# Seminars & Events for 2014-2015

##### On the stability of Prandtl boundary layer expansions of Navier-Stokes in the inviscid limit

I will present two recent results concerning the stability of boundary layer asymptotic expansions of solutions of Navier-Stokes with small viscosity. First, we show that the linearization around an arbitrary stationary shear flow (other than the Couette flow) admits an unstable eigenfunction with small wave number, when viscosity is sufficiently small.

##### On normal crossings symplectic divisors

I will describe purely symplectic notions of normal crossings divisor and configuration. They are compatible with the existence of the desired auxiliary almost Kahler structures, provided ``existence" is suitably interpreted. These notions lead to a multifold version of Gompf's symplectic sum construction.

##### Sign of Green's function of Paneitz operator and the Q curvature equations

I will review some recent works with Paul on the fourth order Paneitz operators and Q curvature equations. Among other things, we will discuss the positivity of Green's function of Paneitz operator and its applications to finding constant Q curvature metrics by variational methods.

##### Analysis, Spectra, and Number Theory: A Conference in Honor of Peter Sarnak on his 61th Birthday

The conference will focus on analytic number theory, with emphasis on its many relationships with analysis and spectral theory. Topics to be highlighted include arithmetic quantum chaos, analysis of families of L-functions, arithmetic statistics, and connections with ergodic theory. It will include a problem session to suggest future directions for the field.

##### TBA - Griffiths

##### TBA - Doran

**Please note special day and time.**

##### TBA - McGibbon

##### Global existence for fully nonlinear stochastic hyperbolic equations : the Nash-Moser approach

In 1979, Klainerman proved the first global existence result for fully nonlinear hyperbolic equations using Nash-Moser's method. The result was a breakthrough in the field.

##### Minimal surfaces in H2xR

**PLEASE NOTE SPECIAL DAY AND LOCATION: Talk #1. **In the classical theory of minimal surfaces of the Euclidean space, the better known ones are those with finite total curvature.

##### A positive mass theorem for asymptotically flat manifolds with a noncompact boundary and an application to a Yamabe-type flow

**PLEASE NOTE SPECIAL DAY AND LOCATION: Talk #2. **First I will discuss a positive mass theorem for noncompact manifolds with boundary and ends asymptotic to the Euclidean half-space.

##### Mean curvature flow without singularities

**Please note special day, time and location. **We study graphical mean curvature flow of complete solutions defined on subsets of Euclidean space. We obtain smooth long time existence. The projections of the evolving graphs also solve mean curvature flow.

##### Endoscopy theory for symplectic and orthogonal similitude groups

The endoscopy theory provides a large class of examples of Langlands functoriality, and it also plays an important role in the classification of automorphic forms. The central part of this theory are some conjectural identities of Harish-Chandra characters between a reductive group and its endoscopic groups.

##### Symplectic forms in algebraic geometry

Imposing the existence of a holomorphic symplectic form on a projective algebraic variety is a very strong condition. After describing various instances of this phenomenon (among which is the fact that so few examples are known!), I will focus on the specific topic of maps from projective varieties admitting a holomorphic symplectic form.

##### A model for studying double exponential growth

We discuss a model for studying spontaneous phenomena in the 2d Euler equations for incompressible fluid flow. We tie the behaviour of the model to the behavior of the actual Euler equations. (Joint work with A. Tapay.)

##### Efficient numerical methods for wave scattering in periodic geometries

A growing number of technologies (communications, imaging, solar energy, etc) rely on manipulating linear waves at the wavelength scale, and accurate numerical modeling is key for device design. I will focus on diffraction problems where time-harmonic scalar waves scatter from piecewise-uniform periodic media. After reviewing the integral equation method, I explain two innovations that allow

##### On the rationality of the logarithmic growth filtration of solutions of $p$-adic differential equations

**Please note special day.** We consider an ordinary linear $p$-adic differential equation Dy=d^ny/dx^n+a_{n-1}d^{n-1}y/dx^{n-1}+\dots+a_0y=0, a_i\in\mathbb{Z}_p[p^{-1}] whose formal solutions in $\mathbb{Q}_p$ converge in the open unit disc $|x|<1$.

##### Height Fluctuations in Interacting Dimers

Perfect matchings of Z^2(also known as non-interacting dimers on the square lattice) are an exactly solvable 2D statistical mechanics model. It is known that the associated height function behaves at large distances like a massless gaussian field, with the variance of height gradients growing logarithmically with the distance. As soon as dimers mutually interact, via e.g.

##### Stabilization of control systems: From the water clocks to the regulation of rivers

A control system is a dynamical system on which one can act by using controls. For these systems a fundamental problem is the stabilization issue: Is it possible to stabilize a given unstable equilibrium by using suitable feedback laws?

##### Mobius randomness and homogeneous dynamics

Qualitative approaches to understanding the randomness of the primes offer a first step toward the extremely difficult quantitative challenge of sharply bounding sums involving the Mobius function.

##### The rare interaction limit in a fast-slow mechanical system

In 2008 Gaspard and Gilbert suggested a two-step strategy to derive the 'macroscopic' heat equation from the 'microscopic' kinetic equation. Their model consisted of a chain of localized and rarely interacting hard disks.