Hausdorff Kolloquium 2018

Date: April 18 - July 11, 2018

Venue: Mathematikzentrum, Lipschitz Lecture Hall, Endenicher Allee 60, Bonn


Eduard Feireisl (University of Prague): On weak solution approach to problems in fluid dynamics

We discuss the concept of dissipative weak solution to the Euler and Navier Stokes systems describing the motion of a general compressible and heat conducting fluid. We present examples of ill-posedness in the context of the Euler system and develop the theory of weak and measure-valued solutions that comply with a certain form of the Second law of thermodynamics. We show applications of the theory to various problems including singular limits, convergence of numerical schemes, and problem driven by stochastic forcing.


Marc Burger (ETH Zürich): Compactifying higher Teichmueller spaces

Let S be a compact surface of negative Euler characteristic and G a semisimple group. Higher Teichmueller theory aims at singling out connected components of the variety of G-representations of the fundamental group of S which are formed of representations with geometric significance, echoing the case of PSL(2,R) where the relevant component is the classical Teichmueller space classifying hyperbolic structures on S. One of the outstanding problems is to construct compactifications of such components on which the mapping class group acts and having good topological properties, like the Thurston compactification does for classical Teichmueller space. In this talk we discuss such compactifications in the case when G=Sp(2n, R),constructed using representations over non-archimedean ordered fields, and relate these compactifications to classical objects like the space of geodesic currents on the surface S. This is joint work with A. Iozzi, A. Parreau, B.Pozzetti.


Wilfried Sieg (Carnegie Mellon University): Proofs as objects: Hilbert’s pivotal thought

The rigor of mathematics lies in its systematic organization that makes for conclusive proofs of assertions on the basis of assumed principles. Proofs are constructed through thinking, but can also be taken as objects of mathematical investigation; that was the key insight underlying Hilbert’s call for a theory of the “specifically mathematical proof” in 1917. This pivotal thought was rooted in revolutionary developments of mathematics and logic, but it also shaped the new field of mathematical logic. It grounded, in particular, Hilbert’s proof theory. The emerging investigations led over time to computability theory, artificial intelligence and cognitive science. Within this broad framework, I will describe first the pursuit of Hilbert’s proof theory with “reductive” foundational goals and then some recent, tentative steps towards a theory of the “specifically mathematical proof”. Those steps have been made possible by a confluence of proof theoretic investigations in the tradition of Gentzen’s work on natural deduction and computer implementations of mechanisms in which proofs can be (automatically) constructed. Here, at the intersection of automated proof search and interactive verification, is a promising avenue for exploring the structure of mathematical thought.


Guido de Philippis (SISSA Trieste): Boundary Regularity for Mass Minimizing currents

Federer and Fleming integral currents allows to solve the Plateau problem  in arbitrary Riemannian manifolds in any dimension and co-dimension. Thanks to the monumental  work of Almgren, recently revised by De Lellis and Spadaro, interior regularity is by now quite well understood. On the other hand, the current literature fails to provide (for the high co-dimension case)  even a single regular point at the boundary unless we require rather restrictive assumptions on the ambient space.

In this talk I will give an overview of the problem and show a  first boundary regularity result for mass minimising currents in any co-dimension.  In particular, I will show that the regular points are dense in the boundary. This, among other things, allows to provide a positive answer to a question of Almgren, namely that for connected boundary data, the solution is actually connected and of multiplicity one.

This is a joint work with C. De Lellis, J. Hirsch and  A. Massaccesi.


Ana Vargas (Universidad Autónoma de Madrid): On the restriction of the Fourier transform to surfaces: the hyperbolic case

The problem of restriction of the Fourier transform to hypersurfaces (or more generally to submanifolds in R^n) was posed by Stein in the seventies. This operator, in its adjoint form, gives the solution of dispersive equations (Schrödinger, wave, etc) in terms of the Fourier transform of the initial data. There are many open problems about dispersive equations for which it can be a powerful tool. Also, the restriction operator can be thought as a model case for more complicated oscillatory integral operators, for instance, the spherical summation operators.

We will make a review of this problem, which is still open. We will present some new results for the case of surfaces with negative curvature. That part is joint work with Stefan Buschenhenke and Detlef Müller.


Peter Koellner (Harvard University): Two futures: pattern and chaos

Set theory is presently at a cross roads, where one is faced with two radically different possible futures.

This is first indicated by Woodin's HOD Dichotomy Theorem, an analogue of Jensen's Covering Lemma with HOD in place of L. The HOD Dichotomy Theorem states that if there is an extendible cardinal, δ, then either HOD is "close'' to V (in the sense that it correctly computes successors of singular cardinals greater than δ) or HOD is "far'' from V (in the sense that all regular cardinals greater than or equal to δ are measurable in HOD). The question is whether the future will lead to the first or the second side of the dichotomy. Is HOD "close'' to V, or "far'' from V?

There are two opposing research programs leading to opposite sides of the dichotomy. The first program is the program of inner model theory. In recent years Woodin has shown that if inner model theory can reach one supercompact cardinal then it "goes all the way", and he has formulated a precise conjecture - the Ultimate-L Conjecture - which, if true, would lead to a fine-structural inner model that can accomodate all of the standard large cardinals. This is the future where pattern prevails.

The second program is the program of large cardinals beyond choice. Kunen famously showed that if AC holds then there cannot be a Reinhardt cardinal. It has remained open whether Reinhardt cardinals are consistent in ZF alone. In recent work - joint with Bagaria and Woodin - the hierarchy of large cardinals beyond choice has been investigated. It turns out that there is an entire hierarchy of choiceless large cardinals of which Reinhardt cardinals are only the beginning, and, surprisingly, this hierarchy appears to be highly ordered and amenable to systematic investigation. Perhaps it is even consistent... The point is that if these choiceless large cardinals are consistent then the Ultimate-L Conjecture must fail. This is the future where there can be no fine-structural understanding of the standard large cardinals. This is the future where chaos prevails.