Robert McCann (University of Toronto)

11/05 at 4:30 pm (CEST)  

A geometric approach to regularity for optimal transport maps  

The first regularity results for maps optimizing an open class of cost were proved by Ma-Trudinger-Wang (2005) using a key inequality which they introduced (based on a classical strategy of Pogorelov).  Away from the boundary, this inequality controls the size of the derivative of any smooth optimal map. After reviewing some history concerning minimal surfaces in signed geometries,  we describe a new derivation of this estimate with Brendle, Leger and Rankin (2024) which relies in part on Kim, McCann and Warren's (2010) observation that the graph of an optimal map becomes a volume maximizing non-timelike submanifold when the product of the source and target domains is endowed with a suitable geometry that combines both the densities being transported and the cost being optimized.  This unexpectedly links optimal transport to the Plateau problem in geometry with split signature (and more philosophically, with observer independence in Einstein's theory of gravity). The key difficulty is to uniformize ellipticity by showing the maximizing non-timelike submanifold is in fact (uniformly) spacelike. J. Reine Angew. Math. 817 (2024) 251-266 doi.org/10.1515/crelle-2024-0071 arXiv 2311.10208

Anna Skorobogatova (ETH Zürich)

25/05 at 4:30 pm (CEST)  

Non-uniqueness of locally minimizing clusters  

Optimal bubble cluster problems concern the study of partitions of n-dimensional Euclidean space into a finite collection of chambers, some with finite volume and some with infinite volume. One looks for local minimizers of interfacial area subject to volume constraints on the finite-volume chambers. The case of one infinite-volume chamber is the classical multiple bubble problem and has received much attention in recent decades, with a well-known conjecture of Sullivan predicting the existence of a unique minimizing configuration when there are not too many chambers, which has been partially verified to be true in low dimensions. 

We study a variant of the multiple bubble problem with more than one infinite-volume chamber, in particular the simplest case of 1 finite-volume chamber and 2 infinite-volume chambers. Here, Bronsard & Novack showed that uniqueness of local minimizers also holds in sufficiently low dimensions. In stark contrast, we show that uniqueness fails in a large number of dimensions n > 7 and we provide some particular surprising phenomena that local minimizers can exhibit in these higher dimensions. This is based on joint work with Lia Bronsard, Robin Neumayer and Michael Novack.

Camillo de Lellis (IAS Princeton)

01/06 at 4:30 pm (CET)  

Boundary Plateau Laws  

Dipping a wire of metal or plastic in soapy water and taking it out is a favorite classroom experiment: typically the soapy water will form a thin film which is attached to the wire. The classical Plateau laws, stated by the Belgian physicist Joseph Plateau in the nineteenth century, assert that, away from the wire, the local geometry of a soap film is described locally by the following list of shapes: a 2-dimensional plane, three halfplanes meeting at a common line with equal angles, and the cone over the 1-dimensional skeleton of a regular tetrahedron. 

Is there a similar list of possible shapes for the points where the film touches its "boundary'', namely the wire of the classroom experiment? The classical Plateau laws were translated into a mathematical theorem by Jean Taylor in the seventies: in a nutshell Taylor's theorem rigorously classifies 2-dimensional conical shapes which minimize the area. In this talk I will illustrate a recent joint work with Federico Glaudo, classifying conical shapes which minimize the area and include a boundary line: the corresponding list suggests an analog of Plateau's laws at the boundary of the soap film, which are very much in agreement with both real-life and numerical experiments.

Yoshihiro Tonegawa (Institute of Science Tokyo)

17/04 at 11:00 am (CEST)  

Mean curvature flow with irregular forcing term

Consider the following initial value problem: suppose that we are given a hypersurface and a time-dependent vector field (such as a flow field). Let the hypersurface evolve by the motion law that the velocity of the hypersurface is equal to the mean curvature plus the given vector field. Obviously, if all of the given quantities are smooth enough, the hypersurface should evolve nicely according to the rule for a while until some singularities occur. For the existence of such evolution, on the other hand, what is the minimal regularity assumption that one should impose on the vector field? Do we have a weak notion of evolution? Do we have some nice regularity theory for such evolving hypersurface?

The problem came from some two-phase flow problem involving the Allen-Cahn equation but it is interesting independent of the origin. In the talk I would like to describe what we can and cannot say currently. The work is joint effort with many people that I mention along the way.

Sylvia Serfaty (NYU Courant)

08/04 at 3:30 pm (CEST) 

Vortex points and vortex lines in the Ginzburg-Landau model 

I will review work with Etienne Sandier and more recent joint work with Carlos Román and Etienne Sandier, where we study the onset of vortices in the two and three-dimensional Ginzburg-Landau model of superconductivity. We discuss the critical field at which the first vortices appear,  the optimal number of lines, and Gamma-limit problems for their effective interaction and  arrangement.

Alessio Figalli (ETH Zürich)

30/03 at 11:00 am (CET) 

The De Giorgi Conjecture for the Free Boundary Allen–Cahn Equation

The Allen–Cahn equation is known to approximate minimal surfaces. This connection led to the conjecture that global stable solutions of the Allen–Cahn equation should be one-dimensional in dimensions up to seven. If true, this statement would imply the celebrated De Giorgi conjecture for monotone solutions.

Motivated by the geometric nature of the Allen–Cahn equation, Jerison has advocated for more than a decade that a free-boundary formulation provides a more natural framework for approximating minimal surfaces. This perspective leads to studying the above conjecture in the free-boundary setting.

In recent joint work with Chan, Fernández-Real, and Serra, we classify all stable global solutions to the one-phase Bernoulli free-boundary problem in three dimensions. As a consequence, we show that global stable solutions to the free-boundary Allen–Cahn equation in three dimensions are one-dimensional.

Mat Langford (ANU Mathematical Sciences Institute)

06/03 at 11 am (CET) 

Breakfast can wait

I will recall the recipe for the construction of the O(n) x O(1)-symmetric "ancient pancake" solution to the mean curvature flow (joint with Bourni and Tinaglia), and then show how to  join two pancakes together with a neck to form a "pancake stack" (joint with Mramor and Yudowitz). Whereas the ancient pancake admits a very precise description, the somewhat different approach employed to construct the pancake stack does not yield very precise information, leaving some interesting questions open.

Enrico Valdinocci (the University Western Australia) 

27/02 at 11 am (CET)

Sheet happens (but only as the root of 1-s)  

We discuss regularity properties of 2-dimensional stable s-minimal surfaces, presenting a robust regularity estimate and an optimal sheet separation bound, according to which the distance between different connected components of the surface must be at least the square root of 1-s. 

Man Chun Lee (The Chinese University of Hong Kong) 

23/02 at 11 am (CET) 

Quantitative Ricci Flow smoothing on manifolds and applications

Local existence theory of Ricci Flow plays an important role in regularizing metrics on both non-compact manifolds and singular metrics. In this talk, we will discuss Ricci Flow smoothing from metrics which is modeled by cone. We will discuss several applications in recent years, concerning both regularity of Gromov-Hausdorff limit and uniformization type problem on non-compact manifolds.