# Seminars & Events for Department Colloquium

##### Ultrametric subsets with large Hausdorff dimensions

##### The black hole stability problem

##### Stability Theorems for some Sharp Inequalities and their Applications

We explain recent results on stability theorems for some classical functional and ge- ometric inequalities, along with two applications: one to evolution equations, and one to statistical mechanics. The inequalities in question include certain Gagliardo-Nirenberg- Sobolev inequalities, the Brun-Minkowski inequality, for example. In these inequalities, all of the cases of equality are known, and indeed, Minkowski's contribution to the Brun- Minkowski inequality was to both determine the cases of equality. One can now ask if, in such an inequality, one almost has equality, is one in some sense near to one of the known cases of equality?

##### On the growth of Betti numbers of arithmetic groups

We study the asymptotic behavior of the Betti numbers of higher rank locally symmetric manifolds as their volumes tend to infinity, and prove a uniform version of the Lueck Approximation Theorem, which is much stronger than the linear upper bounds proved by Gromov. The basic idea is to adapt the theory of local convergence, originally introduced for sequences of graphs of bounded degree by Benjamimi and Schramm, to sequences of Riemannian manifolds. Using rigidity theory we are able to show that when the volume tends to infinity, the manifolds locally converge to the universal cover in a sufficiently strong manner that allows one to derive the convergence of the normalized Betti numbers. Similarly, and more generally, we show that the normalized multiplicity of any unitary representation converges to its Plancherel measure.

##### Littlewood and large forests

Motivated by a classical result of Sz.-Nagy in functional analysis, Dixmier asked in 1950 which group representations can be made unitary. This question is still open, but I will report on some recent progress. We approach the question with ideas borrowed from XIXth century electricity theory as well as from contemporary percolation theory. As a result, we obtain notably non-unitarizable representations for Burnside groups and a new characterization of amenable groups. (The talk will be expository.)

##### Bordered Floer homology

Bordered Floer homology is an invariant for three-manifolds with parameterized boundary. It associates to a parameterized surface a differential graded algebra, to a three-manifold with boundary a module over that algebra. It can be used to recapture the invariant of a closed three-manifold, via the derived tensor product. I will describe this construction and some of its applications. This is joint work with Robert Lipshitz and Dylan Thurston.

##### On the Ma-Trudinger-Wang condition

The Ma-Trudinger-Wang condition, first introduced to prove regularity of optimal transport maps for general cost functions, has turned out to be a useful tool for: - Obtaining geometric informations on the underlying manifold. - Making the principal-agent problem theoretically and computationally tractable, allowing to derive uniqueness and stability of the principal's optimum strategy. In this expository talk I'll give an overview of these results.

##### Some applications of almost mathematics

I will try to explain how, by inserting the word "almost" in appropriate places in a commutative algebra textbook and using the fact that $\mathbb{Q}_p$ an $\mathbb{F}_p((t))$ have the same residue field, Faltings gave a new proof of a result of Fontaine and Wintenberger relating the absolute Galois groups of these two fields. I will also talk about the generalizations of these type of results to higher dimensions.

##### Congruent Numbers and Heegner Points

An anonymous Arab manuscript, written before 972, contains a `` problem of congruent numbers": given an integer $n$, to find a rational square $x^2$ such that $x^2+n and x^2-n$ are both rational squares. For example 1, 2, 3 are not congruent numbers but 5, 6, 7 are. A modern equivalence of this problem is to find a point with infinite order on the elliptic curve: $y^2=x^4-n^2$. A special case of the Birch and Swinnerton--Dyer conjecture assets that any positive integer $n=5$, 6, 7 mod 8 are congruent number while almost all $n=1$, 2, 3 mod 8 are not congruent.

##### How to Raise Harmonic Families?

The talk will present a step towards answering the delicate question in the title! In the colloqium lunch I describe the work of Montgomery on the pair correlation of zeros of the Riemann zeta function. The result is beautifully connected with the eigenvalue statistics of random matrices studied by Gaudin, Mehta and Dyson. This was the beginning of fruitful connections between L-functions and Random Matrix Theory. Katz and Sarnak have emphasized the importance of families and introduced a new perspective in which the existence of a Symmetry Type plays a central role. In joint work with Sug-Woo Shin we establish a general Plancherel equidistribution theorem that is strong enough for such applications. This will be presented along with attached results in harmonic analysis on semisimple groups.

##### On Measures Invariant Under Diagonalizable Groups on Quotients of Semi-simple Groups

Actions of diagonalizable algebraic groups (which are referred to as tori in the theory of algebraic groups, though in cases of interest to us are not compact) on arithmetic quotient spaces play an important role in many number theoretic and other applications. I will present a joint result with Einsiedler (extending earlier work with A.

##### On the L2 Bounded Curvature Conjecture in General Relativity

In order to control locally a space-time which satisfies the Einstein equations, what are the minimal assumptions one should make on its curvature tensor? The L2 bounded curvature conjecture roughly asserts that one should only need L2 bound on the curvature tensor on a given space-like hypersuface. This conjecture has its roots in the remarkable developments of the last twenty years centered around the issue of optimal well-posedness for nonlinear wave equations. In this context, a corresponding conjecture for nonlinear wave equations cannot hold, unless the nonlinearity has a very special nonlinear structure. I will report on recent progress towards the proof of the conjecture, which shed light on the specific null structure of the Einstein equations. This is joint work with S. Klainerman and I. Rodnianski.

##### The Surface Subgroup Theorem and the Ehrenpreis Conjecture

We prove that there is a hyperbolic surface $S$ such that for any closed hy-perbolic 2 or 3-manifold M, and > 0, there is a nite cover ^ S of S, and a map f: ^ S ! M that is locally within of being an isometric immersion. When dimM = 3 this implies that 1(M) has a surface subgroup, and when dimM = 2 this is the Ehrenpreis conjecture. In either case, the surface f(S) is constructed by putting together immersed pairs of pants in M, and in both cases we can construct a collection of good pants that are evenly distributed around every closed geodesic that appears as a boundary. If dimM = 3 then we can immediately assemble these pants, with a twist, to form the desired surface f(S). In the case where dimM = 2, there may be more pants on one side of a geodesic than the other.

##### A Case Study for Critical Non-linear Dispersive quations: The Energy Critical Wave Equation

We will discuss recent work on the energy critical wave equation. The issues studied are global existence, scattering, finite time blow-up, universal profiles at blow-up and soliton resolution. This is viewed not as an isolated series of results, but as a way of approaching many similar critical non-linear dispersive equations.

##### Effective Estimates in the Theory of Indefinite Quadratic Forms and in Unipotent Dynamics

I will describe some recent results related to the size of the smallest solutions of quadratic inequalities , Markoff spectrum, etc. I will also talk briefly about effective $epsilon$-density for orbits of unipotent flows.

##### The Kardar-Parisi-Zhang Equation and its Universality Class

The KPZ equation was introduced in 1986, and has become the default model in physics for random interface growth. It is a member of a large universality class with non-standard fluctuations, including directed random polymers. Even in one dimension, it turned out to be difficult to interpret and analyze mathematically, but at the same time to have a large degree of exact solvability. We will survey the history and recent progress.

##### Nonlocal Maximum Principles for Active Scalars

Active scalar equations play an important role in modeling fluids. Some of the best known examples of active scalars are the 2D Euler equation, the surface quasi-geostrophic (SQG) equation, and the Burgers equation. Most active scalars are nonlocal, making detailed analysis of their solutions challenging. I will review recent progress in this area. In particular, I will talk about a new recently developed method for analysis of active scalars, and its applications to studies of the regularity properties of solutions.

##### Arnold Diffusion via Invariant Cylinders and Mather Variational Method (joint with Ke Zhang)

The famous ergodic hypothesis claims that a typical Hamiltonian dynamics on a typical energy surface is ergodic. However, KAM theory disproves this. It establishes a persistent set of positive measure of invariant KAM tori. The (weaker) quasi-ergodic hypothesis, proposed by Ehrenfest and Birkhoff, says that a typical Hamiltonian dynamics on a typical energy surface has a dense orbit. This question is wide open. In early 60th Arnold constructed an example of instabilities for a nearly integrable Hamiltonian of dimension $n>2$ and conjectured that this is a generic phenomenon, nowadays, called Arnold diffusion. In the last two decades a variety of powerful techniques to attack this problem were developed. In particular, Mather discovered a large class of invariant sets and a delicate variational technique to shadow them.

##### Near-optimal mean value estimates for Weyl sums

Exponential sums of large degree play a prominent role in the analysis of problems spanning the analytic theory of numbers. In 1935, I. M. Vinogradov devised a method for estimating their mean values very much more efficient than the methods available hitherto due to Weyl and van der Corput, and subsequently applied his new estimates to investigate the zero-free region of the Riemann zeta function, in Diophantine approximation, and in Waring?s problem. Recent applications from the 21st century include sum-product estimates in additive combinatorics, and the investigation of the geometry of moduli spaces.

##### Approximate groups and Hilbert's fifth problem

Approximate groups are, roughly speaking, finite subsets of

groups that are approximately closed under the group operations, such

as the discrete interval {-N,...,N} in the integers. Originally

studied in arithmetic combinatorics, they also make an appearance in

geometric group theory and in the theory of expansion in Cayley

graphs.

Hilbert's fifth problem asked for a topological description of Lie

groups, and in particular whether any topological group that was a

continuous (but not necessarily smooth) manifold was automatically a

Lie group. This problem was famously solved in the affirmative by

Montgomery-Zippin and Gleason in the 1950s.