Seminars and Colloquia by Series

Explicit Brauer-Manin obstructions on K3 surfaces

Series
Algebra Seminar
Time
Monday, March 29, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Patrick Corn – Emory University
We will outline some open questions about rational points on varieties, and present the results of some computations on explicit genus-2 K3 surfaces. For example, we'll show that there are no rational numbers w,x,y,z (not all 0) satisfying the equation w^2 + 4x^6 = 2(y^6 + 343z^6).

On a parametrization of positive semidefinite matrices

Series
Algebra Seminar
Time
Monday, March 15, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Josephine Yu – Georgia Tech
We study a class of parametrizations of convex cones of positive semidefinite matrices with prescribed zeros. Each such cone corresponds to a graph whose non-edges determine the prescribed zeros. Each parametrization in the class is a polynomial map associated with a simplicial complex comprising cliques of the graph. The images of the maps are convex cones, and the maps can only be surjective onto the cone of zero-constrained positive semidefinite matrices when the associated graph is chordal. Our main result gives a semi-algebraic description of the image of the parametrizations for chordless cycles. The work is motivated by the fact that the considered maps correspond to Gaussian statistical models with hidden variables. This is joint work with Mathias Drton.

On the arithmetic of modular varieties of D-elliptic sheaves

Series
Algebra Seminar
Time
Monday, March 8, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Mihran Papikian – Penn State
We discuss some arithmetic properties of modular varieties of D-elliptic sheaves, such as the existence of rational points or the structure of their "fundamental domains" in the Bruhat-Tits building. The notion of D-elliptic sheaf is a generalization of the notion of Drinfeld module. D-elliptic sheaves and their moduli schemes were introduced by Laumon, Rapoport and Stuhler in their proof of certain cases of the Langlands conjecture over function fields.

Explicit points on the Legendre elliptic curve

Series
Algebra Seminar
Time
Monday, March 1, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Doug Ulmer – Georgia Tech
It turns out to be very easy to write down interesting points on the classical Legendre elliptic curve y^2=x(x-1)(x-t) and show that they generate a group of large rank. I'll give some basic background, explain the construction, and discuss related questions which would make good thesis projects (both MS and PhD).

State polytopes and GIT

Series
Algebra Seminar
Time
Monday, February 15, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
David Swinarski – University of Georgia
State polytopes in commutative algebra can be used to detect the geometric invariant theory (GIT) stability of points in the Hilbert scheme. I will review the construction of state polytopes and their role in GIT, and present recent work with Ian Morrison in which we use state polytopes to estabilish stability for curves of small genus and low degree, confirming predictions of the minimal model program for the moduli space of curves.

There is no "Theory of Everything" inside E8

Series
Algebra Seminar
Time
Monday, February 8, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Skip Garibaldi – Emory University
The "Exceptionally Simple Theory of Everything" has been the subject of articles in The New Yorker (7/21/08), Le Monde (11/20/07), the Financial Times (4/25/09), The Telegraph (11/10/09), an invited talk at TED (2/08), etc. Despite positive descriptions of the theory in the popular press, it doesn't work. I'll explain a little of the theory, the mathematical reasons why it doesn't work, and a theorem (joint work with Jacques Distler) that says that no similar theory can work. This talk should be accessible to all graduate students in mathematics.

The Geometric Weil Representation

Series
Algebra Seminar
Time
Wednesday, January 27, 2010 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Shamgar Gurevich – Institute for Advanced Study, Princeton –
This is a sequel to my first talk on "group representation patterns in digital signal processing". It will be slightly more specialized. The finite Weil representation is the algebra object that governs the symmetries of Fourier analysis of the Hilbert space L^2(F_q). The main objective of my talk is to introduce the geometric Weil representation---developed in a joint work with Ronny Hadani---which is an algebra-geometric (l-adic perverse Weil sheaf) counterpart of the finite Weil representation. Then, I will explain how the geometric Weil representation is used to prove the main results stated in my first talk. In the course, I will explain the Grothendieck geometrization procedure by which sets are replaced by algebraic varieties and functions by sheaf theoretic objects.

A non-Archimedean Weyl equidistribution theorem

Series
Algebra Seminar
Time
Tuesday, January 19, 2010 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Clay Petsche – Hunter College
Weyl proved that if an N-dimensional real vector v has linearly independent coordinates over Q, then its integer multiples v, 2v, 3v, .... are uniformly distributed modulo 1.  Stated multiplicatively (via the exponential map), this can be viewed as a Haar-equidistribution result for the cyclic group generated by a point on the N-dimensional complex unit torus.  I will discuss an analogue of this result over a non-Archimedean field K, in which the equidistribution takes place on the N-dimensional Berkovich projective space over K.  The proof uses a general criterion for non-Archimedean equidistribution, along with a theorem of Mordell-Lang type for the group variety G_m^N over the residue field of K, which is due to Laurent.

Certified numerical polynomial homotopy continuation

Series
Algebra Seminar
Time
Monday, November 23, 2009 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Anton Leykin – Georgia Tech
This talk will start with an introduction to the area of numerical algebraic geometry. The homotopy continuation algorithms that it currently utilizes are based on heuristics: in general their results are not certified. Jointly with Carlos Beltran, using recent developments in theoretical complexity analysis of numerical computation, we have implemented a practical homotopy tracking algorithm that provides the status of a mathematical proof to its approximate numerical output.

Dynamical Mordell-Lang problems

Series
Algebra Seminar
Time
Thursday, April 23, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Tom Tucker – Univ. of Rochester
Let S be a group or semigroup acting on a variety V, let x be a point on V, and let W be a subvariety of V. What can be said about the structure of the intersection of the S-orbit of x with W? Does it have the structure of a union of cosets of subgroups of S? The Mordell-Lang theorem of Laurent, Faltings, and Vojta shows that this is the case for certain groups of translations (the Mordell conjecture is a consequence of this). On the other hand, Pell's equation shows that it is not true for additive translations of the Cartesian plane. We will see that this question relates to issues in complex dynamics, simple questions from linear algebra, and techniques from the study of linear recurrence sequences.

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