L4

We study a class of non linear integro-differential equations on the Wasserstein space related to the optimal control of McKean-Vlasov jump-diffusions. We develop an intrinsic notion of viscosity solutions that does not rely on the lifting to an Hilbert space and prove a comparison theorem for these solutions. We also show that the value function is the unique viscosity solution. Based on a joint work with V. Ignazio, M. Reppen and H. M. Soner

In the seminar I will present a recent work joint with  S. Suhr (Bochum) giving an optimal transport formulation of the full Einstein equations of general relativity, linking the (Ricci) curvature of a space-time with the cosmological constant and the energy-momentum tensor. Such an optimal transport formulation is in terms of convexity/concavity properties of the Shannon-Bolzmann entropy along curves of probability measures extremizing suitable optimal transport costs. The result gives a new connection between general relativity and  optimal transport; moreover it gives a mathematical reinforcement of the strong link between general relativity and thermodynamics/information theory that emerged in the physics literature of the last years.

Feynman integrals govern the perturbative expansion in quantum field theories. As periods, these integrals generate representations of a motivic Galois group. I will explain this idea and illustrate the 'coaction principle', a mechanism that constrains which periods can appear at any loop order.
 

In this talk, I will describe new mathematical structures in the low-energy  expansion of one-loop string amplitudes. The insertion of external states on the open- and closed-string worldsheets requires integration over punctures on a cylinder boundary and a torus, respectively. Suitable bases of such integrals will be shown to obey simple first-order differential equations in the modular parameter of the surface. These differential equations will be exploited  to perform the integrals order by order in the inverse string tension, similar to modern strategies for dimensionally regulated Feynman integrals. Our method manifests the appearance of iterated integrals over holomorphic  Eisenstein series in the low-energy expansion. Moreover, infinite families of Laplace equations can be generated for the modular forms in closed-string  low-energy expansions.
 

I will present recent results concerning a class of nonlinear parabolic systems of partial differential equations with small cross-diffusion (see doi.org/10.1051/m2an/2018036 and arXiv:1906.08060). Such systems can be interpreted as a perturbation of a linear problem and they have been proposed to describe the dynamics of a variety of large systems of interacting particles. I will discuss well-posedness, regularity, stability and convergence to the stationary state for (strong) solutions in an appropriate Banach space. I will also present some applications and refinements of the above-mentioned results for specific models.

In this talk, I will report about a joint work with D. Ciubotaru, in which we investigate the Dunkl version of the classical Howe-duality (O(k),spo(2|2)). Similar Fischer-type decompositions were studied before in the works of Ben-Said, Brackx, De Bie, De Schepper, Eelbode, Orsted, Soucek and Somberg for other Howe-dual pairs. Our work builds on the notion of a Dirac operator for Drinfeld algebras introduced by Ciubotaru, which was inspired by the analogous theory for Lie algebras, as well as the work of Cheng and Wang on classical Howe dualities.

I will talk about a way of building graded Lie algebras from certain Heisenberg groups. The input for this construction arises naturally when studying families of algebraic curves, and we'll look at some examples in which Lie theory interacts with number theory in an illuminating way. 

The Gibbons-Hawking ansatz is a powerful method for constructing a large family of hyperkaehler 4-manifolds (which are thus Ricci-flat), which appears in a variety of contexts in mathematics and theoretical physics. I will describe work in progress to understand the theory of minimal surfaces and mean curvature flow in these 4-manifolds. In particular, I will explain a proof of a version of the Thomas-Yau Conjecture in Lagrangian mean curvature flow in this setting. This is joint work with G. Oliveira.

The translation method for constructing quasiconvex lower bound of a given function in the calculus of variations and the notion of compensated convex transforms for tightly approximate functions in Euclidean spaces will be briefly reviewed. By applying the upper compensated convex transform to the finite maximum function we will construct computable quasiconvex functions with finitely many point wells contained in a subspace with rank-one matrices. The complexity for evaluating the constructed quasiconvex functions is O(k log k) with k the number of wells involved. If time allows, some new applications of compensated convexity will be briefly discussed.

Localization technique permits to reduce full dimensional problems to possibly easier lower dimensional ones. During the last years a new approach to localization has been obtained using the powerful tools of optimal transport. Following this approach, we obtain quantitative versions of two relevant geometric inequalities  in comparison geometry as Levy-Gromov isoperimetric inequality (joint with F. Maggi and A. Mondino) and the spectral gap inequality (joint with A. Mondino and D. Semola). Both results are also valid in the more general setting of metric measure spaces verifying the so-called curvature dimension condition.