Seminars and Colloquia by Series

A 2-nilpotent real section conjecture

Series
Algebra Seminar
Time
Thursday, March 17, 2011 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Kirsten WickelgrenHarvard University
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.

Post-critically finite polynomials

Series
Algebra Seminar
Time
Monday, March 14, 2011 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Patrick IngramUniversity of Waterloo
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).

Arithmetic of the Legendre curve

Series
Algebra Seminar
Time
Monday, March 7, 2011 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Doug UlmerGeorgia Tech
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.

Four Seemingly Unrelated Problems

Series
Algebra Seminar
Time
Friday, February 25, 2011 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Michael FilasetaUniversity of South Carolina
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?

Tropical elliptic curves

Series
Algebra Seminar
Time
Monday, February 14, 2011 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Matt BakerGeorgia Tech
I will discuss some recent results, obtained jointly with Sam Payne and Joe Rabinoff, on tropicalizations of elliptic curves.

On some invariants of arrangements

Series
Algebra Seminar
Time
Monday, November 15, 2010 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Uli WaltherPurdue University
I will discuss D-module type invariants on hyperplane arrangements and their relation to the intersection lattice (when known).

Quartic Curves and their Bitangents

Series
Algebra Seminar
Time
Wednesday, November 10, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
D.M. Smith Room 015
Speaker
Bernd SturmfelsUniversity of California, Berkeley
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.

The beneficial use of homomorphic images in computer algebra

Series
Algebra Seminar
Time
Friday, June 18, 2010 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Christoph KoutschanRISC Austria
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.

Factorization of Cauchy-Liouville-Mirimanoff polynomials

Series
Algebra Seminar
Time
Monday, May 3, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Pavlos TzermiasUniversity of Tennessee Knoxville
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.

Chern classes identities from weak coupling limits

Series
Algebra Seminar
Time
Monday, April 26, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Paolo AluffiFlorida State University
We generalize a construction of Ashoke Sen of weak couplinglimits' for certain types of elliptic fibrations. Physics argumentsinvolving tadpole anomaly cancellations lead to conjectural identitiesof Euler characteristics. We generalize these identities to identitiesof Chern classes, which we are able to verify mathematically inseveral instances. For this purpose we propose a generalization of theso-called Sethi-Vafa-Witten identity'. We also obtain a typeclassification of configurations of smooth branes satisfying thetadpole condition. This is joint work with Mboyo Esole (Harvard).