L6

This talk will be a survey on the applications of random matrix theory in neuroscience. We will explain why and how we use random matrices to model networks of neurons in the brain. We are mainly interested in the study of neuronal dynamics, and we will present results that cover two parallel directions taken by the field of theoretical neuroscience. First, we will talk about the critical point of transitioning to chaos in cases of random matrices that aim to be more "biologically plausible". And secondly, we will see how a deterministic and a random matrix (corresponding to learned structure and noise in a neuronal network) can interact in a dynamical system.

We construct a cellular decomposition of the
Axelrod-Singer-Fulton-MacPherson compactification of the configuration
spaces in the plane, that is compatible with the operad composition.
Cells are indexed by trees with bi-coloured edges, and vertices are labelled by 
cells of the cacti operad. This answers positively a conjecture stated in 
2000 by Kontsevich and Soibelman.

Stevenhagen conjectured that the density of d such that the negative Pell equation x^2-dy^2=-1 is solvable over the integers is 58.1% (to the nearest tenth of a percent), in the set of positive squarefree integers having no prime factors congruent to 3 modulo 4. In joint work with Peter Koymans, Djordjo Milovic, and Carlo Pagano, we use a recent breakthrough of Smith to prove that the infimum of this density is at least 53.8%, improving previous results of Fouvry and Klüners, by studying the distribution of the 8-rank of narrow class groups of quadratic number fields.

My talk will have two protagonists: (1) Automorphic representations which -- let's be honest -- are very complicated and mysterious, but also (2) Involutions  (=automorphisms of order at most 2) of connected reductive groups -- these are very concrete and can often be represented by diagonal matrices with entries 1,-1 or i, -i. The goal is to explain how difficult questions about (1) can be reduced to relatively easy, concrete questions about (2).
Automorphic representations are representation-theoretic generalizations of modular forms. Like modular forms, automorphic representations are initially defined analytically. But unlike modular forms -- where we have a reinterpretation in terms of algebraic geometry -- for most automorphic representations we currently only have a (real) analytic definition. The Langlands Program predicts that a wide class of automorphic representations admit the same algebraic properties which have been known to hold for modular forms since the 1960's and 70's. In particular, certain complex numbers "Hecke eigenvalues" attached to these automorphic representations are conjectured to be algebraic numbers. This remains open in many cases (especially those cases of interest in number theory and algebraic geometry), in particular for Maass forms -- functions on the upper half-plane which are a non-holomorphic variant of modular forms.
I will explain how elementary structure theory of reductive groups over the complex numbers provides new insight into the above algebraicity conjectures; in particular we deduce that the Hecke eigenvalues are algebraic for an infinite class of examples where this was not previously known. 
After applying a bunch of "big, old theorems" (in particular Langlands' own archimedean correspondence), it all comes down to studying how involutions of a connected, reductive group vary under group homomorphisms. Here I will write down the key examples explicitly using matrices.

In the 1920s Weyl proved the first non-trivial estimate for the Riemann zeta function on the critical line: \zeta(1/2+it) << (1+|t|)^{1/6+\epsilon}. The analogous bound for a Dirichlet L-function L(1/2,\chi) of conductor q as q tends to infinity is still unknown in full generality. In a breakthrough around 2000, Conrey and Iwaniec proved the analogue of the Weyl bound for L(1/2,\chi) when \chi is assumed to be quadratic of conductor q.  Building on the work of Conrey and Iwaniec, we show (joint work with Matt Young) that the Weyl bound for L(1/2,\chi) holds for all primitive Dirichlet characters \chi. The extension to all moduli q is based on aLindelöf-on-average upper bound for the fourth moment of Dirichlet L-functions of conductor q along a coset of the subgroup of characters modulo d when q^*|d, where q^* is the least positive integer such that q^2|(q^*)^3.

Let n be a random integer (sampled from {1,..,X} for some large X). It is a classical fact that, typically, n will have around (log n)^{log 2} divisors. Must some of these be close together? Hooley's Delta function Delta(n) is the maximum, over all dyadic intervals I = [t,2t], of the number of divisors of n in I. I will report on joint work with Kevin Ford and Dimitris Koukoulopoulos where we conjecture that typically Delta(n) is about (log log n)^c for some c = 0.353.... given by an equation involving an exotic recurrence relation, and then prove (in some sense) half of this conjecture, establishing that Delta(n) is at least this big almost surely.

Let G be a finite abelian group, let k be a number field, and let x be an element of k. We count Galois extensions K/k with Galois group G such that x is a norm from K/k. In particular, we show that such extensions always exist. This is joint work with Christopher Frei and Daniel Loughran.

Let $E/k$ be an elliptic curve over a number field and $K/k$ a Galois extension with group $G$. What can we say about $E(K)$ as a Galois module? Not just what complex representations appear, but its structure as a $\mathbb{Z}[G]$-module. We will look at some examples with small $G$.