The wreath product of a finite group, or more generally an algebra, with a symmetric group is a familiar and important construction in representation theory and other areas of Mathematics. I shall present some highlights from my work on the representation theory of wreath products. These will include both structural properties (for example, that the wreath product of a cellular algebra with a symmetric group is again a cellular algebra) and cohomological ones (one 
particular point of interest being a generalisation of the result of Hemmer and Nakano on filtration multiplicities to the wreath product of two symmetric groups). I will also give an outline of some potential applications of this and related theory to important open  problems in algebraic combinatorics.

The nilpotent orbits of a Lie algebra play a central role in modern representation theory notably cropping up in the Springer correspondence and the fundamental lemma. Their behaviour when the base field is algebraically closed is well understood, however the p-adic case which arises in the study of admissible representations of p-adic groups is considerably more subtle. Their classification was only settled in the late 90s when Barbasch and Moy ('97) and Debacker (’02) developed an ‘affine Bala-Carter’ theory using the Bruhat-Tits building. In this talk we combine this work with work by Sommers and McNinch to provide a parameterisation of nilpotent orbits over a maximal unramified extension of a p-adic field in terms of so called dual Springer parameters and outline an application of this result to wavefront sets.

In 2013, Reeder–Yu gave a construction of supercuspidal representations by starting with stable characters coming from the shallowest depth of the Moy–Prasad filtration. In this talk, we will be diving deeper—but not too deep. In doing so, we will construct examples of supercuspidal representations coming from a larger class of “shallow” characters. Using methods similar to Reeder–Yu, we can begin to make predictions about the Langlands parameters for these representations.

Deep convolutional networks have spectacular performances that remain mostly not understood. Numerical experiments show that they classify by progressively concentrating each class in separate regions of a low-dimensional space. To explain these properties, we introduce a concentration and separation mechanism with multiscale tight frame contractions. Applications are shown for image classification and statistical physics models of cosmological structures and turbulent fluids.

Wieler has shown that every irreducible Smale space with totally disconnected stable sets is a solenoid (i.e., obtained via a stationary inverse limit construction). Through examples I will discuss how this allows one to compute the K-theory of the stable algebra, S, and the stable Ruelle algebra, S\rtimes Z. These computations involve writing S as a stationary inductive limit and S\rtimes Z as a Cuntz-Pimsner algebra. These constructions reemphasize the view point that Smale space C*-algebras are higher dimensional generalizations of Cuntz-Krieger algebras. The main results are joint work with Magnus Goffeng and Allan Yashinski.

We have various ways of describing the extent to which two countably infinite groups are "the same." Are they isomorphic? If not, are they commensurable? Measure equivalent? Quasi-isometric? Orbit equivalent? W*-equivalent? Von Neumann equivalent? In this expository talk, we will define these notions of equivalence, discuss the known relationships between them, and work out some examples. Along the way, we will describe recent joint work with Ishan Ishan and Jesse Peterson.

In this talk, we present simple ideas to combine nonparametric approaches based on positive definite kernels with deep learning models. There are many good reasons for bridging these two worlds. On the one hand, we want to provide regularization mechanisms and a geometric interpretation to deep learning models, as well as a functional space that allows to study their theoretical properties (eg invariance and stability). On the other hand, we want to bring more adaptivity and scalability to traditional kernel methods, which are crucially lacking. We will start this presentation by introducing models to represent graph data, then move to biological sequences, and images, showing that our hybrid models can achieves state-of-the-art results for many predictive tasks, especially when large amounts of annotated data are not available. This presentation is based on joint works with Alberto Bietti, Dexiong Chen, and Laurent Jacob.