| BASIC NOTIONS SEMINAR: | Benedict Gross Harvard University |
"Discriminants and root discriminants" |
| on Monday, February 13, 2012, at 3:00 pm in Science Center, room 507 | ||
| HARVARD/MIT GRADUATE STUDENT SEMINAR IN GEOMETRIC REPRESENTATION THEORY: | Dennis Gaitsgory Harvard University |
"Factorizable gerbes" |
| on Tuesday, February 14, 2012, at 5:30 pm in MIT room 2-190 | ||
| NUMBER THEORY SEMINAR: | Keerthi Madapusi Pera Harvard University |
The Tate conjecture for K3 surfaces over finite fields of odd characteristic |
| on , February 15, 202012, at 3:00 pm in Science Center, room 507 | ||
| Using recent advances in the theory of integral models of Shimura varieties, mainly due to Kisin, and the classical Kuga-Satake construction, we show that K3 surfaces over finite fields of odd characteristic p satisfy the Tate conjecture, as long as they admit a polarization of degree prime-to-p. | ||
| RANDOM MATRIX & PROBABILITY SEMINAR: | David R. Nelson Harvard University (Department of Physics) |
Title: Competition and Cooperation at Frontiers |
| on Thursday, February 16, 2012, at 3:00 pm in Science Center, room 232 | ||
| Species often expand from where they first evolved, invade into favorable habitats or move in response to climate changes, or gradients in nutrients, salinity, ambient temperature, etc., in the case of biofilms. Recent microbial experiments on bacteria and yeast have uncovered a remarkable genetic demixing phenomenon at the frontier of a two-dimensional range expansion. Simple models of asexual biological evolution at expanding frontiers can explain both the spatio-genetic correlations that develop for neutral competitions and the effect of natural selection. However, new questions arise when two or more species (or genetic variants of the same species) display cooperative or antagonistic growth strategies. For example, can mutualism prevent demixing? Evidence for a phase transition at a critical degree of cooperativity for mutualists will be presented in a one dimensional stepping stone model. | ||
| GAUGE THEORY & TOPOLOGY: | John Pardon Stanford University |
Totally disconnected groups (not) acting on three-manifolds |
| on Friday, February 17, 2012, at 3:30 in Science Center, room 507 | ||
| Hilbert's Fifth Problem asks whether every topological group which is a manifold is in fact a (smooth!) Lie group; this was solved in the affirmative by Gleason and Montgomery--Zippin. A stronger conjecture is that a locally compact topological group which acts faithfully on a manifold must be a Lie group. This is the Hilbert--Smith Conjecture, which in full generality is still wide open. It is known, however (as a corollary to the work of Gleason and Montgomery--Zippin) that it suffices to rule out the case of the additive group of $p$-adic integers acting faithfully on a manifold. I will present a solution in dimension three. The proof uses tools from low-dimensional topology, for example incompressible surfaces, minimal surfaces, and a property of the mapping class group. | ||
| GAUGE THEORY & TOPOLOGY: | Max Lipyanskiy Columbia University |
Gromov-Uhlenbeck Compactness |
| on Friday, February 24, 2012, at 3:30 pm in Science Center, room 507 | ||
| We introduce an analytic framework that, in special circumstances, unites Yang-Mills theory and the theory of pseudoholomorphic curves. As an application of these ideas, we discuss the relation between instanton Floer homology and Lagrangian Floer homology of representation varieties. | ||
| BASIC NOTIONS SEMINAR: | Barry Mazur Harvard University |
"Statistics in the arithmetic of elliptic curves" |
| on Monday, February 27, 2012, at 3:00 pm in Science Center 507 | ||
| RANDOM MATRIX SEMINAR: | Michael Damron Princeton University |
A simplified proof of the relation between scaling exponents in first - passage percolation. |
| on Thursday, March 1, 2012, at 3:00 pm in Science Center, room 232 | ||
| Abstract: In first passage percolation, we place i.i.d. non-negative weights on the nearest-neighbor edges of Z^d and study the induced random metric. A long-standing conjecture gives a relation between two "scaling exponents": one describes the variance of the distance between two points and the other describes the transversal fluctuations of optimizing paths between the same points. This is sometimes referred to as the "KPZ relation." In a recent breakthrough work, Sourav Chatterjee proved this conjecture using a strong definition of the exponents. I will discuss work I just completed with Tuca Auffinger, in which we introduce a new and intuitive idea that replaces Chatterjee's main argument and gives an alternative proof of the relation. One advantage of our argument is that it does not require a certain non-trivial technical assumption of Chatterjee on the weight distribution. | ||
| BASIC NOTIONS SEMINAR: | Wilfried Schmid Harvard University |
"Mixed Hodge modules" |
| on Monday, March 5, 2012, at 3:00 pm in Science Center, room 507 | ||