This week

In this talk, I will introduce and discuss the spectral properties of transversal field quantum spin glasses.  In the second part of the talk, I will introduce the proof of the convergence of the free energy of the quantum Ising p-spin glass to that of the Quantum Random Energy Model. Finally, I will discuss some conjectures from the physics literature on the 1/p-corrections to the free energy.

In the area of combinatorics known as ‘permutation patterns’, seemingly innocuous questions can conceal a surprising degree of difficulty, giving rise to combinatorial problems that range from trivial to unsolved (despite decades of work). In this talk, we will explore several ways in which permutation pattern questions interface with probability — sometimes giving rise to probabilistic processes, sometimes illuminated by probabilistic reasoning. Some of these will be generalizations of known processes, others new, and still others conjectural.

Based on forthcoming joint work with Slim Kammoun and Einar Steingrimsson.

Consider the set of labelled trees on n vertices. Cayley’s formula tells us that there are n^{n-2} elements in this set; pick one uniformly at random. What can we say about this random tree in the limit as n goes to infinity? There are several different ways in which this (very vague!) question may be approached. One is via a scaling limit, that is, finding a good way to rescale the object with n (in this case, by giving each of its edges length n^{-1/2}) and then sending n to infinity, so that we obtain a continuum limit which is a random tree-like object. The limit in this case was discovered by David Aldous in 1990, and is known as the Brownian continuum random tree. My aim in this talk is to give an introduction to this beautiful area of probability theory.

Effective collective decision-making in human and animal groups requires robust mechanisms to form consensus, typically via feedback loops in which individuals adapt their behaviour based on their perception of others. Such behaviour has been observed and theorised across scales from nucleosomes to entire societies. Of equal importance, but far less well studied, is the question of how consensus is overturned. In many contexts it is vital that group decisions do not remain fixed in the face of new evidence\DSEMIC echo-chamber effects must be suppressed so that the collective preferences which are expressed are not too strongly entrenched. In this talk I will discuss a new mathematical theory for how consensus can be overturned in symmetric binary choice problems, and compare the theoretical predictions to experiments with human and animal groups.

Symmetries of a quantum field theory are implemented by topological operators. These are special extended operators whose correlation functions are insensitive to continuous deformations of their support. The classification of generalized symmetries thus reduces to understanding the spectrum of such topological operators across different codimensions. While a generic QFT may admit infinitely many topological operators, their topological nature imposes strong consistency conditions on their structure. In this talk, I will present a set of such constraints in 2+1 and 3+1 dimensions and highlight how they severely restrict the spectrum of topological operators in higher dimensions, in contrast to 1+1 dimensions. Using these constraints, I will argue that the action of non-invertible symmetries on local operators in higher dimensions is highly restricted. In particular, this action is either invertible or, when non-invertible, admits a description in terms of gauging a finite symmetry.