By a classical theorem of Jordan, every faithful transitive action of a nontrivial finite group admits a derangement (an element with no fixed points). More recently, the existence of derangements with additional properties has attracted much attention, especially for primitive actions of almost simple groups. Surprisingly, there exist almost simple groups with elements that are derangements in every faithful primitive action; we say that these elements are totally deranged. I'll talk about ongoing work to classify the totally deranged elements of almost simple groups, and I'll mention how this solves a question of Garzoni about invariable generating sets for simple groups.

We prove a positive characteristic analogue of the classical result that the centralizer of a nonconstant differential operator in one variable is commutative. This leads to a new, short proof of that classical characteristic zero result, by reduction modulo p. This is joint work with Justin Desrochers available at https://arxiv.org/abs/2201.04606.

Liouville conformal field theory models two-dimensional gravity with a cosmological constant and conformal matter. In its timelike regime, it reproduces the characteristic negative kinetic term of the conformal factor of the metric in the Einstein-Hilbert action, the sign which infamously makes the gravity path integral ill-defined. In this talk, I will first discuss the perturbative computation of the timelike Liouville partition function around the sphere saddle and propose an all-orders result. I will then turn to the disk and present the bulk 1-point functions of this CFT, and discuss possible interpretations in terms of boundary conditions.

Statics and dynamics of droplets on lubricated surfaces

Zhaohe Dai

The abstract is "Slippery liquid infused porous surfaces are formed by coating surface with a thin layer of oil lubricant. This thin layer prevents other droplets from reaching the solid surface and allows such deposited droplets to move with ultra-low friction, leading to a range of applications. In this talk, we will discuss the static and dynamic behaviour of droplets placed on lubricated surfaces. We will show that the layer thickness and the size of the substrate are key parameters in determining the final equilibrium. However, the evolution towards the equilibrium is extremely slow (on the order of days for typical experimental parameter values). As a result, we suggest that most previous experiments with oil films lubricating smooth substrates are likely to have been in an evolving, albeit slowly evolving, transient state.

Modeling and Design Optimization for Pleated Membrane Filters

Yixuan Sun

Membrane filtration is widely used in many applications, ranging from industrial processes to everyday living activities. With growing interest from both industrial and academic sectors in understanding the various types of filtration processes in use, and in improving filter performance, the past few decades have seen significant research activity in this area. Experimental studies can be very valuable, but are expensive and time-consuming, therefore theoretical studies offer potential as a cost-effective and predictive way to improve on current filter designs. In this work, mathematical models, derived from first principles and simplified using asymptotic analysis, are proposed for pleated membrane filters, where the macroscale flow problem of Darcy flow through a pleated porous medium is coupled to the microscale fouling problem of particle transport and deposition within individual pores of the membrane. Asymptotically-simplified models are used to describe and evaluate the membrane performance numerically and filter design optimization problems are formulated and solved for industrially-relevant scenarios. This study demonstrates the potential of such modeling to guide industrial membrane filter design for a range of applications involving purification and separation.