Seminars and Colloquia by Series

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.

Hypergeometric functions: the GKZ-perspective

Series
Algebra Seminar
Time
Monday, March 2, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Uli Walther – Purdue University
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.

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