The seminar meets regularly on Wednesdays at 4pm in Eckhart 202. We also have special seminars during other days. To subscribe or unsubscribe the email list, you may either go to Camp/PDE email list or contact Will Feldman.
I will present some recent results on second-order divergence type equations with piecewise constant coefficients. This problem arises in the study of composite materials with closely spaced interface boundaries, and the classical elliptic regularity theory are not applicable. In the 2D case, we show that any weak solution is piecewise smooth without the restriction of the underling domain where the equation is satisfied. This completely answers a question raised by Li and Vogelius (2000) in the 2D case. Joint work with Hongjie Dong.
Lecture 1: Min-max theory for the area functional - a panorama
Tuesday, January 19, 2016, 4:30pm–5:30pm, Eckhart 133
Abstract: In this talk we will give a current panorama of the min-max theory for the area functional, initially devised by Almgren in the 1960s and improved by Pitts (1981). This is a deep high-dimensional version of the variational theory of closed geodesics. The setting is very general, being that of Geometric Measure Theory, and the main application until very recently was the construction of minimal varieties of any dimension in a compact Riemannian manifold. In the past few years we have discovered new applications of this old theory, including a proof of the Willmore conjecture, of the Freedman-He-Wang conjecture, and of Yau's conjecture (about the existence of infinitely many minimal hypersurfaces) in the positive Ricci curvature setting. We will give an overview of these results and describe open problems and future directions. Most of the material covered in these lectures is based on joint work with Andre Neves.
Lecture 2: Multiparameter sweepouts and a proof of the Willmore conjecture
Wednesday, January 20, 2016, 4pm–5pm, Eckhart 202
Abstract: In 1965, T. J. Willmore conjectured what should be the optimal shape of a torus immersed in three-dimensional Euclidean space. He predicted that the Clifford torus, more precisely a stereographic projection of it, is the minimizer of the Willmore energy - the total integral of the square of the mean curvature. This is a conformally invariant problem. In this lecture we will explain a proof of this conjecture that exploits a connection with the min-max theory of minimal surfaces. A crucial ingredient is the discovery of new five-parameter sweepouts of surfaces in the three-sphere that turn out to be homotopically nontrivial. The solution is based on the study of the geometric and topological properties of such families. We will also describe how to prove the Freedman-He-Wang conjecture (joint with I. Agol and A. Neves) about the Möbius energy of links using similar ideas.
Lecture 3: The case of fluctuations around a global equilibrium.
Thursday, January 21, 2016, 4:30pm–5:30pm, Eckhart 202
Abstract: The space of cycles in a compact Riemannian manifold has very rich topological structure. The space of hypercycles, for instance, taken with coefficients modulo two, is weakly homotopically equivalent to the infinite dimensional real projective space. We will explain how to use this structure, together with Lusternik-Schnirelman theory and work of Gromov and Guth, to prove that every compact Riemannian manifold of positive Ricci curvature contains infinitely many embedded minimal hypersurfaces. Then we will discuss more recent work in which we prove the first Morse index bounds of the theory. The main difficulty comes from the problem of multiplicity, which we are able to settle in the classical one-parameter case.
We discuss variational problems arising in machine learning and their limits as the number of data points goes to infinity. Consider point clouds obtained as random samples of an underlying "ground-truth" measure. Graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points. Many machine learning tasks, such as clustering and classification, can be posed as minimizing functionals on such graphs. We consider functionals involving graph cuts and their limits as the number of data points goes to infinity. In particular we establish under what conditions the minimizers of discrete problems have a well defined continuum limit, and characterize the limit. The question is considered using the Gamma convergence. The Gamma limit, and associated compactness property, are considered with respect to a topology which uses optimal transportation to suitably compare functions defined on graphs with functions defined with respect to the continuum ground-truth measure. The talk is primarily based on joint works with Nicolas Garcia Trillos, as well as on works with Xavier Bresson, Thomas Laurent, and James von Brecht.
I will discuss a class of algorithms known as threshold dynamics for simulating the motion of interfaces, including networks of them, by mean curvature. This evolution arises as gradient descent for the sum of areas of surfaces in the network. It comes up in a number of applications ranging from computer vision to materials science. It plays a particularly prominent role in materials science, where it describes the dynamics of grain boundaries in polycrystalline materials (which include most metals and ceramics) under heat treatment.
The basic version of threshold dynamics, introduced by Merriman, Bence, and Osher in 1992, applies to networks the energy of which weighs the area of each surface equally. It is remarkably elegant: The geometric flow, together with the correct conditions at free boundaries known as junctions (along which three or more interfaces meet) is generated by alternating two simple and efficient operations: Convolution with a kernel, and pointwise thresholding. Moreover, all topological changes, which occur frequently during the flow in the context of networks, appear to be handled reasonably just by these two steps. In joint work with Felix Otto, we describe how to extend threshold dynamics, while preserving its extreme simplicity and efficiency, to networks in which the area of each surface may be weighted by a different constant -- a level of generality that is important in materials science. The key to finding the correct extension to this setting is a new, variational formulation of the original algorithm. If time permits, I will also discuss further extensions to anisotropic surface energies.
This talk is concerned with kinetic Fokker-Planck (FP) equations whose "coefficients" are not regular, merely bounded measurable. We only assume that the diffusion matrix (only involving the velocity variable) is "elliptic". In this sense, the kinetic FP equations we consider are hypoelliptic. We recently proved with Golse, Mouhot and Vasseur that weak solutions to such equations are Hölder continuous. I will explain how classical techniques developed by De Giorgi and Moser can be combined with averaging lemmas and regularity transfers developed in kinetic theory to get such a result.
The Boltzmann equation models the evolution of a rarefied gas, in which particles interact through binary collisions, by describing the evolution of the probability density of particles. The equation balances transport operator with a collision operator, where the latter is a bilinear integral with a non-integrable angular kernel. For a long time the equation was simplified by assuming that this kernel is integrable (so called Grad's cutoff), with a belief that such an assumption does not affect the equation significantly. However, it has recently been observed that a non-integrable singularity carries regularizing properties, which motivates further analysis of the equation in this setting.
We study behavior in time of tails of solutions to the Boltzmann equation in the non-cutoff regime, by examining the generation and propagation in time of $L^1$ and $L^\infty$ exponentially weighted estimates and the relation between them. For this purpose we introduce Mittag-Leffler moments, which can be understood as a generalization of exponential moments. We show how the singularity rate of the angular kernel affects the order of tails that can be propagated in time. This is based on joint works with Alonso, Gamba, Pavlovic and with Gamba, Pavlovic.
We consider a simple model of Quantum Control (cavity quantum electrodynamics). We briefly describe the ingredients of the modeling of open quantum systems and the physical phenomena at play. We then concentrate our attention to the mathematical analysis and the numerical simulation of the model.Techniques combining numerical approximations of PDEs and Monte-Carlo approaches simulating the stochastic dynamics are introduced. We also point out some mathematical questions in the analysis of the models.
The talk is based on a series of joint works with Pierre Rouchon (Ecole des Mines and Inria, Paris).