## Seminars and Colloquia by Series

Monday, November 14, 2016 - 15:05 , Location: Skiles 006 , , University of Minnesota , Organizer: Anton Leykin
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.
Friday, September 30, 2016 - 15:05 , Location: Skiles 005 , Justin Chen , UC Berkeley , Organizer: Anton Leykin

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.

Monday, September 26, 2016 - 15:05 , Location: Skiles 006 , Joe Kileel , UC Berkeley , Organizer: Anton Leykin
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.
Friday, September 16, 2016 - 15:05 , Location: Skiles 005 , , Purdue , Organizer: Anton Leykin
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).
Monday, June 27, 2016 - 11:05 , Location: Skiles 005 , Luke Oeding , Auburn University , Organizer: Anton Leykin
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.
Monday, June 20, 2016 - 11:05 , Location: Skiles 005 , Robert Krone , Queen's University , Organizer: Anton Leykin
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.
Monday, June 13, 2016 - 11:05 , Location: Skiles 005 , Anders Jensen , TU-Kaiserslautern / Aarhus University , Organizer: Anton Leykin
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.
Tuesday, May 31, 2016 - 11:05 , Location: Skiles 006 , Elizabeth Gross , San Jose State University , Organizer: Anton Leykin
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.
Friday, April 15, 2016 - 15:05 , Location: Skiles 249 , , Stanford University , , Organizer:
Monday, April 4, 2016 - 15:00 , Location: Skiles 006 , Alperen Ergur , Texas A&M , Organizer: Greg Blekherman
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.