## Upcoming Seminars

To receive notification of current SMRI seminars and other relevant seminars at the University of Sydney and affiliate organisations, please subscribe to the weekly seminar email update.

SMRI Afternoon Teas take place on Thursday afternoons at 2pm on the Quadrangle Terrace, accessed through A14-04-L4.36 (in the SMRI Common Room in case of bad weather).

## SMRI Seminar

“**State-space models as graphs**“, Victor Elvira, University of Edinburgh**Time and** **date**: 13:00–14:00 AEDT, 29 February 2024**Location**: Law Annex Lecture Theatre 026 and online (Zoom registration link)**Abstract**: Modeling and inference in multivariate time series is central in statistics, signal processing, and machine learning. A fundamental question when analyzing multivariate sequences is the search for relationships between their entries (or the modeled hidden states), especially when the inherent structure is a directed (causal) graph. In such context, graphical modeling combined with sparsity constraints allows to limit the proliferation of parameters and enables a compact data representation which is easier to interpret in applications, e.g., in inferring causal relationships of physical processes in a Granger sense. In this talk, we present a novel perspective consisting on state-space models being interpreted as graphs. Then, we propose novel algorithms that exploit this new perspective for the estimation of the linear matrix operator and also the covariance matrix in the state equation of a linear-Gaussian state-space model. Finally, we discuss the extension of this perspective for the estimation of other model parameters in more complicated models.

## Perspectives on Mathematics Seminar

**“Important Questions**“, presented by Francis Su**Time and** **date**: 14:00–16:00 AEDT, 1 March 2024**Location**: Law Annex Seminar Room 346**Abstract**: In this first session to open the Perspectives Seminar, we will establish important questions we would like to tackle over the course of the term around the humanity, culture, and communication of mathematics. Meet others interested in exploring these topics, hear the organisers talk about the goals of the seminar, and participate in interactive activity and discussion. Suggested readings will enhance one’s experience in the seminar.

### About Our Seminars

SMRI hosts a number of seminar series, and recordings of most SMRI seminars are available on the **SMRI YouTube channel**. The main streams include the following:

**SMRI Seminars**

This 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 under “Past Seminars” below and in the SMRI Seminar YouTube playlist.

To receive notifications of upcoming seminars in this series (excludes other seminar series), please subscribe to the weekly SMRI Seminar email update.

**SMRI What is … ? Seminar**

Each talk in this series is about an idea, concept or method that the speaker has found surprising, useful or intriguing, and which they would like to share with colleagues and students. The talk answers a question of the form “What is…?” and is directed at a broad audience of non-experts and experts alike. There is ample time for discussion, comments and questions. This talk may also serve as a prelude to a more technical talk in a specialised seminar series.

Seminar information and recordings can be found under “Past Seminars” below and in the ‘What is…?’ Seminar YouTube playlist.

**Algebra Seminars** co-presented by SMRI

Algebra Seminars (2022 onwards), and former SMRI Algebra and Geometry Online (SAGO) seminars are specialised research talks by international researchers in algebra and geometry.

Seminar information and recordings can be found under “Past Seminars” below and in the SAGO YouTube playlist (2020-2021 talks).

Related talks will now be hosted under the University of Sydney Algebra Seminar series, for which SMRI occasionally organises online talks.

**One School Seminar**

This seminar series aims to facilitate sharing and learning about the research of our fellow staff members. Early and mid-career researchers will present a broader context of their work which should be accessible and relatable to the entire School community. Seminars will be held in-person, followed by a friendly gathering and refreshments in the SMRI common room or out on the terrace (weather permitting). Everyone is warmly invited.

**Perspectives on Mathematics: Its Humanity, Culture and Communication** (SMRI Special Semester, Semester One 2024)

Perspectives on Mathematics website

Organised by Francis Su and Stephan Tillmann, this interactive seminar series will explore and illuminate the human side of mathematics—the culture(s) of mathematics, how we interact and do research in it, who we include, and the impact of how we communicate maths to students and the broader public. Discussions and invited presentations will help participants envision how they can improve the culture, practices, and communication around maths to better serve their communities, as well as connect with one another.

The seminar comprises five two-week blocks centred on a theme, and will meet 14:00**–**16:00 AEST on Fridays during those weeks (except for the 2nd theme). Each block is followed by an *intermezzo*, a “free week” that allows participants to explore the theme further or to attend aligned events. Visit the website for suggested readings, key presenters and session times.

**Modular Representation Theory** (SMRI Course, Semester One 2023)

Course organised by Geordie Williamson and Chris Hone: **Modular Representation Theory website**.

Seminar information and recordings can be found below and in the **YouTube playlist**.

**D-Modules and Representation Theory** (SMRI Course, Semester Two 2023)

Semester Two, 2023 course presented by Dragan Milicic, Geordie Williamson and others: D-Modules website

D-modules provide an algebraic language for studying systems of linear partial differential equations. In some sense, the idea goes back to Riemann: rather than studying some complicated function directly, study the equations which it satisfies and try relate these equations on different spaces. D-modules have found major applications in representation theory, algebraic analysis, algebraic geometry and mathematical physics.

This course will introduce the basics of the theory of algebraic D-modules (following some famous notes of Bernstein). The localisation theorem of Beilinson and Bernstein will be discussed in detail. We will then move on to other applications in representation theory. Namely, we hope to explain how certain old results of Harish-Chandra and Langlands become quite transparent in the language of D-modules.

Seminar information and recordings can be found below and in the D-Modules YouTube playlist.

**Machine Learning for the Working Mathematician** (SMRI Course, Semester One 2022)

**Machine Learning for the Working Mathematician website**

The Machine Learning for the Working Mathematician seminar is an introduction to ways in which machine learning (and in particular deep learning) has been used to solve problems in mathematics, organised by Joel Gibson, Georg Gottwald, and Geordie Williamson.

We aim for a toolbox of simple examples, where one can get a grasp on what machine learning can and cannot do. We want to emphasise techniques in machine learning as tools that can be used by working mathematics researchers, rather than a source of problems in themselves. The first six weeks or so will be introductory, and the second six weeks will feature talks from experts on applications.

View the **YouTube playlist** and the detailed course overview.

**Mathematical Challenges in AI** (SMRI Course, Semester Two 2023)

Mathematical Challenges in AI website

This series is the successor of Machine Learning for the Working Mathematician.

The main focus will be to explore the mathematical problems that arise in modern machine learning. For example, we aim to cover:

1) Mathematical problems (e.g. in linear algebra and probability theory) whose resolution would assist the design, implementation and understanding of current AI models.

2) Mathematical problems or results resulting from interpretability of ML models.

3) Mathematical questions posing challenges for AI systems.

Our aim is to attract interested mathematicians to what we see as a fascinating and important source of new research directions.

View the YouTube playlist.

###### Other SMRI Courses

Visitors regularly give lecture series and mini courses at SMRI.

**Hilbert schemes**(September–December 2020).**Speakers:**Anthony Henderson, Emily Cliff, Joe Baine, Anthony Licata, Joshua Ciappara, Peter McNamara**Langlands correspondence and Bezrukavnikov’s equivalence**(March 2019–July 2020, recordings April–July 2020).**Speaker:**Geordie Williamson

###### Other seminar series organised by SMRI & staff in the School of Mathematics and Statistics

- Informal Friday Seminars: the IFS is a working group meeting at SMRI
- Sydney Dynamics Group seminars
- Asia-Pacific Analysis and PDE Seminar series
- Australian Geometric Topology Webinar series
- (GT)
^{2}Graduate Talks in Geometry and Topology at the University of Sydney, supported by MATRIX and AMSI - Recent Progress in Mathematics and Statistics
- Seminars of the School of Mathematics and Statistics can be viewed on the School’s upcoming Seminars & Conferences page

###### AMSI-MSRI Winter School 2022

SMRI Director Geordie Williamson was an invited speaker at the AMSI-MSRI Winter School, hosted over two weeks (20 June–1 July 2022) at the University of Queensland. He presented five lectures on Kazhdan-Lusztig Polynomials: Representation, Geometry and Combinatorics.

The **Winter School full playlist** can be viewed on YouTube.

### Past Seminars

**2023 Seminars**

Abstracts and links to YouTube recordings (One School seminars & board talks are not recorded).

**David Treumann** (Boston College): Deligne-Lusztig varieties or irregular connections

**Date**: Thursday 19 October**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.

**Lindon Roberts** (University of Sydney): Blackbox Optimisation Algorithms

One School Seminar

**Date**: Thursday 12 October

**Abstract:** Numerical optimisation—being able to find maxima/minima of functions—is an important part in numerical analysis, with many applications across different disciplines. This problem becomes much harder if only limited information about the functions is available—no analytic expression and/or no exact function values available, for example. In this case, specialised techniques known as ‘blackbox optimisation’ must be used. In this talk, I will give an overview of some useful techniques for blackbox optimisation and some recent work on improving the scalability of these techniques using tools from random matrix theory.

**Renjie Feng** (SMRI): A quick introduction to random matrices and extreme gap problems

**Date**: Thursday 21 September**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.

**Zoe Wyatt** (University of Cambridge): Stability problems in general relativity

**Date**: Thursday 31 August**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.

**Franz Pedit** (University of Massachusetts, Amherst): Minimal Lagrangian surfaces of high genus in CP2

**Date:** Thursday 24 August

**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.

**Iva Halacheva** (Northeastern University): What is a cactus group?

‘What is …?’ Seminar

**Date**: Thursday 3 August

**Abstract**: The braid group is a classical algebraic object with an intuitively natural presentation via stacking pictures of braided strands. One point of view which makes them interesting to topologists is the interpretation of braid groups as the fundamental groups of certain configuration spaces. Braid groups also play a central role in representation theory through the Yang-Baxter equation, where they capture the symmetries of quantum group representations. The cactus group is a close relative of the braid group whose properties are yet to be fully explored. Cactus groups can also be viewed as the fundamental groups of interesting topological spaces and have recently been linked to combinatorial structures associated with quantum groups. I will describe the construction of the cactus group and outline some of the settings in which it appears.

**Peng Lu** (University of Oregon): Conformal Bach flow

**Date**: Thursday 10 August

**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.

**Jose Carrillo** (University of Oxford): Nonlocal Aggregation-Diffusion Equations: fast diffusion and partial concentration

**Date**: Thursday 20 July

**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.

**Bob Rink** (Vrije Universiteit Amsterdam): What is the parametrization method in dynamical systems?

‘What is …?’ Seminar

**Date**: Thursday 11 May

**Abstract:** The parametrization method is a tool to compute invariant manifolds in dynamical systems, such as periodic orbits, (un-)stable manifolds, slow manifolds and invariant tori. The idea behind the method is simple: it works by (algorithmically) finding an embedding of the invariant manifold together with a representation of its dynamics in a coordinate chart. De La Llave et al realized that the method can nicely be combined with ideas from rigorous numerics, to provide computer-assisted proofs for the existence of invariant manifolds. Others, including myself, have used the method to compute high-precision approximations of the dynamics on the invariant manifolds. I will discuss both approaches, starting with the basics and finishing with an unpublished result on high-order phase reduction.

**Speaker bio**: Bob Rink is a professor of mathematical analysis at the Vrije Universiteit Amsterdam, The Netherlands. He obtained his PhD in 2003 from Utrecht University, after which he was an EPSRC postdoctoral fellow at Imperial College London. He came to the Vrije Universiteit Amsterdam in 2007, where he became full professor in 2016. His research is in various directions of dynamical systems, with a focus on network dynamics and bifurcation theory.

**Robert Gray** (University of East Anglia): Subgroups of inverse monoids via the geometry of their Cayley graphs

**Date**: Thursday 4 May

**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.

**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.

**Jonathan James Wylie** (City University of Hong Kong): Unexpected Behaviour in Dilute Granular Materials

**Date**: Thursday 27 April

**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.

**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.

**Sándor Kovács** (University of Washington): What is the parametrization method in dynamical systems?

‘What is …?’ Seminar

**Date**: Thursday 20 April

**Abstract:** Max Noether said that algebraic curves were created by God and algebraic surfaces by the Devil. Unfortunately, that description seems to be also valid for the moduli theory of these objects respectively. I will recall one of the first obstacles one faces when trying to extend the basic results of the moduli theory of curves to that of surfaces and then discuss how one may resolve the arising issue. Time permitting I will also explain the various stability conditions this problem and its resolution led to.

**Speaker bio**: Sándor Kovács is a Professor of Mathematics at the University of Washington. He received his BS degree at Eötvös University in his native country of Hungary, and his PhD at the University of Utah. He held positions at MIT and the University of Chicago before moving to the University of Washington. He received the National Science Foundation’s Faculty Early Career Development Award, the American Mathematical Society’s Centennial Research Fellowship, an Alfred P. Sloan Foundation Research Fellowship, and two Simons Foundation Fellowships, one of which he is currently holding. He is a Fellow of the American Mathematical Society.

**Nathan Duignan** (University of Sydney): Perspectives on Dynamical Systems and Their Application to Nuclear Fusion

One School Seminar

**Date**: Thursday 6 April

**Abstract:** Historically, research of dynamical systems involved finding quantitative, explicit solutions to the defining differential equations. This was an almost purely analytic perspective of dynamical systems. Over the past century, many fields of Mathematics have brought a new perspective on dynamical systems. With these new perspectives has come new fruits; insights and qualitative information about solutions to the system. In this talk I will show how a combination of a geometric, topological, and algebraic perspective of dynamical systems has allowed me (and friends) to make new progress toward nuclear fusion confinement.

**Jie Yen Fan** (University of Sydney): Mimicking: martingales with matching marginals

One School Seminar

**Date**: Thursday 30 March

**Abstract**: Motivated by questions from finance, we are interested in constructing new processes from existing ones while preserving certain desired properties. In particular, starting from a martingale, we construct new martingales that have the same marginal distributions as the original process. We call this mimicking. This allows us to develop alternative (and hopefully better) models for asset price while retaining the (European) option prices. In this talk, I will give an overview of mimicking and some examples

**Greg Yang** (Microsoft Research Lab): The unreasonable effectiveness of mathematics in large scale deep learning

**Date**: Thursday 23 March

**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.

**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.

**Simon Foucart** (Texas A&M University): Nonstatistical Learning Theory: the View from Optimal Recovery

Applied Mathematics Seminar

**Date**: Thursday 2 March

**Abstract:** For a function observed through point evaluations, is there an optimal way to recover it or merely to estimate a dependent quantity? I will give affirmative answers to variations of this data-focused question, especially under the assumption that the function belongs to a model set defined by approximation capabilities.

In fact, I will uncover computationally implementable linear recovery maps that are optimal in the worst-case setting.

I will present some recent and ongoing works extending the theory in several directions, with particular emphasis put on observations that are inexact—adversarially or randomly.

**Michael Griebel** (University of Bonn): Generalized sparse grid methods and applications

**Date**: Wednesday 8 March

**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.

**Marc Burger** (ETH Zürich): On the Real Spectrum Compactification of Character Varieties

Algebra Seminar

**Date**: Friday 10 March

**Abstract:** For a finitely generated group F and a real algebraic semisimple group G, the set of conjugacy classes of G-representations of F is naturally a real semi algebraic set, called the G-character “variety” of F. For instance when F is the fundamental group of a closed surface then, depending on G, certain connected components of the G-character variety of F consist entirely of injective representations with discrete image: these are the so called “higher Teichmueller spaces”.

The real spectrum of areal algebraic set provides then a compactification of all its semialgebraic subsets and leads to interesting compactifications of higher Teichmueller spaces.

I will describe some of the properties of these compactifications and relate them to other known compactifications.

This is part of an ongoing project with Alessandra Iozzi, Anne Parreau and Beatrice Pozzetti.

**D-Modules and Representation Theory**: Semester Two 2023 SMRI Course

Links to YouTube recordings

###### Lecture One: Introduction

**Date**: Friday 11 August**Presenter**: Geordie Williamson

###### Lecture Two: Sheaves of differential operators

###### Lecture Three: Connections and Functors

**Date**: Friday 25 August**Presenter**: Geordie Williamson

###### Lecture Four: Pullbacks, lefts and rights

**Date**: Friday 1 September

Part One presented by Geordie Williamson

Part Two presented by Chris Hone

###### Lecture Five: Borel-Weil-Bott and Localisation

**Date**: Friday 8 September**Presenter**: Dragan Milicic

###### Lecture Six: The Localisation Theorem

**Date**: Friday 15 September**Presenter**: Dragan Milicic

###### Lecture Seven: Borel-Weil-Bott and Localisation

###### D-Modules workshop day 1

**Date**: Tuesday 26 September

**Session 1**: Dougal Davis 1

**Abstract**: In these talks, we will explain some very recent results on mixed Hodge modules and the unitary dual of a real reductive Lie group. (With a little luck, the ink will have dried and there will be a preliminary version on the arXiv by the time the workshop starts.) The main idea behind our work is to upgrade Beilinson-Bernstein localisation from D-modules to mixed Hodge modules, following a proposal made by Schmid and Vilonen over 10 years ago. When it applies, this endows everything in sight with a canonical filtration, the Hodge filtration, which we prove has some extremely nice properties, such as cohomology vanishing and global generation. In the context of real groups, we also prove that the Hodge filtration “sees” exactly which representations are unitary. We hope that this will lead to new progress on the very old problem of determining the unitary dual of a real group. We’ll do our best to put this problem in context, explain what our theorems say, and give the main ideas behind the proof.

**Session 2**: Dragan Milicic 1

###### D-Modules workshop day 2

###### D-Modules workshop day 3

**Date**: Thursday 28 September

**Session 1**: Dougal Davis 3

Session 2: Behrouz Taji

Abstract: My aim in this talk is to discuss how various discoveries in the theory of variation of Hodge structures, including Saito’s Hodge modules, can be used to establish a striking geometric fact (originally due to Arakelov, Migliorini and Kovács): Over C* the only smooth projective family of curves of genus higher than 1, or more generally canonically polarized complex manifolds, is the isotrivial one. This talk is roughly based on works of Popa-Schnell and myself joint with Kovács.

###### Lecture Eight: Borel-Weil-Bott Theorem

###### Lecture Nine: Symplectic stuff, Singular support and Holonomicity

###### Lecture Ten

###### Lecture Eleven

###### Lecture Twelve

###### Lecture Thirteen

###### Lecture Fourteen

###### Lecture Fifteen

**Modular Representation Theory**: Semester One 2023 SMRI Course

Links to YouTube recordings

###### Week One

**Date**: Thursday 23 February**Presenter**: Chris Hone

###### Week Two

**Date**: Thursday 2 March**Presenter**: Chris Hone

###### Week Three

**Date**: Wednesday 8 March**Presenter**: Chris Hone

###### Week Four

**Date**: Thursday 16 March**Presenter**: Chris Hone

###### Week Five

**Date**: Thursday 23 March**Presenter**: Geordie Williamson

###### Week Six

**Date**: Thursday 30 March**Presenter**: Chris Hone

###### Week Seven

**Date**: Thursday 6 April**Presenter**: Geordie Williamson

###### Week Eight

**Date**: Thursday 20 April**Presenter**: Geordie Williamson

###### Week Nine

**Date**: Thursday 27 April**Presenter**: Geordie Williamson

###### Week Ten

**Date**: Thursday 4 May**Presenter**: Geordie Williamson

###### Week Eleven

**Date**: Thursday 11 May**Presenter**: Chris Hone

###### Week Twelve

**Date**: Thursday 18 May**Presenter**: Geordie Williamson

###### Week Thirteen

**Date**: Thursday 1 June**Presenter**: Geordie Williamson

###### Week Fourteen

**Date**: Thursday 8 June**Presenter**: Nick Bridger

**Mathematical challenges in AI** 2023 Seminars

Abstracts and links to YouTube recordings

###### Greg Yang (xAI): The unreasonable effectiveness of mathematics in large scale deep learning

**Date**: Wednesday 13 September

**Abstract:** Recently, the theory of infinite-width neural networks led to the first technology, muTransfer, 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. I’ll end with several concrete key mathematical research questions whose resolutions will have incredible impact on the future of AI.

**Speaker bio**: Greg Yang is a researcher at xAI. 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.

###### Sadhika Malladi (Princeton University): Mathematical Views on Modern Deep Learning Optimization

**Date**: Thursday 28 September

**Abstract**: This talk focuses on how rigorous mathematical tools can be used to describe the optimization of large, highly non-convex neural networks. We start by covering how stochastic differential equations (SDEs) provide a rigorous yet flexible model of how deep networks change over the course of training. We then cover how the SDEs yield practical insights into scaling training to highly distributed settings while preserving generalization performance. In the second half of the talk, we will explore the new deep learning paradigm of pre-training and fine-tuning large language models. We show that fine-tuning can be described by a very simplistic mathematical model, and insights allow us to develop a highly efficient and performant optimizer to fine-tune LLMs at scale. The talk will focus on various mathematical tools and the extent to which they can describe modern day deep learning.

###### Neel Nanda (Deep Mind): Mechanistic Interpretability & Mathematics

**Date**: Thursday 12 October

**Abstract:** Mechanistic Interpretability is a branch of machine learning that takes a trained neural network, and tries to reverse-engineer the algorithms it’s learned. First, I’ll discuss what we’ve learned by reverse-engineering tiny models trained to do mathematical operations, eg the algorithm learned to do modular addition. I’ll then discuss the phenomena of superposition, where models spontaneously learn to use the geometry of high-dimensional spaces to use compression schemes and represent and compute more features than they have dimensions. Superposition is a major open problem in mechanistic interpretability, and I’ll discuss some of the weird mathematical phenomena that come up with superposition, some recent work exploring it, and open problems in the field.

###### Paul Christiano (Alignment Research Center): Formalizing Explanations of Neural Network Behaviors

**Date**: Thursday 26 October

**Abstract:** Existing research on mechanistic interpretability usually tries to develop an informal human understanding of “how a model works,” making it hard to evaluate research results and raising concerns about scalability. Meanwhile formal proofs of model properties seem far out of reach both in theory and practice. In this talk I’ll discuss an alternative strategy for “explaining” a particular behavior of a given neural network. This notion is much weaker than proving that the network exhibits the behavior, but may still provide similar safety benefits. This talk will primarily motivate a research direction and a set of theoretical questions rather than presenting results.

###### François Charton (Meta AI): Transformers for maths, and maths for transformers

**Date**: Thursday 23 November

**Abstract:** Transformers can be trained to solve problems of mathematics. I present two recent applications, in mathematics and physics: predicting integer sequences, and discovering the properties of scattering amplitudes in a close relative of Quantum Chromo Dynamics. Problems of mathematics can also help understand transformers. Using two examples from linear algebra and integer arithmetic, I show that model predictions can be explained, that trained models do not confabulate, and that carefully choosing the training distributions can help achieve better, and more robust, performance.

**Machine Learning for the Working Mathematician** 2022 Seminars

Abstracts and links to YouTube recordings.

**Week One**: Basics of Machine Learning

**Geordie Williamson, University of Sydney****Date:** Thursday 24 February

**Description:** Classic problems in machine learning, kernel methods, deep neural networks, supervised learning, and basic examples.**YouTube video**

**Week Two**: What can and can’t neural networks do

**Joel Gibson, University of Sydney****Date:** Thursday 3 March

**Description:** Universal approximation theorem and convolutional neural networks.**YouTube video**

**Week Three**: How to think about machine learning

**Georg Gottwald, University of Sydney****Date:** Thursday 10 March

**Description:** Borrowing from statistical mechanics, dynamical systems and numerical analysis to better understand deep learning.**YouTube video**

**Week Four**: Recurrent Neural Nets; Regularisation

**Joel Gibson & Georg Gottwald**, University of Sydney**Date:** Thursday 17 March

**Regularisation (Georg Gottwald):** **YouTube video****Recurrent Neural Nets (Joel Gibson): YouTube video**

**Week Five**: Geometric Deep Learning

**Geordie Williamson**, University of Sydney**Title:** Geometric Deep Learning; or never underestimate symmetry**Date:** Thursday 24 March

**Week Six**: Geometric Deep Learning II

**Georg Gottwald & Geordie Williamson**, University of Sydney**Titles:** Geometric Deep Learning II; Saliency + Combinatorial Invariance**Date:** Thursday 31 March

**Geometric Deep Learning II (Georg Gottwald): YouTube video****Saliency + Combinatorial Invariance (Geordie Williamson): YouTube video**

**Week Seven**: A simple RL setup to find counterexamples to conjectures in mathematics

**Adam Zsolt Wagner, Tel Aviv University****Date:** Thursday 7 April

**Description:** In this talk we will leverage a reinforcement learning method, specifically the cross-entropy method, to search for counterexamples to several conjectures in graph theory and combinatorics. We will present a very simplistic setup, in which only minimal changes need to be made (namely the reward function used for RL) in order to successfully attack a wide variety of problems. As a result we will resolve several open problems, and find more elegant counterexamples to previously disproved ones.**YouTube video**

**Week Eight**: No seminar

**Bamdad Hosseini, University of Washington****Title:** Perspectives on graphical semi-supervised learning

Seminar cancelled

**Description:** Semi-supervised learning (SSL) is the problem of extending label information from a small subset of a data set to the entire set. In low-label regimes the geometry of the unlabelled set is a crucial aspect that should be leveraged in order to obtain algorithms that outperform standard supervised learning. In this talk I will introduce graphical SSL algorithms that rely on manifold regularization in order to incorporate this geometric information. I will discuss interesting connections to linear algebra and matrix perturbations, kernel methods, and theory of elliptic partial differential equations.

**Week Nine**: Machine learning for optimizing certain kinds of classification proofs for finite structures

**Carlos Simpson, Université Côte d’Azur****Date:** Thursday 28 April

**Description:** We’ll start by looking at the structure of classification proofs for finite semigroups and how to program these in Pytorch. (That could be the subject of the tutorial.) A proof by cuts generates a proof tree—think of solving Sudoku. Its size depends on the choice of cut locations at each stage. This leads to the question of how to choose the cuts in an optimal way. We’ll discuss the Value-Policy approach to RL for this, and discuss some of the difficulties notably in sampling. Then we’ll look at another approach, somewhat more heuristic, that aims to provide a faster learning process with the goal of obtaining an overall gain in time when the training plus the proof are counted together.**YouTube video**

**Week Ten**: A technical history of AlphaZero

**Alex Davies, DeepMind****Date:** Thursday 4 May

**Description:** In 2016 AlphaGo defeated the world champion go player Lee Sedol in a historic 5 game match. In this lecture we will discuss the research behind this system and the innovations that ultimately lead to AlphaZero, which can learn to play multiple board games, including Go, from scratch without human knowledge.**YouTube video**

**Week Eleven**: Learning selection strategies in Buchberger’s algorithm

**Daniel Halpern-Leinster, Cornell University****Date:** Thursday 12 May

**Description:** Studying the set of exact solutions of a system of polynomial equations largely depends on a single iterative algorithm, known as Buchberger’s algorithm. Optimized versions of this algorithm are crucial for many computer algebra systems (e.g. Mathematica, Maple, Sage). After discussing the problem and what makes it challenging, I will discuss a new approach to Buchberger’s algorithm that uses reinforcement learning agents to perform S-pair selection, a key step in the algorithm. In certain domains, the trained model outperforms state-of-the-art selection heuristics in total number of polynomial additions performed, which provides a proof-of-concept that recent developments in machine learning have the potential to improve performance of algorithms in symbolic computation.**YouTube video**

**Week Twelve**: Deep Learning meets Shearlets

**Gitta Kutyniok, Ludwig-Maximilians-Universität and University of Tromsø****Title:** Deep Learning meets Shearlets: Explainable Hybrid Solvers for Inverse Problems in Imaging Science**Date:** Thursday 19 May

**Description:**

Pure model-based approaches are today often insufficient for solving complex inverse problems in medical imaging. At the same time, methods based on artificial intelligence, in particular, deep neural networks, are extremely successful, often quickly leading to state-of-the-art algorithms. However, pure deep learning approaches often neglect known and valuable information from the modeling world and suffer from a lack of interpretability.

In this talk, we will develop a conceptual approach towards inverse problems in imaging sciences by combining the model-based method of sparse regularization by shearlets with the data-driven method of deep learning. Our solvers pay particular attention to the singularity structures of the data. Focussing then on the inverse problem of (limited-angle) computed tomography, we will show that our algorithms significantly outperform previous methodologies, including methods entirely based on deep learning. Finally, we will also touch upon the issue of how to interpret the results of such algorithms, and present a novel, state-of-the-art explainability method based on information theory.**YouTube video**

**Week Thirteen**: Deep learning for sequence modelling

**Qianxiao Li, National University of Singapore****Date:** Thursday 26 May

**Description:** In this talk, we introduce some deep learning based approaches for modelling sequence to sequence relationships that are gaining popularity in many applied fields, such as time-series analysis, natural language processing, and data-driven science and engineering. We will also discuss some interesting mathematical issues underlying these methodologies, including approximation theory and optimization dynamics.

**Week Fourteen**: Searching for Formulas and Algorithms

**Lars Buesing, Columbia University****Title:** Searching for Formulas and Algorithms: Symbolic Regression and Program Induction**Date:** Thursday 2 June

**Description:** In spite of their enormous success as black box function approximators in many fields such as computer vision, natural language processing and automated decision making, Deep Neural Networks often fall short of providing interpretable models of data. In applications where aiding human understanding is the main goal, describing regularities in data with compact formuli promises improved interpretability and better generalization. In this talk I will introduce the resulting problem of Symbolic Regression and its generalization to Program Induction, highlight some learning methods from the literature and discuss challenges and limitations of searching for algorithmic descriptions of data.**YouTube video**

**2022 Seminars**

Abstracts and links to YouTube recordings. One School seminars & board talks are not recorded.

**Alexandru Suciu** (Northeastern University): Topological invariants of groups and tropical geometry** **

**Alexandru Suciu**

**Recent Progress in Mathematics and Statistics** seminar

**Date**: Thursday 9 December

**Abstract:** There are several topological invariants that one may associate to a finitely generated group G – the characteristic varieties, the resonance varieties, and the Bieri-Neumann-Strebel invariants – which keep track of various finiteness properties of certain subgroups of G. These invariants are interconnected in ways that makes them both more computable and more informative. I will describe one such connection, made possible by tropical geometry, and I will provide examples and applications pertaining to complex geometry and low-dimensional topology.

**Francisco Crespo** (Universidad del Bío-Bío): Relative equilibria in the full N-body problem

**Date**: Thursday 1 December

**Abstract:** The full N-body problem addresses the dynamics of N rigid bodies under mutual gravitational interactions. This physical system has powered the fabric of science and especially mathematics for centuries, having a decisive role in developing geometric mechanics, qualitative theory of dynamical systems, or KAM theory. In this talk, we briefly survey this problem and focus on analyzing special solutions called relative equilibria.

After determining the hamiltonian equations of motion, our approach identifies and uses the existence of translational and rotational symmetries of the N-body problem. In particular, we provide very compact equations characterizing relative equilibria solutions, which become linear by fixing the values of the invariants associated with the action of the symmetry group.

In the existing literature, relative equilibria have been classified into Lagrangian and non-Lagrangian, respectively, corresponding to whether the center of mass of all bodies is in the same plane. Our analysis determines what kind of configurations allow for each type of equilibrium and provides necessary conditions for non-Lagrangian equilibria.

**Jeroen Schillewaert** (University of Auckland): Constructing highly regular expanders from hyperbolic Coxeter groups

**Date**: Thursday 17 November

**Abstract:** Given a string Coxeter system (W,S), we construct highly regular quotients of the 1-skeleton of its universal polytope P, which form an infinite family of expander graphs when (W,S) is indefinite and P has finite vertex links. The regularity of the graphs in this family depends on the Coxeter diagram of (W,S). The expansion stems from superapproximation applied to (W,S). This construction is also extended to cover Wythoffian polytopes. As a direct application, we obtain several notable families of expander graphs with high levels of regularity, answering in particular a question posed by Chapman, Linial and Peled positively.

This talk is based on joint work with Marston Conder, Alexander Lubotzky and Francois Thilmany.

**Changfeng Gui** (**University of Texas at San Antonio**): Some New Inequalities in Analysis and Geometry

**Changfeng Gui****Date**: Thursday 17 November

**Abstract:** The classical Trudinger-Moser inequality is a borderline case of Sobolev inequalities and plays an important role in geometric analysis and PDEs in general. Aubin in 1979 showed that the best constant in the Trudinger-Moser inequality can be improved by reducing to one half if the functions are restricted to the complement of a three dimensional subspace of the Sobolev space $H^1$, while Onofri in 1982 discovered an elegant optimal form of Trudinger-Moser inequality on sphere. In this talk, I will present new sharp inequalities which are variants of Aubin and Onofri inequalities on the sphere with or without mass center constraints.

One such inequality, for example, incorporates the mass center deviation (from the origin) into the optimal inequality of Aubin on the sphere, which is for functions with mass centered at the origin. The main ingredient leading to the above inequalities is a novel geometric inequality: Sphere Covering Inequality.

Efforts have also been made to show similar inequalities in higher dimensions. Among the preliminary results, we have improved Beckner’s inequality for axially symmetric functions when the dimension $n=4, 6, 8$. Many questions remain open.

The talk is based on collaborations with Amir Moradifam, Sun-Yung Alice Chang, Yeyao Hu and Weihong Xie.

**Rafał Kulik** (University of Ottawa): Disjoint and sliding blocks estimators for heavy tailed time series

**Rafał Kulik**

**Date**: Thursday 10 November

**Abstract:** Extreme value theory deals with large values and rare events. These large values tend to cluster in case of temporal dependence. This clustering behaviour is widely observed in practice.

I will start with a mild introduction to extreme value theory, discussing probabilistic and statistical issues. This part will be accessible to a broader audience.

Then, I will talk about a more specific problem of statistical theory for cluster functionals and rare events. Two types of estimators are of a primary importance: disjoint and sliding blocks estimators. It has been conjectured that sliding blocks estimators are “better” (to be made precise in the talk). We proved in a recent series of papers that this is not the case and in fact both disjoint and sliding blocks estimators are asymptotically equivalent. This part will be aimed at probabilistic and statisticians.

I will conclude with recent directions in extreme value theory, such as extremes in high dimension, extremes of graphs and networks.

**Matthew Conder,** (University of Auckland): Discrete two-generator subgroups of PSL(2,Q_p)

**Matthew Conder,**

**Date**: Thursday 10 November

**Abstract**: Discrete two-generator subgroups of PSL(2,R) have been extensively studied by investigating their action by Möbius transformations on the hyperbolic plane. Due to work of Gilman, Rosenberger, Purzitsky and many others, there is a complete classification of such groups by isomorphism type, and an algorithm to decide whether or not a two-generator subgroup of PSL(2,R) is discrete.

Here we completely classify discrete two-generator subgroups of PSL(2,Q_p) over the p-adic numbers Q_p by studying their action by isometries on the corresponding Bruhat-Tits tree. We give an algorithm to decide whether or not a two-generator subgroup of PSL(2,Q_p) is discrete, and discuss how this can be used to decide whether or not a two-generator subgroup of SL(2,Q_p) is dense. This is joint work with Jeroen Schillewaert.

**Geordie Williamson**, (Sydney Mathematical Research Institute): What can the working mathematician expect from deep learning?

**Geordie Williamson**,**Colloquium on Recent Progress in Mathematics and Statistics**

**Date**: Wednesday 2 November

**Abstract**: Deep learning (the training of deep neural nets) is a very simple idea. Yet it has led to many striking applications throughout science and industry over the last decade. It has also become a major tool for applied mathematicians. In pure mathematics the impact has so-far been modest. I will discuss a few instances where it has proved useful, and led to interesting results in pure mathematics. I will also reflect on my experience as a pure mathematician interacting with deep learning.

Finally, I will discuss what can be learned from the successful examples that I understand, and try to guess an answer to the question in the title. (Deep learning also raises interesting mathematical questions, but this talk won’t be about this.)

**Garth Tarr**, (University of Sydney): Working with data

**,****Garth Tarr****One School Seminar**

**Date**: Thursday 27 October

In this talk I’ll provide a brief overview of some of my research interests: understanding the stability of feature selection, robust modelling methods and multi-class modelling.

– Is the model you’ve selected uniquely “best”? (Spoiler: probably not)

– How can we develop methods that are resistant to data corruption?

– How can we take advantage of known structures in our data? For example, discover which groups in our data respond similarly to various inputs.

There are great funding opportunities associated with applied research. I’ve established a long-term association with Meat and Livestock Australia, particularly around predicting beef and sheep eating quality, which has spawned many projects. I’ll share some of my experiences of working with industry partners, interdisciplinary collaborators, and PhD students.

**Bregje Pauwels**, (Sydney Mathematical Research Institute): Categories, approximation, representation theory and algebraic geometry

**,****Bregje Pauwels****One School Seminar**

**Date**: Thursday 20 October

**Abstract**: Category theory takes a bird’s eye view of mathematics, allowing mathematicians to spot new patterns and interconnections. Their abstract nature has proved incredibly useful in mathematics, and its applications have reached areas like neuroscience, chemistry, electrical circuits and computer science. In particular, triangulated categories play a central role in every branch of mathematics that uses homological algebra: representation theory, algebraic geometry and stable homotopy theory. Given a metric on a triangulated category, there is a reasonable notion of Fourier series, which we can use to ‘approximate’ objects. This powerful technical tool, while relatively new, has already been used to powerful effect.

In this talk, I will try to convince you that you should use the tool of approximation in triangulated categories. Failing that, I will at least try to convince you that categories are everywhere, and their language is incredibly useful.

**Will Donovan** (Tsinghua University): Homological comparison of resolution and smoothing

**Will Donovan****Algebra Seminar**

**Date**: Friday 23 September

**Abstract:** A singular space often comes equipped with (1) a resolution, given by a morphism from a smooth space, and (2) a smoothing, namely a deformation with smooth generic fibre. I will discuss work in progress on how these may be related homologically, starting with the threefold ordinary double point as a key example.

**Biography:** Will Donovan is currently an associate professor at Yau MSC, Tsinghua University, Beijing. He is also a member of the adjunct faculty at BIMSA, Yanqi Lake, Huairou, Beijing and a visiting associate scientist at Kavli IPMU, University of Tokyo. He received his PhD in Mathematics in 2011 from Imperial College London. His interests are algebraic geometry, noncommutative geometry, representation theory, string theory and symplectic geometry.

**Kari Vilonen**, (University of Melbourne): What is a Hodge module?

**,****Kari Vilonen****SMRI What is…? Seminar**

**Date**: Thursday 15 September

**Abstract**: I will explain what a Hodge module is starting from Poincare.

**Biography:** Kari Vilonen is a professor of mathematics at the University of Melbourne. His research is in geometric aspects of representation theory and the Langlands program. He has also worked on foundational questions on perverse sheaves and D modules including the microlocal point of view.

**Jana de Wiljes** (University of Potsdam): Sequential Bayesian Learning

**Jana de Wiljes****Date**: Thursday 8 September

**Abstract**: In various application areas it is crucial to make predictions or decisions based on sequentially incoming observations and previous existing knowledge on the system of interest. The prior knowledge is often given in the form of evolution equations (e.g., ODEs derived via first principles or fitted based on previously collected data), from here on referred to as model. Despite the available observation and prior model information, accurate predictions of the „true“ reference dynamics can be very difficult.

Common reasons that make this problem so challenging are: ( i ) the underlying system is extremely complex (e.g., highly nonlinear) and chaotic (i.e., crucially dependent on the initial conditions), (ii) the associate state and/or parameter space is very high dimensional (e.g., worst case 10^8) (iii) Observations are noisy, partial in space and discrete in time.

In practice these obstacles are combated with a series of approximations (the most important ones being based on assuming Gaussian densities and using Monte Carlo type estimations) and numerical tools that work surprisingly well in some settings. Yet the mathematical understanding of the signal tracking ability of a lot of these methods is still lacking. Additionally, solutions of some of the more complicated problems that require simultaneous state and parameter estimation (including control parameters that can be understood as decisions/actions performed) can still not be approximated in a computationally feasible fashion. Here we will try to address the first layer of these issues step by step and discuss the next advances that need to be made in these many layered problems. More specifically a stability and accuracy analysis of a family of the most popular sequential data assimilation methods typically used in practice is presented. Then we will discuss how techniques from the world of machine learning can aid to overcome some of the computational challenges

**Kenneth Ascher** (University of California, Irvine): What is a moduli space?

**Kenneth Ascher**

**SMRI What is…? Seminar**

**Date**: Thursday 25 August**Abstract:** Moduli spaces are geometric spaces which parametrize equivalence classes of algebraic varieties. I will discuss the moduli space of algebraic curves equivalently Riemann surfaces) of genus g, and use this example to motivate some interesting questions in higher dimensions.

**Biography:** Kenneth Ascher is an assistant professor in the department of mathematics at the University of California Irvine. His research area is algebraic and arithmetic geometry, with specific focuses on moduli spaces of higher dimensional varieties and applications to questions in arithmetic. He received his PhD in 2017 from Brown University under the direction of Dan Abramovich , and was a postdoctoral fellow at the Massachusetts Institute of Technology and Princeton University.**YouTube video**

**Henri Guenancia** (Paul Sabatier University): On the invariance of plurigenera

**Henri Guenancia****SMRI Mini-course**

**Date**: Friday 26 August

**Abstract:** In this mini-course, I will talk about a celebrated theorem of Yum-Tong Siu asserting that given a smooth projective family f:X->Y of complex manifolds over an irreducible base and given any positive integer m, the dimension of the space of pluricanonical forms H^0(X_y, mK_{X_y}) is independent of Y. After recasting the result in its historical context, I will mention the Ohsawa-Takegoshi extension theorem which plays a central role of the proof. Finally, I will sketch the main steps following Mihai Paun’s streamlined proof of the theorem.

**Hans Boden** (McMaster University): What is a virtual knot?

**Hans Boden****SMRI What is…? Seminar**

**Date**: Tuesday 31 May

**Abstract:** Virtual knots were introduced by Louis Kauffman in 1999 as a completion of classical knot theory in which planarity is no longer required. Virtual knots have been studied using a variety of approaches, including algebra, combinatorics, and geometric methods. They also have strong connections to quantum topology and finite type invariants. This talk will survey some fascinating results that have been established and present also open problems and directions for future research.

**Biography:** Dr Hans U. Boden is a professor of mathematics at McMaster University in Canada. He is visiting the University of Sydney and SMRI from May 17 to June 11. His research interests are on the geometry and topology of manifolds, especially gauge theory and low-dimensional topology. In recent years, his work has focused on developing geometric methods to understand knotting and linking in 3-dimensional manifolds. While in Sydney, he will be working closely with Dr Zsuzsi Dancso on a collaborative project related to the Tait conjectures in knot theory.

**Jonathan Spreer** (University of Sydney): Studying manifolds in Geometric Topology

**Jonathan Spreer****One School****Seminar**

**Date**: Tuesday 3 May

**Abstract:** Manifolds, that is, spaces that locally look like Euclidean space, occur in many settings and fields of research within and outside mathematics. In dimension two, manifolds are called surfaces.

Geometric topology is the study of manifolds. But unlike in most other settings where manifolds occur, we are not primarily interested in their shapes, but in their properties that remain unchanged under continuous deformations.

I will explain how manifolds can be most conveniently represented; go over some methods to study these representations; and demonstrate how these methods can give rise to deterministic algorithms solving fundamental problems in the field.

The flavour of this research highly depends on an integer: the dimension of the manifolds under investigation. In this talk the flavour will be mostly three.

**Pedram Hekmati** (University of Auckland): What is a cohomological field theory?

**Pedram Hekmati**

**SMRI What is…? Seminar**

**Date**: Tuesday 26 April

**Abstract:**

Many interesting invariants in geometry satisfy certain glueing or factorisation conditions, that are often useful when doing calculations. Topological quantum field theories (TQFTs) emerged in the 1980s as an organising structure for invariants that are governed by bordisms. In 2 dimensions, bordisms are surfaces with boundaries and the TQFT has a simple algebraic description. By remembering the diffeomorphisms of the surfaces, one is lead to the notion of a cohomological field theory.

This talk will give an overview of these ideas and be aimed at a broad audience.

**Clara Grazian** (University of Sydney): Finding structures in observations: consistent(?) clustering analysis

**Clara Grazian****Date**: Tuesday 10 May

**Abstract**: Clustering is an important task in almost every area of knowledge: medicine and epidemiology, genomics, environmental science, economics, visual sciences, among others.

Methodologies to perform inference on the number of clusters have often been proved to be inconsistent and introducing a dependence structure among the clusters implies additional difficulties in the estimation process. In a Bayesian setting, clustering in the situation where the number of clusters is unknown is often performed by using Dirichlet process priors or finite mixture models. However, the posterior distributions on the number of groups have been recently proved to be inconsistent.

This seminar aims at reviewing the Bayesian approaches available to perform via mixture models and give some new insights.

**Pedram Hekmati** (University of Auckland): What is a cohomological field theory?

**Pedram Hekmati**

**SMRI What is…? Seminar**

**Date**: Tuesday 26 April

**Abstract:**

Many interesting invariants in geometry satisfy certain glueing or factorisation conditions, that are often useful when doing calculations. Topological quantum field theories (TQFTs) emerged in the 1980s as an organising structure for invariants that are governed by bordisms. In 2 dimensions, bordisms are surfaces with boundaries and the TQFT has a simple algebraic description. By remembering the diffeomorphisms of the surfaces, one is lead to the notion of a cohomological field theory.

This talk will give an overview of these ideas and be aimed at a broad audience.

**Monica Vazirani** (University of California, Davis): From representations of the rational Cherednik algebra to parabolic Hilbert schemes via the Dunkl-Opdam subalgebra

**SMRI Algebra & Geometry Online Seminar**

**Date**: Thursday 14 April

**Abstract**: Young diagrams and standard tableaux on them parameterize irreducible representations of the symmetric group and their bases, respectively. There is a similar story for the double affine Hecke algebra (DAHA) taking periodic tableaux, or for the rational Cherednik algebra (a.k.a. rational DAHA) with appropriate modifications. This construction of the basis makes use of an alternate presentation of the rational DAHA and the basis diagonalizes the action of its Dunkl-Opdam subalgebra. We make use of the combinatorics to construct explicit maps between standard modules parameterized by hooks, thus recovering the BGG resolution of the simple module parameterized by the trivial hook.

We can also describe this simple module using the geometry of parabolic Hilbert schemes of points on plane curve singularities. The “tableau” basis that diagonalizes the Dunkl-Opdam subalgebra is the basis of equivariant homology that comes from torus fixed points.

This is joint work with Eugene Gorsky and José Simental.

**Bio:** Monica Vazirani is a professor at UC Davis. She received her PhD from UC Berkeley, after which she had an NSF postdoc she spent at UC San Diego and UC Berkeley, as well as postdoctoral positions at MSRI and Caltech. Dr. Vazirani’s research interests center on the representation theory of algebras related to the symmetric group and how to express algebraic phenomena via the combinatorics of partitions, tableaux, crystal graphs and parking functions.

**Clément Canonne** (University of Sydney): What is deterministic amplification?

**Clément Canonne****SMRI What is…? Seminar**

**Date**: Tuesday 12 April

**Abstract:** Suppose we want to solve a given task (say, a decision problem) and have a randomised algorithm for it which is correct; but only with some non-trivial probability, for instance .51. We would like to “amplify” this probability of success to an arbitrarily small amount, as close to 1 as possible: how to do this? And, more importantly, how to do this using as little extra randomness as possible?

I will first discuss why one would want to do this, then how to achieve it naively, and — quite surprisingly — how we can do much better than this naive approach using expander graphs.

**Bio:** Clément Canonne is a Lecturer in the School of Computer Science of the University of Sydney; he obtained his Ph.D. in 2017 from Columbia University, before joining Stanford as a Motwani Postdoctoral Fellow, then IBM Research as a Goldstine Postdoctoral Fellow. His main areas of research are distribution testing (and, broadly speaking, property testing) and learning theory; focusing, in particular, on understanding the computational aspects of learning and statistical inference subject to various resource or information constraints.

**Theodore Vo** (Monash** **University): Canards, Cardiac Cycles, and Chimeras

**Theodore Vo**

**Date**: Tuesday 8 March

**Abstract:** Canards are solutions of singularly perturbed ODEs that organise the dynamics in phase and parameter space. In this talk, we explore two aspects of canard theory: their applications in the life sciences and their ability to generate new phenomena.

More specifically, we will use canard theory to analyse a canonical model of the electrical activity in a heart muscle cell. We demonstrate that pathological heart rhythms, called early afterdepolarisations, are canard-induced phenomena. We use this knowledge to explain the rich set of model behaviours, some of which have also been observed in experiments. Then, we explore a new class of canard-induced patterns in reaction-diffusion PDEs which exhibit coexisting domains of mutually synchronised oscillators and complementary domains of decoherent (asynchronous) oscillators.

**Sang-hyun Kim** (Korea Institute for Advanced Study): Optimal regularity of mapping class group actions on the circle

**SMRI Algebra & Geometry Online Seminar**

**Date**: Wednesday 2 March

**Abstract**: We prove that for each finite index subgroup H of the mapping class group of a closed hyperbolic surface, and for each real number r>1 there does not exist a faithful C^r-action (in Hölder’s sense) of H on a circle. For this, we partially determine the optimal regularity of faithful actions by right-angled Artin groups on a circle. (Joint with Thomas Koberda and Cristobal Rivas)

**Bio:** Sang-hyun Kim works at Korea Institute for Advanced Study as Professor in the School of Mathematics since 2019. Before this, he worked at Seoul National University, KAIST, Tufts University, the University of Texas at Austin and MSRI. He received Ph.D in 2007 at Yale University under the supervision of Andrew Casson. His research interests focus on the interplay between geometric group theory and low–dimensional topology, particularly motivated by right-angled Artin groups and manifold diffeomorphism groups. He was selected as the Scientist of the Month by the Korean Ministry of Science and ICT in 2020.

**Ivan Guo** (Monash University): Stochastic Optimal Transport in Financial Mathematics

**Ivan Guo**

**Date**: Tuesday 22 February

**Abstract**: In recent years, the field of optimal transport has attracted the attention of many high-profile mathematicians with a wide range of applications. In this talk we will discuss some of its recent applications in financial mathematics, particularly on the problems of model calibration, robust finance and portfolio optimisation. Classical topological duality results are extended to probabilistic settings, connecting stochastic control problems with non-linear partial differential equations and providing interesting practical interpretations in finance. We will also look at how numerical methods, including machine learning algorithms, can be implemented to solve these problems.

**2021 Seminars**

Abstracts and links to YouTube recordings (board talks not recorded).

**Alexei Davydov** (Ohio University): Condensation of anyons in topological states of matter and structure theory of E_2-algebras

**Alexei Davydov****Date**: Monday 13 December

**Abstract:** The talk will be on the algebraic structure present in both parts of the title. This algebraic story is most pronounced for E2-algebras in the category of 2-vector spaces (also known as braided fusion categories). Condensation gives rise to an equivalence relation on such E2-algebras (Witt equivalence) with sets of equivalence classes exhibiting nice properties, e.g. being abelian groups (Witt groups). The Witt group of braided fusion categories is a countably generated abelian group with the torsion part annihilated by 32.

**Shane Kelly** (**Tokyo Institute of Technology**): Blowup formulas for nilpotent sensitive cohomology theories

**Shane Kelly****SMRI Algebra and Geometry Online seminar**

**Date**: Thursday 2 December

**Abstract:**

This is joint work in progress with Shuji Saito. Many cohomology theories of interest (l-adic cohomology, de Rham cohomology, motivic cohomology, K-theory…) have long exact sequences associated to blowups. Such a property can be neatly encoded in a Grothendieck topology such as the cdh-topology or the h-topology. These topologies appeared in Voevodsky’s proof of the Bloch-Kato conjecture, and more recently in Beilinson’s simple proof of Fontaine’s CdR conjecture, and in Bhatt and Scholze’s work on projectivity of the affine Grassmanian.

A feature of these topologies which often turns out to be a bug is that the associated sheaves cannot see nilpotents. In this talk I will discuss a modification which can see nilpotents, and which still has long exact sequences for many blowups.

**Biography:** Shane Kelly is an Associate Professor at Tokyo Institute of Technology. His research area is algebraic K-theory and motivic homotopy theory, and more recently he is interested in applications to representation theory. His graduate studies were mostly based in Paris; in 2012 he received a PhD jointly from Université Sorbonne Paris Nord and The Australian National University under the joint supervision of Cisinski and Neeman, respectively.

**Jack Morava** (Johns Hopkins University): On the group completion of the Burau representation

**Jack Morava****SMRI Algebra and Geometry Online**

**Date**: Thursday 11 November

**Abstract:**

On fundamental groups, the discriminant

∏_{i≠k}(*z _{i}* –

*z*) ∈

_{k}^{×}

of a finite collection of points of the plane defines the abelianization homomorphism

sending a braid to its number of overcrossings less undercrossings or writhe.

In terms of diffeomorphisms of the punctured plane, it defnes a kind of

`invertible topological quantum field theory’ associated to the Burau representation,

and in the classical physics of point particles the real part of

its logarithmic derivative is the potential energy of a collection of Coulomb

charges, while its imaginary part is essentially the Nambu-Goto area of a

loop of string in the three-sphere.

Its higher homotopy theory defines a very interesting a double-loop map

× Ω^{2}S^{3} → 𝒫*ic*(*S*^{0})

to the category of lines over the stable homotopy ring-spectrum, related

to Hopkins and Mahowald’s exotic (E2) multiplication on classical integral

homology, perhaps related to the `anyons’ of nonclassical physics.

**Biography:** Jack Johnson Morava, of Czech and Appalachian descent, studied under Eldon Dyer and Sir Michael Atiyah, graduating with a PhD from Rice University in 1968, followed by an Academy of Sciences postdoc in Moscow with Yuri Manin and Sergei Novikov. He joined the Johns Hopkins faculty in 1979 where he was involved in the Japan-US mathematical institute, and from roughly 2003 to 2010 he worked half-time on the DARPA FunBio initiative. He retired in 2017 to live with his anthropological linguist wife in Charlottesville, Virginia and get some work done.

**Vladimir Bazhanov** (The Australian National University): Quantum geometry of 3-dimensional lattices

**Vladimir Bazhanov****Date**: Tuesday 26 October

**Abstract**: In this lecture I will explain a relationship between incidence theorems in elementary

geometry and the theory of integrable systems, both classical and quantum. We will

study geometric consistency relations between angles of 3-dimensional (3D) circular

quadrilateral lattices — lattices whose faces are planar quadrilaterals inscribable

into a circle. We show that these relations generate canonical transformations of a

remarkable “ultra-local” Poisson bracket algebra defined on discrete 2D surfaces

consisting of circular quadrilaterals. Quantization of this structure allowed us to

obtain new solutions of the tetrahedron equation (the 3D analog of the Yang-Baxter

equation) as well as reproduce all those that were previously known. These solutions

generate an infinite number of non-trivial solutions of the Yang-Baxter equation and

also define integrable 3D models of statistical mechanics and quantum field theory. The

latter can be thought of as describing quantum fluctuations of lattice geometry.

**Joel Kamnitzer** (**University of Toronto**): Symplectic duality and (generalized) affine Grassmannian slices

**Joel Kamnitzer****SMRI Algebra and Geometry Online seminar**

**Date**: Thursday 21 October

**Abstract:**

Under the geometric Satake equivalence, slices in the affine Grassmannian give a geometric incarnation of dominant weight spaces in representations of reductive groups. These affine Grassmannian slices are quantized by algebras known as truncated shifted Yangians. From this perspective, we expect to categorify these weight spaces using category O for these truncated shifted Yangians.

The slices in the affine Grassmannian and truncated shifted Yangians can also be defined as special cases of the Coulomb branch construction of Braverman-Finkelberg-Nakajima. From this perspective, we find many insights. First, we can generalize affine Grassmannian slices to the case of non-dominant weights and arbitrary symmetric Kac-Moody Lie algebras. Second, we establish a link with modules for KLRW algebras. Finally, we defined a categorical g-action on the categories O, using Hamiltonian reduction.

**Biography:** Joel Kamnitzer is a Professor of Mathematics at the University of Toronto. His research concerns complex reductive groups and their representations, focusing on canonical bases, categorification, and geometric constructions. His 2005 Ph.D. thesis from UC Berkeley focused on the study of Mirkovic-Vilonen cycles in Affine Grassmannians. He received the 2011 Andre Aisenstadt Prize, a 2012 Sloan Research Fellowship, a 2018 E.W.R. Steacie Memorial Fellowship, a 2018 Poincare Chair, and the 2021 Jeffrey-Williams Prize.

**Giles Gardam** (University of Münster): Solving semidecidable problems in group theory

**Giles Gardam****SMRI Algebra and Geometry Online**

**Date**: Tuesday 5 October

**Abstract:** Group theory is littered with undecidable problems. A classic example is the word problem: there are groups for which there exists no algorithm that can decide if a product of generators represents the trivial element or not. Many problems (the word problem included) are at least semidecidable, meaning that there is a correct algorithm guaranteed to terminate if the answer is “yes”, but with no guarantee on how long one has to wait. I will discuss strategies to try and tackle various semidecidable problems computationally with the key example being the discovery of a counterexample to the Kaplansky unit conjecture.

**Biography:** Giles Gardam is a research associate at the University of Münster working in geometric group theory. He studied mathematics and computer science at the University of Sydney, receiving his Bachelor’s degree in 2012, and completed his doctorate at Oxford in 2017. He was then a postdoc at the Technion before starting at Münster in 2019.

**John Greenlees** (Warwick University): The singularity category of C^*(BG) for a finite group G

**John Greenlees****SMRI Algebra and Geometry Online**

**Date**: **Thursday 16 September**

**Abstract:** The cohomology ring H^*(BG) (with coefficients in a field k of characteristic p)

is a very special graded commutative ring, but this comes out much more clearly if one uses

the cochains C^*(BG), which can be viewed as a commutative ring up to homotopy. For

example C^*(BG) is always Gorenstein (whilst this is not quite true for H^*(BG)).

This leads one to study C^*(BG) as if it was a commutative local Noetherian ring, though of course one has to use homotopy invariant methods. The ring C^*(BG) is regular if and only if G is p-nilpoent and so it is natural to look for ways of deciding where C^*(BG) lies on a the spectrum between regular and Gorenstein rings. For a commutative Noetherian ring, one considers the singularity category D_{sg}(R) (the quotient of finite complexes of finitely generated modules by finitely generated projectives). This is trivial if and only if R is regular, so is the appropriate tool. The talk will describe how to define this for C^*(BG), show it has good basic properties and describe the singularity category in the simplest case it is not trivial (when G has a cyclic Sylow p-subgroup).

**Vladimir Bazhanov** (The Australian National University): SMRI course on Yang-Baxter maps

**Vladimir Bazhanov****Date**s: 3, 9, 16 24 & 31 August

**Abstract**: Vladimir Bazhanov will give a short course of lectures on Yang-Baxter maps.

The topic lies on the intersection of the theory of quantum groups and discrete integrable equations.

YouTube playlist (five lectures)

**Hankyung Ko **(Uppsala University): A singular Coxeter presentation

**SMRI Algebra and Geometry Online seminar**

**Date**: Thursday **26 August**

**Abstract:** A Coxeter system is a presentation of a group by generators and a specific form of relations, namely the braid relations and the reflection relations. The Coxeter presentation leads to, among others, a similar presentation of the (Iwahori-)Hecke algebras and the Kazhdan-Lusztig theory, which provides combinatorial answers to major problems in Lie theoretic representation theory and geometry. Extending such applications to the `singular land’ requires the singular version of the Hecke algebra. Underlying combinatorics of the singular Hecke algebra/category comes from the parabolic double cosets and is the first step in understanding the singular Hecke category. In this talk, I will present a Coxeter theory of the parabolic double cosets developed in a joint work with Ben Elias. In particular, I will explain a generalization of the reduced expressions and describe the braid and non-braid relations.

**Lauren Williams** (Harvard University): Schubert polynomials, the inhomogeneous TASEP, and evil-avoiding permutations

**Lauren Williams****SMRI Algebra and Geometry Online**

**Date**: Tuesday 5 October

**Abstract:** The totally asymmetric simple exclusion process (TASEP) was introduced around 1970 as a model for translation in protein synthesis and traffic flow.

It has interesting physical properties (e.g. boundary-induced phase transitions) and also beautiful mathematical properties. The inhomogeneous TASEP is a Markov chain of weighted particles hopping on a ring, in which the probability that two particles interchange depends on the weight of those particles. If each particle has a distinct weight, then we can think of this as a Markov chain on permutations. In many cases, the steady state probabilities can be expressed in terms of Schubert polynomials. Based on joint work with Donghyun Kim.

**Biography:** Lauren Williams is the Robinson professor of mathematics at Harvard and the Seaver Professor at the Harvard Radcliffe Institute. Her research is in algebraic combinatorics. Williams received her BA in mathematics from Harvard College in 2000, and her PhD from MIT in 2005. Subsequently, she was a postdoc at UC Berkeley and Harvard, then a faculty member at UC Berkeley from 2009 to 2018, before returning to Harvard in 2018. She is the recipient of a Sloan Research Fellowship, an NSF CAREER award, the AWM-Microsoft research prize, and is an Honorary member of the London Mathematical Society.

**Xuhua He** (Chinese University of Hong Kong): Tits groups of Iwahori-Weyl groups and presentations of Hecke algebras

**Xuhua He****SMRI Algebra and Geometry Online**

**Date**: **Thursday 5 August**

**Abstract:** Let G(ℂ) be a complex reductive group and W be its Weyl group. In 1966, Tits introduced a certain subgroup of G(ℂ), which is an extension of W by an elementary abelian 2-group. This group is called the Tits group and provides a nice lifting of W.

In this talk, I will discuss a generalization of the notion of the Tits group T to a reductive p-adic group G. Such T, if exists, gives a nice lifting of the Iwahori-Weyl group of G. I will show that the Tits group exists when the reductive group splits over an unramified extension of the p-adic field and will provide an example in the ramified case that such a Tits group does not exist. Finally, as an application, we will provide a nice presentation of the Hecke algebra of the p-adic group G with ln-level structure.

This talk is based on the recent joint work with Ganapathy.

**Speaker bio:** Xuhua He is the Choh-Ming Professor of Mathematics at the Chinese University of Hong Kong. He works in pure mathematics. His research interests include Arithmetic geometry, Algebraic groups, and representation theory. He received his Bachelor’s degree in mathematics from Peking University in 2001 and a Ph.D. degree from MIT in 2005 under the supervision of George Lusztig. He worked as a member at the Institute for Advanced Study during 2005-2006 and Simons Instructor at Stony Brook University during 2006-2008. He worked at the Hong Kong University of Science and Technology during 2008-2014 as an assistant Professor and associated Professor, and then moved to the University of Maryland during 2014-2019 as a Full Professor of Mathematics before joining CUHK in 2019. He received the Morningside Gold Medal of Mathematics in 2013, the Xplorer Prize in 2020 and is an invited sectional speaker of the International Congress of Mathematicians in 2018.

**Shrawan Kumar** (University of North Carolina): Root components for tensor product of affine Kac-Moody Lie algebra modules

**Shrawan Kumar****SMRI Algebra and Geometry Online**

**Date**: **Thursday 5 August**

**Abstract:** Let G(ℂ) be a complex reductive group and W be its Weyl group. In 1966, Tits introduced a certain subgroup of G(ℂ), which is an extension of W by an elementary abelian 2-group. This group is called the Tits group and provides a nice lifting of W.

In this talk, I will discuss a generalization of the notion of the Tits group T to a reductive p-adic group G. Such T, if exists, gives a nice lifting of the Iwahori-Weyl group of G. I will show that the Tits group exists when the reductive group splits over an unramified extension of the p-adic field and will provide an example in the ramified case that such a Tits group does not exist. Finally, as an application, we will provide a nice presentation of the Hecke algebra of the p-adic group G with ln-level structure.

This talk is based on the recent joint work with Ganapathy.

**Ulrich Thiel** (University of Kaiserslautern): Towards the classification of symplectic linear quotient singularities admitting a symplectic resolution

**Ulrich Thiel**

**SMRI Algebra and Geometry Online seminar**

**Date**: Thursday** **8 July

**Abstract:** Over the past two decades, there has been much progress on the classification of symplectic linear quotient singularities V/G admitting a symplectic (equivalently, crepant) resolution of singularities. The classification is almost complete but there is an infinite series of groups in dimension 4 – the symplectically primitive but complex imprimitive groups – and 10 exceptional groups up to dimension 10, for which it is still open. Recently, we have proven that for all but possibly 39 cases in the remaining infinite series there is no symplectic resolution. We have thereby reduced the classification problem to finitely many open cases. We do not expect any of the remaining cases to admit a symplectic resolution. This is joint work with Gwyn Bellamy and Johannes Schmitt.

**Gus Lonergan** (A Priori Investment Management LLC): Geometric Satake over KU

**Gus Lonergan****SMRI Algebra and Geometry Online**

**Date**: Thursday 24 June

**Abstract:** We describe a K-theoretic version of the equivariant constructible derived category. We state (with evidence!) a ‘geometric Satake’ conjecture relating its value on the affine Grassmannian to representations of the Langlands dual group.

**Biography:** Gus Lonergan is the Chief Mathematician at A Priori Investment Management LLC. He was previously a L.E. Dickson Instructor in the mathematics department at the University of Chicago. He is interested in representation theory.

Lonergan completed his PhD at MIT under Roman Bezrukavnikov; the thesis was an attempt to understand mod p phenomena in algebraic topology in the context of geometric representation theory. He attended Cambridge University for his undergraduate and master’s degree. He plays a little music on the side.

**Magdalena Boos** (Ruhr University Bochum): Advertising symmetric quivers and their representations

**Magdalena Boos****SMRI Algebra and Geometry Online**

**Date**: **Thursday 17 June**

**Abstract:** We introduce the notion of a symmetric quiver as provided by Derksen and Weyman in 2002 and discuss symmetric degenerations in this context (which correspond to orbit closure relations in the symmetric representation variety). After motivating our particular interest in the latter by presenting connections to group actions in algebraic Lie Theory, we explain our main questions: are symmetric degenerations induced by “usual” degenerations in the representation variety of the underlying quiver? We look at (counter)examples and recent results. This is joint work with Giovanni Cerulli Irelli.

**Behrouz Taji** (University of Sydney): Root components for tensor product of affine Kac-Moody Lie algebra modules

**SMRI Seminar**

**Date**: Thursday 10 June

**Abstract:** In the 1920s, building on Fermat’s Last Theorem, Mordell conjectured that the set of rational points of any smooth projective curve of genus at least two, over any number field, is finite. In the 1960s, Shafarevich turned this into a purely algebro-geometric conjecture involving families of smooth projective curves. Parshin, Arakelov and Faltings settled this conjecture by showing that the base spaces of such families are in some sense hyperbolic, as long as there is some variation in the algebraic structure of the fibers. Inspired by recent advances in birational geometry, Kebekus and Kovacs conjectured that these hyperbolicity type properties should hold for a vast class of projective families, with fibers of arbitrary dimension. In this talk I will discuss this conjecture and my solution to it. I will also talk about more recent progress in this area, based on a joint work with Kovacs (University of Washington).

**Paul Zinn-Justin** (University of Melbourne): Generic pipe dreams, conormal matrix Schubert varieties and the commuting variety

**Paul Zinn-Justin**

**SMRI seminar**

**Date**: Wednesday 9 June

**Abstract:** In the first part of the talk, I will review Grobner degenerations of matrix Schubert varieties, following Knutson et al and others. I will interpret this in terms of quantum integrable systems and discuss how this construction is not entirely satisfactory and needs to be generalized. Then I will provide such a generalization; we’ll work with the “lower-upper scheme”; one component of which is closely related to the commuting variety. We’ll discuss applications to the latter.

**Uri Onn** (The Australian National University): Base change and representation growth of arithmetic groups

**Uri Onn****SMRI seminar**

**Date**: Tuesday 3 June

**Abstract:** A group is said to have polynomial representation growth if the sequence enumerating the isomorphism classes of finite dimensional irreducible representations according to their dimension is polynomially bounded. The representation zeta function of such group is the associated Dirichlet generating series. In this talk I will focus on representation zeta functions of arithmetic groups and their properties. I will explain the ideas behind a proof of a variant of the Larsen-Lubotzky conjecture on the representation growth of arithmetic lattices in high rank semisimple Lie groups (joint with Nir Avni, Benjamin Klopsch and Christopher Voll) and analogous results for arithmetic groups of type A_2 in positive characteristic (joint with Amritanshu Prasad and Pooja Singla).

**Stephan Tillmann** (University of Sydney): On the space of properly convex projective structures

**Stephan Tillmann****SMRI seminar**

**Date**:** Thursday 20 May**

**Abstract:** This talk will be in two parts.

I will outline joint work with Daryl Cooper concerning the space of holonomies of properly convex real projective structures on manifolds whose fundamental group satisfies a few natural properties. This generalises previous work by Benoist for closed manifolds. A key example, computed with Joan Porti, is used to illustrate the main results.

**Reinout Quispel** (La Trobe University): How to discover properties of differential equations, and how to preserve those properties under discretization

**Reinout Quispel****SMRI Applied Mathematics**

**seminar**

**Date**: **Thursday 5 August**

**Abstract:** This talk will be in two parts.

The first part will be introductory, and will address the question:

Given an ordinary differential equation (ODE) with certain physical/geometric properties (for example a preserved measure, first and/or second integrals), how can one preserve these properties under discretization?

The second part of the talk will cover some more recent work, and address the question:

How can one deduce hard to find properties of an ODE from its discretization?

**Biography:** Reinout Quispel was an undergraduate at the University of Utrecht (the Netherlands) for nine years, before obtaining a PhD on the discretization of soliton theory from Leiden University in 1983. He moved to Australia for a three-year position in 1986 and is still there 35 years later. His main areas of expertise are in integrable systems and in the geometric numerical integration of differential equations. He was awarded the Onsager Professorship and Medal by the Norwegian University of Science and Technology (NTNU) in 2013.

**Shun-Jen Cheng** (Institute of Mathematics, Academia Sinica): Representation theory of exceptional Lie superalgebras

**Shun-Jen Cheng****SMRI Algebra and Geometry Online**

**Date**: 6 May

**Abstract:** In the first half of the talk we shall introduce the notion of Lie superalgebras, and then give a quick outline of the classification of finite-dimensional complex simple Lie superalgebras.

In the second part of the talk we shall discuss the representation theory of these Lie superalgebras and explain the irreducible character problem in the BGG category. Our main focus will be on our computation of the irreducible characters for two of the exceptional Lie superalgebras. This part is based on recent joint works with C.-W. Chen, L. Li, and W. Wang.

**Marcy Robertson** (University of Melbourne): Expansions, completions and automorphisms of welded tangled foams

**Marcy Robertson****SMRI seminar**

**Date**: 22 April

**Abstract:** Welded tangles are knotted surfaces in R^4. Bar-Natan and Dancso described a class of welded tangles which have “foamed vertices” where one allows surfaces to merge and split. The resulting welded tangled foams carry an algebraic structure, similar to the planar algebras of Jones, called a circuit algebra. In joint work with Dancso and Halacheva we provide a one-to-one correspondence between circuit algebras and a form of rigid tensor category called “wheeled props.” This is a higher dimensional version of the well-known algebraic classification of planar algebras as certain pivotal categories.

This classification allows us to connect these “welded tangled foams” to the Kashiwara-Vergne conjecture in Lie theory. In work in progress, we show that the group of homotopy automorphisms of the (rational completion of) the wheeled prop of welded foams is isomorphic to the group of symmetries KV, which acts on the solutions to the Kashiwara-Vergne conjecture. Moreover, we explain how this approach illuminates the close relationship between the group KV and the pro-unipotent Grothendieck–Teichmueller group.

**Biography:** Marcy Robertson obtained her PhD in Algebraic Topology from the University of Illinois at Chicago in 2010. From there she worked in Canada, France and her native United States before settling down in Australia 2015. She is now a Senior Lecturer of Pure Mathematics at the University of Melbourne.

**Yury Stepanyants** (University of Southern Queensland): The asymptotic approach to the description of two dimensional soliton patterns in the oceans

**Yury Stepanyants****SMRI Applied Mathematics**

**seminar**

**Date**: 15 April

**Abstract:** The asymptotic approach is suggested for the description of interacting surface and internal oceanic solitary waves. This approach allows one to describe a stationary moving wave patterns consisting of two plane solitary waves moving at an angle to each other. The results obtained within the approximate asymptotic theory is validated by comparison with the exact two-soliton solution of the Kadomtsev-Petviashvili equation.

The suggested approach is equally applicable to a wide class of non-integrable equations too. As an example, the asymptotic theory is applied to the description of wave patterns in the 2D Benjamin-Ono equation describing internal waves in the infinitely deep ocean containing a relatively thin one of the layers.

**Biography:** Yury Stepanyants graduated in 1973 with the HD of MSc Diploma from the Gorky State University (Russia) and started to work as the Engineer with the Research Radiophysical Institute in Gorky. He proceeded his career with the Institute of Applied Physics of the Russian Academy of Sciences (Nizhny Novgorod) from 1977 to 1997. In 1983 Yury obtained a PhD in Physical Oceanography, and in 1992 he obtained a degree of Doctor of Sciences in Geophysics. After immigration in Australia in 1998, Yury worked for 12 years as the Senior Research Scientist with the Australian Nuclear Science and Technology Organisation in Sydney. Since July 2009 he holds a position of Full Professor at the University of Southern Queensland in Toowoomba, Australia. Yury has published more than 100 journal papers, three books, several review papers and has obtained three patents.

**Adam Piggott **(Australian National University)** **& **Murray Elder **(University of Technology Sydney): Recent progress on the effective Mordell problem

**Adam Piggott**

**Murray Elder****SMRI Seminar Double-Header**

**Date**: 8 April

**Piggott Abstract:** A program of research, started in the 1980s, seeks to classify the groups that can be presented by various classes of length-reducing rewriting systems. We discuss the resolution of one part of the program (joint work with Andy Eisenberg (Temple University), and recent related work with Murray Elder (UTS).

**Elder Biography:** The growth function of a finitely generated group is a powerful and well-studied invariant. Gromov’s celebrated theorem states that a group has a polynomial growth function if and only if the group is ‘virtually nilpotent’. Of interest is a variant called the ‘geodesic growth function’ which counts the number of minimal-length words in a group with respect to some finite generating set. I will explain progress made towards an analogue of Gromov’s theorem in this case.

I will start by defining all of the terms used in this abstract (finitely generated group; growth function; virtual property of a group; nilpotent) and then give some details of the recent progress made. The talk is based on the papers arxiv.org/abs/1009.5051, arxiv.org/abs/1908.07294 and arxiv.org/abs/2007.06834 by myself, Alex Bishop, Martin Brisdon, José Burillo and Zoran Šunić.

**Jared M. Field** (University of Melbourne): Gamilaraay Kinship Dynamics

**Jared M. Field****SMRI Applied Mathematics**

**seminar**

**Date**: 18 March

**Abstract:** Traditional Indigenous marriage rules have been studied extensively since the mid 1800s. Despite this, they have historically been cast aside as having very little utility. Here, I will walk through some of the interesting mathematics of the Gamilaraay system and show that, instead, they are in fact a very clever construction.

Indeed, the Gamilaraay system dynamically trades off kin avoidance to minimise incidence of recessive diseases against pairwise cooperation, as understood formally through Hamilton’s rule.

**Biography:** Jared Field completed his undergraduate studies at the University of Sydney in Mathematics and French literature, before reading for a DPhil in Mathematical Biology at Balliol College, Oxford. He is now a McKenzie Fellow in the School of Mathematics and Statistics at the University of Melbourne, with broad interests at the intersection of mathematics, evolution and ecology.

**Monica Nevins** (University of Ottawa): Recent progress on the effective Mordell problem

**Monica Nevins****SMRI Algebra and Geometry Online**

**Date**: 26 February

**Abstract:** The theory of complex representations of p-adic groups can feel very technical and unwelcoming, but its central role in the conjectural local Langlands correspondence has pushed us to pursue its understanding.

In this talk, I will aim to share the spirit of, and open questions in, the representation theory of G, through the lens of restricting these representations to maximal compact open subgroups.

Our point of departure: the Bruhat-Tits building of G, a 50-year-old combinatorial and geometric object that continues to reveal secrets about the structure and representation theory of G today.

**Monica Nevins** (University of Ottawa): Recent progress on the effective Mordell problem

**Monica Nevins****SMRI Algebra and Geometry Online**

**Date**: 26 February

**Abstract:** The theory of complex representations of p-adic groups can feel very technical and unwelcoming, but its central role in the conjectural local Langlands correspondence has pushed us to pursue its understanding.

In this talk, I will aim to share the spirit of, and open questions in, the representation theory of G, through the lens of restricting these representations to maximal compact open subgroups.

Our point of departure: the Bruhat-Tits building of G, a 50-year-old combinatorial and geometric object that continues to reveal secrets about the structure and representation theory of G today.

**2020 Seminars**

Abstracts and links to YouTube recordings (where available).

**Minhyong Kim** (University of Ottawa): Recent progress on the effective Mordell problem

**Minhyong Kim****SMRI Algebra and Geometry Online**

**Date**: 9 December

**Abstract:** In 1983, Gerd Faltings proved the Mordell conjecture stating that curves of

genus at least two have only finitely many rational points. This can be understood as

the statement that most polynomial equations (in a precise sense)

f(x,y)=0

of degree at least 4 have at most finitely many solutions. However, the effective

version of this problem, that of constructing an algorithm for listing all rational

solutions, is still unresolved. To get a sense of the difficulty, recall how long it

took to prove that there are no solutions to

x^n+y^n=1

other than the obvious ones. In this talk, I will survey some of the recent progress on

an approach to this problem that proceeds by encoding rational solutions into arithmetic

principal bundles and studying their moduli in a manner reminiscent of geometric gauge

theory.

**Aidan Sims** (University of Wollongong): Homotopy of product systems, K-theory of k-graph algebras, and the Yang-Baxter equations

**Aidan Sims****SMRI Seminar**

**Date**: 18 December

**Abstract:** Higher-rank graphs, or k-graphs, are higher-dimensional analogues of directed graphs.

Each k-graph can be described in terms of a coloured graph, called its skeleton, and some factorisation rules that describe how 2-coloured paths pair up into commuting squares. C*-algebras of k-graphs generalise Cuntz-Krieger algebras, and have been the subject of sustained interest essentially because questions about crossed products of C*-algebras by higher-rank free abelian groups are hard, and k-graph algebras constitute a comparably tractable class of examples that could point the way to general theorems.

A particularly obstinate question in this vein is that of determining the K-theory of a k-graph algebra, or even just whether the K-theory depends on the factorisation rules, or only on the skeleton. I’ll outline some joint work with James Fletcher and Elizabeth Gillaspy that uses a homotopy argument to establish a surprising link between this question and the question of connectedness (or otherwise) of the space of solutions to a Yang-Baxter-like equation. I won’t assume any background about C*-algebras, k-graphs, or the Yang-Baxter equations, and all are welcome—and people who might know about connectedness (or otherwise) of the spaces of solutions to Yang-Baxter-like equations are especially welcome!

**David Robertson** (University of New England): Piecewise full groups of homeomorphisms of the Cantor set

**David Robertson****SMRI Seminar**

**Date**: 11 November

**Abstract:** A group G acting faithfully by homeomorphisms of the Cantor set is called piecewise full if any homeomorphism assembled piecewise from elements of G is itself an element of G.

They first appeared in the work of Giordano, Putnam and Skau in the context of Cantor minimal systems. Recently they have received significant attention as a source of new examples of finitely generated infinite simple groups. I will present a number of results about these groups obtained in joint work with Alejandra Garrido and Colin Reid.A group G acting faithfully by homeomorphisms of the Cantor set is called piecewise full if any homeomorphism assembled piecewise from elements of G is itself an element of G.

They first appeared in the work of Giordano, Putnam and Skau in the context of Cantor minimal systems. Recently they have received significant attention as a source of new examples of finitely generated infinite simple groups. I will present a number of results about these groups obtained in joint work with Alejandra Garrido and Colin Reid.

**James Borger and Lance Gurney** (Australian National University): The geometric approach to cohomology

**James Borger and Lance Gurney****SMRI Course**

**Date**: 11 November

**Abstract:** The aim of these two talks is to give an overview of the geometric aka stacky approach to various cohomology theories for schemes: de Rham, Hodge, crystalline and prismatic (due to Simpson and later Drinfel’d). The basic observation is that interesting cohomology theories for schemes can be realised as the (humble) coherent cohomology an associated stack. Interesting aspects of the cohomology theories e.g. comparison theorems, theories of coefficients, perfectness etc can then be naturally expressed and proven in terms of the geometry of the associated stacks.**YouTube videos**

**Peng Shan** (Tsinghua University): Coherent categorification of quantum loop sl(2)

**Peng Shan****SMRI Algebra and Geometry Online**

**Date**: 26 October

**Abstract:** We explain an equivalence of categories between a module category of quiver Hecke algebras associated with the Kronecker quiver and a category of equivariant perverse coherent sheaves on the nilpotent cone of type A. This provides a link between two different monoidal categorifications of the open quantum unipotent cell of affine type A₁, one given by Kang–Kashiwara–Kim–Oh–Park in terms of quiver Hecke algebras, the other given by Cautis–Williams in terms of equivariant perverse coherent sheaves on affine Grassmannians. The first part of the talk will be devoted to introduction to quiver Hecke algebras and categorification of quantum cluster algebras. The main result will be explained in the second part. This is a joint work with Michela Varagnolo and Eric Vasserot.

**Anthony Licata** (Australian National University): Stability conditions and automata

**Anthony Licata****SMRI Algebra and Geometry Online**

**Date**: 21 October

**Abstract:** Autoequivalences of triangulated categories are an interesting and understudied class of groups. In large part due to the development of the theory of Bridgeland stability conditions, there are suggestive parallels between these groups and mapping class groups of surfaces. The goal of this talk will be to explain how some of the geometric group theory which appears in the study of mapping class groups also arises in the study of triangulated autoequivalences.

**Shamgar Gurevich** (University of Wisconsin, Madison): Harmonic analysis on GLₙ over finite fields

**Shamgar Gurevich****SMRI Algebra and Geometry Online**

**Date**: 8 October

**Abstract:** There are many formulas that express interesting properties of a finite group G in terms of sums over its characters. For estimating these sums, one of the most salient quantities to understand is the character ratio: Trace(ρ(g)) / dim(ρ), for an irreducible representation ρ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of the mentioned type for analyzing certain random walks on G.

Recently, we discovered that for classical groups G over finite fields there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant rank.

Rank suggests a new organization of representations based on the very few “Small” ones. This stands in contrast to Harish-Chandra’s “philosophy of cusp forms”, which is (since the 60s) the main organization principle, and is based on the (huge collection) of “Large” representations.

This talk will discuss the notion of rank for the group GLₙ over finite fields, demonstrate how it controls the character ratio, and explain how one can apply the results to verify mixing time and rate for random walks.

This is joint work with Roger Howe (Yale and Texas A&M). The numerics for this work was carried with Steve Goldstein (Madison) and John Cannon (Sydney).

**Sam Raskin** (University of Texas at Austin): Tate’s thesis in the de Rham setting

**Sam Raskin****SMRI Algebra and Geometry Online**

**Date**: 4 August

**Abstract**: This is joint work with Justin Hilburn. We will explain a theorem showing that D-modules on the Tate vector space of Laurent series are equivalent to ind-coherent sheaves on the space of rank 1 de Rham local systems on the punctured disc equipped with a flat section. Time permitting, we will also describe an application of this result in the global setting. Our results may be understood as a geometric refinement of Tate’s ideas in the setting of harmonic analysis. They also may be understood as a proof of a strong form of the 3d mirror symmetry conjectures in a special case.

**Eugen Hellmann** (University of Münster): On the derived category of the Iwahori–Hecke algebra

**Eugen Hellmann****SMRI Algebra and Geometry Online**

**Date**: 9 December

**Abstract:** In this talk I will state a conjecture which predicts that the derived category of smooth representations of a p-adic split reductive group admits a fully faithful embedding into the derived category of (quasi-)coherent sheaves on a stack of L-parameters. We will make the conjecture precise in the case of the principal block of GLn and relate it to the construction of a family of representations on the stack of L-parameters that interpolates a modified version of the local Langlands correspondence. The existence of this family is suggested by the work of Helm and Emerton–Helm. I will explain why the derived tensor product with this “Emerton–Helm family” should realize the expected embedding of derived categories and discuss some explicit examples.

**Victor Ostrik** (University of Oregon): Incompressible symmetric tensor categories

**Victor Ostrik****SMRI Algebra and Geometry Online**

**Date**: 25 June

**Abstract:** This talk is based on joint work with Benson and Etingof. We say that a symmetric tensor category is incompressible if there is no symmetric tensor functor from this category to a smaller tensor category. Our main result is a construction of new examples of incompressible tensor categories in positive characteristic.

**David Ben-Zvi** (University of Texas at Austin): Boundary conditions and hamiltonian actions in geometric Langlands

**David Ben-Zvi****SMRI Algebra and Geometry Online**

**Date**: 3 June

**Abstract**: I will discuss some of the structures suggested by the physics of supersymmetric gauge theory of relevance to the geometric Langlands program. The discussion will include defects of various dimensions in field theory, with an emphasis on the role of boundary conditions and how they lead to the suggestion of a duality between hamiltonian actions of Langlands dual groups.

**Tom Bridgeland** (University of Sheffield): Introduction to derived categories of coherent sheaves

**Tom Bridgeland****SMRI Algebra and Geometry Online**

**Date**: 26 February–11 March 2020

**Abstract**: These lectures will cover some basic results about derived categories of coherent sheaves (e.g. the structure of the derived category of a curve, Fourier-Mukai transforms and how to construct them, tilting bundles, auto-equivalence groups, perhaps spaces of stability conditions). I will assume that the audience is vaguely familiar with the definition of derived and triangulated categories, but I will spend a fair amount of time in the first few lectures trying to give some intuitive feel for these general constructions, and explaining how one makes calculations in practice. I will also need to assume some familiarity with basic algebraic geometry and sheaf theory.

**Nancy Reid** (University of Toronto): In praise of small data

**Nancy Reid****SMRI Colloquium**

**Date**: 30 January

**Abstract**: Statistical science has a 200-year history of advances in theory and application. Data science is a relatively newly defined area of enquiry developing from the explosion in the ubiquitous collection of data. The interplay between these fields, and their interactions with science, are a topic of lively discussion among statisticians. This talk will overview some of the current research in statistical science that is motivated by new developments in data science.

**Biography:** Nancy Reid is University Professor and Canada Research Chair in Statistical Theory and Applications at the University of Toronto. Her research interests include statistical theory, likelihood inference, design of studies, and statistical science in public policy. Her main research contributions have been to the field of theoretical statistics. Professor Reid is a Fellow of the Royal Society, the Royal Society of Canada, the American Association for the Advancement of Science, and a Foreign Associate of the National Academy of Sciences. In 2014 she was appointed Officer of the Order of Canada.