Anomalies in quantum systems are present when a classical symmetry is broken by quantum effects. They give rise to physical predictions and constraints. This talk will focus on the mathematical features of anomalies of continuous, ordinary, symmetries. In the first part, we will review the topological nature of anomalies, in particular the connection to the Atiyah-Singer index theorem and its non-perturbative path-integral computation by Fujikawa. We will then discuss how anomalies and their associated (topological) Chern-Simons polynomials are related to BRST cohomology via the Stora-Zumino chain of descent equations, explaining the connection to the two-step descent procedure reviewed in the talk by Alice Lüscher last term.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.

I will present some contributions to the quasiisometry classification of solvable Lie groups of exponential growth that we obtain using sublinear bilipschitz equivalences, which are generalized quasiisometries. This is joint work with Ido Grayevsky.

Discrete and continuous energy problems that arise in a variety of scientific contexts are introduced, along with their fundamental existence and uniqueness results. Particular emphasis will be on Riesz and Gaussian pair potentials and their connections with best-packing and the discretization of manifolds. The latter application leads to the asymptotic theory (as N → ∞) for N-point configurations that minimize energy when the potential is hypersingular (short-range). For fixed N, the determination of such minimizing configurations on the d-dimensional unit sphere S d is especially significant in a range of contexts that include coding theory, discrete geometry, and physics. We will review linear programming methods for proving the optimality of configurations on S d , including Cohn and Kumar’s theory of universal optimality. The following reference will be made available during the short course: Discrete Energy on Rectifiable Sets, by S. Borodachov, D.P. Hardin and E.B. Saff, Springer Monographs in Mathematics, 2019.

Sessions:

Friday, 24 January 14:00-16:00

Friday, 31 January 14:00-16:00