Seminars and Colloquia by Series

Cellular Binomial Ideals

Series
Algebra Seminar
Time
Friday, April 3, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Laura Felicia MatusevichTexas A&M
Primary decomposition is a fundamental operation in commutative algebra. Although there are several algorithms to perform it, this remains a very difficult undertaking in general. In cases with additional combinatorial structure, it may be possible to do primary decomposition "by hand". The goal of this talk is to explain in detail one such example. This is joint work with Zekiye Eser; no prerequisites are assumed beyond knowing the definitions of "polynomial ring" and "ideal".

Concrete Chern classes, the cyclic quantum dilogarithm and the Bloch group

Series
Algebra Seminar
Time
Wednesday, April 1, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Stavros GaroufalidisGatech
The talk involves an explicit formula for the Chern class on K_3(F), F=number field, givenin terms of the cyclic quantum dilogarithm on the Bloch group of F. Such a formula constructsexcplicitly units in number fields, given a complete hyperbolic 3-manifold, and a complex root ofunity, and those units fit in the asymptotic expansion of quantum knot invariants. The existence ofsuch a formula was conjectured 4 years ago by Zagier (and abstractly follows from Voevodsky's work),and the final solution to the problem was given in recent joint work of the speaker with FrankCalegari and Don Zagier. The key ingredient to the concrete formula is a special function, thecyclic quantum dilogarithm, from a physics 1993 paper of Kashaev and others. The connection of thisformula with physics, and with the Quantum Modular Form Conjecture of Zagier continues with jointwork with Tudor Dimofte. But this is the topic of another talk.

Sparse sum-of-squares certificates on finite abelian groups

Series
Algebra Seminar
Time
Monday, March 30, 2015 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hamza FawziMIT
We consider functions on finite abelian groups that are nonnegative and also sparse in the Fourier basis. We investigate conditions under which such functions admit sparse sum-of-certificates certificates of nonnegativity, i.e., certificates where the functions in the sum of squares decomposition have a small common sparsity pattern. Our conditions are purely combinatorial in nature, and are based on finding particularly nice chordal covers of a certain Cayley graph. These techniques allow us to show that any nonnegative quadratic function in binary variables is a sum of squares of functions of degree at mostceil(n/2), resolving a conjecture of Laurent. After discussing the connection with semidefinite programming lifts of polytopes, we also see how our techniques provide an example of separation between sizes ofsemidefinite programming lifts and linear programming lifts. This is joint work with James Saunderson and Pablo Parrilo.

Furstenberg sets and Furstenberg schemes over finite fields

Series
Algebra Seminar
Time
Friday, March 27, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jordan EllenbergUniversity of Wisconsin, Madison

Please Note: Useful background:The paper I’m discussing: http://arxiv.org/abs/1502.03736Terry Tao’s blog post on Dvir’s theorem: https://terrytao.wordpress.com/2008/03/24/dvirs-proof-of-the-finite-fiel...My earlier paper with Terry and Richard Oberlin about Kakeya restriction over finite fields: http://arxiv.org/abs/0903.1879

The study of extremal configurations of points and subspaces sits at the boundary between combinatorics, harmonic analysis, and number theory; since Dvir’s 2008 resolution of the Kakeya conjecture over finite fields, it has been clear that algebraic geometry is also part of the story.We prove a theorem of Kakeya type for the intersection of subsets of n-space over a finite field with k-planes. Let S be a subset of F_q^n with the "k-plane Furstenberg property": for every k-plane V, there is a k-plane W parallel to V which intersects S in at least q^c points. We prove that such a set has size at least a constant multiple of q^{cn/k}. The novelty is the method; we prove that the theorem holds, not only for subsets of the plane, but arbitrary 0-dimensional subschemes, and reduce the problem by Grobner methods to a simpler one about G_m-invariant non-reduced subschemes supported at a point. The talk will not assume that everyone in the room is an algebraic geometer. It will, however, try to convince everyone in the room that it can be useful to be an algebraic geometer.This is joint work with Daniel Erman.

The Euclidean Distance Degree

Series
Algebra Seminar
Time
Wednesday, March 25, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Bernd SturmfelsUC Berkeley
The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. The Euclidean distance degree is the number of critical points for this optimization problem. We focus on projective varieties seen in engineering applications, and we discuss tools for exact computation. Our running example is the Eckart-Young Theorem which relates the nearest point map for low rank matrices with the singular value decomposition. This is joint work with Jan Draisma, Emil Horobet, Giorgio Ottaviani, Rekha Thomas.

Birational Models of Moduli of Sheaves on Surfaces via the Derived Category

Series
Algebra Seminar
Time
Wednesday, March 11, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Aaron BertramUniversity of Utah
Jacobians aren't particularly interesting from the point of view of the minimal model program, and neither are the moduli spaces of vector bundles on curves. But once we pass to vector bundles of higher rank (or torsion-free sheaves) on surfaces, then the birational geometry becomes very interesting. In this talk, I want to describe some recent results that rely on "tilting" the category of coherent sheaves on a surface to produce birational models of moduli that are themselves moduli spaces that come up naturally in the minimal model program.

Inclusion of Spectrahedra, the Matrix Cube Problem and Beta Distributions.

Series
Algebra Seminar
Time
Monday, March 9, 2015 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Igor KlepUniversity of Auckland
Given a tuple A=(A_1,...,A_g) of symmetric matrices of the same size, the affine linear matrix polynomial L(x):=I-\sum A_j x_j is a monic linear pencil. The solution set S_L of the corresponding linear matrix inequality, consisting of those x in R^g for which L(x) is positive semidefinite (PsD), is called a spectrahedron. It is a convex basic closed semialgebraic subset of R^g. Given a spectrahedron S_L, the matrix cube problem of Nemirovskii asks for the biggest cube [-r,r]^g included in S_L. We solve a relaxation of this problem based on``matricial’’ spectrahedra and estimate the error inherent in this relaxation. The talk is based on joint work with B. Helton, S. McCullough and M. Schweighofer.

Uniform bounds on rational points on curves of low Mordell-Weil rank

Series
Algebra Seminar
Time
Wednesday, February 18, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Eric KatzUniversity of Waterloo
In this talk, I discuss our recent proof that there is a uniform bound forthe number of rational points on genus g curves of Mordell-Weill rank atmost g-3, extending a result of Stoll on hyperelliptic curves. I outlinethe Chabauty-Coleman for bounding the number of rational points on a curveof low Mordell-Weil rank and discuss the challenges to making the bounduniform. These challenges involving p-adic integration and Newton polygonestimates, and are answered by employing techniques in Berkovich spaces,tropical geometry, and the Baker-Norine theory of linear systems on graphs.

An Equidistribution Result in Non-Archimedean Dynamics

Series
Algebra Seminar
Time
Monday, January 26, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Kenny JacobsUniversity of Georgia
Let K be a complete, algebraically closed, non-Archimedean field, and let $\phi$ be a rational function defined over K with degree at least 2. Recently, Robert Rumely introduced two objects that carry information about the arithmetic and the dynamics of $\phi$. The first is a function $\ord\Res_\phi$, which describes the behavior of the resultant of $\phi$ under coordinate changes on the projective line. The second is a discrete probability measure $\nu_\phi$ supported on the Berkovich half space that carries arithmetic information about $\phi$ and its action on the Berkovich line. In this talk, we will show that the functions $\ord\Res_\phi(x)$ converge locally uniformly to the Arakelov-Green's function attached to $\phi$, and that the family of measures $\nu_{\phi^n}$ attached to the iterates of $\phi$ converge to the equilibrium measure of $\phi$.​

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