# School of Mathematics

## Values of L-Functions and Modular Forms

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.

## Non-abelian Lubin-Tate Theory Modulo $\ell$

## Spectral Geometry of Random Graphs

we will describe various models of sparse and planar graphs and the associated distributions of eigenvalues (and eigenvalue spacings) which come up. The talk will be light on theorems, and heavy on experimental data.

## Global Stringy Orbifold Cohomology, K-Theory and de Rham theory with Possible Applications to Landau-Ginzburg Theory

## Splitting of Iwasawa Modules and Leopoldt Conjecture

Let p be an odd prime number and let F be a totally real field. Let F_cyc be the cyclotomic extension of F generated by the roots of unity of order a power of p . From the maximal abelian extension of F_cyc which is unramified (resp. unramified outside auxiliary primes), we get exact sequences of Iwasawa modules. We will discuss how splitting of these exact sequences are linked to Leopoldt conjecture for F and p . (JW with C. Khare)

## GEOMETRY/DYNAMICAL SYSTEMS

## Rank Bounds for Design Matrices with Applications to Combinatorial Geometry and Locally Correctable Codes

A (q,k,t)-design matrix is an m x n matrix whose pattern of zeros/non-zeros satisfies the following design-like condition: each row has at most q non-zeros, each column has at least k non-zeros and the supports of every two columns intersect in at most t rows. We prove that for $m\geq n$, the rank of any $(q,k,t)$-design matrix over a field of characteristic zero (or sufficiently large finite characteristic) is at least $n - (qtn/2k)^2$ .

Using this result we derive the following applications:

## Invariants of Graphs, Their Associated Clique Complexes and Right-Angled Coxeter Groups

Associated to any simplicial graph there is a right-angled Coxeter group. Invariants of the Coxeter group such as its growth series or its weighted L^2 Betti numbers can be computed from the graph's clique complex (i.e., its flag complex).

## A Unified Framework for Testing Linear-Invariant Properties

In a sequence of recent papers, Sudan and coauthors have investigated the relation between testability of properties of Boolean functions and the invariance of the properties with respect to transformations of the domain. Linear-invariance is arguably the most common such symmetry for natural properties of Boolean functions on the hypercube. Hence, it is an important goal to find necessary and sufficient conditions for testability of linear-invariant properties.