity, in which we utilize the circuit polynomials to find symbolic certificates of nonnegativity for 2site phosphorylation cycle in a recent work of the speaker joint with Elisenda Feliu, Nidhi Kaihnsa and Timo de Wolff. Moreover, we will give a description of all possible certificates that can arise from 2site phosphorylation cycle, and provide further insight into the number of positive steady states of the nsite phosphorylation cycle model. 

Siddarth Kannan (Brown University)  03.02.2021  16:30  Online  Symmetries of tropical moduli spaces of curves 
Abstract: I will briefly introduce the moduli space of nmarked stable tropical curves of genus g, and then discuss the combinatorial calculation of its automorphism group. I will conclude by talking about natural followup questions to this result, motivated by connections to the DeligneMumford moduli stack of marked stable algebraic curves and the mapping class group. 

Abhishek Rathod (TU Munich)  27.01.2021  16:30  Online  Parameterized inapproximability of Morse matching 
Abstract: In this talk, we look at the problem of minimizing the number of critical simplices (MinMorse) from the point of view of inapproximability and parameterized complexity. Let n denote the size of a simplicial complex. We first show that MinMorse can not be approximated within a factor of 2^{\log^{(1\epsilon)}n} \delta, for any \epsilon,\delta>0, unless NP \subseteq QP. Our second result shows that MinMorse is WPhard with respect to the standard parameter. Next, we show that MinMorse with standard parameterization has no FPTapproximation algorithm for any approximation factor \rho, unless WP = FPT. Since the gadgets involved in our reduction are 2complexes, the above hardness results are applicable for all complexes of dimension \geq 2. 

Christian Steinert (RWTH Aachen University)  20.01.2021  16:30  Online  A Diagrammatic Approach to String Polytopes 
Abstract: It is a common principle of mathematics to translate problems from one
area of research to another and solve them there. A very particular
example of this approach is the study of string polytopes, originally
introduced and studied by Berenstein and Zelevinsky as well as
Littelmann. Their original motivation stems from representation theory
but they also provide fruitful tools in the algebraic geometry of flag
varieties and more as studied by Caldero as well as Alexeev and Brion.
Although these string polytopes have been around for more than 30
years, results on their combinatorial properties are scarce. 

Dhruv Ranganathan (Universtiy of Cambridge)  13.01.2021  16:30  Online  DonaldsonThomas theory and the secondary polytope 
Abstract: DonaldsonThomas theory extracts numerical invariants from an algebraic variety X by examining the geometry of the Hilbert schemes of X. While the sister subject of GromovWitten theory has had a long and rich interaction with tropical and polyhedral methods, the relationship between the Hilbert scheme and tropical geometry is less developed. I will give an introduction to tropical and logarithmic aspects of DonaldsonThomas theory, from the point of view of the GKZ secondary polytope construction, and generalizations thereof. This is based on joint work with Davesh Maulik (MIT). 

Michael Joswig (TU Berlin)  09.12.2020  16:30  Online  Moduli of tropical curves  data and algorithms 
Abstract: In joint work with Sarah Brodsky, Ralph Morrison and Bernd Sturmfels in 2015 we computed the moduli of tropical plane curves of genus $\leq 4$. After a brief summary of the mathematical results the focus of this presentation is about the computational methods and the 400MB of data produced. We will see how this is being used for further research (ongoing joint work with Dominic Bunnett). On the way we will touch upon general questions concerning data in mathematics. 

Melody Chan (Brown University)  02.12.2020  16:30  Online  Tropical abelian varieties 
Abstract: I'll give an introduction to the moduli space of tropical abelian varieties, assuming no background in tropical geometry. Lots of different combinatorics arises, including the beautiful centuryold combinatorics of Voronoi reduction theory, perfect quadratic forms, regular matroids, and metric graphs. On the geometric side, it relates to toroidal compactifications of the classical moduli space A_g of abelian varieties. I'll explain how, and, time permitting, I'll report on workinprogress in which we use tropical techniques to find new rational cohomology classes in A_g in a previously inaccessible range. Joint work with Madeline Brandt, Juliette Bruce, Margarida Melo, Gwyneth Moreland, and Corey Wolfe. 

Kemal Rose (TU Berlin)  25.11.2020  16:30  Online  Certifying roots of polynomial systems using interval arithmetic 
Abstract: We report on an implementation of the Krawczyk method which establishes interval arithmetic as a practical tool for certification in numerical algebraic geometry. Our software HomotopyContinuation.jl now has a builtin function certify, which proves the correctness of an isolated solution to a square system of polynomial equations. This implementation dramatically outperforms earlier approaches to certification. 

Benjamin Schröter (KTH Stockholm)  18.11.2020  16:30  Online  Reconstructing Matroid Polytopes 
Abstract: This talk deals with two fundamental objects of discrete mathematics
that are closely related  (convex) polytopes and matroids.
Both appear in many areas of mathematics, e.g., algebraic geometry,
topology and optimization. 

Nick Early (Perimeter Institute for Theoretical Physics)  11.11.2020  16:30  Online  Combinatorial Geometries: From Polyhedral Subdivisions to Scattering Amplitudes 
Abstract: In 1948, Richard Feynman introduced a formalism in Quantum Field Theory to organize the expansion of a scattering amplitude as a sum of elementary rational functions, labeled by graphs. These graphs, now called Feynman diagrams, specify which singularities of the amplitude are compatible and can appear simultaneously. 

Dominic Bunnett (TU Berlin)  04.11.2020  16:30  Online  Embedded stacky fans and tropical moduli spaces 
Abstract: Stacky fans are orbifold objects in discrete geometry
with a rich combinatorial structure. They arise naturally
as the moduli spaces of tropical curves. Explicitly, a
stacky fan is built from polyhedral cones quotiented out
by finite groups. 

Rainer Sinn (University of Leipzig)  28.10.2020  16:30  Online  Realization spaces of polytopes 
Abstract: We study a simple model of the realization space of a polytope that lends itself to the application of the implicit function theorem. We explore how far this basic tool can carry us. Combined with quite elementary combinatorial arguments, this recovers most of the known results of “nice” realization spaces. 

Federico Castillo (MPI Leipzig)  21.10.2020  16:30  Online  The space of Minkowski Summands 
Abstract: Given a polytope P we study the set of all possible Minkowski summands. Being a Minkowski summand is equivalent to normal (or strongly combinatorial) equivalence. Different points of views provide different looking (but equivalent) parametrizations of the space of Minkowski summands as a polyhedral cone. We will go over several examples like polygons, permutohedra, cubes, and more. 

Leonid Monin (University of Bristol)  14.10.2020  16:30  Online  Inversion of matrices, MLdegrees and the space of complete quadrics 
Abstract: In my talk I will discuss the following questions: (1) How many quadrics go through *k* generic points and are tangent to *m * generic hyperplanes? (2) What is the maximum likelihood degree of a general linear concentration model? (3) What is the degree of the variety *L1* obtained as the closure of the set of inverses of matrices from a generic linear subspace *L* of Sym*n*? In fact, all three questions are equivalent! First I will briefly explain why this is the case, and then I will describe several approaches to the above questions and possible answers they lead to. This is joint work in progress with Laurent Manivel, Mateusz Michalek, Tim Seynnaeve, Martin Vodicka, Andrzej Weber, and Jaroslaw A. Wisniewski. 

Karin Schaller (FU)  07.10.2020  16:30  Online  Mirror symmetry constructions for quasismooth CalabiYau hypersurfaces in weighted projective spaces 
Abstract: We consider a general combinatorial framework for constructing mirrors of quasismooth CalabiYau hypersurfaces defined by weighted homogeneous polynomials. Our mirror construction shows how to obtain mirrors being CalabiYau compactifications of nondegenerate affine hypersurfaces associated to certain Newton polytopes. This talk is based on joint work with Victor Batyrev. 

Lukas Kühne (Hebrew Universiy of Jerusalem)  30.09.2020  16:30  Online  The Resonance Arrangement 
Abstract: The resonance arrangement is the arrangement of hyperplanes which has all nonzero 0/1vectors in R^n as normal vectors. It is also called the allsubsets arrangement. Its chambers appear in algebraic geometry, in mathematical physics and as maximal unbalanced families in economics. 

Jaeho Shin (Seoul National University)  26.08.2020  16:30  Online  Polytropes and Tropical Linear Spaces 
Abstract: A polytrope is a convex polytope that is expressed as the tropical convex hull of a finite number of points. It is well known that every bounded cell of a tropical linear space is a polytrope, and its converse statement has been a conjecture. We develop an elementary approach to the relationship between tropical convexity and tropical linearity, and show that the conjecture holds in dimension up to 3 and fails in every higher dimension. Thus, tropical convexity is strictly bigger than tropical linearity. 

Yue Ren (Swansea University)  29.07.2020  16:30  Online  TBA 
Abstract: 

Stephane Gaubert (INRIA and CMAP, Ecole polytechnique, CNRS)  08.07.2020  16:30  Online  Understanding and monitoring the evolution of the Covid19 epidemic from medical emergency calls: the example of the Paris area 
Abstract: We portray the evolution of the Covid19 epidemic during the crisis of MarchApril 2020 in the Paris area, by analyzing the medical emergency calls received by the EMS of the four central departments of this area (Centre 15 of SAMU 75, 92, 93 and 94). Our study reveals strong dissimilarities between these departments. We show that the logarithm of each epidemic observable can be approximated by a piecewise linear function of time. This allows us to distinguish the different phases of the epidemic, and to identify the delay between sanitary measures and their influence on the load of EMS. This also leads to an algorithm, allowing one to detect epidemic resurgences. We rely on a transport PDE epidemiological model, and we use methods from PerronFrobenius theory and tropical geometry. 

Mara Belotti (SISSA)  24.06.2020  16:30  Online  Topology of rigid isotopy classes of geometric graphs 
Abstract: We study the topology of rigid isotopy classes of geometric graphs on n vertices in a ddimensional Euclidean space. We consider in particular two different regimes: a large number of vertices for the graphs (n large) and a large dimension of the space in which the graphs live (d large). In both cases we make considerations on the number of such classes and study their Betti numbers. In particular, for the second case we register a "shift to infinity of the topology". This is joint work with A.Lerario and A.Newman. 

Raman Sanyal (Goethe University Frankfurt)  17.06.2020  16:30  Online  Inscribable fans, type cones, and reflection groupoids 
Abstract: Steiner posed the question if any 3dimensional polytope had a realization with vertices on a sphere. Steinitz constructed the first counter examples and Rivin gave a complete resolution. In dimensions 4 and up, Mnev's universality theorem renders the question for inscribable combinatorial types hopeless. In this talk, I will address the following refined question: Given a polytope P, is there a normally equivalent polytope with vertices on a sphere? That is, can P be deformed into an inscribed polytope while preserving its normal fan? It turns out that the answer gives a rich interplay of geometry and combinatorics, involving local reflections and type cones. This is based on joint work with Sebastian Manecke (who will continue the story in the FU discrete geometry seminar next week). 

Taylor Brysiewicz (Texas A&M University)  03.06.2020  16:30  Online  The degree of Stiefel manifolds 
Abstract: The Stiefel manifold is the set of orthonormal bases for kplanes in an ndimensional space. We compute its degree as an algebraic variety in the space of kbyn matrices using techniques from classical algebraic geometry, representation theory, and combinatorics. We give a combinatorial interpretation of this degree in terms of nonintersecting lattice paths. (This is joint work with Fulvio Gesmundo) 

Thorsten Theobald (Goethe University Frankfurt)  27.05.2020  16:30  Online  Conic stabilility of polynomials and spectrahedra 
Abstract: In the geometry of polynomials, the notion of stability is of prominent importance. The purpose of the talk is to discuss its recent generalization to conic stability. Given a convex cone K in real nspace, a multivariate polynomial f in C[z] is called Kstable if it does not have a root whose vector of the imaginary parts is contained in the interior of $K$. If $K$ is the nonnegative orthant, then $K$stability specializes to the usual notion of stability of polynomials. In particular, we focus on $K$stability with respect to the positive semidefinite cone and develop stability criteria building upon the connection to the containment problem for spectrahedra, to positive maps and to determinantal representations. These results are based on joint work with Papri Dey and Stephan Gardoll. 

Luis Carlos Garcia Lirola (Kent State University)  20.05.2020  16:30  Online  Volume product and metric spaces 
Abstract: Given a finite metric space M, the set of Lipschitz functions on M with Lipschitz constant at most 1 can be identified with a convex polytope in R^n. In this talk, we will show that there is a strong connection between the geometric properties of this polytope (as the vertices and the volume product) and the properties of the metric space M. We will also relate this study with a famous open problem in Convex Geometry, the Mahler conjecture, on the product of the volume of a convex body and its polar. This is a joint work with M. Alexander, M. Fradelizi, and A. Zvavitch. 

Georg Loho (London School of Economics)  13.05.2020  16:30  Online  Oriented Matroids from Triangulations of Products of Simplices 
Abstract: Classically, there is a rich theory in algebraic combinatorics surrounding the various objects associated with a generic real matrix. Examples include regular triangulations of the product of two simplices, coherent matching fields, and realizable oriented matroids. In this talk, we will extend the theory by skipping the matrix and starting with an arbitrary triangulation of the product of two simplices instead. In particular, we show that every polyhedral matching field induces oriented matroids. The oriented matroid is composed of compatible chirotopes on the cells in a matroid subdivision of the hypersimplex. Furthermore, we give a corresponding topological construction using Viro’s patchworking. This talk will also sketch the relationship between BakerBowler’s matroids over hyperfields and our work. This is joint work with Marcel Celaya and ChiHo Yuen. 

Martin Winter (TU Chemnitz)  06.05.2020  16:30  Online  EdgeTransitive Polytopes 
Abstract: Despite the long history of the study of symmetric polytopes, aside
from two extreme cases, the general transitivity properties of convex
polytopes are still badly understood.
For example, it has long been known that there are five
flagtransitive (aka. regular) polyhedra (called, Platonic solids),
six flagtransitive 4polytopes and exactly three flagtransitive
dpolytopes for any d>=5. The symmetry requirement of
flagtransitivity is therefore quite restrictive for convex polytopes.
On the other end of the spectrum, the class of vertextransitive (aka.
isogonal) polytopes is as rich as the category of finite groups.
There seems to be little known about intermediate symmetries, like
transitivity on edges or kfaces for any k>=2, and their interactions. 

Marek Kaluba  29.04.2020  17:00  Online  Random polytopes in Machine Learning 
Abstract: In this talk I will describe very recent work on analyzing different
models of autoencoder neural networks using random polytopes. In general, a
class of neural networks known as autoencoders can be understood as a low
dimensional (piecewise linear) embedding ℝᴺ → ℝⁿ (with N >> n), where points
in the domain are e.g. images (in their pixel form) and each dimension the
range extracts a single "feature". 

Francisco Criado  29.04.2020  16:30  Online  Borsuk’s problem 
Abstract: In 1932, Borsuk proved that every 2dimensional convex body can be
divided in three parts with strictly smaller diameter than the
original. He also asked if the same would hold for any dimension d: is
it true that every ddimensional convex body can be divided in (d+1)
pieces of strictly smaller diameter? 

Jonathan Spreer (University of Sydney)  22.04.2020  16:30  MA 621  Linking topology and combinatorics: Widthtype parameters of 3manifolds 
Abstract: Many fundamental topological problems about 3manifolds are algorithmically solvable in theory, but continue to withstand practical computations. In recent years some of these problems have been shown to allow efficient solutions, as long as the input 3manifold comes with a sufficiently "thin" presentation. 

Ayush Kumar Tewari  15.04.2020  17:00  Online  Forbidden patterns in tropical planar curves and Panoptagons 
Abstract: 

Dominic Bunnett  15.04.2020  16:30  Online  Geometric discriminants and moduli 
Abstract: A discriminant is a subset of a function space consisting of singular functions. We shall consider discriminants of finite dimensional vector spaces of polynomials defining hypersurfaces in a toric variety. In the best case scenario (for plane curves for example), the discriminant is an irreducible hypersurface, however this is not the case for almost any other toric variety. We discuss positive results on the geometry of discriminants of weighted projective spaces and how to compute them via the Adiscriminants of Gelfand, Kapranov and Zelevinsky. 

Robert Loewe  08.04.2020  16:30  Online  Minkowski decompositions of generalized associahedra 
Abstract: We give an explicit subword complex description of the rays of the type cone of the gvector fan of an finite type cluster algebra with acyclic initial seed. In particular, this yields a nonrecursive description of the Newton polytopes of the Fpolynomials as conjectured by Brodsky and Stump. We finally show that for finite type cluster algebras, the cluster complex is isomorphic to the totally positive part of the tropicalization of the cluster variety as conjectured by Speyer and Williams. 

Lars Kastner  01.04.2020  16:30  Online  Hyperplane arrangements in polymake 
Abstract: I will report on the implementation of hyperplane arrangements in polymake.
Hyperplane arrangements appear in a wide variety of applications in tropical
and algebraic geometry. An important construction is the induced subdivision.
We will discuss the new HyperplaneArrangement object, its properties and our
simple algorithm for constructing the cell subdivision. 

Paul Vater  25.03.2020  17:00  Online  Patchworking real tropical hypersurfaces 
Abstract: We take a look at the tropical version of Viro's combinatorial patchworking method, as well as a recent implementation in polymake. In particular we investigate an efficient way of computing the homology with Z_2 coefficients of such a patchworked hypersurface. 

Andrew Newman  25.03.2020  16:30  Online  A twostep random polytope model 
Abstract: We consider a model for generating nonsimple, nonsimplicial random polytopes. The first step of the random process generates a random polytope P via random hyperplanes tangent to the ddimensional sphere; the second step is given by the convex hull of a binomial sample of the vertices of P. In this talk we will discuss some ongoing work establishing results about the expected complexity of such polytopes. 

Oguzhan Yuruk  11.03.2020  16:30  MA 621  Understanding the Parameter Regions of Multistationarity in Dual Phosporylation Cycle via SONC 
Abstract: Parameterized ordinary differential equation systems are crucial for modeling in biochemical reaction networks under the assumption of massaction kinetics. Existence of multiple positive solutions in systems arising from biochemical reaction network are crucial since it is linked to cellular decision making and memory related on/off responses. Recent developments points out that the multistationarity, along with some other qualitative properties of the solutions, is related to various questions concerning the signs of multivariate polynomials in positive orthant. In this work, we provide further insight to the set of kinetic parameters that enable or preclude multistationarity of dual phosphorylation cycle by utilizing circuit polynomials to find symbolic certificates of nonnegativity. This is a joint work with Elisenda Feliu, Nidhi Kaihnsa and Timo de Wolff. 

Ralph Morrison (Williams College)  26.02.2020  16:30  MA 621  Tropically planar graphs: geometry and combinatorics 
Abstract: Tropically planar graphs are graphs that arise as the skeletons of smooth tropical plane curves, which are dual to regular unimodular triangulations of lattice polygons. In this talk we study what geometric properties such graphs have in the moduli space of metric graphs, as well as the combinatorial properties these graphs satisfy. This talk will include older work with Brodsky, Joswig, and Sturmfels; newer work with Coles, Dutta, Jiang, and Scharf; and ongoing work with Tewari. 

Antonio Macchia (FU Berlin)  12.02.2020  16:30  MA 621  Binomial edge ideals of bipartite graphs 
Abstract: Binomial edge ideals are ideals generated by binomials corresponding to the
edges of a graph, naturally generalizing the ideals of 2minors of a generic
matrix with two rows. They also arise naturally in the context of
conditional independence ideals in Algebraic Statistics. 

Amy Wiebe (FU Berlin)  05.02.2020  16:30  MA 621  Combining Realization Space Models of Polytopes 
Abstract: In this talk I will present a model for the realization space of a polytope which represents a polytope by its slack matrix. This model provides a natural algebraic relaxation for the realization space, and comes with a defining ideal which can be used as a computational engine to answer questions about the realization space. We will see how this model is related to more classical realization space models (representing realizations by Gale diagrams or points of the Grassmannian). In particular, we will see these relationships can be used to improve computational efficiency of the slack model. 

Matias Villagra (Pontifical Catholic University of Chile)  29.01.2020  16:30  MA 621  On a canonical symmetry breaking technique for polytopes 
Abstract: Given a group of symmetries of a polytope, a Fundamental Domain is a set of R^n that aims to select a unique representative of symmetric vectors, i.e. such that each point in the set is a unique representative under its Gorbit, effectively eliminating all isomorphic points of the polytope. The canonical Fundamental Domain found in the literature, which can be constructed for any permutation group, is NPhard to separate even for structurally simple groups whose elements are disjoint involutions (Babai & Luks 1983). 

Andres R. Vindas Melendez (University of Kentucky)  22.01.2020  16:30  MA 621  The Equivariant Ehrhart Theory of the Permutahedron 
Abstract: In 2010, Stapledon described a generalization of Ehrhart theory with group actions. In 2018, Ardila, Schindler, and I made progress towards answering one of Stapledon�s open problems that asked to determine the equivariant Ehrhart theory of the permutahedron. We proved some general results about the fixed polytopes of the permutahedron, which are the polytopes that are fixed by acting on the permutahedron by a permutation. In particular, we computed their dimension, showed that they are combinatorially equivalent to permutahedra, provided hyperplane and vertex descriptions, and proved that they are zonotopes. Lastly, we obtained a formula for the volume of these fixed polytopes, which is a generalization of Richard Stanley?s result of the volume for the standard permutahedron. Building off of the work of the aforementioned, we determine the equivariant Ehrhart theory of the permutahedron, thereby resolving the open problem. This project presents combinatorial formulas for the Ehrhart quasipolynomials and Ehrhart Series of the fixed polytopes of the permutahedron, along with other results regarding interpretations of the equivariant analogue of the Ehrhart series. This is joint work with Federico Ardila (San Francisco State University) and Mariel Supina (UC Berkeley). 

Simon Telen (KU Leuven)  15.01.2020  16:30  MA 621  Numerical Root Finding via Cox Rings 
Abstract: In this talk, we consider the problem of solving a system of (sparse) Laurent polynomial equations defining finitely many nonsingular points on a compact toric variety. The Cox ring of this toric variety is a generalization of the homogeneous coordinate ring of projective space. We work with multiplication maps in graded pieces of this ring to generalize the eigenvalue, eigenvector theorem for root finding in affine space. We use numerical linear algebra to compute the corresponding matrices, and from these matrices a set of homogeneous coordinates of the solutions. Several numerical experiments show the effectiveness of the resulting method, especially for solving (nearly) degenerate, high degree systems in small numbers of variables. 

Maria Angelica Cueto (Ohio State University)  11.12.2019  16:30  MA 621  Anticanonical tropical del Pezzo cubic surfaces contain exactly 27 lines. 
Abstract: Since the beginning of tropical geometry, a persistent
challenge has been to emulate tropical versions of classical results
in algebraic geometry. The wellknown statement "any smooth surface of
degree three in P^3 contains exactly 27 lines'' is known to be false
tropically. Work of Vigeland from 2007 provides examples of tropical
cubic surfaces with infinitely many lines and gives a classification
of tropical lines on general smooth tropical surfaces in TP^3. 

Roser Homs Pons (TU Berlin)  20.11.2019  16:30  MA 621  HilbertBurch matrices of ideals in k[x,y] and k[[x,y]] 
Abstract: In this talk we will discuss HilbertBurch matrices of ideals of codimension two and compare global monomial orders in k[x,y] with local orders in k[[x,y]]. We will see how these tools can be used to give a parametrization of zerodimensional ideals in the ring of polynomials and the ring of formal power series in two variables. 

Sascha Timme (TU Berlin)  13.11.2019  16:30  MA 621  3264 Conics in a Second 
Abstract: Enumerative algebraic geometry counts the solutions to certain geometric constraints. Numerical nonlinear algebra determines these solutions for any given instance. This talk illustrates how these two fields complement each other, especially in the light of emerging new applications. We start with a wonderful piece of 19th century geometry, namely the 3264 conics that are tangent to five given conics in the plane. Thereafter we turn to current problems in statistics and data science, with focus on the maximum likelihood estimation for linear Gaussian covariance models. 

Jesus Yepes Nicolas (University of Murcia)  30.10.2019  16:30  MA 406  TBA 
Abstract: TBA 

Laura Brustenga i Moncusi (TU Berlin, UAB)  23.10.2019  16:30  MA 621  Numerically computing the local dimension of an algebraic set. 
Abstract: Symbolic computations based on Gr�bner basis are widely used, with great success, to compute several algebraic invariants. Nevertheless, such symbolic computations are expensive in time. Numerical Algebraic Geometry, an emerging field, specialises algorithms form Numerical Analysis to work on polynomial systems defined over the complex numbers, improving its performance by means of Algebraic Geometry. In this talk, we will outline an algorithm to compute the local dimension of an algebraic set. 

Scott Kemp (Queen Mary University of London)  16.10.2019  16:30  MA 621  An Alternative Characterisation of Valuated Matroids 
Abstract: We give an alternative way of defining a valuated matroid in terms of a rank function. The characterisation we give comes from looking at the valuated matroid basis polytope in an analogous way to the matroid basis polytope and how we obtain the rank function for a matroid from it. 

Robert Loewe (TU Berlin)  02.10.19  16:30  MA 621  On the discriminant of a cubic quaternary form 
Abstract: We determine the 166104 extremal monomials of the discriminant of a quaternary cubic form. These are in bijection with Dequivalence classes of regular triangulations of the 3dilated tetrahedron. We describe how to compute these triangulations and their Dequivalence classes in order to arrive at our main result. The computation poses several challenges, such as dealing with the sheer amount of triangulations effectively, as well as devising a suitably fast algorithm for computation of a Dequivalence class. 

Daniel Galicer (University of Buenos Aires)  11.09.19  16:30  MA 406  TBA 
Abstract: TBA 

Dominic Bunnett  04.09.19  16:30  MA 621  Stability of hypersurfaces and Newton Polytopes 
Abstract: Constructing moduli spaces is a fundamental problem in algebraic geometry. In this context, the stability of an object determines whether or not it fits into a moduli space and often becomes an important notion in its own right (for example stable sheaves and stable curves). The moduli of hypersurfaces in toric varieties is constructed using geometric invariant theory and an analysis of stability using the Newton polytope uncovers discrete geometry as a powerful tool to study the moduli of such hypersurfaces. In this talk we show how one relates stability of hypersurfaces and certain discrete geometric conditions on their Newton Polytopes. 

Victor Magron (CNRS Grenoble)  31.07.19  16:30  MA 621  Certified Semidefinite Approximations of Reachable Sets 
Abstract: We consider the problem of approximating the reachable set of
a continuous or discretetime polynomial system from a semialgebraic set
of initial conditions under general semialgebraic set constraints.
Assuming inclusion in a given simple set like a box or an ellipsoid, we
provide a method to compute certified outer approximations of the
reachable set. 

Andrew Newman, Francisco Criado (TU Berlin)  24.07.19  16:30  MA 621  Randomness in Algebra and Topology / Minkowski decompositions of polytopes via geometric graphs: followup 
Abstract: I will give a short talk about the recent Summer School at MPI Leipzig on Randomness and Learning in NonLinear Algebra. I will focus particularly on models of random monomial ideals and their connections with random graphs and simplicial complexes. Abstract:This talk is a followup on Guillermo Pineda's previous DMG talk: "Minkowski decompositions of polytopes via geometric graphs". In his work (PinedaVillacicencio et al (2018)), there is an example of a 3dimensional polytope that admits both a decomposable (i.e.: Minkowski sum of nonhomothetic polytopes) realisation and a indecomposable realisation. This shows that decomposability of polytopes cannot be decided by the combinatorics alone. In this talk, we will show why his example does not extend to higher dimensions. 

Elisabeth Werner (Case Western Reserve University)  10.07.19  16:30  MA 406  On the affine surface area 
Abstract: Given a convex body K in R^n, we study the quantity AS(K) = sup_{K'\subseteq K}as(K'), where as(K') denotes the affine surface area of K', and the supremum is taken over all convex?subsets of K.?We study ?continuity properties of AS(K) and give asymptotic estimates. Based on joint work with Ohad Giladi, Han Huang and Carsten Schuett. 

Marcel Celaya (Georgia Tech)  05.07.19  12:00  MA 406  A chirotopebased proof of the BohneDress theorem 
Abstract: The celebrated BohneDress theorem characterizes tilings of a given zonotope $\mathcal{Z}$ by zonotopes in terms of singleelement liftings of the oriented matroid associated to $\mathcal{Z}$. In this talk I will show how one might prove this result using a certain reinterpretation of the chirotope of an oriented matroid. I will also discuss several related open questions. 

Stephan Tillmann (University of Sydney)  03.07.19  16:30  MA 621  Minimal triangulations of cusped hyperbolic 3manifolds 
Abstract: Thurston observed that ideal triangulations are a useful tool to
study the geometry and topology of cusped hyperbolic 3?manifolds. Geometric
triangulations are useful to study geometric properties of a manifold. Min
imal triangulations, i.e. topological ideal triangulations using the least
number of ideal 3simplices, are used in census enumeration, and as a
platform to study the topology of the manifold using normal surface theory. 

Sarah Morell  26.06.19  16:30  MA 621  On the singlesource unsplittable flow problem 
Abstract: The singlesource unsplittable flow problem has been introduced by Kleinberg (1996) and generalizes several NPcomplete problems from various areas in combinatorial optimization such as packing, partitioning, scheduling and load balancing. Commodities must be routed simultaneously in a given arccapacitated graph. Each of the commodities has a common source, a destination and a requested flow demand. The problem consists in routing each commodity through a single path without violating the given arc capacities. 

Rekha Thomas (University of Washington)  19.06.19  16:30  MA 621  The Slack Realization Space of a Polytope 
Abstract: We introduce a new model of a realization space of a polytope that arises as the positive part of a real variety. The variety is determined by the slack ideal of the polytope, a saturated determinantal ideal of a sparse generic matrix that encodes the combinatorics of the polytope. The slack ideal offers a uniform computational framework for several classical questions about polytopes such as rational realizability, projectively uniqueness, nonprescribability of faces, and realizability of combinatorial polytopes. The simplest slack ideals are toric. We identify the toric ideals that arise from projectively unique polytopes. New and classical examples illuminate the relationships between projective uniqueness and toric slack ideals. 

Alexander Koldobsky (University of Missouri)  12.06.19  16:30  MA 406  An estimate for the distance from a convex body to subspaces of L_p 
Abstract: TBA 

Guillermo Pineda Villavicencio (Federation University Australia)  29.05.19  16:30  MA 621  Minkowski decompositions of polytopes via geometric graphs 
Abstract: Minkowski decomposition of polytopes is presented via geometric graphs. This is due to Kallay (1982), who reduced the decomposability of a realisation of polytope to that of its geometric graph, and in this way, introduced the decomposability of geometric graphs. One advantage of this approach is its versatility. The decomposability of polytopes reduces to the decomposability of geometric graphs, which are not necessarily polytopal. And statements on decomposability of geometric graphs often revolve around the existence of suitable subgraphs or useful properties in the graphs. 

Giulia Codenotti (FU Berlin)  22.05.19  16:30  MA 406  The covering minima of lattice polytopes 
Abstract: Covering minima of a convex body were introduced by Kannan and Lovasz to give a better bound on the constant in the flatness theorem, which states that the width of hollow convex bodies in a fixed dimension is bounded by a constant. These minima are similar in flavor to Minkowski's successive minima, and on the other hand generalize the covering radius of a convex body. I will speak about recent joint work with Francisco Santos and Matthias Schymura, where we investigate extremal values of these covering minima for nonhollow lattice polytopes. 

Boasz Slomka (Weizmann Institute)  15.05.19  16:30  MA 406  On Hadwiger�s covering conjecture 
Abstract: https://www3.math.tuberlin.de/combi/wp_henk/event/seminartalkbyboaszslomka/ 

Alheydis Geiger (Tuebingen)  08.05.19  16:30  MA 621  Realisability of infinite families of tropical lines on general smooth tropical cubic surfaces 
Abstract: A tropical line on a smooth tropical cubic surface can be realised, if
there exists a line on a smooth cubic surface, such that the
tropicalisation of the surface and the line coincide with the given
tropical cubic surface and tropical line. (This is called relative realisability.)


Andreas Paffenholz  17.04.19  16:30  MA 621  tropical cubic surfaces 
Abstract: The Ehrhart polynomial counts lattice points in multiples of a lattice polytope P. It is a polynomial e(t) of degree d, and we know that its constant coefficient and the coefficients of t^d and t^(d1) are positive, as those coefficients have a geometric interpretation. Much less is known about the other coefficients. 

Erika Roldan  10.04.19  16:30  MA 621  Evolution of the homology and related geometric properties of the Eden Growth Model. 
Abstract: In this talk, we study the persistent homology and related geometric properties of the evolution in time of a discretetime stochastic process defined on the 2dimensional regular square lattice. This process corresponds to a cell growth model called the Eden Growth Model (EGM). It can be described as follows: start with the cell square of the 2dimensional regular square lattice of the plane that contains the origin; then make the cell structure grow by adding one cell at each time uniformly random to the perimeter. We give a characterization of the possible change in the rank of the first homology group of this process (the "number of holes"). Based on this result we have designed and implemented a new algorithm that computes the persistent homology associated to this stochastic process and that also keeps track of geometric features related to the homology. Also, we present obtained results of computational experiments performed with this algorithm, and we establish conjectures about the asymptotic behavior of the homology and other related geometric random variables. The EGM can be seen as a First Passage Percolation model after a proper timescaling. This is the first time that tools and techniques from stochastic topology and topological data analysis are used to measure the evolution of the topology of the EGM and in general in FPP models. 

Andrew Newman, Manuel Radons, and Oguzhan Yuruk (TU Berlin)  03.04.19  16:30  MA 621  Radons: Shellability, Yuruk: Lattice Polytopes and Face Enumeration of Certain Classes of Simplicial Polytopes, Newman: Random polytopes 
Abstract: Radons: Following Tim Roehmer's third lecture at the Bochum spring
school, I will introduce the notion of shellability of a pure polytopal
complex. A simplification of the concept for pure simplicial complexes
and apllications of shellings in inductive proofs will be discussed. 

Holger Eble and Ayush Tewari (TU Berlin)  27.03.19  16:30  MA 621  Eble: Epistatic weights and polyhedral geometry Tewari: Moduli of tropical plane curves of genus 6 
Abstract: Eble:
Epistatic weights and cluster filtrations provide a mathematical
tool for studying biological data sets. In this informal talk I will
present the geometric motivation behind these notions. 

Jonathan Kliem (FU Berlin)  20.03.19  16:30  MA 621  A face iterator for polyhedra 
Abstract: We discuss a new algorithm that iterates over all elements of a meetsemilattice, where every interval is coatomic. 

Jens Forsgaard (Universiteit Utrecht)  13.02.19  16:30  MA 621  Nonegativity and Discriminants 
Abstract: We study the class of nonnegative polynomials obtained from the inequality of arithmetic and geometric means, called \emph{agiforms} or \emph{nonnegative circuit polynomials}. They generate a full dimensional subcone $S$ of the cone of all nonnegative polynomials, which is distinct from the cone of sums of squares. Let $\mathbb{R}^A$ denote the space of all real polynomials with support $A$. We describe the boundary of the cone $S \cap \mathbb{R}^A$ as a space stratified in real semialgebraic varieties. In order to describe the strata, we take a journey through discriminants, polytopes, and triangulations, oriented matroids, and tropical geometry. Based on joint work with Timo de Wolff. 

Christian Krattenthaler  13.02.19  15:30  MA 621  Discrete analogues of MacdonaldMehta integrals 
Abstract: The Mehta integral and its generalisations due to Macdonald originate from random matrix theory, but are as well important objects in the theory of (multivariable) orthogonal polynomials and in combinatorics. These (now) classical integrals are certain multidimensional integrals which  suprisingly?  can be evaluated into closed form product formulae. I shall consider certain discretisations of the MacdonaldMehta integrals. There are ten families of such discretisations which can be evaluated in closed form. I shall sketch the ideas which go into the proofs of these identities, which come from combinatorics of nonintersecting lattice paths, identities for classical group characters, and a transformation formula for elliptic hypergeometric series, respectively. No prior knowledge of any of these is required to follow the talk as everything will be explained during the talk. This is joint work with Richard Brent and Ole Warnaar. 

Deane Yang (NYU Courant)  06.02.19  16:30  MA 406  Information theoretic inequalities and their convex geometric analogues 
Abstract: It will be shown how, using the homogeneous contour integral, information theoretic invariants and inequalities involving entropy, moments, Fisher information can be translated into convex geometric invariants and inequalities. 

Johannes Rau (Tuebingen)  23.01.19  16:30  MA 621  Tropical patchworking for nodal curves in the plane (joint with Ilia Itenberg and Grigory Mikhalkin) 
Abstract: Given a real algebraic curve in the plane, probably the most elementary questions you can ask are: What is its number of connected components? How are they arranged in the plane? This leads to the porblem of topological classification of such curves which received ongoing attention ever since being included as Hilbert's 16th problem in his famous list from 1900. While most results deal with the case of smooth curves, I will present a classification for a certain type of singular curves of degree 5 in the plane. To construct such curves, we will use a "tropical" version of Viro's patchworking construction which is better suited for the singular case. 

Timo de Wolff  09.01.19  16:30  MA 621  Nondegenerate Multistationarity in Small Reaction Networks 
Abstract: Much attention has been focused in recent years on the following algebraic problem arising from applications: which chemical reaction networks, when taken with massaction kinetics, admit multiple positive steady states? The interest behind this question is in steady states that are stable. As a step toward this difficult question, here we address the question of multiple nondegenerate positive steady states. Mathematically, this asks whether certain families of parametrized, real, sparse polynomial systems ever admit multiple positive real roots that are simple. Our main results settle this problem for certain types of small networks, and our techniques point the way forward for larger networks. This is joint work with Anne Shiu. 

Tanka Nath Dhamala (Tribhuvan University)  05.12.18  16:30  MA 406  Flow Models and Solution Strategies for Evacuation Planning Problems 
Abstract: Unavoidable human created or natural disasters worldwide highly demand an efficient and reliable evacuation planning strategy to save the people and property in emergency periods. There exist a variant of mathematical modeling approaches and solution techniques covering from wide range of mathematical fields, engineering and management sciences. Although solution techniques with fluid dynamical equations using differential equations or cell based simulation approaches seek to address the problems more accurately and also taking care of the individual behaviors, they could not handle large scale problems because of high computational costs involved. On the other hand, flow models with different optimization techniques that seek to model the real life with time varying or flow dependent attributes also yield nonlinearity and high computational inefficiency. As a compromise, one can model these problems within the framework of evacuation network optimization which can handle relatively larger size instances with compromised solution quality in discrete time settings. We consider the models in latter approach and illustrate some algorithms that are more interesting theoretically and also from practical view points. 

Milena Wrobel (MPI Leipzig)  21.11.18  16:30  MA 621  On the anticanonical complex 
Abstract: Toric Fano varieties are in one to one correspondence to so called Fano polytopes. The lattice points in the Fano polytope determine the singularity type of the corresponding toric Fano variety and thus allows classification with purely combinatorial methods. For varieties with a torus action of complexity one, i.e. the general torus orbit is of dimension one less than the dimension of the variety, the anticanonical complex has been introduced as a natural generalisation of the toric Fano polytope. We give a sufficient condition on Fano varieties with torus action of arbitrary complexity that admit an anticanonical complex, i.e. a polyhedral complex whose lattice points determine their singularity type. We show that the possibility to apply the anticanonical complex to these varieties is connected to certain properties of their quotients and use this fact to construct first example classes. 

Anne FruehbisKrueger (Leibniz Universitaet Hannover)  14.11.18  16:30  MA 621  Some massively parallel computations in algebraic geometry 
Abstract: While massively parallel computations are ubiquitous in numerics and simulation, they have rarely ever been thought about in computational algebraic geometry. Already the Gr�bner Basis Algorithm, which is the workhorse behind numerous computations, does not have a natural parallelization and thus seems a huge obstacle. But there are tasks in algebraic geometry, which are accessible to a very coarse grained, massively parallel approach due to the local nature of the problem itself. In this talk, I will show a few examples, in which such an approach proved fruitful such as e.g. a smoothness test based on the termination criterion of Hironaka�s resolution of singularities. 

Iskander Aliev (Cardiff University)  31.10.18  16:30  MA 406  On the distance to lattice points in knapsack polyhedra 
Abstract: In this talk we will present some recent results on the distance from a vertex of an integer feasible knapsack polyhedron \(P\) to its nearest lattice point in \(P\) (referred to as a vertex distance). We give a sharp upper bound for the vertex distance that only depends on the maximum norm of the vector \(a\). In a randomised setting, we show that the vertex distance for a typical knapsack polyhedron is drastically smaller than the vertex distance that occurs in a worst case scenario. This is a joint work with Martin Henk and Timm Oertel. 

Akiyoshi Tsuchiya  24.10.18  16:30  MA 621  Reflexive polytopes arising from (0,1)polytopes 
Abstract: A (0,1)polytope is a lattice subpolytope of [0,1]^d. Several classes of (0,1)polytopes arise from some combinatorial objects and many properties of the (0,1)polytopes are characterized in terms of the combinatorial objects. In this talk, we introduce construction of reflexive polytopes arising (0,1)polytopes and discuss their properties. A reflexive polytope is one of the keywords belonging to the current trends on the research of lattice polytopes. In fact, many authors have studied reflexive polytopes from viewpoints of combinatorics, commutative algebra and algebraic geometry. We also present a linear algebraic technique to show a lattice polytope is reflexive and we give a family of reflexive polytopes arising from the edge polytopes of finite simple graphs. This talk is based on joint work with Takahiro Nagaoka. 

Marta Panizzut  17.10.18  16:30  MA 621  Introduction to padic geometry 
Abstract: In this survey talk we will introduce padic fields and more generally nonarchimedean valued fields. We will point out that the topology induced by the absolute value is totally disconnected. This make it more difficult to define manifolds and geometric theories over these fields. We will see how Berkovich's approach of analytic spaces deal with these problems. 

Leon Zhang  29.08.18  16:30  MA 621  Constructing an enveloping membrane for a minconvex hull 
Abstract: Joswig, Sturmfels, and Yu (2007) describe a correspondence between a minconvex hull in the affine building of SL_d(K) and a tropical convex hull in tropical projective space TP^{d1}, using a membrane containing the minconvex hull. We describe an algorithm for constructing such a membrane, which gives a bound on the number of points needed to span the tropical convex hull. 

Charles Wang  22.08.18  16:30  MA 621  Cluster Algebras and Grassmannians 
Abstract: We will study the interactions of combinatorics and geometry via cluster algebras. We will begin with the simple case of the Grassmannians Gr(2,n), where this connection is completely determined by the type A_n associahedron, corresponding to triangulations of an n+3gon. Next, we will discuss the case for general Gr(k,n) and the Lagrangian Grassmannians LG(n), where will focus on combinatorial objects called plabic graphs. We will discuss how the cluster structure in these cases gives a method to understand the NewtonOkounkov bodies of these varieties, and give a connection to Khovanskii bases and toric degenerations. Time permitting, we will also the study relation to the tropical Grassmannians and positivity. 

Stephan Tillmann  18.07.18  16:30  MA 621  What is the Thurston norm? 
Abstract: Thurston�s norm on (co)homology of a 3manifold reveals interesting topological properties of the 3manifold. At the centre of its study is its unit ball, which is a finite polytope. Associated to some of its top dimensional faces are special triangulations of the manifold defined by Agol, called veering triangulations. After a general introduction to these topics, I will outline algorithms developed in collaboration with Daryl Cooper and Will Worden to compute the unit ball, and ask the audience for input on interesting properties of the polytopes that one might want to address. 

Marek Kaluba  11.07.18  16:30  MA 621  Noncommutative positivity and property (T) 
Abstract: The underlying theme of the talk is the reduction of the (hard analytic) ques tion of Kazhdan�s property (T) to an (algebraic, computable) problem of non commutative optimisation. In this formulation, property (T) is equivalent to the positivity of the element ? 2 ? ? ? in (full) C * group algebra (where ? is the group Laplacian). This problem will serve as a motivation to explore the topic of noncommutative positivity. I will present certain results on positivity for free * algebras and group al gebras. It turns out that, due to much richer representation theory of non commutative objects, the Positivestellens�tze in the noncommutative world are much stronger than in the commutative world: the positive cone is in many cases equal to the cone of sums of (hermitian) squares. Moreover one of the characterisations of cones in the �algebraic topology� of * algebra allows turning a numerical approximation of the the sum of squares decomposition (of ? 2 ? ? ? ) into a mathematical proof of the positivity. I will finish by presenting the very concrete case of the application of the theory to S Aut( F n ), the (special) automorphism group of the free group on five generators. Using Gramm method we formulated The successful execution of the described programme was obstructed by the sheer size of the optimisation problem. To overcome these difficulties we exploited the symmetry of ? 2 ? ? ? and used the theory of representations of finite groups to significantly reduce the complexity of the optimisation problem, wich might be of interest of its own. 

Ngoc Tran (UT Austin/Hausdorff Center for Mathematics, Bonn)  04.07.18  16:30  MA 621  A computational approach to tropical semigroups 
Abstract: In many semigroups it is difficult to construct and verify identities. For the semigroups of Tropical uppertriangular matrices, we show that each word has a signature, which is a sequence of polytopes, and two words form an identity if and only if their signatures agree. This leads to fast algorithms for constructing semigroup identities. Our algorithms allow us to disprove conjectures, prove new theorems on the structure of the semigroup, and form new conjectures at the intersection of probability, combinatorics and semigroup theory. In this talk, we will focus on explaining these conjectures from the viewpoint of polyhedral computations. Joint work with Marianne Johnson 

Timur Sadykov  28.06.18  14:15  MA 316  Amoebas of multivariate hypergeometric polynomials 
Abstract: Polynomial instances of hypergeometric functions in one and
several variables are very diverse. They comprise the classical
Chebyshev polynomials of the first and the second kind, the
Gegenbauer, Hermite, Jacobi, Laguerre and Legendre polynomials as
well as their numerous multivariate analogues. 

Christian Stump  20.06.18  16:30  MA 621  Reflection groups and singularity theory 
Abstract: I aim to motivate, as simple as possible and limited by my own understanding, Arnold's problem "Find applications of the complex reflection groups in singularity theory". I start with discussing the connection between the discriminant of a general polynomial in one variable and the realization of the braid group using the braid arrangement. I then intend to give a detailed introduction to reflection groups in rational / real / complex vector spaces, their reflection arrangements, and their discriminants, and in particular discuss the connection to Kleinian singularities associated with symmetry groups of platonic solids and binary polyhedral groups. 

Kalina Mincheva (Yale University)  13.06.18  16:30  MA 621  The Picard group of a tropical toric scheme 
Abstract: Through the process of tropicalization one obtains from an algebraic variety $X$ a combinatorial object called the tropicalization of $X$, $trop(X)$, that retains a lot of information about the original variety. Following the work of J. Giansiracusa and N. Giansiracusa, one can endow $trop(X)$ with more structure, to obtain a tropical scheme. Loosely speaking, we consider more equations than the ones needed to determine the tropical variety. We are interested what information about the original variety $X$ is preserved by the tropical scheme $X_\mathbb{T}$ (but possibly not by the tropical variety). In particular, we study the relation between the Picard group of $X$ and $X_\mathbb{T}$. We solve the problem in the case when $X$ is a toric variety. 

Jonathan Spreer  06.06.2018  16:30  MA 621  Separationtype combinatorial invariants for triangulations of manifolds 
Abstract: In my talk I will propose and discuss a set of combinatorial invariants of simplicial complexes which, in some sense, can be seen as a refinement of the fvector. The invariants are very elementary and defined by counting connected components and/or homological features of induced subcomplexes. 

Alperen Erguer  30.05.18  16:30  MA 406  Multihomogenous Nonnegative Polynomials and Sums of Squares 
Abstract: It is due to Blekherman that we now know there are significantly more nonnegative forms than sums of squares for any fixed degree \(2d\) and large number of variables \(n\). What if we ask the same comparison for polynomials with a prescribed Newton polytope? I will present some general ideas from (modern) convex geometry that handles the case of multihomogeneous polynomials. These results refine the estimates of Blekherman and happens to have some implications in quantum information theory. I will also try to indicate missing ingredient(s) for handling general Newton polytopes, which might be of interest to the lovers of toric geometry and/or Lie algebras. 

Travis Scrimshaw  29.05.18  17:00  MA 621  On the Combinatorics of Crystals 
Abstract: In algebra, a crystal base or canonical base is a base of a representation, such that generators of a quantum group or semisimple Lie algebra have a particularly simple action on it. Crystal bases were introduced by Kashiwara (1990) and Lusztig (1990) (under the name of canonical bases). 

Grigoris Paouris (Texas A&M University)  24.05.18  16:15  MA 313  The "Small ball" property in high dimensional measures 
Abstract: I will discuss some methods/techniques to prove "small ball probabilities" and I will review some connections with other problems in Asymptotic Geometric Analysis. The emphasis will be on how highdimensional geometry and convexity is crucial to understand the concentration behavior in the "small ball regime". 

Grigoris Paouris (Texas A&M University)  23.05.18  16:30  MA 313  Almost Euclidean section of highdimensional convex bodies 
Abstract: The celebrated theorem of Dvoretzky states that for any n and \(\varepsilon \in (0,1)\) there exists a function \(k(n,?)\) with the property that any ndimensional symmetric convex body has a kdimensional section at a distance at most \((1+\varepsilon)\) from the Euclidean ball and moreover \(k(n,\varepsilon)\) tends to infinity as \(n \to \infty\) for fixed \(\varepsilon\). The asymptotic behavior of this function remains an open problem 64 years after Grothendieck proposed it. I will discuss some recent developments on the problem. In particular I will present how the notion of superconcentration is central in the latest improvements. Based on joint work with P. Valettas. 

Chris O'Neill (UC Davis)  09.05.18  16:30  MA 621  Computing the delta set of an affine semigroup: a status report 
Abstract: An affine semigroup \(S\) is a subset of \(\mathbb Z_{\ge 0}^k\) that is closed under vector addition, and a factorization of \(a \in S\) is an expression of \(a\) as a sum of generators of \(S\). The delta set of \(a\) is a set of positive integers determined by the 'missing factorization lengths' of \(a\), and the delta set of \(S\) is the union of the delta sets of its elements. Although the delta set of any affine semigroup is finite, its definition as an infinite union makes explicit computation difficult. In this talk, we explore algebraic and geometric properties of the delta set, and survey the history of its computation for affine semigroups. The results presented here span the last 20 years, ranging from a first algorithm for a small class of semigroups that is impractical for even basic examples, to recent joint work with GarciaSanchez and Webb expressing the delta set of any affine semigroup in terms of Groebner bases, and include results from numerous undergraduate research projects. 

Vijaylaxmi Trivedi (Tata Institute of Fundamental Research, Mumbai)  02.05.18  16:30  MA 406  HilbertKunz functions and HK density functions 
Abstract: We give a brief survey of HilbertKunz function and its leading coefficient \(e_{HK}\) called HK multiplicity, for a commutative Noetherian ring. 

Victor Magron (CNRS Grenoble)  25.04.18  16:30  MA 621  On exact polynomial optimization 
Abstract: To compute certificates of nonnegativity, an approach based on sums of
squares (SOS) decompositions has been popularized by Lasserre and Parillo. 

Ben Smith (Queen Mary University of London)  18.04.18  16:30  MA 621  Matching fields and lattice points of simplicies 
Abstract: An (n,d)matching field is a collection of matchings such that there is a unique matching for each dsubset of [n]. They naturally arise as minimal matchings of a weighted complete bipartite graph, and can be thought of as the tropical analogue to matroids. They have relationships to multiple combiatorial objects, in particular triangulations of the product of two simplices. In this talk, we will use this relationship to introduce a construction that associates to the matching field a collection of bipartite graphs and a bijection to the lattice points of a scaled simplex. We will then use this construction to prove two outstanding conjectures of Sturmfels and Zelevinsky on matching fields. 

Michael Joswig (TU Berlin)  11.04.18  16:30  MA 621  Coxeter groups, BNpairs, and buildings 
Abstract: Buildings form a fundamental geometric concept which generalize the Coxeter complex of a Coxeter group. The theory was mostly developed by Tits (and Borel) in the 1960s and 1970s. While its origins are in Lie theory one particularly striking application was its contribution to the classification of the finite simple groups (completed around 1980). In this talk (of historical survey type) I will define the three notions in the title and look into one family of examples. In the end I will briefly try to sketch the impact on group theory in general. 

Sascha Timme  04.04.18  16:30  MA 621  Fast Computation of Amoebas in low Dimensions 
Abstract: The concept of an amoeba was introduced by Gelfand, Kapranov and Zelevinsky
in 1994 to study the relationship between the zero locus of a polynomial and
its Newton polytope. Amoebas have a lot of fascinating structural properties.
In particular, there exists for each amoeba a tropical hypersurface, called
the spine, which is a deformation retract of the amoeba. 

Adam Kurpisz  28.03.18  16:30  MA 621  On the hierarchies of relaxations for 0/1 optimization problems 
Abstract: Constructing efficient algorithms for combinatorial optimization problems is a task that requires a lot of knowledge, experience and creativity. The goal of this talk is to present the automatizable hierarchies of algorithms (Lasserre/SumofSquares (SoS), Sherali Adams (SA)) and the recent development in this field, the SumofNonnegativeCircuit (SoNC) algorithm. 

Francisco Criado  07.02.18  16:30  MA 621  Packing algorithms applied to linear programming 
Abstract: The YamnistskyLevin algorithm (1982) is a variation of the ellipsoid method that uses a simplex instead of an ellipsoid. Unlike the ellipsoid method, it does not require square root computations, and it still runs in weakly polynomial time. However, its running time is still not practical. We introduce a variation of this method that examines a subset of cutting planes, and uses the Multiplicative Weights framework developed by Plotkin, Shmoys and Tardos for packing linear programs. We will show how our algorithm improves the original YL algorithm for a simplification of the problem, and what questions remain open to give a complete proof of convergence. 

Romanos Malikiosis  24.01.18  16:30  MA 406  Fuglede's spectral set conjecture on cyclic groups 
Abstract: Fuglede's conjecture (1974) states that a bounded measurable subset in \(\mathbb{R}^d\) accepts an orthogonal basis of exponential functions (i.e. it is spectral) if and only if it tiles the space with a discrete set of translations. This conjecture turned out to be false by Tao's counterexample in 2003. Using Tao's ideas, counterexamples in finite Abelian groups such as \(\mathbb{Z}_N^d\) can be lifted to counterexamples in \(\mathbb{R}^d\), thus shifting the interest on this conjecture to this setting in recent years. This has been successful for \(d\geq3\), but the conjecture is still open for \(d=1,2\). 

Linda Kleist  20.12.17  16:30  MA 621  Rainbow cycles in flip graphs 
Abstract: The vertices of flip graph are combinatorial or geometric objects, and its edges link two of these objects whenever they can be obtained from one another by an elementary operation called a flip. With each edge, we associate the type of a flip (thought of as a color) and seek for rrainbow cycles, i.e., cycles where every flip type occurs exactly r times. 

Franz Schuster (TU Wien)  06.12.17  16:30  MA 406  Affine vs. Euclidean Sobolev inequalities 
Abstract: In this talk we explain how every even, zonal measure on the Euclidean unit sphere gives rise to a sharp Sobolev inequality for functions of bounded variation which directly implies the classical Euclidean Sobolev inequality. The strongest member of this large family of inequalities is shown to be the only affine invariant one among them � the affine ZhangSobolev inequality. We discuss in some detail the geometry behind these analytic inequalities and also relate our new Sobolev inequalities to the sharp GromovGagliardoNirenberg Sobolev inequality for general norms and discuss further improvements of special cases. 

Gennadiy Averkov (OvGU Magdeburg)  29.11.17  16:30  MA 406  Latest news on Hensley's conjecture 
Abstract: The talk is about the maximum volume of ddimensional lattice polytopes with a fixed positive number of interior lattice points. Several researchers conjectured that the maximum is attained on a nice family of polytopes arising from the socalled Sylvester sequence. I will report on the recent progress on Hensley's conjecture. This is joint work with Benjamin Nill and Jan Kr�mpelmann. 

Didier Henrion (LAASCNRS Univ Toulouse, FR and Czech Tech Univ Prague, CZ)  22.11.17  16:30  MA 621  Convergence rates of momentsumofsquares hierarchies for volume approximation of semialgebraic sets 
Abstract: Momentsumofsquares hierarchies of semidefinite programs can be used to approximate the volume of a given compact basic semialgebraic set K. The idea consists of approximating from above the indicator function of K with a sequence of polynomials of increasing degree d, so that the integrals of these polynomials generate a convergence sequence of upper bounds on the volume of K. In this talk we show how we could derive an asymptotic rate of convergence for this approximation scheme. Joint work with Milan Korda, see arXiv:1609.02762. 

Henning Seidler  15.11.17  16:30  MA621  Circuit Polynomials for Simplex Newton Polytopes 
Abstract: Finding the minimum of a multivariate real polynomial is a wellknown hard problem with various applications. We present an implementation to approximate such lower bounds via sums of nonnegative circuit polynomials (SONCs). We provide a testsuite, where we compare our approach, using different solvers, with several solvers for sums os squares (SOS), including sostools and gloptipoly. It turns out that the circuit polynomials yield bounds competitive to SOS in several cases, but using much less time and memory. 

Samuli Lepp�nen (Dalle Molle Institute for Artificial Intelligence Research)  08.11.17  16:30  MA621  Some integrality gap results for the 0/1 SoS Hierarchy 
Abstract: In this talk we present the SoS hierarchy in the context of 0/1 optimization along with some of our recent integrality gap/lower bound results. We introduce the relevant concepts and give a brief derivation of the hierarchy. As for our results, we characterize the possible gap instances on the round n1 of the hierarchy. Then we show an unbounded integrality gap after nonconstant number of rounds for a problem that admits a polynomial time algorithm. Finally we present a simplification of the positive semidefiniteness constraints when a high degree of symmetry can be assumed, and some applications to integrality gaps and lower Bounds. 

Daryl Cooper  13.09.17  16:30  MA621  Limits of Algebraic varieties: towards a continuous Nullstellansatz 
Abstract: The map that sends a polynomial ideal to an algebraic variety in complex affine space is not continuous with respect to the obvious topology on the domain. However it is continuous on the closure of the set of ideals generated by polynomials of degree at most d. Moreover the closure of this subspace is compact. This is work in progress, and is joint with Ricky Demer. 

Dr. Iskander Aliev (Cardiff, Wales)  05.09.17  16:30  MA 406  Sparse solutions to the systems of linear Diophantine equations 
Abstract: We present structural results on sparse nonnegative integer solutions to underdetermined systems of linear equations. 

Marta Panizzut  30.08.17  16:30  MA 621  The hunt for K3 polytopes 
Abstract: The connected components of the complement of a tropical hypersurface are called regions, and they are convex polyhedra. A smooth tropical quartic surface of genus one has exaclty one bounded region. We say that a 3 dimensional polytope is a K3 polytope if it arises as the bounded region of a smooth tropical quartic surface. In a joint project with Gabriele Balletti, we are investigating properties of K3 polytopes. In this talk I will report on the progress of this work. 

Timo de Wolff  23.08.17  16:30  MA 621  Intersections of Amoebas 
Abstract: 

Georg Loho  16.08.17  16:30  MA 621  Monomial Tropical Cones for Multicriteria Optimization 
Abstract: We introduce a special class of tropical cones which we call 'monomial tropical cones'. They arise as a crucial tool in the description of discrete multicriteria optimization problems. We employ them to derive an algorithm for computing all nondominated points. Furthermore, monomial tropical cones are intimately related with monomial ideals. We sketch directions for further work. This is based on joint work with Michael Joswig. 

Johannes Hofscheier  19.07.17  16:30  MA 621  A Combinatorial Smoothness Criterion for Spherical Varieties 
Abstract: In this talk we will present algorithms to check smoothness for the class of spherical varieties which contains those of toric varieties, flag varieties and symmetric varieties, and which forms a remarkable class of algebraic varieties with an action by an algebraic group having an open dense orbit. In the toric case there is a wellknown simple combinatorial smoothness criterion whereas in the spherical case Brion, Camus and Gagliardi have shown smoothness criteria for spherical varieties which either are rather involved or rely on certain classification results. We suggest a purely combinatorial smoothness criterion by introducing a rational invariant which only depends on the combinatorics of the spherical variety. We have a conjectural inequality this invariant should satisfy where the equality case would imply that the spherical variety has a toric torus action. Our conjecture would also imply the generalized Mukai conjecture for spherical varieties. We complete our talk by summarizing in which cases the above mentioned approach is known to be true. This is joint work with Giuliano Gagliardi. 

Sinai Robins (University of S�o Paulo)  06.07.17  16:30  MA 406  The Ehrhart quasipolynomials of polytopes that come from trivalent graphs, and a discrete version of Hilbert's 3rd problem for the unimodular group 
Abstract: A graph all of whose nodes have degree \(1\) or \(3\) is called a \(\{1,3\}\)graph. Liu and Osserman associated to each \(\{1, 3\}\)graph a polytope. They studied these polytopes and their Ehrhart quasipolynomials. We prove a conjecture of Liu and Osserman stating that the associated polytopes of all connected \(\{1,3\}\)graphs with the same number of nodes and edges have the same Ehrhart quasipolynomial. We also present some structural properties of these polytopes and we show a relation between them and the wellknown metric polytopes. This study is related to a question of Hasse and McAllister on a discrete version of Hilbert's 3rd problem for the unimodular group. 

Matthias Lenz  28.06.17  16:30  MA 621  On powers of Pl�cker coordinates and representability of arithmetic matroids 
Abstract: Given \(k\in \mathbb{R}_{\ge 0}\) and a vector \(v\) of Plücker
coordinates of a point in the real Grassmannian, is the vector obtained by
taking the \(k\)th power of each entry of \(v\) again a vector of
Plücker coordinates? We will show that for \(k\neq 1\), this is true if
and only if the corresponding matroid is regular. Similar results hold over
other fields. We will also describe the subvariety of the Grassmannian that
consists of all the points that define a regular matroid. 

JeanPhilippe Labb� / Michael Joswig  21.06.17  16:30  MA 621  polymake/sage 
Abstract: JeanPhilippe will give an introduction to the new polymake interface in sage. Michael will report basic design principles underlying polymake. This is followed by a common coding session and beer. 

H�ctor Andrade Loarca  14.06.17  16:30  MA 621  Fast Multidimensional Signal Processing with Shearlab.jl. 
Abstract: We live in the age of data, huge amount of Data its been generated and acquired everyday in different forms and flavors. One would like to have a technique to store and process optimally this data; this is possible since the relevant information in almost all data found in typical applications is sparse due the high correlation of its elements. The challenge will lie in the search of appropriate dictionary that can represent optimally this information. Along the history very talented scientist have proposed different signal transforms based on certain representation dictionaries, starting with the Fourier Transform and Short Time Fourier Transform; in 1980s the wavelet transform was proposed representing a breakthrough in one dimensional signal representation, with the ability to not just optimally represent the data but also capture certain singularities, regularities and other features; the flaw of the wavelets transforms is its lack of directional sensitivity which does not aloud it to represent optimally multidimensional singularities like curves that are predominant in general multidimensional signals (for instance images and videos). The Shearlet transform is a product of almost 10 years of research on sparsifying transform with directional sensitivity that attains the optimal best Nterm approximation error, it was proposed by G. Kutyniok, D. Labate and K Guo in 2015 and since then it's been known to be the best of its type. 

Hiroshi Hirai  31.05.17  16:30  MA 621  Maximum vanishing subspace problem, CAT(0)space relaxation, and blocktriangularization of partitioned matrix 
Abstract: In this paper, we address the following algebraic generalization of the bipartite stable set problem. We are given a blockstructured matrix (partitioned matrix) \(A=(A_{\alpha\beta})\), where \(A_{\alpha\beta}\) is an \(m_{\alpha}\) by \(n_{\beta}\) matrix over field F for \(\alpha=1,2,\ldots,\mu\) and \(\beta=1,2,\ldots,\nu\). The maximum vanishing subspace problem (MVSP) is to maximize \(\sum_{\alpha}\mbox{dim}X_{\alpha}+\sum_{\beta}\mbox{dim}X_{\beta}\) over vector subspaces \(X_{\alpha} \subseteq F^{m_{\alpha}}\) for \(\alpha=1,2,\ldots,\mu\) and \(Y_{\beta}\subseteq F^{n_{\beta}}\) for \(\beta=1,2,\ldots,\nu\) such that each \(A_{\alpha\beta}\) vanishes on \(X_{\alpha}\times Y_{\beta}\) when \(A_{\alpha\beta}\) is viewed as a bilinear form \(F^{m_{\alpha}}\times F^{n_{\beta}}\rightarrow F\). This problem arises from a study of a canonical blocktriangular form of \(A\) by Ito, Iwata, and Murota (1994). We prove that MVSP can be solved in polynomial time. Our proof is a novel combination of submodular optimization on modular lattice and convex optimization on CAT(0)space. We present implications of this result on blocktriangularization of partitioned matrix. 

Ted Bisztriczky (University of Calgary)  31.05.17  16:30  MA 406  ErdosSzekeres type theorems for planar convex sets 
Abstract: A family \(\mathcal{F}\) of sets is in convex position if none of its members is contained in the convex hull of the union of the others. The members of \(\mathcal{F}\) are ovals (compact convex sets) in the plane that have a certain property. An ErdosSzekeres type theorem concerns the existence, for any integer \(n\geq 3\), of a smallest positive integer \(N(n)\) such that if \(\mathcal{F}?N(n)\) then there are \(n\) ovals of \(\mathcal{F}\) in convex position. 

Dimitrios Dais (University of Crete)  29.05.17  17:00  MA 406  Toric log del Pezzo surfaces with a unique singularity 
Abstract: The toric log del Pezzo surfaces are constructed by means of the socalled LDPlattice polygons. In the talk it will be explained how one classifies the subclass of surfaces of this kind which have a unique singularity. 

Richard Gardner (Western Washington University)  17.05.17  16:30  MA 406  Open problems in Geometric Tomography 
Abstract: This talk will focus on open problems in Geometric Tomography, which aims to retrieve information about a geometric object (such as a convex body, star body, finite set, etc.) from data concerning its intersections with planes or lines and/or projections (i.e., shadows) on planes or lines. The problems, which span nearly a hundred years of mathematics, are diverse. Many have a common thread, however, since they are linked to various integral transforms: the Xray transform, divergent beam transform, circular Radon transform, cosine transform, or spherical Radon transform. 

Michael Joswig  03.05.17  16:30  MA 621  Computing Invariants of Simplicial Manifolds 
Abstract: This is a brief survey on algorithms for computing basic algebraic invariants such as homology, cup products, and intersection forms. 

Romanos Malikiosis  26.04.17  16:30  MA 406  Formal duality in finite cyclic groups 
Abstract: Numerical computations by Cohn, Kumar, and Schu?rmann in energy minimizing periodic configurations of density \(\rho\) and \(1/\rho\) revealed an impressive kind of symmetry, the so�called formal duality. Formal dual subsets of \(\mathbb{R}^n\) satisfy a generalized version of Poisson summation formula, and it is conjectured that the only periodic subsets of \(\mathbb{R}\) of density \(1\) possessing a formal dual subset, are \(\mathbb{Z}\) and \(2\mathbb{Z}\cup(2\mathbb{Z}+\frac12)\). 

Manuel Radons  12.04.17  16:30  MA 621  Piecewise linear methods in nonsmooth optimization 
Abstract: Several techniques in numerical analysis, e.g. Newton's methods and ODE solvers, are based on local linear approximations of a smooth function \(f: \mathbb R^n\rightarrow \mathbb R^n\). If \(f\) is piecewise smooth such approximations may be arbitrarily bad near nondifferentiabilites. Piecewise linear approximations  or, briefer, piecewise linearizations  restore the structural correspondence between approximation and underlying function. We give an overview of the generalized numerical methods based on piecewise linearizations and their properties, e.g. convergence results, as well as the techniques used in their investigation, which include degree theory for piecewise affine functions and nonsmooth analysis. 

Philipp Jell  05.04.17  16:30  MA 621  Differential forms on Berkovich spaces and their relation to tropical geometry 
Abstract: 

Takayuki Hibi (Osaka University)  29.03.17  16:30  MA 621  Order Polytopes and Chain Polytopes 
Abstract: Given a finite partially ordered set P, we associate two poset polytopes, viz, the order polytope O(P) and the chain polytope C(P). In my talk, after reviewing fundamental materials on O(P) and C(P), the question when O(P) and C(P) are unimodularly equivalent will be discussed. Then, in the frame of Gr�bner bases, various reflexive polytopes arising from O(P) and C(P) will be presented. Finally, some questions and conjectures will be summarized. No special knowledge will be required to understand my talk. 

Luc�a L�pez de Medrano (UNAM Cuernavaca)  22.03.17  17:00  MA 621  On the genus of tropical curves 
Abstract: 

Felipe Rinc�n (University of Oslo)  22.03.17  16:15  MA 621  Tropical Ideals 
Abstract: 

Skip Jordan  22.03.17  15:30  MA 621  Parallel Vertex and Facet Enumeration with mplrs 
Abstract: We introduce mplrs, a new parallel vertex/facet enumeration program based on the reversesearch code lrs. The implementation uses MPI and can be used on single machines, clusters and supercomputers. We describe the budgeted parallel tree search implemented in mplrs and compare performance with other sequential and parallel programs for vertex/facet enumeration. In some instances, mplrs achieves almost linear scaling with more than 1000 cores. The approach used in mplrs can be easily applied to various other problems. This is joint work with David Avis. 

Alex Fink  22.02.17  16:30  MA 621  The characteristic polynomial two ways 
Abstract: The theory of hyperplane arrangements and matroids derives great utility from the application of algebraic and algebrogeometric methods. The characteristic polynomial often appears in familiar guise in these methods, as an enumerator of the no broken circuit sets. The characteristic polynomial also appears, on the face of it unrelatedly, in the recent magisterial resolution of Rota's conjecture by Huh, Katz, and Adiprasito. In joint work with David Speyer and Alex Woo we explain how these two manifestations are related after all. 

Akiyoshi Tsuchiya  08.02.17  16:30  MA 621  Gorenstein simplices and the associated finite abelian groups 
Abstract: A lattice polytope is a convex polytope each of whose vertices has integer coordinates. It is known that a lattice simplex of dimension \(d\) corresponds to a finite abelian subgroup of \((\mathcal{R}/\mathcal{Z})^{d+1}\). Conversely, given a finite abelian subgroup of \((\mathcal{R}/\mathcal{Z})^{d+1}\) such that the sum of all entries of each element is an integer, we can obtain a lattice simplex of dimension \(d\). In this talk, we discuss a characterization of Gorenstein simplices in terms of the associated finite abelian groups. Gorenstein polytopes are of interest in combinatorial commutative algebra, mirror symmetry and tropical geometry. In particular, we present complete characterizations of Gorenstein simplices whose normalized volume equal \(p\), \(p^2\) and \(pq\), where \(p,q\) are prime integers. 

Robert L�we  25.01.17  16:30  MA 621  Secondary fans of a punctured Riemann surface 
Abstract: The secondary fan of a finite point configuration in \(R^n\) stratifies the space of height functions by the combinatorial types of regular subdivisions. Now given a punctured Riemann surface, similar techniques yield a polyhedral fan whose cones correspond to horocyclic Delaunay tessellation in the sense of Penner's convex hull construction. The purpose of the talk is to give an introduction to this topic and the corresponding problems that arise naturally. 

Eva Maria Feichtner (Universit�t Bremen)  18.01.17  16:30  MA 406  A Leray model for OrlikSolomon algebras 
Abstract: Although hyperplane arrangement complements are rationally formal,
we note that they have nonminimal rational (CDGA) models which are topologically and combinatorially significant. We construct a family of CDGAs which interpolates between the OrlikSolomon algebra and the cohomology algebras of arrangement compactifications. Our construction is combinatorial and extends to all matroids, regardless of their (complex) realizability. 

Benjamin Schr�ter  04.01.17  16:30  MA 621  Fundamental Polytopes of Metric Trees via Hyperplane Arrangements 
Abstract: In this talk I will present a summary of the recent paper 'Fundamental Polytopes of Metric Trees via Hyperplane Arrangements' (arXiv:1612.05534) of Emanuele Delucchi and Linard Hossly. In this paper they develop a formula for the fvector of the fundamental polytope of a finite metric space and its dual the Lipschitz polytope. 

Georg Loho  04.01.17  17:15  MA 621  Slicing and Dicing Polytopes 
Abstract: I present the recent work 'Slicing and dicing polytopes' (arXiv:1608.05372) by Patrik Nor�n on cellular resolutions. I introduce diced and sharp polytopes which are necessary for the construction and give a brief introduction to cellular resolutions. The main result, introducing a new class of polyhedral subdivisions which support cellular resolutions, is illustrated along some examples. 

Marta Panizzut  26.10.16  13:30  MA 621  Linear systems on metric graphs, gonality sequence and lifting problems 
Abstract: A combinatorial theory of linear systems on graphs and metric graphs has been introduced in analogy with the one on algebraic curves. The interplay is given by the Specialization Lemma. Let \(X\) be a smooth curve over the field of fractions of a complete discrete valuation ring and let \(\mathfrak{X}\) be a strongly semistable regular model of \(X\). It is possible to specialize a divisor on the curve to a divisor on the dual graph of the special fiber of \(\mathfrak{X}\); through this process the rank of the divisor can only increase. The complete graph \(K_d\) pops up if we take a model of a smooth plane curve of degree d degenerating to a union of d lines. Moreover omitting edges from \(K_d\) can be interpreted as resolving singularities of a plane curve. In this talk we present some results on linear systems on complete graphs and complete graphs with a small number of omitted edges, and we will compare them with the corresponding results on plane curves. In particular, we compute the gonality sequence of complete graphs and the gonality of graphs obtained by omitting edges. We explain how to lift these graphs to curves with the same gonality using models of plane curves with nodes and harmonic morphisms. This is partially a joint work with Filip Cools. 

Fei Xue (BMS)  19.10.16  16:30  MA 406  On Lattice Coverings by Simplices 
Abstract: By studying the volume of a generalized difference body, this talk presents the first nontrivial lower bound for the lattice covering density by \(n\)dimensional simplices. 

Ngoc Tran  05.10.16  14:45  MA 621  Tropical geometry and mechanism design 
Abstract: How to ensure an outcome based on collective decisions is 'better for all' when a) everyone acts in their personal interest only, and b) we do not know the exact 'motives' of each person? Crudely, this is the central problem in mechanism design, a branch of economics. In this talk, I show how insights from tropical geometry can solve such problems. 

Thomas Kahle  05.10.16  13:30  MA 621  The geometry of rankone tensor completion 
Abstract: tba 

Andr� Wagner  28.09.16  16:30  MA 621  Veronesean almost binomial almost complete intersections 
Abstract: The second Veronese ideal I_n contains a natural complete intersection J_n generated by the principal 2minors of a symmetric (n � n)matrix. In this talk I show, how to determine the subintersections of the primary decomposition of J_n where one intersectand is omitted. These subintersections can be described via combinatorial method and yield interesting insights into binomial ideals. This talk is based on joint work with Thomas Kahle. 

Laura Silverstein (TU Wien)  27.07.16  16:30  MA 406  Tensor Valuations on Lattice Polytopes 
Abstract: A classification of symmetric tensor valuations on lattice polytopes in \(\mathbb{R}^n\) that intertwine the special linear group over the integers is established for the cases in which the rank of the tensor is at most \(n\). The scalarvalued case was classified by Betke and Kneser where it was shown that the only such valuations are the coefficients of the Ehrhart polynomial. Extending this result, the coefficients of the discrete moment tensor form a basis for the symmetric tensor valuations. 

Apostolos Giannopoulos (University of Athens)  20.07.16  16:30  MA 406  Inequalities about sections and projections of convex bodies 
Abstract: We discuss lower dimensional versions of the slicing problem and of the BusemannPetty problem, both in the classical setting and in the generalized setting of arbitrary measures in place of volume. We introduce an alternative approach which is based on the generalized BlaschkePetkantschin formula, on asymptotic estimates for the dual affine quermassintegrals and on some new LoomisWhitney type inequalities in the spirit of the uniform cover inequality of Bollobas and Thomason. 

Gerassimos Barbatis (University of Athens)  13.07.16  16:30  MA 406  On the Hardy constant of some nonconvex planar domains 
Abstract: The Hardy constant of a simply connected domain \(\Omega\subset\mathbb{R}^2\) is the best constant for the inequality \[ \int_\Omega \nabla u^2 dx \geq c\int_\Omega \frac{u^2}{\operatorname{dist}(x,\partial\Omega)^2} dx, \quad u\in C_c^\infty(\Omega). \] After the work of Ancona where the universal lower bound 1/16 was obtained, there has been a substantial interest on computing or estimating the Hardy constant of planar domains. In this talk we determine the Hardy constant of an arbitrary quadrilateral as well as of some other planar domains (joint work with Achilles Tertikas). 

Matthew Trager  15.06.16  16:30  MA 621  Geometric models for computer vision 
Abstract: The goal of computer vision is to recover realworld information, either semantic (e.g., object recognition) or geometric (3D reconstruction), from sets of photographs or videos. In a broad sense, my research aims at investigating the fundamental geometric models, both synthetic and analytic, that can be used to describe shapes, cameras, and image contours, in processes of visual inference. In this talk, I will present some properties of 'silhouettes' that are obtained as projections of objects in space. I will also discuss some ongoing work that recasts traditional multiview vision in terms of the Grassmannian of lines in P^3. This is joint work with JeanCharles Faugere, Xavier Goaoc, Martial Hebert, Jean Ponce, Mohab Safey El Din and Bernd Sturmfels. 

Axel Flinth  18.05.16  16:30  MA 406  High Dimensional Geometry in Compressed Sensing 
Abstract: The main aim of compressed sensing is to recover a lowdimensional object (e.g. a sparse vector, a lowrank matrix, ...) from few linear measurements. Many algorithms for this task (e.g. Basis Pursuit, nuclear normminimization) consists of minimizing a structurepromoting function \(f\) over the set of possible solutions. 

Amanda Cameron  04.05.16  16:30  MA 621  A lattice point counting generalisation of the Tutte polynomial 
Abstract: The Tutte polynomial for matroids is not directly applicable to polymatroids. For instance, deletioncontraction properties do not hold. We construct a polynomial for polymatroids which behaves similarly to the Tutte polynomial of a matroid, and in fact contains the same information as the Tutte polynomial when we restrict to matroids. This polynomial is constructed using lattice point counts in the Minkowski sum of the base polytope of a polymatroid and scaled copies of the standard simplex. We also show that, in the matroid case, our polynomial has coefficients of alternating sign, with a combinatorial interpretation closely tied to the Dawson partition. 

Paul Breiding  27.04.16  16:30  MA 621  The spectral Atheory of tensors 
Abstract: The spectral theory of tensors aims at generalizing the concept of eigenvalues and eigenvectors of matrices to higher dimensional arrays. In this talk I will focus on the definition of Eeigenvalues and Zeigenvalues given by Qi. After giving the definition itself I will explain various applications, give the higher dimensional equivalent of the characteristic polynomial and talk about the number of eigenvalues of both real and complex tensors. The last part of the talk will be about an algorithm to compute eigenvalues of tensors using homotopy methods and the complexity of this algorithm. 

Shiri ArtsteinAvidan (Tel Aviv University)  27.04.16  11:00  MA 406  On Godbersen�s conjecture and related inequalities 
Abstract:  

Stefan Weltge (OttovonGuericke Universit�t Magdeburg)  20.04.16  16:30  MA 406  Tight bounds on discrete quantitative Helly numbers 
Abstract: Given a subset S of R^n, let c(S,k) be the smallest number t such that whenever finitely many convex sets have exactly k common points in S, there exist at most t of these sets that already have exactly k common points in S. For S = Z^n, this number was introduced by Aliev et al. [2014] who gave an explicit bound showing that c(Z^n,k) = O(k) holds for every fixed n. Recently, Chestnut et al. [2015] improved this to c(Z^n,k) = O(k (log log k)(log k)^{1/3} ) and provided the lower bound c(Z^n,k) = Omega(k^{(n1)/(n+1)}). We provide a combinatorial description of c(S,k) in terms of polytopes with vertices in S and use it to improve the previously known bounds as follows: We strengthen the bound of Aliev et al. [2014] by a constant factor and extend it to general discrete sets S. We close the gap for Z^n by showing that c(Z^n,k) = Theta(k^{(n1)/(n+1)}) holds for every fixed n. Finally, we determine the exact values of c(Z^n,k) for all k <= 4. 

Anna Lena Birkmeyer  16.03.16  16:30  MA 621  Realizability of Tropical Hypersurfaces in Matroid Fans 
Abstract: In tropical geometry, an important aim is to understand which tropical varieties arise as tropicalizations of algebraic varieties. We investigate this question in a relative setting: Given a matroid fan F coming from a linear space W, we ask if a tropical hypersurface in F is the tropicalization of an algebraic subvariety of W. We present an algorithm able to decide for any tropical hypersurface in F if it is realizable in W. Moreover, we use this algorithmic approach to describe the structure of the realization space of a tropical hypersurface H in F, i.e. the space of algebraic subvarieties of W tropicalizing to H, and show that the space of all realizable tropical hypersurfaces in F is an abstract polyhedral set. 

Martin Genzel  02.03.16  16:30  MA 406  Convex Recovery of Structured Signals from NonLinear Observations 
Abstract: In this talk, we study the problem of estimating a structured signal \(x_0\in\mathbb{R}\) linear Gaussian observations. Supposing that \(x_0\) belongs to a certain convex subset \(K\subset\mathbb{R}\) we will see that an accurate recovery is possible as long as the number of observations exceeds the effective dimension of \(K\), which is a common measure for the complexity of signal classes. Interestingly, it will turn out that the (possibly unknown) nonlinearity of our model affects the error rate only by a multiplicative constant. This achievement is based on recent works by Yaniv Plan and Roman Vershynin, who have suggested to treat the nonlinearity rather as noise which perturbs a linear measurement process. Using the concept of restricted strong convexity, we show that their results for the generalized Lasso can be extended to a fairly large class of convex loss functions. This is especially appealing for practical applications, since in many realworld scenarios, adapted loss functions empirically perform better than the classical square loss. To this end, the presented results provide a unified and general framework for signal reconstruction in high dimensions, covering various challenges from the fields of compressed sensing, signal processing, and statistical learning. 

Mihalis Kolountzakis (University of Crete)  03.02.16  16:30  MA 406  Periodicity for tilings and spectra 
Abstract: We will talk about periodicity (and structure, more generally) in the study of tilings by translation, where the tile is a set or a function in an Abelian group, and also in the study of spectra of sets (sets of characters which form an orthogonal basis for L^2 of the set). There are connections to harmonic analysis, number theory, combinatorics and computation, and these make this subject so fascinating. Starting from the Fuglede conjecture, now disproved in dimension at least 3, which would connect tilings with spectra, we will go over cases where periodicity always holds, cases where it is optional and cases where it's never true, in one dimension (most positive results) and higher dimension (most interesting questions). 

Simon Hampe  20.01.16  16:30  MA 621  Matroids over hyperfields 
Abstract: This talk is a summary of the very recent paper by Matthew Baker of the same title (http://arxiv.org/pdf/1601.01204.pdf). Various flavours of ''matroids with additional structure'', such as valuated and oriented matroids have been around for quite a while, with often very little connection between the different matroid species. Matroids over hyperfields are a generalization of all those concepts  thus allowing a unified treatment. In this talk I will first give a short introduction to hyperfields. I will then define what matroids over hyperfields are, give examples and discuss generalizations of familiar matroid operations such as duality and minors. 

Silouanos Brazitikos (University of Athens)  13.01.16  16:30  MA 406  Quantitative versions of Helly's theorem 
Abstract: We provide a new quantitative version of Helly's theorem: there exists an absolute constant\(\alpha >1\)with the following property: if\(\{P_i: i\in I\}\)is a finite family of convex bodies in\({\mathbb R}^n\)with\({\rm int}\left (\bigcap_{i\in I}P_i\right)\neq\emptyset\), then there exist\(z\in {\mathbb R}^n\),\(s\ls \alpha n\)and\(i_1,\ldots i_s\in I\)such that \begin{equation*} z+P_{i_1}\cap\cdots\cap P_{i_s}\subseteq cn^{3/2}\left(z+\bigcap_{i\in I}P_i\right), \end{equation*} where\(c>0\)is an absolute constant. This directly gives a version of the 'quantitative' volume and diameter theorem of > B\'{a}r\'{a}ny, Katchalski and Pach, with a polynomial dependence on the dimension. 

Kristin Shaw  06.01.16  16:30  MA 621  What is �tale cohomology? 
Abstract: Wished for by Weil and defined by Grothendieck, �tale cohomology is to varieties over fields of finite characteristic what usual cohomology is to complex or real algebraic varieties when they are equipped with the analytic, respectively Euclidean topology. This very short introduction will motivate and try to explain the basic ingredients of �tale cohomology. Throughout I will make analogies to the complex and real situations and stick to simple examples in hopes to provide some intuition into this deep theory. 

Benjamin Schr�ter  02.12.15  16:30  MA 621  The degree of a tropical basis 
Abstract: In this talk I will present my work with Michael Joswig. I will give an explicit bound on the degree of a tropical basis. Our result is derived from the algorithm of Bogart, Jensen, Speyer, Sturmfels and Thomas that computes a tropical variety via Groebner bases and saturation. Furthermore I will give examples that illustrate the difference between tropical and Groebner bases. 

Bernardo Gonz�lez Merino (TU M�nchen)  18.11.15  16:30  MA 406  On the Minkowski measure of symmetry 
Abstract: The Minkowski measure of symmetry s(K) of a convex body K, is the smallest positive dilatation of K containing a translate of K. In this talk we will explain some of its basic properties in detail. 

Andr� Wagner  04.11.15  16:30  MA 621  Introduction to singular value decomposition 
Abstract: In this short talk I will give a brief introduction to singular value decomposition, some of it's applications and efficient ways to compute it. 

Jonathan Spreer  28.10.15  16:30  MA 621  Algorithms and complexity for TuraevViro invariants 
Abstract: The TuraevViro invariants are a powerful family of topological invariants for distinguishing between different 3manifolds. They are invaluable for mathematical software, but current algorithms to compute them require exponential time. I will discuss this family of invariants, and present an explicit fixedparameter tractable algorithm for arbitrary r which is practical  and indeed preferable  to the prior state of the art for real computation. 

Alexander Gamkrelidze  11.08.15  15:00  MA 621  Algorithms for Convex Topology: Computation of HomComplexes 
Abstract: After a brief introduction to the field of polytopal complexes and affine maps between them, we define the set Hom(P,Q) of all possible affine mappings between two polytopal complexes P and Q that builds a polytopal complex in a higher dimensional space. There are efficient software packages like Polymake that deal (among other things) with the computation of Homcomplexes between two polytopes, but no complexes. In this talk, we introduce algorithms that compute the above mentioned structures and their homotopies. At the end of the talk we introduce a conjectured geometric analog to a well known formula Hom(P,ker(f))=kerHom(P,f): Hom(P,\(\Sigma\)(f))=\(\Sigma\) Hom(P,f) and discuss some ideas to compute the Sigma operator and the possible ways to prove the equation. In case the above conjecture holds, the wide area of the algebraic theory of polytopes can be established. 
Konferenzen/Workshops
Titel  Datum  Ort  Homepage 

8th polymake workshop  February 24, 2017  TU Berlin  8th polymake workshop 
7th polymake workshop  January 2830, 2016  TU Berlin  7th polymake workshop 
Meeting on Algebraic Vision 2015  October 89, 2015  TU Berlin  MAV 
Tropical geometry in Europe  March 3031, 2015  Berlin  TGE 
6th polymake workshop  December 5,2014  TU Berlin  6th polymake workshop 
5th polymake workshop  March 28, 2014  TU Berlin  5th polymake workshop 
4th polymake conference and developer meeting at TU Berlin  November 12, 2013  TU Berlin  4th polymake conference and developer meeting 
Delaunay Geometrie: Polytopes, Triangulations and Spheres  October 79,2013  FU Berlin  Second ERC "SDModels" Workshop 