Seminars and Colloquia by Series

Monday, January 9, 2017 - 15:05 , Location: Sklles 005 , Rachel Pries , Colorado State University , , Organizer:
We prove a result about the Galois module structure of the Fermat curve using commutative algebra, number theory, and algebraic topology.  Specifically, we extend work of Anderson about the action of the absolute Galois group of a cyclotomic field on a relative homology group of the Fermat curve.  By finding explicit formulae for this action, we determine the maps between several Galois cohomology groups which arise in connection with obstructions for rational points on the generalized Jacobian.  Heisenberg extensions play a key role in the result. This is joint work with R. Davis, V. Stojanoska, and K. Wickelgren.
Monday, December 5, 2016 - 16:15 , Location: Skiles 005 , Padma Srinivasan , Georgia Tech , Organizer: Matt Baker
Conductors and minimal discriminants are two measures of degeneracy of the singular fiber in a family of hyperelliptic curves. In the case of elliptic curves, the Ogg-Saito formula shows that (the negative of) the Artin conductor equals the minimal discriminant. In the case of genus two curves, equality no longer holds in general, but the two invariants are related by an inequality. We investigate the relation between these two invariants for hyperelliptic curves of arbitrary genus.
Monday, November 28, 2016 - 15:05 , Location: Skiles 005 , Oliver Lorscheid , IMPA , Organizer: Matt Baker
Recent work of Jeff and Noah Giansiracusa exhibits a scheme theoretic structure for tropicalizations of classical varieties in terms of so-called semiring schemes. This works well in the framework of closed subvarieties of toric varieties, and Maclagan and Rincon recover the structure of a weighted polyhedral complex from the scheme theoretic tropicalization of a variety embedded into a torus.In this talk, I will review these ideas and show how these results can be extended by using blue schemes. This leads to an intrinsic notion of a tropicalization, independent from an embedding into an ambient space, and generalizes the above mentioned results to the broader context of log-schemes.
Monday, November 14, 2016 - 15:05 , Location: Skiles 006 , Peter Olver , 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 , Saugata Basu , 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.