Talks (titles and abstracts)
Alberto Bressan: A time dependent Dido's problem
We initially consider a controlled reaction-diffusion equation, modeling the spreading of an invasive population. By formally taking a sharp interface limit, a simpler model is derived, describing the controlled evolution of the contaminated region.
This leads to a family of optimization problems for a moving set. Results on controllability, existence of optimal strategies, and necessary conditions for optimality will be given. Roughly speaking, optimal strategies are those that shrink the initial set to the empty set, keeping the perimeter as short as possible over time. Some open questions will be discussed.
Antonin Chambolle: On the jump set of minimizers of Total-Variation based regularization problems
I will review old results on the regularity of minimizers of Total-Variation regularized inverse (denoising) problems. Addressing then the issue of the localization of the jump set of minimizers, for data with bounded variation, I will introduce a new and much simpler technique which provides some information for a much broader class of problems and regularizers (joint work with M. Lasica, mathematical institute of the polish academy of science).
Camillo De Lellis: Area-minimizing integral currents: singularities and structure
Area-minimizing integral currents are a natural generalization of area-minimizing oriented surfaces, a concept pioneered by De Giorgi and Federer-Fleming. Famous examples of singular 7-dimensional minimizers in \(\mathbb R^8\) and of singular \(2\)-dimensional minimizers in \(\mathbb R^4\) are known since long. Moreover a theorem which summarizes the work of several mathematicians in the 60es and 70es (De Giorgi, Fleming, Almgren, Simons, and Federer) and a celebrated work by Almgren in the 80es give dimension bounds for the singular set which match the one of the examples, in codimension 1 and in general codimension respectively.
In codimension higher than 1 a recent result of Liu shows that the singular set can in fact be a fractal of any Hausdorff dimension \(\alpha \leq m-2\). In joint works with Anna Skorobogatova and Paul Minter we prove that the singular set is \(m-2\). A proof of the same result has been discovered at the same time and independently by Brian Krummel and Neshan Wickramasekera. The theorem is the counterpart of a celebrated work of Leon Simon in the nineties for the hypersurface case, and indeed a byproduct of our proof is the uniqueness of the tangent cone at \(\mathcal{H}^{m-2}\)-a.e. point.
Maria Esteban: Open problems about Dirac eigenvalues in molecular configurations
In this talk I will present recent results and open problems concerning the lowest eigenvalue of a Dirac operator with a general multi-pole external electrostatic potential. They describe a relativistic quantum electron moving in the field of some (point-wise or extended) nuclei, possibly in a molecule. One of the main questions we ask is whether the eigenvalue is minimal when the nuclear charge is concentrated at one single point. This well-known property in non-relativistic quantum mechanics (involving the Schrödinger operator) has escaped all attempts of proof in the relativistic case.
This is work in collaboration with M. Lewin and E. Séré.
Craig Evans: Streamlines and shocks for the infinity Laplacian PDE
The two dimensional infinity Laplacian PDE, the Euler-Lagrange equation for a sup-norm variational problem, can be formally converted into the heat equation by the Legendre transform. I will discuss how to reverse this process, to help understand the geometry of "soft'' shocks for solutions, along which first derivatives are continuous but second derivatives are unbounded.
Gero Friesecke: Can we compute high-dimensional multi-marginal optimal transport plans accurately and certifiably?
High-dimensional multi-marginal optimal transport problems arise in many applications (many-electron physics, fluid dynamics, data interpolation), but their numerical computation is extremely challenging due to the curse of dimensionality: even just storing a discretized N-marginal plan requires exponentially many (w.r.to N) components.
We discuss our recent GenCol algorithm which solves the OT linear program on a dynamically updated low-dimensional subset of sparse plans, at linear (w.r.to N) storage cost. We prove the storage cost reduction for any N (without any loss in accuracy), and convergence of the algorithm to a global minimizer for N=2. Based on joint work with Daniela Voegler, Andreas Schulz, Maximilian Penka.
Inwon Kim: Tumor growth with nutrients: geometry of the tumor patch
We study a tumor growth model driven by nutrition and the pressure variable generated by the density height constraint. Our focus is on the free boundary regularity of the tumor patch that holds beyond topological changes, starting from general initial data. We will discuss the problem and our analysis, which centers around an elliptic obstacle problem and a Hopf-Lax formula for the pressure variable. Building on these ingredients, we show that the patch boundary is space-time regular in \(\R^d \times (0,\infty)\) except on a set of Hausdorff dimension less or equal to \(d-\alpha\).
Bernd Kirchheim: Convexity and Uniqueness in the Calculus of Variations
Whereas general existence results for minimizers of (vectorial) variational problems are clearly related to (coercivity) and Morreys quasiconvexity, the situation becomes much more constrained if also uniqueness of the minimizers is required for all linear pertubation of the energy. In this case a rather natural notion of functional convexity arises in a general Banach space context. We will discuss what are the specific implications for energy densities of integral cost functions.
This is joint work with J. Campos Cordero (Mexico), J. Kollar (Prag) and J.Kristensen (Oxford).
Bo'az Klartag: Isoperimetry and slices of convex sets
The slicing problem by Bourgain is an innocent-looking question in convex geometry. It asks whether any convex body of volume one in an n-dimensional Euclidean space admits a hyperplane section whose (n-1)-dimensional volume is at least some universal constant. There are several equivalent formulations and implications of this conjecture, which occupies a rather central role in the field. The slicing conjecture would follow from the isoperimetric conjecture of Kannan, Lovasz and Simonovits, which suggests that the most efficient way to partition a convex body into two parts of equal volume so as to minimize their interface, is a hyperplane bisection, up to a universal constant. In this lecture we will discuss progress from the last two years, showing that these two conjectures hold true up to factors that increase logarithmically with the dimension.
Fanghua Lin: Critical Point Sets of Solutions in Elliptic Homogenization
The quantitative uniqueness and the geometric measure estimates for the nodal and critical point sets of solutions of second order elliptic equations depend crucially on the bound of the associated Almgren's frequency function. The latter is possible (only) when the leading coefficients of equations are uniformly Lipschitz. One does not have this uniform Lipschitz continuity for coefficients of equations in elliptic homogenization. Instead, by using quantitative homogenization, successive harmonic approximation and suitable L^2-renormalization, we shall see how one can get a uniform estimate (independent of a small parameter characterizing the nature of homogenization) of co-dimension two Hausdorff measure as well as the Minkowski content of the critical point sets. A key element is an estimate of "turning" for the projection of a non-constant solution onto the subspace of spherical harmonics of order N, when the doubling index of solution on annular regions is trapped near N.
Svitlana Mayboroda: Free boundary problems for the partially reflected Brownian motion: the structure of the Robin harmonic measure
Robert McCann: A Nonsmooth Approach to Einstein's Theory of Gravity
While Einstein's theory of gravity is formulated in a smooth setting, the celebrated singularity theorems of Hawking and Penrose describe many physical situations in which this smoothness must eventually breakdown. In positive-definite signature, there is a highly successful theory of metric and metric-measure geometry which includes Riemannian manifolds as a special case, but permits the extraction of nonsmooth limits under curvature and dimension bounds analogous to the energy conditions in relativity: here sectional curvature is reformulated through triangle comparison, while Ricci curvature is reformulated using entropic convexity along geodesics of probability measures. This lecture explores recent progress in the development of an analogous theory in Lorentzian signature, whose ultimate goal is to provide a nonsmooth theory of gravity. We highlight how the null energy condition of Penrose admits a nonsmooth formulation as a variable lower bound on timelike Ricci curvature.
Felix Otto: A variational regularity theory for Optimal Transportation, and its application to matching
A couple of years ago, with M.~Goldman we devised a new approach to the regularity theory for Optimal Transportation that mimics De Giorgi's approach to the regularity theory of minimal surfaces in the sense that a harmonic approximation result is at its center: Under a non-dimensional smallness condition, the displacement is close to the gradient of a harmonic function.
Probably the main advantage of this variational regularity theory over the one based on maximum principle -- and attached to the name of Caffarelli -- is that it does not require any regularity of the involved measures. Hence it can be applied to the matching problem, where it allows for a regularity theory on microscopically large scales, and thus makes Parisi's linearization quantitative down to these scales. It thus provides a mesoscopic counterpart to macroscopic theory pioneered by Ambrosio and coworkers.
Tristan Rivière: Area Variations under Legendrian Constraint
In the first part of the talk we will adress the problem of studying area variations of surfaces under pointwise Lagrangian constraint in \({\mathbb C}^2\) (or any arbitrary Kähler Surface). We will explain the challenges of performing analysis (well posedness, existence, regularity...) with the associated Euler-Lagrange Equation. Then in trying to find conserved quantities and monotonicity formula for this problem we will naturally be invited to "lift" our problem to 5 dimensions by introducing a fifth Legendrian coordinate and to work in the Heisenberg group (or any Sasakian 5-manifold). The Lagrangian constraint is then converted into a Legendrian one. The area variation under pointwise Legendrian constraint consists in looking for critical points of the area among surfaces which are horizontal. This is a model of "extreme anisotropic" variational problem where one direction is forbidden while total isotropy holds in the remaining 4 directions (which are not integrable). We will derive a new monotonicity formula for this problem. Ultimately the main result we would like to explain is the following : In any 5 dimensional closed Sasakian manifold \(N^5\) (e.g. \(S^5\), \(S^3\times S^2\), Heisenberg group \({\mathbb H}^2\)...etc) we prove that any minmax operation on the area among Legendrian surfaces is achieved by a continuous conformal Legendrian map from a closed riemann surface \(S\) into \(N^5\) equipped with an integer multiplicity bounded in \(L^\infty\). Moreover this map, equipped with this multiplicity, satisfies a weak version of the Hamiltonian Minimal Equation. We conjecture that any solution to this equation is a smooth branched Legendrian immersion away from isolated Schoen-Wolfson conical singularities with non zero Maslov class.
If time permits we will explain our motivation for studying such question in relation with the Willmore conjecture in arbitrary co-dimensions.
Mete Soner: Eikonal Equations on Wasserstein Spaces
Mean-field or McKean-Vlasov type optimal control is closely related to the exciting program of mean-field games as initiated by Larry and Lions. Dynamic programming approach to these control problems result in nonlinear partial differential equations on the space of probability measures. These equations not only require the solution to be differentiable but impose further regularity on the derivatives which are being on the dual of the set of measures are also functions themselves. Despite these difficulties, several approaches to characterize the value function of the control problems as the unique appropriate weak solutions have been developed. In this talk, I will first introduce the mean field games through an interesting example of Kuramoto type synchronization. Then, I will extend this example to a general setting and prove uniqueness of for a class of equations that are analogous to classical Eikonal equations. This talk is based on joint works with Rene Carmona and Qinxin Yan of Princeton, and Quentin Cormier of INRIA.
Karl-Theodor Sturm: Spectral estimates under variable and distributional Ricci bounds
We discuss the role of variable and distributional (synthetic lower) bounds for the Ricci curvature in studying spectral gap and gradient estimates for the heat semigroup. In particular, we present a novel sharp lower bound for the spectral gap on a nonnegatively curved Riemannian manifold or, more generally, on an RCD\((k,N)\) space with a variable Ricci bound \(k: X to R_+\). Our estimate in terms of the \(L^{-p}\) norm of \(k\) for \(p=1-1/N\) improves upon the celebrated Lichnerowicz estimate (1958, case \(p=\infty\)) and the estimate of Veysseire (2010, case \(p=1\)). Also, spectral gap estimates with negative Ricci curvature in the Kato class will be briefly discussed.
Moreover, we present gradient estimates for the Neumann heat semigroup on non-convex domains which leads to a negative, distribution-valued Ricci curvature in the Kato class.
Vladimir Sverak: Vanishing viscosity limit for vortex rings
We consider the Cauchy problem for the Navier-Stokes equation with viscosity \(\nu\) in \(R^3\), and the initial vorticity given by an idealized vortex ring of a given radius and zero thickness. It can be shown the problem has a unique solution in its natural symmetry class. The goal is to study the behavior of the solutions for \(\nu\to 0\). Joint work with Thierry Gallay.
László Székelyhidi: Magnetohydrodynamic turbulence : weak solutions and conserved quantities
The ideal magnetohydrodynamic system in three space dimensions consists of the incompressible Euler equations coupled to the Faraday system via Ohm’s law. This system has a wealth of interesting structure, including three conserved quantities : the total energy, cross-helicity and magnetic helicity. Whilst the former two are analogous (and analytically comparable) to the total kinetic energy for the Euler system, magnetic helicity is known to be more robust and of a different nature. In particular, when studying weak solutions, Onsager-type conditions for all three quantities are known, and are basically on the same level of 1/3-differentiability as the kinetic energy in the ideal hydrodynamic case for the former two. In contrast, magnetic helicity does not require any differentiability, only L^3 integrability. From the physical point of view this difference lies at the heart of the Taylor-Woltjer relaxation theory. From the mathematical point of view it turns out to be closely related to the div-curl structure of the Faraday system. In the talk we present and compare some recent constructions of weak solutions and along the way highlight some of the hidden structures in the ideal magnetohydrodynamic system.
This is joint work with Daniel Faraco and Sauli Lindberg.
Tatiana Toro: Geometry of Measures
The works of Preiss and Preiss & Mattila provide criteria for the rectifiability of measures in terms of the existence of densities and principal values for the Riesz transform. Whether these results depend on the underlying metric in Euclidean space in the case of the density has been a puzzling problem for years. In this talk we will discuss some recent results in this direction as well as what happens under the assumption of the existence of principal values for the gradient of the fundamental solution to general divergence form operators. This is joint work with M. Goering and B. Wilson.