An important invariant in the complex representation theory of reductive p-adic groups is the wavefront set, because it contains information about the character of such a representation. In this talk, Mick Gielen will introduce a new invariant called the canonical dimension, which can be said to measure the size of a representation and which has a close relation to the wavefront set.  He will then state some results he has obtained about the canonical dimensions of compactly induced representations and show how they teach us something new about the wavefront set. This illustrates a completely new approach to studying the wavefront set, because the methods used to obtain these results are very different from the ones usually used.

The simplex of traces of a unital C*-algebra has long been regarded as a central invariant in the theory. Likewise, from the group-theoretic perspective, the simplex of traces of a discrete group (namely, the simplex of traces of its maximal C*-algebra) is a fundamental object in harmonic analysis, and the study of this simplex led to many applications in recent years.

Itamar Vigdorovich , UCSD, will discuss several results describing the simplex of traces in concrete and significant cases. These include Property (T) groups and especially higher rank lattices, for which the simplex of traces is as tame as possible. In contrast, for free products, the simplex is typically as wild as possible, yet still admits a canonical and universal structure—the Poulsen simplex. In ongoing work, an analogous result is obtained for the space of traces on the fundamental group of a closed surface of genus g≥2.

Itamar presents these results, outlines the main ideas behind the proofs, and gives an overview of the central concepts. The talk is based on joint works with Gao, Ioana, Levit, Orovitz, Slutsky, and Spaas.

Fracture is usually treated as an outcome to be avoided; here we see it as something we may write into a lattice's microstructure. Maxwell lattices sit at the edge of mechanical stability, where robust topological properties provide a way on how stress localises and delocalises across the structure with directional preference. Building on this, we propose a direct relationship between lattice topology and damage propagation. We identify a set of topology- and geometry-dependent parameters that gives a simple, predictive framework for nonideal Maxwell lattices and their damage processes. We will discuss how topological polarisation and domain walls steer and arrest damage in a repeatable way. Experiments confirm the theoretical predicted localisation and the resulting tuneable progression of damage and show how this control mechanism can be used to enhance dissipation and raise the apparent fracture energy.

The mitigation of bacterial adhesion to surfaces and subsequent biofilm formation is a key challenge in healthcare and manufacturing processes. To accurately predict biofilm formation you must determine how changes to bacteria behaviours and dynamics alter their ability to adhere to surfaces. In this talk, I will present a framework for incorporating microscale behaviour into continuum models using techniques from statistical mechanics at the microscale combined with boundary-layer theory at the macroscale.

We will examine the flow of a dilute suspension of motile bacteria over a flat absorbing surface, developing an effective model for the bacteria density near the boundary inspired by the classical Lévêque boundary layer problem. We use our effective model to derive analytical solutions for the bacterial adhesion rate as a function of fluid shear rate and individual motility parameters of the bacteria, validating against stochastic numerical simulations of individual bacteria. We find that bacterial adhesion is greatest at intermediate flow rates, since at higher flow rates shear-induced upstream swimming limits adhesion.

Rare and extreme events are notoriously hard to handle in any complex stochastic system: They are simultaneously too rare to be reliably observable in experiments or numerics, but at the same time often too impactful to be ignored. Large deviation theory provides a classical way of dealing with events of extremely small probability, but generally only yields the exponential tail scaling of rare event probabilities. In this talk, I will discuss theory, and algorithms based upon it, that improve on this limitation, yielding sharp quantitative estimates of rare event probabilities from a single computation and without fitting parameters. Notably, these estimates require the computation of determinants of differential operators, which in relevant cases are not traceclass and require appropriate renormalization. We demonstrate that the Carleman--Fredholm operator determinant is the correct choice. Throughout, I will demonstrate the applicability of these methods to high-dimensional real-world systems, for example coming from atmosphere and ocean dynamics.