Christopher Marks (U of California, Santa Cruz/MPI)
The main purpose of this talk is to explain the connection between vector-valued modular forms for SL(2,Z) and certain ordinary differential equations in the punctured unit disk, whose coefficient functions are holomorphic modular forms. An important tool arising in this context is a modular version of the familiar Wronskian from the classical ODE theory; I will discuss, for example, how this modular Wronskian provides a lower bound for the weight of a nonzero holomorphic form associated to a given representation of SL(2,Z). In the remaining time, I will describe my own work, which attempts to use this ODE theory -- in conjunction with some recent joint work with G. Mason, and a classification theorem of Tuba and Wenzl concerning representations of Artin's Braid group -- to give an algebraic classification of vector-valued modular forms, for irreducible representations of SL(2,Z) of dimension less than six.
Categories of integrable $sl(\infty)$, $o(\infty)$, $sp(\infty)$ - modules
In this talk we discuss two categories of integrable modules: modules with finite-dimensional weight spaces, and tensor modules. The latter modules form an interesting category on which the functor of algebraic dualization preserves the property of a module to have finite Loewy length. This is an unusual situation. Both categories are natural analogs of the category of finite-dimensional modules.
Complete invariants of t-structures
A t-structure is a kind of truncation on a triangulated category that is analogous to the Postnikov approximation in homotopy theory. Using the thick subcategory theorem of Hopkins and Smith, work of Bousfield gives a classification of these truncations in the category of finite p-torsion topological spaces. After a gentle introduction, we will look at the analogous situation in the derived category of a commutative Noetherian ring D(R) and construct complete invariants of t-structures on a subcategory of D(R). Together with work of Deligne and Bezrukavnikov, this allows a classification of these t-structures when R has a dualizing complex.
Target categories for the quantum Tate realization
A Subconvexity Bound for Automorphic $L$-functions for $SL(3,Z)$
In this joint work with Stephan Baier, we prove a subconvexity bound for Godement-Jacquet $L$-functions associated with Maass forms for $SL(3,Z)$ The bound arrives from extending a method of M. Jutila (with new ingredients and innovations) on exponential sums with Fourier coefficients of holomorphic cusp forms for $SL(2,Z)$ to a $GL(3)$ setting.
Triangle groups, finite simple groups and applications
In this talk we will discuss the following problem: Given a triple of integers (r,s,t), which finite simple groups are quotients of the triangle group T(r,s,t)? This problem has many applications, especially concerning Riemann surfaces and Beauville surfaces. In the talk we will focus on the group theoretical aspects of this problem.
Detecting motives vs detecting quantum motives
The Chowla-Selberg phenomenon
Determinantal differential operators: reduction from the big to the small.
A numerical test of the generalized Birch and Swinnerton-Dyer conjecture
In its crudest form the generalized Birch and Swinnerton-Dyer relates the order of vanishing of an L-function at the center of the critical strip to the rank of a Chow group. This talk describes the conjecture and an attempt to put it to a modest test in the case of motives of the form Sym$^3H^1(E)$, where E is an elliptic curve over Q. This is joint work with Joe Buhler and Jaap Top.
Universal family for subgroups of an algebraic group
I describe the construction of a moduli space for the connected subgroups of an algebraic group, and of a universal family. I give a quick illustration of the notion of universal families, trough Grassmann varieties, then discuss in turn the construction of a moduli space and of a universal family, balancing general statements and examples.
Recent progress on the local Langlands conjecture for $G_2$
I will describe recent work, joint with G. Savin, in which we prove a dichotomy result for generic cuspidal representations of the exceptional group G_2, over a p-adic field. This result reduces the local Langlands conjectures for generic cuspidal represenations G_2 to the corresponding conjectures on the classical groups PGL_3 and PGSp_6. These corresponding conjectures are proven and "almost proven", respectively. Our methods include a study of the theta correspondence in the groups E_6 and E_7, a uniqueness result for "Shalika periods" in PGSp_6, and various incarnations of Spin L-functions. The talk will include much background on exceptional groups and the local Langlands conjecture, accessible to a general number theoretic audience. I aim to present a precise summary of our results, with hints about the method of proof.
Stabilization hypothesis and higher braided operads
The stabilization hypothesis of Breen,Baez and Dolan states that k-fold monoidal n-category is "the same" as (k+1)-fold (and therefore $\infty$-fold) monoidal n-category if k is grater or equal to n+2. In the first half of my talk I will explain this hypothesis in an informal manner and I will relate it to the geometry of configuration spaces of k points in $R^n$. In somewhat more technical second half I will introduce n-braided operads and will give a sketch of a proof of a stabilization theorem for n-braided operads. The BBD stabilization hypothesis is an easy corollary from this statement.
Irreducibility of the moduli space $I_n$ of mathematical instanton vector bundles on the projective space $P_3$ for arbitrary odd second Chern class $n$
The problem of description of the moduli space $I_n$ of mathematical instanton vector bundles on the projective space $P_3$ has been a challenging problem since 70's. It was conjectured by R.Hartshorne in 1976 that $I_n$ is irreducible for an arbitrary second Chern class $n>0$. This problem has an affirmative solution for small values of $n$, up to $n=5$. Namely, the cases $n=1,2,3,4$ and 5 were settled by Barth (1977), Hartshorne (1978), Ellingsrud-Stromme (1981), Barth (1981) and Coanda-Tikhomirov-Trautmann (2003), respectively. The aim of this talk is to give a proof of the irreducibility of $I_n$ for arbitrary odd second Chern class $n$".
Homotopy theory of spaces of representations
Circle actions on certain symplectic manifolds with minimal even Betti numbers
Suppose that the circle acts on a symplectic manifold in a Hamiltonian fashion. Under certain "minimal'' conditions of the fixed point set of the action, we classify the integral cohomology ring and the Chern classes of the manifold, and we classify the circle action.
Harder's conjecture and ratios of standard L-values
I will explain how the Bloch-Kato conjecture leads to the following conclusion: any large prime dividing a critical value of the L-function of a classical Hecke eigenform of level 1, should also divide a certain ratio of critical values for the standard L-function of a related genus 2 (and in general vector-valued) Hecke eigenform F. This can be proved in the scalar-valued case, and there is experimental evidence in the vector-valued case (where the relation between f and F is a congruence of Hecke eigenvalues conjectured by Harder).
Transformations between classical geometric structures
How to find all diffeomorphisms between an open subset of a real projective space and an open subset of a sphere that take all straight line segments to arcs of circles? We will discuss this and similar questions - these questions look classical but they are open in higher dimensions. In dimension 4, these questions have unexpected answers.
Moduli of abelian varieties and congruences of Siegel mod. forms
Congruences between modular forms and mixed motives
English
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