# Seminars & Events for 2014-2015

##### 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.

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

**Please note special location. **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.

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

**Please note special location and time. **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.

##### 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.

##### TBA - Nori

##### TBA - Huh

##### The topology of proper toric maps

**Please note special day and time. ** I will discuss some of the topology of the fibers of proper toric maps and a combinatorial invariant that comes out of this picture. Joint with Luca Migliorini and Mircea Mustata.

##### 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.

##### The 15-theorem & the 290-theorem

The citation for Manjul Bhargava's recent Fields Medal mentions his improvement of my proof (partly with Will Schneeberger) of the 15- theorem, and his proof (with John Hanke) of the 290-theorem. I shall talk about the history of universal quadratic forms, which was started in 1770 by Lagrange's four squares theorem, and culminated about 20 years ago in these two theorems.

##### Uniform strong primeness in matrix rings

A ring $R$ is uniformly strongly prime if some finite $S \subseteq R$ is such that for $a,b \in R$, $aSb = \{0\}$ implies $a$ or $b$ is zero, in which case the bound of uniform strong primeness of $R$ is the smallest possible size of such an $S$. The case of matrix rings $R$ is considered.

##### On well-posedness and small data global existence for a damped free boundary fluid-structure model

We address a fluid-structure system which consists of the incompressible Navier-Stokes equations and a damped linear wave equation defined on two dynamic domains. The equations are coupled through transmission boundary conditions and additional boundary stabilization effects imposed on the free moving interface separating the two domains.

##### 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.

##### Bipartite decomposition of graphs

For a graph G, let bc(G) denote the minimum possible number of pairwise edge disjoint complete bipartite subgraphs of G so that each edge of G belongs to (exactly) one of them. The study of this quantity and its variants received a considerable amount of attention and is related to problems in communication complexity and geometry.

##### 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.

##### Dynamics and polynomial invariants of free-by-cyclic groups

The theory developed by Thurston, Fried and McMullen provides a near complete picture of the various ways a hyperbolic 3-manifold M can fiber over the circle.

##### 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.

##### 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.

##### On two extremal problems for the Fourier transform

One of the most fundamental facts about the Fourier transform is the Hausdorff-Young inequality, which states that for any locally compact Abelian group, the Fourier transform maps $L^p$ boundedly to $L^q$, where the two exponents are conjugate and $p \in [1,2]$.