# Seminars & Events for Department Colloquium

##### Distribution of primes in arithmetic progressions with application

We discuss the recent developments on the distribution of primes in arithmetic progressions which are regarded as stronger versions of the Bombieri-Vinogradov theorem. First we explain why the arguments based on the dispersion method, Fourier analysis and Kloosterman sums make it possible to obtain mean value results for arithmetic prgressions to moduli beyond x1=2. In particular, stronger results can be derived if certain

constraints on the moduli are imposed. we also describe how to apply these results to study the small gaps between consecutive primes.

##### Ricci Curvature and Infinite Dimensional Analysis on Path Space

In this talk we discuss recent connections between the Ricci curvature of a Riemannian manifold and the analysis on the path space of the manifold. We will see that bounded Ricci curvature controls the analysis on the path space P(M) of a manifold in much the same way that lower Ricci curvature controls the analysis on M itself. PLEASE CLICK ON COLLOQUIUM TITLE FOR COMPLETE ABSTRACT.

##### Dynamics near criticality

PLEASE CLICK ON COLLOQUIUM TITLE FOR COMPLETE ABSTRACT. Heuristically, one can give arguments why the fluctuations of classical models of statistical mechanics near criticality are typically expected to be described by nonlinear stochastic PDEs.

##### Multiple Dirichlet Series

We review the theory of multiple Dirichlet series which are Dirichlet series in several complex variables having analytic continuation with finitely many polar divisors and satisfying a finite group of functional equations. Converse theorems state that if a Dirichlet series and all its twists satisfy functional equations of the right type then the Dirichlet series is modular, i.e., it is the Mellin transform of an automorphic form. We shall present a new converse theorem for Dirichlet series in two complex variables. As an application we will show that the Shintani zeta function associated to a certain pre homogeneous vector space is in fact a Weyl group multiple Dirichlet series of type A_2. This is joint work with Nikos Diamantis.

##### Tales of Our Forefathers

This is not a mathematics talk but it is a talk for mathematicians. Too often, we think of historical mathematicians as only names assigned to theorems. With vignettes and anecdotes, I'll convince you they were also human beings and that, as the Chinese say, "May you live in interesting times" really is a curse.

##### Picard-Lefschetz theory and hidden symmetries

Picard-Lefschetz theory studies algebraic varieties by induction on their dimension. It can be used to determine their topology, and in more modern terms their symplectic geometry. We will apply this theory to describe extra structure which appears for a special class of algebraic varieties. This is part geometry and part homological algebra; the latter appears because the outcome is formulated in terms of actions of algebraic groups on categories.

##### The triangulation conjecture

The triangulation conjecture stated that any n-dimensional topological manifold is homeomorphic to a simplicial complex. It is true in dimensions at most 3, but false in dimension 4 by the work of Casson and Freedman. In this talk I will explain the proof that the conjecture is also false in higher dimensions. This result is based on previous work of Galewski-Stern and Matumoto, who reduced the problem to a question in low dimensions (the existence of elements of order 2 and Rokhlin invariant one in the 3-dimensional homology cobordism group). The low-dimensional question can be answered in the negative using a variant of Floer homology, Pin(2)-equivariant Seiberg-Witten Floer homology.

##### Universal spaces for birational invariants

Anabelian geometry techniques allow the construction of explicit universal spaces which capture birational properties of algebraic varieties. I will describe this theory and its applications (joint with F. Bogomolov).

##### Euler’s Amicable Numbers

In this talk, I’ll first sketch the life and work of Leonhard Euler (1707 – 1783), one of the great figures from the long and glorious history of mathematics.

##### Small gaps between primes

It is believed that there should be infinitely many pairs of primes which differ by 2; this is the famous twin prime conjecture. More generally, it is believed that for every positive integer $m$ there should be infinitely many sets of $m$ primes, with each set contained in an interval of size roughly $m\log{m}$. Although proving these conjectures seems to be beyond our current techniques, recent progress has enabled us to obtain some partial results. We will introduce a refinement of the `GPY sieve method' for studying these problems. This refinement will allow us to show (amongst other things) that $\liminf_n(p_{n+m}-p_n)<\infty$ for any integer $m$, and so there are infinitely many bounded length intervals containing $m$ primes.

##### Torsion in the cohomology of arithmetic groups, and the Langlands program

In this talk I will discuss some recent work related to torsion in the cohomology of arithmetic groups, and its connection to the Langlands program. I will focus primarily on stability phenomena in the cohomology of SL_n (joint work of the speaker with Frank Calegari). I will also discuss some other examples involving cohomology in low degree (joint work of the speaker with Toby Gee), as well as other recent work on this topic (Calegari--Venkatesh, Scholze) . As I hope to explain, these examples suggest that the deep harmonic analytic theory of automorphic forms, due to Langlands, Arthur, etc., should have an analogue in the torsion world, the precise nature of which remains to be understood.

##### Contact structures on high dimensional manifolds

PLEASE CLICK ON COLLOQUIUM TITLE FOR COMPLETE ABSTRACT. Contact structures on manifolds are very natural objects. Born over two centuries ago, in the work of Huygens, Hamilton and Jacobi on geometric optics, they have been studied by many mathematicians and seem to touch on diverse areas of mathematics and physics, but only in the last couple of decades have they moved into the foreground of mathematics.

##### Some special functions and their special values

In his last letter to G. H. Hardy, Ramanujan wrote about his so-called "mock theta functions". These functions are now largely understood (by work of Zwegers) in the context of nonholomorphic Jacobi forms. Here we discuss recent work which pertains to the roles of these mock theta functions in Zagier's theory of quantum modular forms, and we construct new mock theta functions (one for each rational elliptic curve) from the Weierstrass zeta-function and we explain their relevance in the study of the arithmetic of elliptic curves.

##### Hyperbolic triangles with no nonconstant Neumann eigenfunctions

The Neumann Laplacian acts on a dense subspace of square-integrable functions on a Riemannian manifold. If the manifold is compact then the spectrum is a countable discrete subset of the nonnegative reals. If the manifold is not compact then the spectrum can be mixture of eigenvalues and essential spectrum. In joint work with Luc Hillairet, we show that the generic triangle in the hyperbolic plane having a cusp has no nonconstant Neumann eigenfunctions, and hence there are no eigenvalues embedded in its essential spectrum. The proof involves a singular analytic perturbation, `asymptotic separation of variables', and an analysis of potential eigenvalue crossings.

##### Periodicity and Complexity

A beautiful example of a global property being determined by a local one is the Morse-Hedlund Theorem; it gives the relation between the global property of periodicity for an infinite word in a finite alphabet and local information on the complexity of the word. I will discuss higher dimensional versions of this problem, using local conditions on complexity to determine global properties of configurations. This is joint work with Van Cyr.

##### Modular forms for noncongruence subgroups: an overview

The two most important tools used to study the arithmetic of modular forms for congruence subgroups are the Hecke theory and l-adic Galois representations. Unlike their congruence counterpart, the arithmetic for noncongruence modular forms remains mysterious. A main reason is the lack of efficient Hecke operators. Based on their numerical evidence, Atkin and Swinnerton-Dyer proposed a substitute for the Hecke eigenform at good primes, expressed in terms of 3-term congruence relations. Later Scholl attached l-adic Galois representations to the space of noncongruence forms. These representations are motivic, and hence should correspond to automorphic representations according to the Langlands philosophy. The automorphy of Scholl representations is established only for very special cases.

##### Solving Boltzmann Equation, Green's Function Approach.

PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT. We will describe an quantitative approach for solving the Boltzmann equation in the kinetic theory. The approach has been developed, with Shih-Hsien Yu, in the past decade and proven effective in understanding some of the important physical phenomena.