S1.37

Come along for free pizza and to hear about the Mathematrix events this term.

Electric 1-form symmetries are generically broken in gauge theories with Chern-Simons terms. In this talk we discuss how infinite subsets of these symmetries become non-invertible topological defects. Time permitting we will also discuss generalizations and applications to the Swampland program in relation to the completeness hypothesis.

Fermat’s two-squares theorem is an elementary theorem in number theory that readily lends itself to a classification of the positive integers representable as the sum of two squares. Given this, a natural question is: what is the minimal number of squares needed to represent any given (positive) integer? One proof of Fermat’s result depends on essentially a buffed pigeonhole principle in the form of Minkowski’s Convex Body Theorem, and this idea can be used in a nearly identical fashion to provide 4 as an upper bound to the aforementioned question (this is Lagrange’s four-square theorem). The question of identifying the integers representable as the sum of three squares turns out to be substantially harder, however leaning on a powerful theorem of Dirichlet and a handful of tricks we can use Minkowski’s CBT to settle this final piece as well (this is Legendre’s three-square theorem).

An informal session for DPhil students, ECRs and undergraduates with an interest in probability. The aim is to gain exposure to areas outside of your own research interests in an informal and accessible way.

By way of shameless advertising for a TCC course I hope to give next term on the theory of totally disconnected locally compact groups, I will present two interesting and illuminating examples of such groups: the full automorphism group of a regular tree, and Neretin's group of spheromorphisms
 

Expansion, rank gradient and virtual splitting are all concepts of great interest in asymptotic group theory. We discuss a result of Marc Lackenby which demonstrates a surprising relationship between then, and give examples exhibiting different combinations of asymptotic behaviour.