# Upcoming Seminars & Events

## Primary tabs

##### A modulated two-soliton with transient turbulent regime for a focusing cubic nonlinear half-wave equation on the real line

In this talk we discuss work in progress regarding a nonlocal focusing cubic half-wave equation on the real line. Evolution problems with nonlocal dispersion naturally arise in physical settings which include models for weak turbulence, continuum limits of lattice systems, and gravitational collapse.

##### Asymptotics beyond all orders: the devil's invention?

"Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever." --- N. H. Abel. The lecture will introduce the concept of an asymptotic series, showing how useful divergent series can be, despite Abel's reservations. We will then discuss Stokes' phenomenon, whereby the coefficients in the series appear to change discontinuously.

##### Statistical Mechanics on Sparse Random Graphs: Mathematical Perspective

Theoretical physics studies of disordered materials lead to challenging mathematical problems with applications to random combinatorial problems and coding theory. The underlying structure is that of many discrete variables that are strongly interacting according to a mean field model determined by a random sparse graph.

##### Symplectic fillings and star surgery

**This is a joint Topology-Symplectic Geometry seminar. Please note special time and location.** Although the existence of a symplectic filling is well-understood for many contact 3-manifolds, complete classifications of all symplectic fillings of a particular contact manifold are more rare.

##### Symplectic fillings and star surgery

**This is a joint Symplectic Geometry-Topology seminar. Please note special time and location.** Although the existence of a symplectic filling is well-understood for many contact 3-manifolds, complete classifications of all symplectic fillings of a particular contact manifold are more rare.

##### A blow-up result for dyadic models of fluid dynamics

Dyadic models in fluid dynamics are toy models for Euler and Navier-Stokes equations. Among many interesting results that can be proved in these models, we will focus on blow-up results; that is, some Sobolev norm can become infinite in finite time. This is joint work with Dong Li.

##### Excluding topological minors and well-quasi-ordering

Robertson and Seymour proved that graphs are well-quasi-ordered by the minor relation and the weak immersion relation. In other words, given infinitely many graphs, one graph contains another as a minor (or a weak immersion, respectively). Unlike the relation of minor and weak immersion, the topological minor relation does not well-quasi-order graphs in general.

##### Some results on singular transport equations arising in fluid mechanics

We will discuss a few recent results in the study of fluid equations which stem from studying the dynamics of transport equations with non-local forcing. These are equations of the form: $f_t +u\cdot\nabla f =R(f)$ where $R$ is a singular integral operator and $u$ is a divergence-free vector field possibly depending upon $f$. These types of equations arise in a variety of physical scenarios.

##### Umbilicity and characterization of Pansu spheres in the Heisenberg group

For n≥2 we define a notion of umbilicity for hypersurfaces in the Heisenberg group H_{n}. We classify umbilic hypersurfaces in some cases, and prove that Pansu spheres are the only umbilic spheres with positive constant p(or horizontal)-mean curvature in H_{n} up to Heisenberg translations. This is joint work with Hung-Lin Chiu, Jenn-Fang Hwang, and Paul Yang.

##### A geometric approach for sharp Local well-posedness of quasilinear wave equations

**Please note special location. **The commuting vector fields approach, devised for Strichartz estimates by Klainerman, was employed for proving the local well-posedness in the Sobolev spaces $H^s$ with $s>2+\frac{2-\sqrt{3}}{2}$ for general quasi-linear wave equation in ${\mathbb R}^{1+3}$ by him and Rodnianski.

##### Focal points and sup-norms of eigenfunctions

If (M,g) is a compact real analytic Riemannian manifold, we give a necessary and sufficient condition for there to be a sequence of quasimodes saturating sup-norm estimates. The condition is that there exists a self-focal point x_0\in M for the geodesic flow at which the associated Perron-Frobenius operator U: L^2(S_{x_0}^*M) \to L^2(S_{x_0}^*M) has a nontrivial invariant function.

##### TBA - Wang

##### Applications of diffusion maps in dynamical systems

There is great current interest in the use of diffusion maps for dimension reduction. We discuss some examples of diffusion methods applied to understanding dynamical data, in particular combining spectral approaches with delay coordinates. In addition, we extend the usual diffusion map construction by introducing local kernels, a generalization of the standard isotropic kernel.

##### Increasing subsequences on the plane and the Slow Bond Conjecture

For a Poisson process in R^2 with intensity 1, the distribution of the maximum number of points on an oriented path from (0,0) to (N,N) has been studied in detail, culminating in Baik-Deift-Johansson's celebrated Tracy-Widom fluctuation result.

##### Finiteness properties for the fundamental groups of complex algebraic varieties

We describe some relations obtained in joint work with S.Papadima and A. Suciu between finiteness properties of fundamental groups and resonance and characteristic varieties.

##### TBA - Scott

##### TBA

##### Onsager's Conjecture

In 1949, Lars Onsager in his famous note on statistical hydrodynamics conjectured that weak solutions to the Euler equation belonging to Hölder spaces with Hölder exponent greater than 1/3 conserve energy; conversely, he conjectured the existence of solutions belonging to any Hölder space with exponent less than 1/3 which dissipate energy.

##### The standard L-function for G_2: a "new way" integral

We consider a Rankin-Selberg integral representation of a cuspidal (not necessarily generic) representation of the exceptional group G2. Although the integral unfolds with a non-unique model, it turns out to be Eulerian and represents the standard L-function of degree 7. We discuss a general approach to the integrals with non-unique models.

##### Regularity of manifolds with bounded Ricci curvature and the codimension $4$ conjecture

This talk will concern joint work with Aaron Naber on the regularity of noncollapsed Riemannian manifolds $M^n$ with bounded Ricci curvature $|{\rm Ric}_{M^n}|\leq n-1$, as well as their Gromov-Hausdorff limit spaces, $(M^n_j,d_j)\stackrel{d_{GH}}{\longrightarrow}(X,d)$, where $d_j$ denotes the Riemannian distance.