# Seminars & Events for 2015-2016

##### Bigeodesics in first-passage percolation

In first-passage percolation, we place i.i.d. continuous weights at the edges of Z^2 and consider the weighted graph metric. A distance minimizing path between points x and y is called a geodesic, and a bigeodesic is a doubly-infinite path whose segments are geodesics. It is a famous conjecture that almost surely, there are no bigeodesics.

##### Lecture II: Dynamics on moduli spaces of hyperbolic surfaces

In the second lecture, I will discuss several natural geometric flows defined on bundles over the moduli spaces of curves. I will describe basic ergodic properties of these flows. I will discuss some open questions and some of the progress made in recent years.

##### Applications of the Chebotarev density theorem

The Chebotarev density theorem is a basic and useful theorem in number theory. I will introduction some applications of CDT to modular forms based on Deligne’s theorem on l-adic representation attached to modular forms, and make connections with conjectures of Maeda and Lang-Trotter.

##### Regularity theory for fully nonlinear integro-differential equations

**Please note special day (Thursday) and room (Fine1001). **Integro-differential equations appear naturally when studying discontinuous stochastic processes, and we are interested in the regularity properties of their solutions.

##### Large deviations and random polynomials

We consider large deviation principles (LDP) in the context of random polynomials. In one direction, we obtain a large deviations principle for the empirical measure of zeroes of random polynomials with i.i.d. exponential coefficients. One of the key challenges here is the fact that the coefficients are a.s.

##### Constructing equivariant spectra

Equivariant spectra determine cohomology theories that incorporate a group action on spaces. Such spectra are increasingly important in algebraic topology but can be difficult to understand or construct. I will discuss recent work with Angelica Osorno, in which we build such spectra out of purely algebraic data based on symmetric monoidal categories.

##### A Riemannian structure on the space of conformal metrics

**Please note special day (Thursday) and room (Fine 601). ** I will describe a Riemannanian structure on the space of conformal metrics satisfying a certain positivity condition. This metric is inspired by the Riemannian of the space of Kahler metrics, and shares many of the same properties.

##### Convection enhanced mixing and spectral properties of the advection-diffusion equation in the semi-classical limit for vanishing diffusivity

We consider the two-dimensional advection-diffusion equation on a bounded domain subject to Dirichlet or von Neumann boundary conditions involving a Liouville integrable Hamiltonian. Transformation to action-angle coordinates permits averaging in time and angle, resulting in an equation that allows for separation of variables.

##### Heegaard Floer homology for tangles and cobordisms between them

Heegaard Floer homology was generalized for non-closed 3-manifolds with certain boundary decoration called sutured manifolds, by Juhasz, Eftekhary and I. Sutured manifolds can be described as a generalization of oriented tangles. We use this description to define a notion of cobordism between sutured manifolds.

##### Hasse principle for Kummer varieties

The existence of rational points on the Kummer variety associated to a 2-covering of an abelian variety A over a number field can sometimes be established through the variation of the 2-Selmer group of quadratic twists of A.

##### Excluding theta graphs

A theta graph, denoted T(a,b,c), consists of a pair of vertices together with three disjoint paths between the vertices of lengths a, b, and c. In this talk, we characterize graphs which exclude certain theta graphs as a minor. We begin with small theta graphs, in particular those with at most 7 edges.

##### Legendrian Fronts in Contact Topology

In this talk we focus on Legendrian presentations of Weinstein manifolds; in particular, we discuss loose Legendrian embeddings and their absolute analogues, flexible Weinstein manifolds and overtwisted contact structures. The required definitions and necessary results will be provided.

##### Lecture III: Counting mapping class group orbits on hyperbolic surfaces

Let $X$ be a complete hyperbolic metric on a surface of genus $g$ with $n$ punctures. In this lecture I will discuss the problem of the growth of $s^{k}_{X}(L)$, the number of closed curves of length at most $L$ on $X$ with at most $k$ self-intersections.

##### Absolute continuity and rectifiability of harmonic measure

**Please note special location. **The properties of harmonic measure (most importantly, absolute continuity and rectifiability) are key to many problems in Analysis, Probability, Geometric Measure Theory, as well as PDEs.

##### On the Complexity of Detecting Planted Solutions

Many combinatorial problems appear to be hard even when inputs are drawn from simple, natural distributions, e.g., SAT for random formulas, clique in random graphs etc. To understand their complexity, we consider random problems with planted solutions, e.g., planted k-SAT/k-CSP (clauses are drawn at random from those satisfying a fixed assignment), planted clique (a large clique is added to a r

##### Almost optimal local existence for radially symmetric time like minimal surface equation in 1+3 dimensional Minkowski space

**Please note special time and location.**

##### Needle decomposition and Ricci curvature

**Please note special day and time. **Needle decomposition is a technique in convex geometry, which enables one to prove isoperimetric and spectral gap inequalities, by reducing an n-dimensional problem to a 1-dimensional one. This technique was promoted by Payne-Weinberger, Gromov-Milman and Kannan-Lovasz-Simonovits.

##### Syzygies on abelian surfaces, construction of singular divisors, and Newton-Okounkov bodies

Constructing divisors with prescribed singularities is one of the most powerful techniques in modern projective geometry, leading to proofs of major results in the minimal model program and the strongest general positivity theorems by Angehrn-Siu and Kollár-Matsusaka. We present a novel method for constructing singular divisors on surfaces based on infinitesimal Newton-Okounkov bodies.

##### 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.

##### Birkhoff Conjecture for convex planar billiards and deformational spectral rigidity of planar domains

**Please note special time and location. ** The classical Birkhoff conjecture states that the only integrable convex planar domains are circles and ellipses. In a joint work with A. Avila and J. De Simoi we show that this conjecture is true for perturbations of ellipses of small eccentricity.