*The SMRI seminar series gives visitors and staff members the opportunity to explain the context and aims of their work. These research talks cover any field in the mathematical sciences, and should be presented in a way that is understandable and interesting to a broad audience.* *Seminar information and recordings can be found below and in the SMRI Seminar YouTube playlist.*

##### SMRI Seminar ‘Deligne-Lusztig varieties or irregular connections’

David Treumann, Boston College

**Thursday 19 October 2023**

*Abstract:* I will give an introduction to Deligne-Lusztig theory, and a second introduction to the theory of irregular singularities of linear ODEs, and make some comparisons. Deligne-Lusztig theory organizes most of the irreducible characters of a finite group G of Lie type of into “series,” that are indexed by conjugacy classes of maximal abelian subgroups T of G. The representations in one series are those that appear in the cohomology of an F_p-bar-variety X equipped with an action of the finite group G x T. A basic result of Deligne and Lusztig is “orthogonality”, which tells e.g. that representations in the series corresponding to T are different from representations in the series corresponding to T-prime, when T is not conjugate to T-prime. It is proved by analyzing a stratification of the quotient (X times X-prime)/G. I will explain how the varieties X and (X times X-prime)/G, and this stratification, arise as moduli spaces of constructible sheaves on a topological annulus. They have a lot in common with moduli spaces of connections on C^* with irregular singularities at zero and infinity.

##### SMRI Seminar ‘A quick introduction to random matrices and extreme gap problems’

**Renjie Feng**, SMRI

**Thursday 21 September 2023**

*Abstract:* Random matrices are studied in various mathematical areas, including statistics, high-energy physics, statistical physics, number theory, (quantum) information theory, numerical analysis, integrable systems, string theory, and more. I will first introduce two types of random matrices and discuss classical results such as the semicircle law and the Tracy-Widom law. Then I will provide several examples in statistical physics and representation theory where the Tracy-Widom law surprisingly emerges. Finally, I will present our recent findings regarding extreme gap problems in classical random matrices and propose several conjectures.

##### SMRI Seminar ‘Stability problems in general relativity’

Zoe Wyatt, University of Cambridge

**Thursday 31 August 2023**

*Abstract:* Einstein’s theory of general relativity makes spectacular predictions, like gravitational waves, about our universe. For the mathematician, the analysis of the hyperbolic Einstein equations is one of the most powerful ways to understand conceptual questions of the theory. In this talk, I will explain some of the contributions of mathematics to general relativity, highlighting a recent joint work showing the stability of Kaluza-Klein spacetimes. These are important models in supergravity and their stability is connected to claims of Penrose and Witten. Watch the recording.

##### SMRI Seminar ‘Minimal Lagrangian surfaces of high genus in CP2’

Franz Pedit, University of Massachusetts Amherst

**Thursday 24 August 2023**

*Abstract:* The study of properties of surfaces in space has historically been a fertile ground for advances in topology, analysis, geometry, Lie theory, and mathematical physics. The most important surface classes are those which arise form variational problems, for example, minimal surfaces which are critical points of the area functional. The Euler Lagrange equations are PDEs which serve as model cases for developments in geometric analysis. Often these equations exhibit large (sometimes infinite dimensional) symmetry groups which puts the theory into the realm of integrable systems, that is, PDEs which allow for an infinte hierarchy of conserved quantities. This theory has been studied extensively over the past 40 years and led to significant advances in the classification of (minimal, constant mean curvature, Willmore etc.) surfaces of genus one. The higher genus case has been more illusive and examples are usually constructed using non-linear perturbation theory and gluing techniques.

In this talk I will explain how one can use ideas from integrable systems to construct examples of high genus minmal Lagrangian surfaces without recourse to hard analysis.

This approach is more explicit than PDE existence results and one is able to obtain more quantitative information about the constructed examples, for instance, asymptotic area/energy estimates. I will also give a brief overview of the historical developments and the significance of minimal Lagrangian surfaces in mathematical physics. Watch the recording.

##### SMRI Seminar ‘Conformal Bach flow’

Peng Lu, University of Oregon

**Thursday 10 August 2023**

*Abstract:* We introduce conformal Bach flow and establish its well-posedness on closed manifolds. We also obtain its backward uniqueness. To give an attempt to study the long-time behavior of conformal Bach flow, assuming that the curvature and the pressure function are bounded, global and local Shi’s type $L^2$-estimate of derivatives of curvatures are derived. To make the talk more accessible, we will spend some time to survey on high order parabolic curvature flow. This is a joint work with Jiaqi Chen of Xiamen University and Jie Qing of UCSC. Watch the recording.

##### SMRI Seminar ‘Nonlocal Aggregation-Diffusion Equations: fast diffusion and partial concentration’

Jose Carrillo, University of Oxford

**Thursday 20 July 2023**

*Abstract:* We will discuss several recent results for aggregation-diffusion equations related to partial concentration of the density of particles. Nonlinear diffusions with homogeneous kernels will be reviewed quickly in the case of degenerate diffusions to have a full picture of the problem. Most of the talk will be devoted to discuss the less explored case of fast diffusion with homogeneous kernels with positive powers. We will first concentrate in the case of stationary solutions by looking at minimisers of the associated free energy showing that the minimiser must consist of a regular smooth solution with singularity at the origin plus possibly a partial concentration of the mass at the origin. We will give necessary conditions for this partial mass concentration to and not to happen. We will then look at the related evolution problem and show that for a given confinement potential this concentration happens in infinite time under certain conditions. We will briefly discuss the latest developments when we introduce the aggregation term. This talk is based on a series of works in collaboration with M. Delgadino, J. Dolbeault, A. Fernandez, R. Frank, D. Gomez-Castro, F. Hoffmann, M. Lewin, and J. L. Vazquez.

Watch the recording.

##### SMRI Seminar ‘Subgroups of inverse monoids via the geometry of their Cayley graphs’

Robert Gray, University of East Anglia

**Thursday 4 May 2023**

*Abstract:* In the 1960s Higman was able to characterize the finitely generated subgroups of finitely presented groups, that is, groups defined using a finite set of generators and finite set of defining relations. His result, which is called the Higman Embedding Theorem, is a key result in combinatorial group theory which makes precise the connection between group presentations and logic. In this talk I will present a result of a similar flavour, proved in recent joint work with Mark Kambites (Manchester), in which we characterise the groups of units of inverse monoids defined by presentation where all the defining relators are of the form w=1. I will explain what an inverse monoid is, the motivation for studying this class of inverse monoids, and also outline some of the geometric ideas that we developed in order to prove our results. Watch the recording.

*Speaker bio*: Robert Gray is a Reader in Pure Mathematics and an EPSRC Research Fellow at the University of East Anglia in the UK. His research lies at the interface of algebra, logic, and theoretical computer science. A central theme in his recent research has been the study of certain fundamental algorithmic questions for infinite groups, monoids and inverse semigroups, using methods from infinite combinatorics, topology, geometry and theoretical computer science.

##### SMRI Seminar ‘Unexpected Behaviour in Dilute Granular Materials’

Jonathan James Wylie, City University of Hong Kong

**Thursday 27 April 2023**

*Abstract:* The phrase ‘granular material’ is used to describe a large number of discrete solid, macroscopic particles that lose energy whenever the particles collide. One might naively imagine that such systems would exhibit similar behaviour to traditional fluid and solid mechanics. However, we present two problems that superficially appear to be extremely simple but yield surprisingly rich dynamics that have no analogue in traditional mechanics. Firstly, we consider a dilute stream of particles that collides with an oblique planar wall. Secondly, we show several surprising phenomena that occur in an extremely simple system of a single frictionless, inelastic, spherical particle falling under gravity through a symmetric funnel. Watch the recording.

*Speaker bio*: Jonathan Wylie obtained his PhD and was subsequently awarded a Junior Research Fellowship from King’s College, Cambridge. He then held research fellow positions in Woods Hole Oceanographic Institution and the University of Toronto before joining the City University of Hong Kong. His research interests include fluid mechanics, granular materials, ion transport and the mathematical modelling of geophysical systems.

##### SMRI Seminar ‘The unreasonable effectiveness of mathematics in large scale deep learning’

Greg Yang, Microsoft Research Lab

**Thursday 23 March 2023**

*Abstract:* Recently, the theory of infinite-width neural networks led to the first technology, mu Transfer, for tuning enormous neural networks that are too expensive to train more than once. For example, this allowed us to tune the 6.7 billion parameter version of GPT-3 using only 7% of its pretraining compute budget, and with some asterisks, we get a performance comparable to the original GPT-3 model with twice the parameter count. In this talk, I will explain the core insight behind this theory. In fact, this is an instance of what I call the *Optimal Scaling Thesis*, which connects infinite-size limits for general notions of “size” to the optimal design of large models in practice, illustrating a way for theory to reliably guide the future of AI. I’ll end with several concrete key mathematical research questions whose resolutions will have incredible impact on how practitioners scale up their NNs. Watch the recording.

*Speaker bio:* Greg Yang is a researcher at Microsoft Research in Redmond, Washington. He joined MSR after he obtained Bachelor’s in Mathematics and Master’s degrees in Computer Science from Harvard University, respectively advised by ST Yau and Alexander Rush. He won the Hoopes prize at Harvard for best undergraduate thesis as well as Honourable Mention for the AMS-MAA-SIAM Morgan Prize, the highest honour in the world for an undergraduate in mathematics. He gave an invited talk at the International Congress of Chinese Mathematicians 2019.

##### SMRI Seminar ‘Generalized sparse grid methods and applications’

Michael Griebel, University of Bonn

**Thursday 8 March 2023**

*Abstract:* High-dimensional problems appear in various mathematical models. Their numerical approximation involves the well-known curse of dimension, which renders any direct discretization obsolete. One approach to circumvent this issue, at least to some extent, is the use of generalized sparse grid methods, which can exploit additional smoothness properties if present in the underlying problem. In this talk, we will discuss the main principles and basic features of generalized sparse grids and show their application in such diverse areas as econometrics, fluid dynamics, quantum chemistry, uncertainty quantification and machine learning.