## Seminars and Colloquia by Series

### Twists of elliptic curves with a large set of integral points over function fields

Series
Algebra Seminar
Time
Monday, April 12, 2010 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Ricardo ConceicaoOxford College of Emory University
We will explicitly construct twists of elliptic curves with an arbitrarily large set of integral points over $\mathbb{F}_q(t)$. As a motivation to our main result, we will discuss a conjecture of Vojta-Lang concerning the behavior of integral points on varieties of log-general type over number fields and present a natural translation to the function field setting. We will use our construction to provide an isotrivial counter-example to this conjecture. We will also show that our main result also provides examples of elliptic curves with arbitrarily large set of independent points and of function fields with large $m$-class rank.

### Orbitopes

Series
Algebra Seminar
Time
Monday, April 5, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Frank SottileTexas A&amp;amp;M
An orbitope is the convex hull of an orbit of a compact group acting linearly on a vector space. Instances of these highly symmetric convex bodies have appeared in many areas of mathematics and its applications, including protein reconstruction, symplectic geometry, and calibrations in differential geometry.In this talk, I will discuss Orbitopes from the perpectives of classical convexity, algebraic geometry, and optimization with an emphasis on motivating questions and concrete examples. This is joint work with Raman Sanyal and Bernd Sturmfels.

### 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 CornEmory 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 YuGeorgia 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 PapikianPenn 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 UlmerGeorgia 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 SwinarskiUniversity 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 GaribaldiEmory 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 GurevichInstitute 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 PetscheHunter 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.