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Series: Algebra Seminar

Real sub-varieties and more generally semi-algebraic subsets of $\mathbb{R}^n$
that are stable under the action of the symmetric group on $n$ elements acting
on $\mathbb{R}^n$ by permuting coordinates, are expected to be topologically
better behaved than arbitrary semi-algebraic sets. In this talk I will
quantify this statement by showing polynomial upper bounds on the
multiplicities of the irreducible $\mathfrak{S}_n$-representations that
appear in the rational cohomology groups of such sets.
I will also discuss some algorithmic results on the complexity
of computing the equivariant Betti numbers of such sets and sketch some
possible connectios with the recently developed theory of FI-modules.
(Joint work with Cordian Riener).

Series: Algebra Seminar

In Multiview Geometry, a field of Computer Vision one is interested in reconstructing 3-dimensional scenes from 2-dimensional images. I will review the basic concepts in this area from an algebraic viewpoint, in particular I'll discuss epipolar geometry, fundamental matrices, and trifocal and quadrifocal tensors. I'll also highlight some in open problems about the algebraic geometry that arise.This will be an introductory talk, and only a background in basic linear algebra should be necessary to follow.

Series: Algebra Seminar

The Macaulay dual space offers information about a polynomial ideal localized at a point such as initial ideal and values of the Hilbertfunction, and can be computed with linear algebra. Unlike Gr\"obner basis methods, it is compatible with floating point arithmetic making it anatural fit for the toolbox of numerical algebraic geometry. I willpresent an algorithm using the Macaulay dual space for computing theregularity index of the local Hilbert function.

Series: Algebra Seminar

Deciding if a polynomial ideal contains monomials is a problem which can be solved by standard Gr\"obner basis techniques. Deciding if a polynomial ideal contains binomials is more complicated. We show how the general case can be reduced to the case of a zero-dimensional ideals using projections and stable intersections in tropical geometry. In the case of rational coefficients the zero-dimensional problem can then be solved with Ge's algorithm relying on the LLL lattice basis reduction algorithm. In case binomials exists, one will be computed.This is joint work with Thomas Kahle and Lukas Katthän.

Series: Algebra Seminar

Systems biology focuses on modeling complex biological systems, such as
metabolic and cell signaling networks. These biological networks are
modeled with polynomial dynamical systems. Analyzing these systems at
steady-state results in algebraic varieties that live in
high-dimensional spaces. By understanding these varieties, we can
provide insight into the behavior of the models. Furthermore, this
algebro-geometric framework yields techniques for model selection and
parameter estimation that can circumvent challenges such as limited or
noisy data. In this talk, we will introduce biochemical reaction
networks and their resulting steady-state varieties. In addition, we
will discuss the questions asked by modelers and their corresponding
geometric interpretation, particularly in regards to model selection and
parameter estimation.

Series: Algebra Seminar

Series: Algebra Seminar

We define a variant of tropical varieties for exponential sums.
These polyhedral complexes can be used to approximate, within an explicit
distance bound, the real parts of complex zeroes of exponential sums. We
also discuss the algorithmic efficiency of tropical varieties in relation
to the computational hardness of algebraic sets. This is joint work with
Maurice Rojas and Grigoris Paouris.

Series: Algebra Seminar

In the 90s, generalizing the classical Chowla-Selberg formula, P. Colmez formulated a conjectural formula for the Faltings heights of abelian varieties with multiplication by the ring of integers in a CM field, which expresses them in terms of logarithmic derivatives at 1 of certain Artin L-functions. Using ideas of Gross, he also proved his conjecture for abelian CM extensions. In this talk, I will explain a proof of Colmez's conjecture in the average for an arbitrary CM field. This is joint work with F. Andreatta, E. Goren and B. Howard.

Series: Algebra Seminar

Hurwitz correspondences are certain multivalued self-maps of the moduli space M0,N parametrizing marked genus zero curves. We study the dynamics of these correspondences via numerical invariants called dynamical degrees. We compare a given Hurwitz correspondence H on various compactifications of M0,N to show that, for k ≥ ( dim M0,N )/2, the k-th dynamical degree of H is the largest eigenvalue of the pushforward map induced by H on a comparatively small quotient of H2k(M0,N). We also show that this is the optimal result of this form.

Series: Algebra Seminar