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Series: Algebra Seminar

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).

Series: Algebra Seminar

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.

Series: Algebra Seminar

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.

Series: Algebra Seminar

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.

Series: Algebra Seminar

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.

Series: Algebra Seminar

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.

Series: Algebra Seminar

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.

Series: Algebra Seminar

Starting with some classical hypergeometric functions, we explain how to derive their classical univariate differential equations. A severe change of coordinates transforms this ODE into a system of PDE's that has nice geometric aspects. This type of system, called A-hypergeometric, was introduced by Gelfand, Graev, Kapranov and Zelevinsky in about 1985. We explain some basic questions regarding these systems. These are addressed through homology, combinatorics, and toric geometry.