Seminars and Colloquia by Series

NumericalImplicitization for Macaulay2

Series
Algebra Seminar
Time
Friday, September 30, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Justin Chen – UC Berkeley

Please Note: 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.

Algebraic Geometry for Computer Vision

Series
Algebra Seminar
Time
Monday, September 26, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Joe Kileel – UC Berkeley
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.

On the isotypic decomposition of cohomology modules of symmetric semi-algebraic sets

Series
Algebra Seminar
Time
Friday, September 16, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Saugata Basu – Purdue
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).

An Algebraic Introduction to Multiview Geometry and Tensors

Series
Algebra Seminar
Time
Monday, June 27, 2016 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Luke Oeding – Auburn University
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.

Macaulay dual spaces and local Hilbert function

Series
Algebra Seminar
Time
Monday, June 20, 2016 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Robert Krone – Queen's University
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.

Finding binomials in polynomial ideals

Series
Algebra Seminar
Time
Monday, June 13, 2016 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Anders Jensen – TU-Kaiserslautern / Aarhus University
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.

Algebraic Systems Biology, Model Selection and Parameter Estimation

Series
Algebra Seminar
Time
Tuesday, May 31, 2016 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Elizabeth Gross – San Jose State University
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.

Tropical Varieties for Exponential Sums

Series
Algebra Seminar
Time
Monday, April 4, 2016 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alperen Ergur – Texas A&M
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.

On the average height of abelian varieties with complex multiplication

Series
Algebra Seminar
Time
Monday, March 28, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Keerthi Madapusi Pera – University of Chicago –
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.

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