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Inhalt des Dokuments


Jaeho Shin (Seoul National University)12.08.202016:30Online Polytropes and Tropical Linear Spaces

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)29.07.202016:30Online TBA

Stephane Gaubert (INRIA and CMAP, Ecole polytechnique, CNRS)08.07.202016:30Online Understanding and monitoring the evolution of the Covid-19 epidemic from medical emergency calls: the example of the Paris area

We portray the evolution of the Covid-19 epidemic during the crisis of March-April 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 Perron-Frobenius theory and tropical geometry.

Mara Belotti (SISSA)24.06.202016:30Online Topology of rigid isotopy classes of geometric graphs

We study the topology of rigid isotopy classes of geometric graphs on n vertices in a d-dimensional 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)17.06.202016:30Online Inscribable fans, type cones, and reflection groupoids

Steiner posed the question if any 3-dimensional 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)03.06.202016:30Online The degree of Stiefel manifolds

The Stiefel manifold is the set of orthonormal bases for k-planes in an n-dimensional space. We compute its degree as an algebraic variety in the space of k-by-n matrices using techniques from classical algebraic geometry, representation theory, and combinatorics. We give a combinatorial interpretation of this degree in terms of non-intersecting lattice paths. (This is joint work with Fulvio Gesmundo)

Thorsten Theobald (Goethe University Frankfurt)27.05.202016:30Online Conic stabilility of polynomials and spectrahedra

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 n-space, a multivariate polynomial f in C[z] is called K-stable if it does not have a root whose vector of the imaginary parts is contained in the interior of $K$. If $K$ is the non-negative 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)20.05.202016:30Online Volume product and metric spaces

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)13.05.202016:30Online Oriented Matroids from Triangulations of Products of Simplices

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 Baker-Bowler’s matroids over hyperfields and our work. This is joint work with Marcel Celaya and Chi-Ho Yuen.

Martin Winter (TU Chemnitz)06.05.202016:30Online Edge-Transitive Polytopes

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 flag-transitive (aka. regular) polyhedra (called, Platonic solids), six flag-transitive 4-polytopes and exactly three flag-transitive d-polytopes for any d>=5. The symmetry requirement of flag-transitivity is therefore quite restrictive for convex polytopes. On the other end of the spectrum, the class of vertex-transitive (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 k-faces for any k>=2, and their interactions.
In this talk we discuss the next most accessible transitivity, namely, edge-transitivity of convex polytopes. That is, we ask for a classification of convex polytopes in which all edges are identical under the symmetries of the polytope. Despite this restriction feeling more similar to vertex-transitivity than to flag-transitivity, we will see that the contrary seems to be the case: the class of edge-transitive polytopes appears to be quite restricted. We give, what we believe to be, a complete list of all edge-transitive polytopes, as well as a full classification for certain interesting sub-classes. Thereby, we show how edge-transitive polytopes can be studied with the tools of spectral graph theory.

Marek Kaluba29.04.202017:00Online Random polytopes in Machine Learning

In this talk I will describe very recent work on analyzing different models of auto-encoder neural networks using random polytopes. In general, a class of neural networks known as auto-encoders 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".
Although understanding the properties of these embedings is crucial for explainability (e.g. interpreting the decisions taken by the network), the research in this direction has been undertaken only from the statistical point of view. We introduce `random polytope descriptors` which try to approximate datasets in the feature space by possibly tight convex bodies. This is the first attempt at understanding the very coarse geometry of embeddings provided by neural networks. Using such descriptors one can answer e.g. if the natural clusters of of images has been preserved or distorted by the neural network, and if the embedding actually separates them (the question of entanglement).
We use random polytope descriptors to examine the behaviour of auto-encoder networks in both vanilla and variational form, and provide first evidence for susseptibility of the latter to out-of-distribution attacks.
This is joint work with L.Ruffs (TUB, Department of Electrical Engineering and Computer Science) and M.Joswig (TUB, Chair of Discrete Mathematics/Geometry; MPI for Mathematics in the Sciences, Leipzig).

Francisco Criado29.04.202016:30Online Borsuk’s problem

In 1932, Borsuk proved that every 2-dimensional 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 d-dimensional convex body can be divided in (d+1) pieces of strictly smaller diameter?
The answer to this question is positive in 3 dimensions (Perkal 1947), but in 1993, Kahn and Kalai constructed polytopal counterexamples in dimension 1325. Nowadays, we know that the answer is "no" for all dimensions d>=64 (Bondarenko and Jenrich 2013). It remains open for every dimension between 4 and 63
In this talk we introduce the problem, and we show the proofs for dimension 2 and 3. plus computational ideas on how to extend these solutions to dimension 4.

Jonathan Spreer (University of Sydney)22.04.202016:30MA 621 Linking topology and combinatorics: Width-type parameters of 3-manifolds

Many fundamental topological problems about 3-manifolds 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 3-manifold comes with a sufficiently "thin" presentation.
More specifically, a 3-manifold given as a triangulation is considered thin, if the treewidth of its dual graph is small. I will show how this combinatorial parameter, defined on a triangulation, can be linked back to purely topological properties of the underlying manifold. From this connection it can then be followed that, for some 3-manifolds, we cannot hope for a thin triangulation.
This is joint work with Krist�f Husz�r and Uli Wagner

Ayush Kumar Tewari15.04.202017:00Online Forbidden patterns in tropical planar curves and Panoptagons

Dominic Bunnett15.04.202016:30Online Geometric discriminants and moduli

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 A-discriminants of Gelfand, Kapranov and Zelevinsky.

Robert Loewe08.04.202016:30Online Minkowski decompositions of generalized associahedra

We give an explicit subword complex description of the rays of the type cone of the g-vector fan of an finite type cluster algebra with acyclic initial seed. In particular, this yields a non-recursive description of the Newton polytopes of the F-polynomials 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 Kastner01.04.202016:30Online Hyperplane arrangements in polymake

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.
This is joint work with Marta Panizzut.

Paul Vater25.03.202017:00Online Patchworking real tropical hypersurfaces

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 Newman25.03.202016:30Online A two-step random polytope model

We consider a model for generating non-simple, non-simplicial random polytopes. The first step of the random process generates a random polytope P via random hyperplanes tangent to the d-dimensional 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 Yuruk11.03.202016:30MA 621 Understanding the Parameter Regions of Multistationarity in Dual Phosporylation Cycle via SONC

Parameterized ordinary differential equation systems are crucial for modeling in biochemical reaction networks under the assumption of mass-action 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)26.02.202016:30MA 621 Tropically planar graphs: geometry and combinatorics

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)12.02.202016:30MA 621 Binomial edge ideals of bipartite graphs

Binomial edge ideals are ideals generated by binomials corresponding to the edges of a graph, naturally generalizing the ideals of 2-minors of a generic matrix with two rows. They also arise naturally in the context of conditional independence ideals in Algebraic Statistics.
We give a combinatorial classification of Cohen-Macaulay binomial edge ideals of bipartite graphs providing an explicit construction in graph-theoretical terms. In the proof we use the dual graph of an ideal, showing in our setting the converse of Hartshorne�s Connectedness theorem.
As a consequence, we prove for these ideals a Hirsch-type conjecture of Benedetti-Varbaro.
This is a joint work with Davide Bolognini and Francesco Strazzanti. Organizer: DM/G

Amy Wiebe (FU Berlin)05.02.202016:30MA 621 Combining Realization Space Models of Polytopes

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)29.01.202016:30MA 621 On a canonical symmetry breaking technique for polytopes

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 G-orbit, 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 NP-hard to separate even for structurally simple groups whose elements are disjoint involutions (Babai & Luks 1983).
We consider a recent set of inequalities that has been implemented in CPLEX as a symmetry breaking technique for arbitrary finite permutation groups (Salvagnin 2018). We show a strong connection of this set with the set of lexicographically maximal vectors (LEX), and show that it defines a Fundamental Domain with quadratic many facets on the dimension, yielding a polynomial time separation algorithm. Moreover, we study when LEX defines a closed set, which suggests a stronger way of breaking symmetries.
This is joint work with Jos� Verschae and L�onard von Niederh�usern.

Andres R. Vindas Melendez (University of Kentucky)22.01.202016:30MA 621 The Equivariant Ehrhart Theory of the Permutahedron

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)15.01.202016:30MA 621 Numerical Root Finding via Cox Rings

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)11.12.201916:30MA 621 Anticanonical tropical del Pezzo cubic surfaces contain exactly 27 lines.

Since the beginning of tropical geometry, a persistent challenge has been to emulate tropical versions of classical results in algebraic geometry. The well-known 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.
In this talk I will explain how to correct this pathology by viewing the surface as a del Pezzo cubic and considering its embedding in P^44 via its anticanonical bundle. The combinatorics of the root system of type E_6 and a tropical notion of convexity will play a central role in the construction. This is joint work with Anand Deopurkar (arXiv: 1906.08196).

Roser Homs Pons (TU Berlin)20.11.201916:30MA 621 Hilbert-Burch matrices of ideals in k[x,y] and k[[x,y]]

In this talk we will discuss Hilbert-Burch 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 zero-dimensional ideals in the ring of polynomials and the ring of formal power series in two variables.

Sascha Timme (TU Berlin)13.11.201916:30MA 621 3264 Conics in a Second

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)30.10.201916:30MA 406 TBA


Laura Brustenga i Moncusi (TU Berlin, UAB)23.10.201916:30MA 621 Numerically computing the local dimension of an algebraic set.

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)16.10.201916:30MA 621 An Alternative Characterisation of Valuated Matroids

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)02.10.1916:30MA 621 On the discriminant of a cubic quaternary form

We determine the 166104 extremal monomials of the discriminant of a quaternary cubic form. These are in bijection with D-equivalence classes of regular triangulations of the 3-dilated tetrahedron. We describe how to compute these triangulations and their D-equivalence 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 D-equivalence class.

Daniel Galicer (University of Buenos Aires)11.09.1916:30MA 406 TBA


Dominic Bunnett04.09.1916:30MA 621 Stability of hypersurfaces and Newton Polytopes

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)31.07.1916:30MA 621 Certified Semidefinite Approximations of Reachable Sets

We consider the problem of approximating the reachable set of a continuous or discrete-time 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.
The proposed method consists of building a hierarchy of relaxations for an infinite-dimensional moment problem. Under certain assumptions, the optimal value of this problem is the volume of the reachable set and the optimum solution is the restriction of the Lebesgue measure on this set. Then, one can outer approximate the reachable set as closely as desired with a hierarchy of super level sets of increasing degree polynomials. For each fixed degree, finding the coefficients of the polynomial boils down to computing the optimal solution of a convex semidefinite program. When the degree of the polynomial approximation tends to infinity, we provide strong convergence guarantees of the super level sets to the reachable set.
We also present some application examples together with numerical results.

Andrew Newman, Francisco Criado (TU Berlin)24.07.1916:30MA 621 Randomness in Algebra and Topology / Minkowski decompositions of polytopes via geometric graphs: follow-up

I will give a short talk about the recent Summer School at MPI Leipzig on Randomness and Learning in Non-Linear Algebra. I will focus particularly on models of random monomial ideals and their connections with random graphs and simplicial complexes.


This talk is a follow-up on Guillermo Pineda's previous DMG talk: "Minkowski decompositions of polytopes via geometric graphs". In his work (Pineda-Villacicencio et al (2018)), there is an example of a 3-dimensional polytope that admits both a decomposable (i.e.: Minkowski sum of non-homothetic 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)10.07.1916:30MA 406 On the affine surface area

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)05.07.1912:00MA 406 A chirotope-based proof of the Bohne-Dress theorem

The celebrated Bohne-Dress theorem characterizes tilings of a given zonotope $\mathcal{Z}$ by zonotopes in terms of single-element 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)03.07.1916:30MA 621 Minimal triangulations of cusped hyperbolic 3-manifolds

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 3-simplices, are used in census enumeration, and as a platform to study the topology of the manifold using normal surface theory.
In this talk, I introduce the basic notions, describe basic facts about minimal ideal triangulations of cusped hyperbolic 3-manifold, and some sharp lower bounds on the number of ideal tetrahedra in a minimal ideal triangulation coming from homology. This is joint work with Bus Jaco, Hyam Rubinstein and Jonathan Spreer.

Sarah Morell26.06.1916:30MA 621 On the single-source unsplittable flow problem

The single-source unsplittable flow problem has been introduced by Kleinberg (1996) and generalizes several NP-complete problems from various areas in combinatorial optimization such as packing, partitioning, scheduling and load balancing. Commodities must be routed simultaneously in a given arc-capacitated 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.
Twenty years ago, in a landmark paper, Dinitz, Garg and Goemans proved that any multiple-path routing can be turned into a single-path routing with an increase of flow on each arc bounded by the value of the maximum demand. In the first part of the talk, we present a completely new and considerably simpler proof for this result, with the goal of making progress towards the related but still unresolved Conjecture of Goemans for minimum-cost unsplittable flows. In the second part of the talk, we focus on known relations between the single-source unsplittable flow problem and polytopes.
This is joint work with Martin Skutella.

Rekha Thomas (University of Washington)19.06.1916:30MA 621 The Slack Realization Space of a Polytope

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, non-prescribability 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.

Joint work with Joao Gouveia, Antonio Macchia and Amy Wiebe

Alexander Koldobsky (University of Missouri)12.06.1916:30MA 406 An estimate for the distance from a convex body to subspaces of L_p


Guillermo Pineda Villavicencio (Federation University Australia)29.05.1916:30MA 621 Minkowski decompositions of polytopes via geometric graphs

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.
As applications, we revisit many of the known results in this area, including the classifications into decomposable and decomposable, of d-polytopes with at most 2d vertices due to Kallay (1979), and of d-polytopes with 2d+1 vertices due to Pineda-Villacicencio et al (2018). Classifying d-polytopes with 2d+2 vertices seems not possible at least for d=3; there exists a 3-polytope with both a decomposable realisation and an indecomposable realisation, showing in the strongest possible way that decomposability is not a combinatorial property.
The talk concludes with the main open problems in the area.

Giulia Codenotti (FU Berlin)22.05.1916:30MA 406 The covering minima of lattice polytopes

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 non-hollow lattice polytopes.

Boasz Slomka (Weizmann Institute)15.05.1916:30MA 406 On Hadwiger�s covering conjecture


Alheydis Geiger (Tuebingen)08.05.1916:30MA 621 Realisability of infinite families of tropical lines on general smooth tropical cubic surfaces

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.)
The tropical lines on a smooth tropical cubic surface can be classified by their combinatorial position. Thus we can distinguish isolated lines and two-point families. The latter are infinite families of tropical lines on the same surface of the same combinatorial position. There are two combinatorial positions of lines of general smooth tropical cubic surfaces that allow infinite families. The question is, whether all the lines in these families can be realised on some lift of the tropical cubic surface. This problem is only solved for one of the two positions and only if char(k)?2 I will present the solution.

Andreas Paffenholz17.04.1916:30MA 621 tropical cubic surfaces

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^(d-1) are positive, as those coefficients have a geometric interpretation. Much less is known about the other coefficients.
In the talk I first introduce some families of polytopes where it is known that all coefficients of the Ehrhart polynomial is positive, either via a suitable construction or because the coefficients have a combinatorial interpretation. In the second part I will introduce constructions for polytopes where at least one coefficient is negative. In particular I will discuss smooth polytopes. Here, it has been conjectured that such polytopes have an Ehrhart polynomial with positive coefficients. I will disprove this conjecture by providing explicit examples, also in the much smaller class of smooth reflexive polytopes.
This is joint work with F. Castillo, F. Liu, B. Nill.

Erika Roldan10.04.1916:30MA 621 Evolution of the homology and related geometric properties of the Eden Growth Model.

In this talk, we study the persistent homology and related geometric properties of the evolution in time of a discrete-time stochastic process defined on the 2-dimensional 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 2-dimensional 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 time-scaling. 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)03.04.1916:30MA 621 Radons: Shellability, Yuruk: Lattice Polytopes and Face Enumeration of Certain Classes of Simplicial Polytopes, Newman: Random polytopes

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.
Yuruk: This talk will summarize a certain part of Spring School on Polytopes in Ruhr University Bochum. We will first cover a lattice polytopes part of the lecture on geometry of polytopes. Then, we talk about the face enumeration of certain classes of simplicial polytopes that were discussed in the spring school.
Newman: Following the lectures of Pierre Calka at the Bochum meeting, I will discuss the history of the study of random polytopes going back to Sylvester's Four Point Problem. From there I will also discuss recent results of Werner, Reitzner, and others on the subject.

Holger Eble and Ayush Tewari (TU Berlin)27.03.1916:30MA 621 Eble: Epistatic weights and polyhedral geometry Tewari: Moduli of tropical plane curves of genus 6

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.
Tewari: In the paper 'Moduli of tropical plane curves Brodsky', etc (2015) compute the moduli space of tropical curves of genus g, with g = 3, 4 and 5. In this talk, we would delve into computations trying to extend that computation to the case of g = 6.

Jonathan Kliem (FU Berlin)20.03.1916:30MA 621 A face iterator for polyhedra

We discuss a new algorithm that iterates over all elements of a meet-semilattice, where every interval is coatomic.

Jens Forsgaard (Universiteit Utrecht)13.02.1916:30MA 621 Nonegativity and Discriminants

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 semi-algebraic 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 Krattenthaler13.02.1915:30MA 621 Discrete analogues of Macdonald-Mehta integrals

The Mehta integral and its generalisations due to Macdonald originate from random matrix theory, but are as well important objects in the theory of (multi-variable) orthogonal polynomials and in combinatorics. These (now) classical integrals are certain multi-dimensional integrals which - suprisingly? - can be evaluated into closed form product formulae. I shall consider certain discretisations of the Macdonald-Mehta 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 non-intersecting 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)06.02.1916:30MA 406 Information theoretic inequalities and their convex geometric analogues

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)23.01.1916:30MA 621 Tropical patchworking for nodal curves in the plane (joint with Ilia Itenberg and Grigory Mikhalkin)

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 Wolff09.01.1916:30MA 621 Nondegenerate Multistationarity in Small Reaction Networks

Much attention has been focused in recent years on the following algebraic problem arising from applications: which chemical reaction networks, when taken with mass-action 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)05.12.1816:30MA 406 Flow Models and Solution Strategies for Evacuation Planning Problems

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.
In this talk, we are particularly interested in lane reversal (also a contraflow reconfiguration) techniques that significantly increases the flow values with decreased evacuation time. These problems are computationally challenging and there exist many heuristics in literature and practice, however, they are also polynomially solvable in many cases and investigated analytically recently. We present the strength of these techniques and also give some case examples that demonstrate the relevances of these models.

Milena Wrobel (MPI Leipzig)21.11.1816:30MA 621 On the anticanonical complex

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 Fruehbis-Krueger (Leibniz Universitaet Hannover)14.11.1816:30MA 621 Some massively parallel computations in algebraic geometry

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)31.10.1816:30MA 406 On the distance to lattice points in knapsack polyhedra

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 Tsuchiya24.10.1816:30MA 621 Reflexive polytopes arising from (0,1)-polytopes

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 Panizzut17.10.1816:30MA 621 Introduction to p-adic geometry

In this survey talk we will introduce p-adic fields and more generally non-archimedean 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 Zhang29.08.1816:30MA 621 Constructing an enveloping membrane for a min-convex hull

Joswig, Sturmfels, and Yu (2007) describe a correspondence between a min-convex hull in the affine building of SL_d(K) and a tropical convex hull in tropical projective space TP^{d-1}, using a membrane containing the min-convex 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 Wang22.08.1816:30MA 621 Cluster Algebras and Grassmannians

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+3-gon. 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 Newton-Okounkov 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 Tillmann18.07.1816:30MA 621 What is the Thurston norm?

Thurston�s norm on (co)homology of a 3-manifold reveals interesting topological properties of the 3-manifold. 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 Kaluba11.07.1816:30MA 621 Non-commutative positivity and property (T)

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 non-commutative 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 non-commutative 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)04.07.1816:30MA 621 A computational approach to tropical semigroups

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 Sadykov28.06.1814:15MA 316 Amoebas of multivariate hypergeometric polynomials

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.
Despite the diversity of families of hypergeometric polynomials, most of them share the following key properties that justify the usage of the term "hypergeometric":
1. The polynomials are dense (possibly after a suitable monomial change of variables).
2. The coefficients of a hypergeometric polynomial are related through some recursion with polynomial coefficients.
3. For univariate polynomials, there is typically a single representative (up to a suitable normalization) of a given degree within a family of hypergeometric polynomials.
4. All polynomials in the family satisfy a differential equation of a fixed order with polynomial coefficients (or a system of such equations) whose parameters encode the degree of a polynomial.
5. In the case of one dimension, the absolute values of the roots of a classical hypergeometric polynomial are all different (possibly after a suitable monomial change of variables).
6. Many of hypergeometric polynomials enjoy various extremal properties.
In the talk, we will introduce a definition of a multivariate hypergeometric polynomial in several complex variables that is coherent with the properties 1-6 listed above. Namely, with any integer convex polytope P we associate a multivariate hypergeometric polynomial whose set of exponents is P.
The hypergeometric polynomial associated with the polytope P is defined uniquely up to a constant multiple and satisfies a holonomic system of partial differential equations of Horn's type. We prove that under certain nondegeneracy conditions the zero locus of any such polynomial is optimal in the sense of Forsberg-Passare-Tsikh. Generally speaking, this means that the topology of the amoeba of such a polynomial is as complicated as it could possibly be. This property is the multivariate counterpart of the property of having different absolute values of the roots for a polynomial in a single variable.

Christian Stump20.06.1816:30MA 621 Reflection groups and singularity theory

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)13.06.1816:30MA 621 The Picard group of a tropical toric scheme

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 Spreer06.06.201816:30MA 621 Separation-type combinatorial invariants for triangulations of manifolds

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 f-vector. The invariants are very elementary and defined by counting connected components and/or homological features of induced subcomplexes.
I will first define these invariants, state some identities they satisfy, and connect them to natural operations on triangulated manifolds and spheres. Then I will present the very natural connjecture that, for a given f-vector, the invariants are maximised for the Billera-Lee-spheres.
This is joint work with Giulia Codenotti and Francisco Santos.

Alperen Erguer30.05.1816:30MA 406 Multihomogenous Nonnegative Polynomials and Sums of Squares

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 Scrimshaw29.05.1817:00MA 621 On the Combinatorics of Crystals

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)24.05.1816:15MA 313 The "Small ball" property in high dimensional measures

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 high-dimensional geometry and convexity is crucial to understand the concentration behavior in the "small ball regime".

Grigoris Paouris (Texas A&M University)23.05.1816:30MA 313 Almost Euclidean section of high-dimensional convex bodies

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 n-dimensional symmetric convex body has a k-dimensional 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)09.05.1816:30MA 621 Computing the delta set of an affine semigroup: a status report

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 Garcia-Sanchez 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)02.05.1816:30MA 406 Hilbert-Kunz functions and HK density functions

We give a brief survey of Hilbert-Kunz function and its leading coefficient \(e_{HK}\) called HK multiplicity, for a commutative Noetherian ring.
In order to study \(e_{HK}\) we have introduced a compactly supported continuous function \(HKd\) (in the graded set up). This idea of replacing a point (\(e_{HK}\)) by a function (\(HKd\)) seems to be an effective technique to handle the notoriously difficult invariant like \(e_{HK}\). We apply this theory to projective toric varieties and, as a corollary, get an algebraic characterization of the tiling property of a rational convex polytope. Similarly we define a density function for the second coefficient of a HK function (in toric case). Here we use a refined version of a result by Henk-Linke on the boundedness of the coefficients of rational Ehrhart quasi-polynomials of convex rational polytopes.

Victor Magron (CNRS Grenoble)25.04.1816:30MA 621 On exact polynomial optimization

To compute certificates of nonnegativity, an approach based on sums of squares (SOS) decompositions has been popularized by Lasserre and Parillo.
In the first part of this talk, we consider the problem of finding SOS decompositions of nonnegative univariate polynomials. It is well-known that every non-negative univariate real polynomial can be written as the sum of two polynomial squares with real coefficients. When one allows a weighted sum of finitely many squares instead of a sum of two squares, then one can choose all coefficients in the representation to lie in the field generated by the coefficients of the polynomial. We describe, analyze and compare two algorithms computing such a weighted sums of squares decomposition for univariate polynomials with rational coefficients. - The first algorithm, due to Schweighofer, relies on real root isolation, quadratic approximations of positive polynomials and square-free decomposition but its complexity was not analyzed. We provide bit complexity estimates, both on runtime and output size of this algorithm. They are exponential in the degree of the input univariate polynomial and linear in the maximum bitsize of its complexity. This analysis is obtained using quantifier elimination and root isolation bounds. - The second algorithm, due to Chevillard, Harrison, Joldes and Lauter, relies on complex root isolation and square-free decomposition and has been introduced for certifying positiveness of polynomials in the context of computer arithmetics. Again, its complexity was not analyzed. We provide bit complexity estimates, both on runtime and output size of this algorithm, which are polynomial in the degree of the input polynomial and linear in the maximum bitsize of its complexity. This analysis is obtained using Vieta's formula and root isolation bounds. While the second algorithm is, as expected from the complexity result, more efficient on most of examples, we exhibit families of non-negative polynomials for which the first algorithm is better.
In the second part of this talk, we consider the problem of finding exact sums of squares (SOS) decompositions for certain classes of nonnegative multivariate polynomials, while relying on semidefinite programming (SDP) solvers. - We start by providing a hybrid symbolic-numeric algorithm computing exact rational SOS decompositions for polynomials lying in the interior of the SOS cone. This algorithm computes an approximate SOS decomposition for a perturbation of the input polynomial with an arbitrary-precision SDP solver. An exact SOS decomposition is obtained thanks to the perturbation terms. We prove that bit complexity estimates on output size and runtime are both polynomial in the degree of the input polynomial and simply exponential in the number of variables. This analysis is based on quantifier elimination as well as bounds on the cost of the ellipsoid method and Cholesky's decomposition. - Next, we apply this algorithm to compute exact Polya and Putinar's representations respectively for positive definite forms and polynomials positive over basic compact semialgebraic sets. We compare the implementation of our algorithms with existing methods based on CAD or critical points.
This is joint work with Mohab Safey El Din and Markus Schweighofer.

Ben Smith (Queen Mary University of London)18.04.1816:30MA 621 Matching fields and lattice points of simplicies

An (n,d)-matching field is a collection of matchings such that there is a unique matching for each d-subset 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)11.04.1816:30MA 621 Coxeter groups, BN-pairs, and buildings

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 Timme04.04.1816:30MA 621 Fast Computation of Amoebas in low Dimensions

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.
In this talk we investigate how the structural properties of amoebas can be used to enable the fast approximation of amoebas in low dimensions as well as to compute the spine of 2-dimensional amoebas. We also present the Julia package Amoebas.jl in which these results are implemented.

Adam Kurpisz28.03.1816:30MA 621 On the hierarchies of relaxations for 0/1 optimization problems

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/Sum-of-Squares (SoS), Sherali Adams (SA)) and the recent development in this field, the Sum-of-Nonnegative-Circuit (SoNC) algorithm.
We start with a motivation and a gentle introduction to the hierarchies of relaxations for 0/1 optimization problems. Next we discuss the strength and weaknesses of this approach and present a novel method in this field, the SoNC hierarchy.
Finally, in this talk we show that key results for SOS/SA hierarchy remain valid for SoNC hierarchy. More precisely we show that for a polynomial optimization problem over the $n$-variate boolean hypercube with constraints of degree $d$, the SoNC hierarchy at level $n+d$ converges to the convex hull of integral solutions.

Francisco Criado07.02.1816:30MA 621 Packing algorithms applied to linear programming

The Yamnistsky-Levin 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 Malikiosis24.01.1816:30MA 406 Fuglede's spectral set conjecture on cyclic groups

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 \(d=1,2\).
Some recent results in the cyclic group setting will be presented in this talk, which are connected to the work of Coven-Meyerowitz and Laba on tiling subsets of \(\mathbb{Z}\), as well as the structure of vanishing sums of roots of unity. This is joint work with Mihalis Kolountzakis & work in progress.

Linda Kleist20.12.1716:30MA 621 Rainbow cycles in flip graphs

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 r-rainbow cycles, i.e., cycles where every flip type occurs exactly r times.
For instance, the flip graph of triangulations has as vertices all triangulations of a convex n-gon, and an edge between any two triangulations that differ in exactly one edge. An r-rainbow cycle in this graph is a cycle in which every inner edge of the triangulation appears exactly r times.
In this talk we investigate the existence of rainbow cycles in some of the following flip graphs of triangulations of a convex n-gon, plane trees on a set of n points, non-crossing perfect matchings on points in convex position, permutations of [n], and the flip graph of k-element subsets of [n].
The talk is based on joint work with Stefan Felnser, Torsten M�tze and Leon Sering.

Franz Schuster (TU Wien)06.12.1716:30MA 406 Affine vs. Euclidean Sobolev inequalities

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 Zhang-Sobolev inequality. We discuss in some detail the geometry behind these analytic inequalities and also relate our new Sobolev inequalities to the sharp Gromov-Gagliardo-Nirenberg Sobolev inequality for general norms and discuss further improvements of special cases.

Gennadiy Averkov (OvGU Magdeburg)29.11.1716:30MA 406 Latest news on Hensley's conjecture

The talk is about the maximum volume of d-dimensional 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 so-called 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 (LAAS-CNRS Univ Toulouse, FR and Czech Tech Univ Prague, CZ) 22.11.1716:30MA 621 Convergence rates of moment-sum-of-squares hierarchies for volume approximation of semialgebraic sets

Moment-sum-of-squares 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 Seidler15.11.1716:30MA621 Circuit Polynomials for Simplex Newton Polytopes

Finding the minimum of a multivariate real polynomial is a well-known hard problem with various applications. We present an implementation to approximate such lower bounds via sums of non-negative circuit polynomials (SONCs). We provide a test-suite, 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)08.11.1716:30MA621 Some integrality gap results for the 0/1 SoS Hierarchy

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 n-1 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 13.09.1716:30MA621 Limits of Algebraic varieties: towards a continuous Nullstellansatz

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)05.09.1716:30MA 406 Sparse solutions to the systems of linear Diophantine equations

We present structural results on sparse nonnegative integer solutions to underdetermined systems of linear equations.
Our proofs are algebraic and number theoretic in nature and make use of Siegel's Lemma and classical tools from the Geometry of Numbers.
These results have some interesting consequences in discrete optimisation.

Marta Panizzut30.08.1716:30MA 621 The hunt for K3 polytopes

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 Wolff23.08.1716:30MA 621 Intersections of Amoebas

Georg Loho16.08.1716:30MA 621 Monomial Tropical Cones for Multicriteria Optimization

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 Hofscheier19.07.1716:30MA 621 A Combinatorial Smoothness Criterion for Spherical Varieties

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 well-known 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)06.07.1716:30MA 406 The Ehrhart quasipolynomials of polytopes that come from trivalent graphs, and a discrete version of Hilbert's 3rd problem for the unimodular group

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 quasi-polynomials. 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 quasi-polynomial. We also present some structural properties of these polytopes and we show a relation between them and the well-known 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.
This is joint work with Cristina Fernandez, Jose? C. de Pina, and Jorge Luis Rami?rez Alfonsi?n.

Matthias Lenz28.06.1716:30MA 621 On powers of Pl�cker coordinates and representability of arithmetic matroids

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.
An arithmetic matroid is a matroid equipped with a function that assigns a multiplicity (a positive integer) to each subset of the ground set. We will discuss a discrete version of the problem described above that deals with the representability of the arithmetic matroid obtained by taking the \(k\)th power of the multiplicity function. This setting is motivated by some combinatorial questions on CW complexes.

Jean-Philippe Labb� / Michael Joswig21.06.1716:30MA 621 polymake/sage

Jean-Philippe 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 Loarca14.06.1716:30MA 621 Fast Multidimensional Signal Processing with Shearlab.jl.

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 N-term 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.
In this talk I will explain the upsides and downsides of the already mentioned transforms, why the shearlet transform is great solution for anisotropic representation of multidimensional signals, how can we implement wavelet and shearlet transform in Julia, and I will show you some applications and demos of the Shearlet Transform Library that I implemented with Professor Gitta Kutyniok based on the matlab version developed by her group two years ago; I will also explain why Julia is the best option to implement algorithms in Signal and Image Processing and show some benchmarks that show its great performance.

Hiroshi Hirai31.05.1716:30MA 621 Maximum vanishing subspace problem, CAT(0)-space relaxation, and block-triangularization of partitioned matrix

In this paper, we address the following algebraic generalization of the bipartite stable set problem. We are given a block-structured 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 block-triangular 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 block-triangularization of partitioned matrix.

Ted Bisztriczky (University of Calgary)31.05.1716:30MA 406 Erdos-Szekeres type theorems for planar convex sets

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 Erdos-Szekeres 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.
In this talk, I survey some known theorems, introduce a new one based upon work with Gabor Fejes Toth and examine the relation between \(N(n)\) and Ramsey numbers of the type \(R_k(n_1,n_2)\).

Dimitrios Dais (University of Crete)29.05.1717:00MA 406 Toric log del Pezzo surfaces with a unique singularity

The toric log del Pezzo surfaces are constructed by means of the so-called LDP-lattice 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)17.05.1716:30MA 406 Open problems in Geometric Tomography

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 X-ray transform, divergent beam transform, circular Radon transform, cosine transform, or spherical Radon transform.
The talk will be illustrated by plenty of pictures.

Michael Joswig03.05.1716:30MA 621 Computing Invariants of Simplicial Manifolds

This is a brief survey on algorithms for computing basic algebraic invariants such as homology, cup products, and intersection forms.

Romanos Malikiosis26.04.1716:30MA 406 Formal duality in finite cyclic groups

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 \(2\mathbb{Z}\cup(2\mathbb{Z}+\frac12)\).
In this talk, we will present recent progress on this conjecture by the speaker, utilizing several tools from different areas of mathematics, such as (a) the field descent method by Bernhard Schmidt, used chiefly on the circulant Hadamard matrix conjecture, (b) the 'polynomial method' by Coven & Meyerowitz used in the characterization of sets tiling \(\mathbb{Z}\) or \(\mathbb{Z}_N\), and (c) basic arithmetic of cyclotomic fields.

Manuel Radons12.04.1716:30MA 621 Piecewise linear methods in nonsmooth optimization

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 Jell05.04.1716:30MA 621 Differential forms on Berkovich spaces and their relation to tropical geometry

Takayuki Hibi (Osaka University)29.03.1716:30MA 621 Order Polytopes and Chain Polytopes

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)22.03.1717:00MA 621 On the genus of tropical curves

Felipe Rinc�n (University of Oslo)22.03.1716:15MA 621 Tropical Ideals

Skip Jordan22.03.1715:30MA 621 Parallel Vertex and Facet Enumeration with mplrs

We introduce mplrs, a new parallel vertex/facet enumeration program based on the reverse-search 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 Fink22.02.1716:30MA 621 The characteristic polynomial two ways

The theory of hyperplane arrangements and matroids derives great utility from the application of algebraic and algebro-geometric 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 Tsuchiya08.02.1716:30MA 621 Gorenstein simplices and the associated finite abelian groups

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�we25.01.1716:30MA 621 Secondary fans of a punctured Riemann surface

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)18.01.1716:30MA 406 A Leray model for Orlik-Solomon algebras

Although hyperplane arrangement complements are rationally formal, we note that they have non-minimal rational (CDGA) models which are topologically and combinatorially significant. We construct a family of CDGAs which interpolates between the Orlik-Solomon algebra and the cohomology algebras of arrangement compactifications. Our construction is combinatorial and extends to all matroids, regardless of their (complex) realizability.
This is joint work with Christin Bibby and Graham Denham.

Benjamin Schr�ter04.01.1716:30MA 621 Fundamental Polytopes of Metric Trees via Hyperplane Arrangements

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 f-vector of the fundamental polytope of a finite metric space and its dual the Lipschitz polytope.

Georg Loho04.01.1717:15MA 621 Slicing and Dicing Polytopes

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 Panizzut26.10.1613:30MA 621 Linear systems on metric graphs, gonality sequence and lifting problems

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)19.10.1616:30MA 406 On Lattice Coverings by Simplices

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 Tran05.10.1614:45MA 621 Tropical geometry and mechanism design

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 Kahle05.10.1613:30MA 621 The geometry of rank-one tensor completion


Andr� Wagner28.09.1616:30MA 621 Veronesean almost binomial almost complete intersections

The second Veronese ideal I_n contains a natural complete intersection J_n generated by the principal 2-minors 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)27.07.1616:30MA 406 Tensor Valuations on Lattice Polytopes

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 scalar-valued 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)20.07.1616:30MA 406 Inequalities about sections and projections of convex bodies

We discuss lower dimensional versions of the slicing problem and of the Busemann-Petty 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 Blaschke-Petkantschin formula, on asymptotic estimates for the dual affine quermassintegrals and on some new Loomis-Whitney type inequalities in the spirit of the uniform cover inequality of Bollobas and Thomason.

Gerassimos Barbatis (University of Athens)13.07.1616:30MA 406 On the Hardy constant of some non-convex planar domains

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 Trager15.06.1616:30MA 621 Geometric models for computer vision

The goal of computer vision is to recover real-world 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 multi-view vision in terms of the Grassmannian of lines in P^3. This is joint work with Jean-Charles Faugere, Xavier Goaoc, Martial Hebert, Jean Ponce, Mohab Safey El Din and Bernd Sturmfels.

Axel Flinth18.05.1616:30MA 406 High Dimensional Geometry in Compressed Sensing

The main aim of compressed sensing is to recover a low-dimensional object (e.g. a sparse vector, a low-rank matrix, ...) from few linear measurements. Many algorithms for this task (e.g. Basis Pursuit, nuclear norm-minimization) consists of minimizing a structure-promoting function \(f\) over the set of possible solutions.

An elegant approach to finding recovery guarantees of such programs relies on high-dimensional geometry of convex cones. The techniques were developed relatively recently, but have quickly gained popularity. In the first half of this talk, this approach will be presented in relative high detail.

In the second half of the talk, we will discuss the connection between geometry and stability of compressed sensing algorithms. In particular, we will present a new geometric criterion on \(A\) implying stability both with respect to model inexactness of the signal as well as to additive noise in the measurements.

Amanda Cameron04.05.1616:30MA 621 A lattice point counting generalisation of the Tutte polynomial

The Tutte polynomial for matroids is not directly applicable to polymatroids. For instance, deletion-contraction 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 Breiding27.04.1616:30MA 621 The spectral Atheory of tensors

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 E-eigenvalues and Z-eigenvalues 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 Artstein-Avidan (Tel Aviv University)27.04.1611:00MA 406 On Godbersen�s conjecture and related inequalities


Stefan Weltge (Otto-von-Guericke Universit�t Magdeburg)20.04.1616:30MA 406 Tight bounds on discrete quantitative Helly numbers

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^{(n-1)/(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^{(n-1)/(n+1)}) holds for every fixed n. Finally, we determine the exact values of c(Z^n,k) for all k <= 4.

Anna Lena Birkmeyer16.03.1616:30MA 621 Realizability of Tropical Hypersurfaces in Matroid Fans

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 Genzel02.03.1616:30MA 406 Convex Recovery of Structured Signals from Non-Linear Observations

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) non-linearity 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 non-linearity 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 real-world 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)03.02.1616:30MA 406 Periodicity for tilings and spectra

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 Hampe20.01.1616:30MA 621 Matroids over hyperfields

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)13.01.1616:30MA 406 Quantitative versions of Helly's theorem

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 Shaw06.01.1616:30MA 621 What is �tale cohomology?

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�ter02.12.1516:30MA 621 The degree of a tropical basis

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)18.11.1516:30MA 406 On the Minkowski measure of symmetry

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.

Afterwards, we will show how s(.) can be used to strengthen, smoothen, and join different geometric inequalities, as well as its connections to other concepts such as diametrical compleness, Jung's inequality, or Banach-Mazur distance.

Andr� Wagner04.11.1516:30MA 621 Introduction to singular value decomposition

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 Spreer28.10.1516:30MA 621 Algorithms and complexity for Turaev-Viro invariants

The Turaev-Viro invariants are a powerful family of topological invariants for distinguishing between different 3-manifolds. 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 fixed-parameter tractable algorithm for arbitrary r which is practical -- and indeed preferable -- to the prior state of the art for real computation.
This is joint work with Ben Burton and Clément Maria

Alexander Gamkrelidze11.08.1515:00MA 621 Algorithms for Convex Topology: Computation of Hom-Complexes

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 Hom-complexes 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.


8th polymake workshop
February 2-4, 2017
TU Berlin
8th polymake workshop [1]
7th polymake workshop
January 28-30, 2016
TU Berlin
7th polymake workshop [2]
Meeting on Algebraic Vision 2015
October 8-9, 2015
TU Berlin
MAV [3]
Tropical geometry in Europe
March 30-31, 2015
TGE [4]
6th polymake workshop
December 5,2014
TU Berlin
6th polymake workshop [5]
5th polymake workshop
March 28, 2014
TU Berlin
5th polymake workshop [6]
4th polymake conference and developer meeting at TU Berlin
November 1-2, 2013
TU Berlin
4th polymake conference and developer meeting [7]
Delaunay Geometrie: Polytopes, Triangulations and Spheres
October 7-9,2013
FU Berlin
Second ERC "SDModels" Workshop [8]



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