# Seminars & Events for Department Colloquium

##### Minimal graphs and harmonic diffeomorphisms.

In 1952, Heinz gave another proof of Bernstein's Theorem ( an entire minimal graph over the euclidean plane is a plane) by showing there is no harmonic diffeomorphism from the disc onto the euclidean plane. Since then the existence theory of harmonic diffeomorphisms between closed surfaces has been developed and useful. In this talk I will use minimal graphs to obtain harmonic diffeomorphisms between complete surfaces of finite topology.

##### Singular Sets of Geometric PDE's

Given a solution of a nonlinear pde the two primary issues regarding the regularity theory are a priori estimates and the structure of the singular set. We will discuss new techniques in the analysis of these issues, which have been particularly successful in the study of geometrically motivated equations. To pick an example: in the context of a stationary harmonic map f:M->N between Riemannian manifolds we will see that the singular set S(f) may be stratified into pieces S^k(M) which are k-rectifiable. If f is minimizing harmonic map then we will see that the singular set S(f) has finite n-3 measure, and has apriori estimates in weak L^3 , both of which are sharp estimates.

##### Stochastic Arnold diffusion of deterministic systems

In 1964 V. Arnold constructed an example of nearly integrable deterministic system exhibiting instabilities. In the 1970s physicist B. Chirikov coined the term for this phenomenon ``Arnold diffusion'', where diffusion refers to stochastic nature of instability. One of most famous example of stochastic instabilities for nearly integrable systems is dynamics of Asteroids in Kirkwood gaps in the Asteroid belt. They were discovered numerically by astronomer J. Wisdom. During the talk I describe a class of nearly integrable deterministic systems with stochastic diffusive behaviour. Namely, we show that each system from this class has a probability measure \mu in the phase space such that the distributions given by the deterministic evolution \mu_t of \mu converge to an Ito diffusion process.

##### Veering triangulations and pseudo-Anosov flows

We’ll discuss veering triangulations associated to pseudo-Anosov mapping tori, and how they arise dynamically. We’ll survey some of the results obtained regarding these triangulations. Then we’ll discuss a new construction of these triangulations associated to certain pseudo-Anosov flows, which is joint work with François Guéritaud.

##### Number-theoretic algorithms in quantum computing

In quantum computation, one considers groups of unitary operators that are generated by some finite set of operators called "gates". Words in these generators are called "circuits". An important problem is the so-called approximate synthesis problem: to find a quantum circuit, preferably as short as possible, that approximates a given unitary operator up to given epsilon. Moreover, the solution should be computed by an efficient algorithm. For nearly two decades, the standard solution to this problem was the Solovay-Kitaev algorithm, which is based on geometric ideas. This algorithm produces circuits of size O(log^c(1/epsilon)), where c is approximately 3.97. It was a long-standing open problem whether this exponent c could be reduced to 1.

##### Completing the Square

L-functions (e.g., Riemann zeta function) constitute a special class of functions in one complex variable. It is “natural" to look at the Taylor expansion of L-functions (suitably normalized) at their “centers”. Assuming one of the standard conjectures on L-functions (i.e., the generalized Riemann hypothesis), I will explain how to deduce that all Taylor coefficients are positive (if they are all real). How can one prove the positivity unconditionally? A natural idea is by completing the square. We present some successful examples where the “square roots” bear interesting geometric meanings.

##### Effective dynamics of the dispersive equations and the Hardy-Littlewood circle method

The long-time behavior of small amplitude solutions to nonlinear dispersive equations on $\mathbb{R}^n$ is relatively well understood. However, the situation is markedly different when these equations are considered on a bounded domain. In this talk, we will consider nonlinear Schrodinger equations on rational tori and show how to derive limiting equations that govern the long-time dynamics of solutions. The derivation of the limiting equations and the proofs rely heavily on results from analytic number theory.

##### TBA - Jacob Tsimerman

##### Mass in Kähler Geometry

Given a complete Riemannian manifold that looks enough like Euclidean space at infinity, physicists have defined a quantity called the “mass” which measures the asymptotic deviation of the geometry from the Euclidean model. In this lecture, I will explain a simple formula, discovered in joint work with Hajo Hein, for the mass of any asymptotically locally Euclidean (ALE) Kähler manifold. For ALE scalar-flat Kähler manifolds, the mass turns out to be a topological invariant, depending only on the underlying smooth manifold, the first Chern class of the complex structure, and the Kähler class of the metric. When the metric is actually AE (asymptotically Euclidean), our formula not only implies a positive mass theorem for Kähler metrics, but also yields a Penrose-type inequality for the mass

##### Mean curvature flow

A mean curvature flow is an evolving submanifold M_t whose velocity is equal to its mean curvature. Mean curvature flow is in some respects the most natural evolution equation for a moving submanifold: it is the gradient flow of the area functional, as well as the analog of the heat equation for submanifolds. The lecture will survey mean curvature flow for a general mathematical audience.

##### Intersection numbers and higher derivatives of L-functions for function fields, II

In joint work with Wei Zhang, we prove a higher derivative analogue of the Waldspurger formula and the Gross-Zagier formula in the function field setting under the assumption that the relevant objects are everywhere unramified. Our formula relates the self-intersection number of certain cycles on the moduli of Shtukas for GL(2) to higher derivatives of automorphic L-functions for GL(2). This second talk does not assume the first talk, and I will give motivations and the complete statement of the formula.

##### Geometry, topology and arithmetic of canonical curves

Let K be a knot in S^3 with hyperbolic complement. The seminal work of Thurston, and Culler-Shalen established the SL(2,C)-character variety X(K) as a powerful tool in the study of the topology of M. The canonical

component C is a component of X(K) that contains the character of a faithful discrete representation. Thurston proved that the canonical component is a curve. In this talk we will discuss some recent work in the

direction of trying to understand what “these curves look like” as well as properties of these curves, their relation to the topology of S^3\K and arithmetic properties of Dehn surgeries on K.

##### Integrability versus Wave Turbulence in Hamiltonian partial differential equations

In the world of Hamiltonian partial differential equations, complete integrability and wave turbulence are often considered as opposite paradigms. The purpose of this talk is to give a rough idea of these different notions, and to discuss the example of a nonlinear wave toy model which surprisingly displays both properties. The key is a Lax pair structure involving Hankel operators from classical analysis, and is connected to a surprisingly explicit inverse spectral method.