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Series: Algebra Seminar

In this talk, I will refine the concept of the symmetry group of a
geometric object through its symmetry groupoid, which incorporates both
global and local symmetries in a common framework. The symmetry
groupoid is related to the weighted differential invariant signature of a
submanifold, that is introduced to capture its fine grain equivalence
and symmetry properties. The groupoid/signature approach will be
connected to recent developments in signature-based recognition and
symmetry detection of objects in digital images, including jigsaw puzzle
assembly.

Series: Algebra Seminar

Many varieties of interest in algebraic geometry and applications

are given as images of regular maps, i.e. via a parametrization.

Implicitization is the process of converting a parametric description of a

variety into an intrinsic (i.e. implicit) one. Theoretically,

implicitization is done by computing (a Grobner basis for) the kernel of a

ring map, but this can be extremely time-consuming -- even so, one would

often like to know basic information about the image variety. The purpose

of the NumericalImplicitization package is to allow for user-friendly

computation of the basic numerical invariants of a parametrized variety,

such as dimension, degree, and Hilbert function values, especially when

Grobner basis methods take prohibitively long.

Series: Algebra Seminar

This talks presents two projects at the interface of computer vision
and algebraic geometry. Work with Zuzana Kukelova, Tomas
Pajdla and Bernd Sturmfels introduces the distortion varieties
of a given projective variety. These are parametrized by
duplicating coordinates and multiplying them with monomials.
We study their degrees and defining equations. Exact formulas
are obtained for the case of one-parameter distortions, the
case of most interest for modeling cameras with image distortion.
Single-authored work determines the algebraic degree of
minimal problems for the calibrated trifocal variety. Our techniques
rely on numerical algebraic geometry, and the homotopy
continuation software Bertini.

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.