SCHEDULE

* Camillo De Lellis will give the regular Warwick Mathematics Department Colloquium on Friday at 4pm.

COURSE DETAILS

Tristan Rivière

Title: Integrability by Compensation in the Analysis of Conformally Invariant Problems

Course Details: A set of notes for the course can be found here

Igor Rodnianski

Title: Linear and nonlinear waves

Vladimir Sverak

Title: PDE aspects of the Navier-Stokes and Euler's equations

Abstract: The solutions of the equations of fluid mechanics can exhibit a wide variety of behaviour. After explaining some of the basics, we will investigate some of the interesting flow regimes. These will include parts of the Kolmogorov theory of turbulence and parts of the geometric approach to Euler's equations.

Background Material:
A.Chorin, J.Marsden: A Mathematical Introduction to Fluid Mechanics
Landau, Lifschitz: Fluid Mechanics
V.I.Arnold: Classical Mechanics
V.I.Arnold, B.A. Khesin: Topological Methods in Fluid Mechanics

Slides: The slides for the course are available for download.

Lecture 1

Lecture 2

Lecture 3

Lecture 4

SINGLE-LECTURE DETAILS

James Colliander

Title: Nonlinear Schrödinger Evolutions from Low Regularity Initial Data

Abstract: This talk will be aimed at PhD students of mathematics. The talk will motivate and describe studies of the nonlinear Schrödinger (NLS) equation with low regularity initial data. In particular, I will prove Bourgain's bilinear refinement of the Strichartz estimate and explain its role in some recent studies of the NLS dynamics.

Background and more details:

The breakthrough advance establishing global well-posedness below the level of conserved regularity was made by J. Bourgain in:

Bourgain, J. Refinements of Strichartz' inequality and applications to $2$D-NLS with critical nonlinearity. Internat. Math. Res. Notices 1998, no. 5, 253--283. ams link

Further developments along this research line are exposed in the papers appearing at:
http://arxiv.org/abs/math/0203218
http://arxiv.org/abs/0704.2730
http://arxiv.org/abs/0811.1803

Slides: The slides for the course are available for download.

Lecture

Camillo De Lellis

Title: Genus bounds for min–max constructions
Abstract: In 1983 Smith, in his PhD thesis, proved the existence of a minimal embedded 2-sphere in any (sufficiently smooth) Riemannian 3-sphere, building upon arguments of Pitts and Simon. Later Pitts and Rubinstein claimed more general genus bounds for surfaces generated by a certain type of min–max argument. However a proof of these claims has never appearead. In a survey article with Tobias Colding we gave an account of the regularity theory needed in these constructions. In a recent joint work with Filippo Pellandini we have proved a general genus bound, which however is slightly weaker than the one claimed by Pitts and Rubinstein.

Background and more details:

The following two papers contain the results of the seminar:

http://arxiv.org/abs/math/0303305

http://arxiv.org/abs/0905.4035

This paper is related to it and might be useful/interesting:

http://arxiv.org/abs/0905.4192

Alessio Figalli

Title: Monge-Ampere type equations and regularity of optimal transport maps on Riemannian manifolds

Asbtract: The issue of regularity of optimal transport maps in the case ``cost=squared distance'' on R^n was solved by Caffarelli in the 1990s. However, a major open problem in the theory was the question of regularity for more general cost functions, or for the case ``cost=squared distance'' on a Riemannian manifold. A breakthrough to this problem has been achieved by Ma-Trudinger-Wang and Loeper, who found a necessary and sufficient condition on the cost function in order to ensure the regularity of the optimal map.
In the special case ``cost=squared distance'' on a Riemannian manifold, this condition corresponds to the non-negativity of a new curvature tensor on the manifold, which implies strong geometric consequences on the geometry of the manifold and on the structure of its cut-locus.

Background and more details:

The following notes from the Bourbaki seminar cover the subject of the talk

http://cvgmt.sns.it/papers/fig09b/

Inwon Kim

Title: Well-posedness of a free boundary problem with non-standard sources.

Abstract: We will investigate a 1-D free boundary problem modelling price formulation, derived by J.M. Lasry and P.L. Lions. The free boundary is given as the zero set of a diffusion equation, with dynamically evolving, non-standard sources. We prove the global existence and uniqueness of the solutions. This is joint work with L. Chayes, M. Gonzalez and M. Gualdani.

Background and more details:

The main observations come from careful analysis on the behavior of the zero set and the amount of flux it produces in various settings using heat kernels. This does not need any background material but our preprint is on the webpage www.math.ucla.edu/~ikim/research

Herbert Koch

Title: Rough initial data for geometric parabolic flows

Abstract: The initial value problem for parabolic equations with rough initial data is studied. The techniques are applicable for example to a graph with small Lischitz constant as initial data for the Willmore flow.

Background and more details:

Koch, H. and Tataru, D. Well-posedness for the Navier-Stokes equations. Adv. Math. 157 (2001), no. 1, 22--35. ams link

Koch, H and Lamm, T. Geometric flows with rough initial data. arxiv link

Ovidiu Savin

Title: Parabolic Monge-Ampere equations

Abstract: In this talk we describe interior regularity of viscosity solutions of certain parabolic Monge-Ampere equations. Equations of this form appear in geometric evolution problems and in particular in the motion of a convex $n$-dimensional hyper-surface embedded in $\R^{n+1}$ under Gauss curvature flow.

Background and more details:

The relevant paper (C^{1,\alpha} regularity for parabolic Monge-Ampere equations) can be found on my web-page:

www.math.columbia.edu/~savin