# On the linear forced Euler and Navier-Stokes equations: damping and modified scattering

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We study the long-time asymptotic behavior of the linearized Euler and nonlinear Navier-Stokes equations close to Couette flow. As a main result we show that suitable forcing breaks asymptotic stability results at the level of the vorticity, but that solutions never the less exhibit convergence of the velocity field. Thus, here linear inviscid damping persists despite instability of the vorticity equations.

# Geometry of real hypersurfaces meets Subelliptic PDEs

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In his seminal work from 1979,

Joseph J. Kohn invented

his theory of multiplier ideal sheaves

connecting a priori estimates for the d-bar problem

with local boundary invariants

constructed in purely algebraic way.

I will explain the origin and motivation of the problem,

and how Kohn's algorithm reduces it

to a problem in local geometry

of the boundary of a domain.

I then present my recent work with Sung Yeon Kim

based on the technique of jet vanishing orders,

and show how it can be used to

control the effectivity of multipliers in Kohn's algorithm,

subsequently leading to precise a priori estimates.

# Singularity Formation in Incompressible Fluids

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We will discuss recent progress on finite-time singularity formation for solutions to the incompressible Euler equation and related models.

# Singular Abreu equations and minimizers of convex functionals with a convexity constraint

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Abreu type equations are fully nonlinear, fourth order, geometric PDEs which can be viewed as systems of two second order PDEs, one is a Monge-Ampere equation and the other one is a linearized Monge-Ampere equation. They first arise in the constant scalar curvature problem in differential geometry and in the affine maximal surface equation in affine geometry. This talk discusses the solvability and convergence properties of second boundary value problems of singular, fourth order equations of Abreu type arising from approximation of convex functionals whose Lagrangians depend on the gradient variable, subject to a convexity constraint. These functionals arise in different scientific disciplines such as Newton's problem of minimal resistance in physics and the Rochet-Choné model of monopolist's problem in economics. This talk explains how minimizers of the 2D Rochet-Choné model can be approximated by solutions of singular Abreu equations.

# Mean field games on graphs

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Mean field game theory is the study of the limit of Nash

equilibria of differential games when the number of players tends to infinity. It

was introduced by J.-M. Lasry and P.-L. Lions, and independently by P.

Caines, M. Huang and R. Malhame. A fundamental object in the theory is the

master equation, which fully characterizes the limit equilibrium. In this

talk, we will introduce Mean field game and master equations on graphs. We will

construct solutions to both equations and link them to the solution to

a Hamilton-Jacobi equation on graph.

# TBA

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This is a joint Nonlinear PDEs seminar with Analysis seminar

# Bergman-Einstein metrics on Strongly pseudoconvex domains in a complex space.

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I will give a proof of S.Y. Cheng's conjecture that a bounded strongly pseudoconvex domain in C^n has its Bergman metric being Einstein if and only if it is holomorphically equivalent to the ball.