Veranstaltungen

Reduction of Poisson manifolds by coisotropic submanifolds formalizes symmetry reduction of classical mechanical systems, and therefore plays an important role in Poisson geometry. Constraint algebras encode the additional structure on the algebra of functions needed for reduction and their deformations yield (formal) star products compatible with reduction. I will discuss the deformation theory of these constraint algebras and present some results about an adapted Hochschild-Kostant-Rosenberg Theorem computing the cohomology controlling this deformation theory.

In this talk, based on joint work with Marco Mazzucchelli, I will present some new results on the dynamics of geodesic flows of closed Riemannian surfaces, proved using the curve shortening flow. The first result is a forced existence theorem for orientable closed Riemannian surfaces of positive genus, asserting that the existence of a contractible simple closed geodesic γ forces the existence of infinitely many closed geodesics in every primitive free homotopy class of loops and intersecting γ. I will then explain how this type of result can be used to show the existence of Birkhoff sections for the geodesic flow of any closed orientable Riemannian surface.

The conformal dimension of a metric space $(X,d)$ measures its optimal shape from the perspective of quasiconformal geometry. It is defined by infimizing dimension over metrics in the quasisymmetric equivalence class of $d$. Introduced by Pierre Pansu in 1989, conformal Hausdorff dimension played an important early role in the development of analysis on metric spaces. Subsequently a variant notion, conformal Assouad dimension, gained prominence. Assouad dimension—which bounds Hausdorff dimension from above—is a scale-invariant, quantitative measurement of optimal coverings of a space. Dimension interpolation is an emerging program of research in fractal geometry which identifies geometrically natural one-parameter dimension functions interpolating between existing concepts. Two exemplars are the Assouad spectrum (Fraser-Yu, 2015), which interpolates between box-counting and Assouad dimension, and the intermediate dimensions (Falconer-Fraser-Kempton, 2020), which interpolate between Hausdorff and box-counting dimension. In this talk, I’ll discuss the mapping-theoretic properties of intermediate dimensions and the Assouad spectrum, with applications to the quasiconformal classification of sets and to the range of conformal Assouad spectrum. The latter results are based on a recent joint project with Efstathios Chrontsios Garitsis (Univ of Tennessee) and an ongoing collaboration with Jonathan Fraser (Univ of St. Andrews).

The laws of two continuous martingales will typically be singular to each other and hence have infinite relative entropy. But this does not need to happen in discrete time. This suggests defining a new object, the specific relative entropy, as a scaled limit of the relative entropy between the discretized laws of the martingales. This definition goes all the way back to Nina Gantert's PhD thesis, and in recent time Hans Foellmer has rekindled the study of this object by for instance obtaining a novel transport-information inequality. Independently, this object has made sporadic appearances in finance over the years. In this talk I will first discuss the existence of a closed-form formula for the specific relative entropy, depending on the quadratic variation of the involved martingales. Next I will describe an application of this object to prediction markets. Concretely, David Aldous asked in an open question to determine the 'most exciting game', i.e. the prediction market with the highest entropy. With M. Beiglböck we give an answer to this question by solving a stochastic control problem whose cost criterion is the specific relative entropy. If time permits I will also discuss the multidimensional version of this question, based on a joint work with Wang and Zhang.