The usual Katz-Mazur model for the modular curve $X(p^n)$ has horribly singular reduction. For large n there isn't any model of $X(p^n)$ which has good reduction, but after extending the base one can at least find a semistable model, which means that the special fiber only has normal crossings as singularities. We will reveal a new picture of the special fiber of a semistable model of the entire tower of modular curves. We will also indicate why this problem is important from the point of view of the local Langlands correspondence for $GL(2)$ .
In this talk I will describe a real-variable method to extract long-time asymptotics for solutions of many nonlinear equations (including the Schrodinger and mKdV equations). The method has many resemblances to the classical stationary phase method in the theory of oscillatory integrals.
In the talk, I will describe recent attempts to understand the mysterious and beautiful geometry of nodal lines of random spherical harmonics and of random plane waves. If time permits, I will also discuss asymptotic statistical topology of other natural polynomial-like ensembles of random functions. The talk is based on a joint work with Fedja Nazarov.
This will be an introduction to special value formulas for L-functions and especially the uses of modular forms in establishing some of them -- beginning with the values of the Riemann zeta function at negative integers and hopefully arriving at some more recent work on the Birch-Swinnerton-Dyer formula.
I will discuss the problem of determining the number of infinite-volume ground states in the Edwards-Anderson (nearest neighbor) spin glass model on $Z^D$ for $D \geq 2$. There are no complete results for this problem even in $D=2$. I will focus on this case and explain recent results which go some way toward proving that (with zero external field, so that ground states come in pairs, related by a global spin flip) there is only a single ground state pair (GSP).
There is a way to specify any smooth, closed oriented four-manifold using a surface decorated with simple closed curves, something I call a surface diagram. In this talk I will describe three moves on these objects, two of which are reminiscent of Heegaard diagrams for three-manifolds. These may form part of a uniqueness theorem for such diagrams that is likely to be useful for understanding Floer theories for non-symplectic four-manifolds.
ANALYSIS/MATHEMATICAL PHYSICS SEMINAR
In recent research it has become clear that there are fascinating connections between constructive mathematics, especially as formulated in the type theory of Martin-Löf, and homotopy theory, especially in the modern treatment in terms of Quillen model categories and higher-dimensional categories. This talk will survey some of these developments.