In ’95 Lusztig gave a local Langlands correspondence for unramified representations of inner to split adjoint groups combining many deep results from type theory and geometric representation theory. In this talk, I will present a gentle reformulation of his construction revealing some interesting new structures, and with a view toward proving functoriality results in this framework. 

This seminar is organised jointly with the Junior Algebra and Representation Theory Seminar - all are very welcome!

There is an idea, going back to work of Krasner, that p-adic fields tend to function fields as absolute ramification tends to infinity. We will present a new way of rigorizing this idea, as well as give applications to the local Langlands correspondence of Fargues–Scholze.

The "index of hypocoercivity" is defined via a coercivity-type estimate for the self-adjoint/skew-adjoint parts of the generator, and it quantifies `how degenerate' a hypocoercive evolution equation is, both for ODEs and for evolutions equations in a Hilbert space. We show that this index characterizes the polynomial decay of the propagator norm for short time and illustrate these concepts for the Lorentz kinetic equation on a torus. Discrete time analogues of the above systems (obtained via the mid-point rule) are contractive, but typically not strictly contractive. For this setting we introduce "hypocontractivity" and an "index of hypocontractivity" and discuss their close connection to the continuous time evolution equations.

This talk is based on joint work with F. Achleitner, E. Carlen, E. Nigsch, and V. Mehrmann.

References:
1) F. Achleitner, A. Arnold, E. Carlen, The Hypocoercivity Index for the short time behavior of linear time-invariant ODE systems, J. of Differential Equations (2023).
2) A. Arnold, B. Signorello, Optimal non-symmetric Fokker-Planck equation for the convergence to a given equilibrium, Kinetic and Related Models (2022).
3) F. Achleitner, A. Arnold, V. Mehrmann, E. Nigsch, Hypocoercivity in Hilbert spaces, J. of Functional Analysis (2025).
 

Anomalies characterize the breaking of a classical symmetry at the quantum level. They play an important role in quantum field theories, and constitute robust observables which appear in various contexts from phenomenological particle physics to black hole microstates, or to classify phases of matter. The anomalies of a d-dimensional QFT are naturally encoded via descent equations into the so-called anomaly polynomial in (d+2)-dimensions. The aim of this seminar is to review the descent procedure, anomaly polynomial, anomaly inflow, and in particular their realisation in M-theory. While this is quite an old story, there has been some more recent developments involving holography that I'll describe if time permits. 

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.