## Seminars and Colloquia by Series

Thursday, March 17, 2011 - 16:00 , Location: Skiles 006 , Joe Rabinoff , Harvard University , Organizer: Matt Baker
An elliptic curve over the integer ring of a p-adic field whose special fiber is ordinary has a canonical line contained in its p-torsion.  This fact has many arithmetic applications: for instance, it shows that there is a canonical partially-defined section of the natural map of modular curves X_0(Np) -> X_0(N).  Lubin was the first to notice that elliptic curves with "not too supersingular" reduction also contain a canonical order-p subgroup.  I'll begin the talk by giving an overview of Lubin and Katz's theory of the canonical subgroup of an elliptic curve.  I'll then explain one approach to defining the canonical subgroup of any abelian variety (even any p-divisible group), and state a very general existence result.  If there is time I'll indicate the role tropical geometry plays in its proof.
Thursday, March 17, 2011 - 15:00 , Location: Skiles 006 , Kirsten Wickelgren , Harvard University , Organizer: Matt Baker
Grothendieck's anabelian conjectures say that hyperbolic curves over certain fields should be K(pi,1)'s in algebraic geometry. It follows that points on such a curve are conjecturally the sections of etale pi_1 of the structure map. These conjectures are analogous to equivalences between fixed points and homotopy fixed points of Galois actions on related topological spaces. This talk will start with an introduction to Grothendieck's anabelian conjectures, and then present a 2-nilpotent real section conjecture: for a smooth curve X over R with negative Euler characteristic, pi_0(X(R)) is determined by the maximal 2-nilpotent quotient of the fundamental group with its Galois action, as the kernel of an obstruction of Jordan Ellenberg. This implies that the set of real points equipped with a real tangent direction of the smooth compactification of X is determined by the maximal 2-nilpotent quotient of Gal(C(X)) with its Gal(R) action, showing a 2-nilpotent birational real section conjecture.
Monday, March 14, 2011 - 15:00 , Location: Skiles 005 , Patrick Ingram , University of Waterloo , Organizer: Matt Baker
In classical holomorphic dynamics, rational self-maps of the Riemann sphere whose critical points all have finite forward orbit under iteration are known as post-critically finite (PCF) maps. A deep result of Thurston shows that if one excludes examples arising from endomorphisms of elliptic curves, then PCF maps are in some sense sparse, living in a countable union of zero-dimensional subvarieties of the appropriate moduli space (a result offering dubious comfort to number theorists, who tend to work over countable fields). We show that if one restricts attention to polynomials, then the set of PCF points in moduli space is actually a set of algebraic points of bounded height. This allows us to give an elementary proof of the appropriate part of Thurston's result, but it also provides an effective means of listing all PCF polynomials of a given degree, with coefficients of bounded algebraic degree (up to the appropriate sense of equivalence).
Monday, March 7, 2011 - 15:00 , Location: Skiles 005 , Doug Ulmer , Georgia Tech , Organizer: Matt Baker
Let k be a field (not of characteristic 2) and let t be an indeterminate.  Legendre's elliptic curve is the elliptic curve over k(t) defined by y^2=x(x-1)(x-t).  I will discuss the arithmetic of this curve (group of solutions, heights, Tate-Shafarevich group) over the extension fields k(t^{1/d}).  I will also mention several variants and open problems which would make good thesis topics.
Friday, February 25, 2011 - 13:05 , Location: Skiles 006 , , University of South Carolina , , Organizer: Stavros Garoufalidis
We begin this talk by discussing four different problems that arenumber theoretic or combinatorial in nature.  Two of these problems remainopen and the other two have known solutions.  We then explain how these seeminglyunrelated problems are connected to each other.  To disclose a little more information,one of the problems with a known solution is the following:  Is it possible to find anirrational number $q$ such that the infinite geometric sequence $1, q, q^{2}, \dots$has 4 terms in arithmetic progression?
Monday, February 14, 2011 - 15:00 , Location: Skiles 005 , Matt Baker , Georgia Tech , Organizer: Matt Baker
I will discuss some recent results, obtained jointly with Sam Payne and Joe Rabinoff, on tropicalizations of elliptic curves.
Monday, November 15, 2010 - 10:00 , Location: Skiles 255 , Uli Walther , Purdue University , Organizer: Anton Leykin
I will discuss D-module type invariants on hyperplane arrangements and their relation to the intersection lattice (when known).
Wednesday, November 10, 2010 - 14:00 , Location: D.M. Smith Room 015 , , University of California, Berkeley , Organizer: Anton Leykin
A smooth quartic curve in the projective plane has 36 representations as a symmetric determinant of linear forms and 63 representations as a sum of three squares. We report on joint work with Daniel Plaumann and Cynthia Vinzant regarding the explicit computation of these objects. This lecture offers a gentle introduction to the 19th century theory of plane quartics from the current perspective of convex algebraic geometry.
Friday, June 18, 2010 - 15:05 , Location: Skiles 171 , Christoph Koutschan , RISC Austria , , Organizer: Stavros Garoufalidis
In this talk we recall some modular techniques (chinese remaindering,rational reconstruction, etc.) that play a crucial role in manycomputer algebra applications, e.g., for solving linear systems over arational function field, for evaluating determinants symbolically,or for obtaining results by ansatz ("guessing"). We then discuss howmuch our recent achievements in the areas of symbolic summation andintegration and combinatorics benefited from these techniques.
Monday, May 3, 2010 - 14:00 , Location: Skiles 171 , Pavlos Tzermias , University of Tennessee Knoxville , Organizer: Matt Baker
The polynomials mentioned in the title were introduced by Cauchy and Liouville in 1839 in connection with early attempts at a proof of Fermat's Last Theorem. They were subsequently studied by Mirimanoff who in 1903 conjectured their irreducibility over the rationals. During the past fifteen years it has become clear that Mirimanoff's conjecture is closely related to properties of certain special functions and to some deep results in diophantine approximation. Apparently, there is also a striking connection to hierarchies of certain evolution equations (which this speaker is not qualified to address). We will present and discuss a number of recent results on this problem.