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In unserem Kolloquium algorithmische Mathematik und Komplexitätstheorie tragen Mitarbeiter und Gäste über aktuelle Themen ihrer Forschung vor.

Das Oberseminar findet dienstags von 10:15 bis etwa 11:45 Uhr im Raum MA 316 statt.


Geplante Vorträge
Andre Wagner
Rigid- and critical-configurations
Kathlen Kohn
Singular Loci of Coincident Root Loci
Paul Breiding
The condition of join decomposition
Antonio Lerario
The Kac-Rice formula and the enumerative geometry of lines on real hypersurfaces
Alperen Ergür
Condition Number of Random Polynomial Systems


Rigid- and critical-configurations

This talk will focus on two different topics in 'algebraic vision'. First of all we will have a look at images of two world points with a fixed distance. This defines a generalization of the multiview variety. The additional rigid constraint on the world points gives octic constraints on the image points, additional to the multilinear generators of the multiview variety. Secondly, we will study the behavior of the famous 8-point algorithm, if the eight world points are the vertices of a combinatorial cube. 

Singular Loci of Coincident Root Loci

The projective variety $X_\lambda\,$ of binary forms of degree $d\,$ whose linear factors are distributed according to the partition $\lambda\,$ of $d\,$ is called coincident root locus. 

Hence it describes $d\,$ points on the projective line that occur with certain multiplicities given by $\lambda\,$. 

I will present results of Kurmann (2011) and Chipalkatti (2003) that describe the singular locus of $X_\lambda\,$ in purely combinatorial terms. 

I would like to generalize this result to general enough projective hypersurfaces, since intersections of hypersurfaces with lines are also points occurring with certain multiplicities. 

The condition number of join decomposition

Joins of real manifolds appear in a wide area of application and often one is interested in decomposing a point on a join into its factors. Naturally, one may expect that the performance of any numerical algorithm that solves the join decomposition problem is governed by the condition number of the problem. In this talk I will define the condition number of the join decomposition problem and describe is as the inverse distance to some ill-posed set in a product of Grassmannians. Moreover, I will model the join decomposition as an optimization problem, provide an algorithm for it and explain how the condition number influences the performance of this algorithm. Finally, I will show you the behaviour of the condition number close to boundary points of the join. This is joint work with Nick Vannieuwenhoven. 

The Kac-Rice formula and the enumerative geometry of lines on real hypersurfaces

The problem of counting lines on a generic hypersurface of degree $2n-3\;$ in $n\;$-dimensional space is classical. For example, there are 27 lines on a generic cubic in 3-space. If the hypersurface is defined by a real equation, these lines do not need to be real; in fact the number of real lines on the hypersurface strongly depends on the coefficients of the defining polynomial -- this is a typical problem when counting real solutions of real equations. 

In this talk I will adopt a probabilistic approach and count the expected number of real lines, when the hypersurface is defined by a random Kostlan polynomial. For example, on a random cubic there are on average $6\sqrt{2}-3\;$ real lines. More generally I will show that a "square root law" (in the sense of Edelman-Kostlan-Shub-Smale) holds for this problem when $n\;$ goes to infinity (this was already conjectured by P. Bürgisser around 10 years ago). 

Joint work with S. Basu, E. Lundberg and C. Peterson

Condition Number of Random Polynomial Systems

The condition number of a polynomial system measures the sensitivity of its roots to perturbations in the coefficients. We study the condition number of random polynomial systems for a broad family of distributions.  We do not impose algebraic invariance assumptions on the distribution which allows us to address condition number polynomial system with prescribed monomial structure.  This is joint work with Grigoris Paouris and J. Maurice Rojas. 

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