Iwasawa algebras are completions of group algebras for p-adic Lie groups, and have applications for studying the representations of these groups. It is an ongoing project to study the prime ideals, and more generally the two-sided ideals, of these algebras.

In the case of Iwasawa algebras corresponding to a simple Lie algebra with a Chevalley basis, we aim to prove that all non-zero two-sided ideals have finite codimension. To prove this, it is sufficient to show faithfulness of modules arising from highest-weight modules for the corresponding Lie algebra.

I have proved two main results in this direction: firstly, I proved the faithfulness of generalised Verma modules over the Iwasawa algebra. Secondly, I proved the faithfulness of all infinite-dimensional highest-weight modules in the case where the Lie algebra has type A. In this talk, I will outline the methods I used to prove these cases.

Recent joint work with Konstantin Ardakov has been devoted to classifying equivariant line bundles with flat connection on the Drinfeld p-adic half-plane defined over F, a finite extension of Q_p, and proving that their global sections yield admissible locally analytic representations of GL_2(F) of finite length. In this talk we will discuss this work and invite reflection on how it might be extended to equivariant vector bundles with connection on the p-adic half-plane and, if time permits, to higher dimensional analogues of the half-plane.

Apart from the binomial coefficients which are ubiquitous in many counting problems, the Catalan and Fibonacci sequences seem to appear almost as frequently. There are also well-known interpretations of the Catalan numbers as lattice paths, or as the number of ways of connecting 2n points on a circle via non-intersecting lines. We start by obtaining some identities for sums involving the Catalan sequence. In addition, we use the beautiful binomial transform which allows us to obtain several pretty identities involving Fibonacci numbers, Catalan numbers, and trigonometric sums.

Course Overview
The course gives an introduction to a topic of current interest in Lorentzian geometic analysis and mathematical General Relativity: an approach to nonregular spacetimes based on a “metric” point of view.
 

Learning Outcomes
Becoming acquainted with Lorentzian length spaces, sectional and Ricci curvature bounds for non-regular Lorentzian spaces and the appropriate techniques.
 

Course Synopsis
Lecture 1a: Review of Lorentzian geometry, spaces of constant curvature, causality theory, singularity theorems.
Lecture 1b: Introduction to Lorentzian length spaces, timelike sectional curvature bounds.

Lecture 2a: Optimal transport, timelike Ricci curvature bounds
Lecture 2b: Sobolev calculus for time functions. Literature: [O’N83, KS18, CM20].
 

Reading group: Depending on student’s interest one could discuss the papers [GKS19, AGKS21, ABS22].

References
[ABS22] L. Aké Hau, S. Burgos, and D. A. Solis. Causal completions as Lorentzian pre-length spaces. General Relativity and Gravitation, 54(9), 2022. doi:10.1007/s10714-022-02980-x.
[AGKS21] S. B. Alexander, M. Graf, M. Kunzinger, and C. Sämann. Generalized cones as Lorentzian length spaces: Causality, curvature, and singularity theorems. Comm. Anal. Geom., to appear, 2021. doi:10.48550/arXiv.1909.09575. arXiv:1909.09575 [math.MG].
[CM20] F. Cavalletti and A. Mondino. Optimal transport in Lorentzian synthetic spaces, synthetic timelike Ricci curvature lower bounds and applications. Cambridge Journal of Mathematics, to appear, arXiv:2004.08934 [math.MG], 2020. doi:10.48550/arXiv.2004.08934.
[GKS19] J. D. E. Grant, M. Kunzinger, and C. Sämann. Inextendibility of spacetimes and Lorentzian length spaces. Ann. Global Anal. Geom., 55(1):133–147, 2019. doi:10.1007/s10455-018-9637-x.
[KS18] M. Kunzinger and C. Sämann. Lorentzian length spaces. Ann. Glob. Anal. Geom., 54(3):399–447, 2018. doi:10.1007/s10455-018-9633-1.
[O’N83] B. O’Neill. Semi-Riemannian geometry with applications to relativity, volume 103 of Pure and Applied Mathematics. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983.

Should you be interested in taking part in the course, please send an email to @email  by 10 May 2024. 

We study “bottom-up” models for the collective motion of large groups of animals. Similar models can be encoded into (micro)robotic matter, capable of sensing light and processing information. Agents are endowed only with visual sensing and information processing. We study a model in which moving agents reorientate to maximise the path-entropy of their visual environment over paths into the future. There are general arguments that principles like this that are based on retaining freedom in the future may confer fitness in an uncertain world. Alternative “top-down” models are more common in the literature. These typically encode coalignment and/or cohesion directly and are often motivated by models drawn from physics, e.g. describing spin systems. However, such models can usually give little insight into how co-alignment and cohesion emerge because these properties are encoded in the model at the outset, in a top-down manner. We discuss how our model leads to dynamics with striking similarities with animal systems, including the emergence of coalignment, cohesion, a characteristic density scaling anddifferent behavioural phenotypes. The dynamics also supports a very unusual order-disorder transition in which the order (coalignment) initially increases upon the addition of sensory or behavioural noise, before decreasing as the noise becomes larger.