**Wednesdays** at **4pm** in **Eckhart 202**.

- April 1
**Monica Visan,**University of Chicago

The focusing energy-critical nonlinear Schrodinger equation- April 22
**at 3pm in E202** **Sijue Wu,**University of Michigan

Almost global well-posedness of the 2-D full water wave problem- May 6
**Ovidiu Savin,**Columbia University

Minimizers of convex functionals arising in random surfaces- May 13
**Alexander Kiselev,**University of Wisconsin

Nonolocal maximum principles for active scalars- June 3
**John Gibson,**Georgia Institute of Technology

Equilibria, traveling waves, and periodic orbits in plane Couette flow

We introduce the defocusing and focusing energy-critical nonlinear Schrodinger equations. We then present joint work with Rowan Killip on the focusing problem.

We consider the problem of global in time existence and uniqueness of solutions of the 2-D infinite depth full water wave equation. It is known that this equation has a solution for a time period $[0, T/\epsilon]$ for initial data of type $\epsilon\Phi$, where $T$ depends only on $\Phi$. We show that for such data there exists a unique solution for a time period $[0, e^{T/\epsilon}]$. This is achieved by better understandings of the nature of the nonlinearity of the water wave equation.

We discuss $C^1$ regularity of minimizers to $\int F(\nabla u)dx$ in two dimensions for certain classes of non-smooth convex functionals $F$. In particular our results apply to the surface tensions that appear in recent works on random surfaces and random tilings of Kenyon, Okounkov and others. This is a joint work with D. De Silva.

I will discuss nonlocal maximum principles for a family of dissipative active scalar equations. This technique gives proofs of global regularity for the critical surface quasi-geostrophic equation, Burgers equation and some other related models. The talk is based on works joint with Fedya Nazarov, Roman Shterenberg and Sasha Volberg.

Equilibrium, traveling wave, and periodic orbit solutions of pipe and plane Couette flow can now be computed precisely at Reynolds numbers above the onset of turbulence. These invariant solutions capture the complex dynamics of coherent roll-streak structures in wall-bounded flows and provide a framework for connecting wall-bounded turbulence to dynamical systems theory. We present a number of newly computed solutions of plane Couette flow and observe how they are visited by the turbulent flow. What emerges is a picture of low-Reynolds turbulence as a walk among a set of weakly unstable invariant solutions.

For questions, contact

**Andrej Zlatos**, az@math.uchicago.edu