ETH Zurich

The general problem of shock formation in three space dimensions was solved by Christodoulou in 2007. In his work also a complete description of the maximal development of the initial data is provided. This description sets up the problem of continuing the solution beyond the point where the solution ceases to be regular. This problem is called the shock development problem. It belongs to the category of free boundary problems but in addition has singular initial data because of the behavior of the solution at the blowup surface. In my talk I will present the solution to this problem in the case of spherical symmetry. This is joint work with Demetrios Christodoulou.

Lawler, Schramm and Werner studied 2-point chordal restriction measures and gave several constructions using SLE tools.

It is possible to characterize 3-point chordal restriction measures in a similar manner. Their boundaries are SLE(8/3)-like curves with a slightly different drift term.

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More than twenty years ago Erdős conjectured that a triangle-free graph $G$ of chromatic number $k$ contains cycles of at least $k^{2−o(1)}$ different lengths. In this talk we prove this conjecture in a stronger form, showing that every such $G$ contains cycles of $ck^2\log k$ consecutive lengths, which is tight. Our approach can be also used to give new bounds on the number of different cycle lengths for other monotone classes of $k$-chromatic graphs, i.e.,  clique-free graphs and graphs without odd cycles.

Joint work with A. Kostochka and J. Verstraete.

We consider the numerical approximation of Kolmogorov equations arising in the context of option pricing under L\'evy models and beyond in a multivariate setting. The existence and uniqueness of variational solutions of the partial integro-differential equations (PIDEs) is established in Sobolev spaces of fractional or variable order.

Most discretization methods for the considered multivariate models suffer from the curse of dimension which impedes an efficient solution of the arising systems. We tackle this problem by the use of sparse discretization methods such as classical sparse grids or tensor train techniques. Numerical examples in multiple space dimensions confirm the efficiency of the described methods.

We will start with a recollection on factorization algebras and factorization homology. We will then explain what fully extended TFTs are, after Jacob Lurie. And finally we will see how factorization homology can be turned into a fully extended TFT. This is a joint work with my student Claudia Scheimbauer.

We will explain how the result of Pantev-Toën-Vaquié-Vezzosi, about shifted symplectic structures on mapping stacks, can be extended to relative mapping stacks and Lagrangian structures. We will also provide applications in ordinary symplectic geometry and topological field theories.

In 2000 Eliashberg-Polterovich introduced the natural notion of orderability of contact manifolds; that is, the existence of a natural partial order on the group of contactomorphisms. I will explain how one can study orderability questions using the machinery of Rabinowitz Floer homology. We establish a link between orderable and hypertight contact manifolds, and show that the Weinstein Conjecture holds (i.e. there exists a closed Reeb orbit) whenever there exists a positive (not necessarily contractible) loop of contactomorphisms.

Joint work with Peter Albers and Urs Fuchs.

Rabinowitz Floer homology (RFH) is the Floer theory associated to the Rabinowitz action functional. One can think of this functional as a Lagrange multiplier functional of the unperturbed action functional of classical mechanics. Its critical points are closed orbits of arbitrary period but with fixed energy.

This fixed energy problem can be transformed into a fixed period problem on an enlarged phase space. This provides a way to see RFH as a "standard" Hamiltonian Floer theory, and allows one to treat RFH on an equal footing to other related Floer theories. In this talk we explain how this is done and discuss several applications.

Joint work with Alberto Abbondandolo and Alexandru Oancea.

The original transport problem is to optimally move a pile of soil to an excavation.

Mathematically, given two measures of equal mass, we look for an optimal bijection that takes

one measure to the other one and also minimizes a given cost functional. Kantorovich relaxed

this problem by considering a measure whose marginals agree with given two measures instead of

a bijection. This generalization linearizes the problem. Hence, allows for an easy existence

result and enables one to identify its convex dual.

In robust hedging problems, we are also given two measures. Namely, the initial and the final

distributions of a stock process. We then construct an optimal connection. In general, however,

the cost functional depends on the whole path of this connection and not simply on the final value.

Hence, one needs to consider processes instead of simply the maps S. The probability distribution

of this process has prescribed marginals at final and initial times. Thus, it is in direct analogy

with the Kantorovich measure. But, financial considerations restrict the process to be a martingale

Interestingly, the dual also has a financial interpretation as a robust hedging (super-replication)

problem.

In this talk, we prove an analogue of Kantorovich duality: the minimal super-replication cost in

the robust setting is given as the supremum of the expectations of the contingent claim over all

martingale measures with a given marginal at the maturity.

This is joint work with Yan Dolinsky of Hebrew University.