In the classical setting of real semisimple Lie groups, the Dirac inequality (due to Parthasarathy) gives a necessary condition that the infinitesimal character of an irreducible unitary representation needs to satisfy in terms of the restriction of the representation to the maximal compact subgroup. A similar tool was introduced in the setting of representations of p-adic groups in joint work with Barbasch and Trapa, where the necessary unitarity condition is phrased in terms of the semisimple parameter in the Kazhdan-Lusztig parameterization and the hyperspecial parahoric restriction. I will present several consequences of this inequality to the problem of understanding the unitary dual of the p-adic group, in particular, how it can be used in order to exhibit several isolated "extremal" unitary representations and to compute precise "spectral gaps" for them.

Any binary classifier (or score-function) can be used to define a dissimilarity between two distributions of points with positive and negative labels. Actually, many well-known distribution-dissimilarities are classifier-based dissimilarities: the total variation, the KL- or JS-divergence, the Hellinger distance, etc. And many recent popular generative modelling algorithms compute or approximate these distribution-dissimilarities by explicitly training a classifier: eg GANs and their variants. After a brief introduction to these classifier-based dissimilarities, I will focus on the influence of the classifier's capacity. I will start with some theoretical considerations illustrated on maximum mean discrepancies --a weak form of total variation that has grown popular in machine learning-- and then focus on deep feed-forward networks and their vulnerability to adversarial examples. We will see that this vulnerability is already rooted in the design and capacity of our current networks, and will discuss ideas to tackle this vulnerability in future.

We develop random graph models where graphs are generated by connecting not only pairs of vertices by edges but also larger subsets of vertices by copies of small atomic subgraphs of arbitrary topology. This allows the for the generation of graphs with extensive numbers of triangles and other network motifs commonly observed in many real world networks. More specifically we focus on maximum entropy ensembles under constraints placed on the counts and distributions of atomic subgraphs and derive general expressions for the entropy of such models. We also present a procedure for combining distributions of multiple atomic subgraphs that enables the construction of models with fewer parameters. Expanding the model to include atoms with edge and vertex labels we obtain a general class of models that can be parametrized in terms of basic building blocks and their distributions that includes many widely used models as special cases. These models include random graphs with arbitrary distributions of subgraphs, random hypergraphs, bipartite models, stochastic block models, models of multilayer networks and their degree corrected and directed versions. We show that the entropy for all these models can be derived from a single expression that is characterized by the symmetry groups of atomic subgraphs.

In this talk I will consider model-independent pricing problems in a stochastic interest rates framework. In this case the usual tools from Optimal Transport and Skorokhod embedding cannot be applied. I will show how some pricing problems in a fixed-income market can be reformulated as Weak Optimal Transport (WOT) problems as introduced by Gozlan et al. I will present a super-replication theorem that follows from an extension of WOT results to the case of non-convex cost functions.
This talk is based on joint work with M. Beiglboeck and G. Pammer.