This seminar series gives visitors and staff members the opportunity to explain the context and aims of their work. During semester, the SMRI seminar is usually on Thursdays from 1 pm — 2 pm, followed by afternoon tea. These research talks cover any field in the mathematical sciences, and should be presented in a way that is understandable and interesting to a broad audience. Seminar information and recordings can be found below and in the SMRI Seminar YouTube playlist.
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Special SMRI Seminar, ‘Gradient optimization methods: the benefits of instability’ (Part of Mathematical Science of AI Safety Focus Period)
Peter Bartlett, UC Berkeley
Wednesday 10th December 2025
Abstract: Deep learning, the technology underlying the recent progress in AI, has revealed some major surprises from the perspective of theory. These methods seem to achieve their outstanding performance through different mechanisms from those of classical learning theory, mathematical statistics, and optimization theory. Optimization in deep learning relies on simple gradient descent algorithms that are traditionally viewed as a time discretization of gradient flow. However, in practice, large step sizes – large enough to cause oscillation of the loss – exhibit performance advantages. This talk will review recent results on gradient descent with logistic loss with a step size large enough that the optimization trajectory is at the “edge of stability.” We show the benefits of this initial oscillatory phase for linear functions and for multi-layer networks. Based on joint work with Pierre Marion, Matus Telgarsky, Jingfeng Wu, and Bin Yu. Watch on YouTube.
About the speaker: Peter Bartlett is Professor of the Graduate School in Statistics and Computer Science at UC Berkeley and Principal Scientist at Google DeepMind. At Berkeley, he is the Machine Learning Research Director at the Simons Institute for the Theory of Computing, Director of the Foundations of Data Science Institute, and Director of the Collaboration on the Theoretical Foundations of Deep Learning, and he has served as Associate Director of the Simons Institute. He is President of the Association for Computational Learning and co-author with Martin Anthony of the book Neural Network Learning: Theoretical Foundations. He was awarded the Malcolm McIntosh Prize for Physical Scientist of the Year, and has been an Institute of Mathematical Statistics Medallion Lecturer, an IMS Fellow and Australian Laureate Fellow, a Fellow of the ACM, a recipient of the UC Berkeley Chancellor’s Distinguished Service Award, and a Fellow of the Australian Academy of Science.
SMRI Seminar ‘The rich world of Fano geometry‘
Caucher Birkar, Tsinghua University
Thursday 27th November 2025
Abstract: Fano varieties and more general Fano structures on algebraic varieties are of fundamental importance in modern mathematics. In fact, they have been central objects throughout the history of algebraic geometry. In this lecture I will discuss some fascinating aspects of the Fano world in various contexts mentioning recent results as well as open problems.
SMRI Seminar ‘Geometry of Prediction in Large Language Models‘
Daniel Murfet, University of Melbourne
Thursday 13th November 2025
Abstract: Our current best models of human language are neural networks with billions of parameters, trained by stochastic gradient descent. This training process somehow manages to take patterns from the training data and embed them into the parameters of these networks, but why and how this works remains quite mysterious. Part of the mathematical picture involves the (algebraic) geometry of the loss function that drives this gradient descent, according to deep work in Bayesian statistics by Sumio Watanabe. I’ll explain some of these connections between neural networks and geometry, and how at Timaeus (an AI-safety non-profit) we are applying these ideas to interpretability, an exciting new field that aims to discover the underlying “algorithms” driving neural network computation.
SMRI Seminar ‘Anatomy of big algebras‘
Tamás Hausel, Institute of Science and Technology Austria
Thursday 16th October 2025
Abstract: Starting with Schrodinger’s critique of the difficulty of visualising Heisenberg’s matrix quantum physics we will discuss the notion of non-commutative matrices and its use in particle physics via Gell-Mann’s eightfold way describing heavy particles in terms of quarks.
Then I will use these backgrounds to motivate my recent work on big algebras — commutative avatars of non-commutative matrix representations — and show some visualisation of them to highlight their complex anatomy, shedding some new light on the quantum numbers of the heavy particles in the baryon octet and decuplet. Watch on YouTube.
SMRI Seminar ‘Mobility vs Epidemics‘
Fakhteh Ghanbarnejad, SRH University
Thursday 9th October 2025
Abstract: In this talk, I will explore “Gravity and the Big Bang in the Context of Epidemics” and discuss how human mobility shapes the spread of epidemics—and how epidemics, in turn, reshape mobility.
The COVID-19 pandemic reshaped daily life, altering not only health but also how people move through space. Using aggregated mobility data from parks in Washington State, we examined how visitation patterns changed before and during the pandemic. We found that the gravity model— classic tool in spatial interaction modeling—accurately predicted park visitation across different income groups and geographic scales. We observed that higher-income residents tended to broaden their recreational activities, while lower-income residents restricted their options [1].
In another study, we developed a mathematical framework for modeling epidemic dynamics that accounts for both local and long-distance mobility within a metapopulation network. By linearizing the system, we can identify the “Big Bang” of an epidemic—its origin and initiation time. Using a new concept of effective distance, we demonstrate that epidemic spread follows a universal geometric pattern. This prediction was validated using data from both the COVID-19 and H1N1 outbreaks. Our framework, relying only on mobility data and active case counts, provides new insights into contagion dynamics and offers a foundation for designing public health policies that better anticipate and mitigate epidemic spread [2]. Watch on YouTube.
1. Ghadiri, Z., Mashhadi, A., Timme, M. et al. Recreational mobility prior and during the COVID-19 pandemic. Commun Phys 7, 55 (2024).
2. Babazadeh Maghsoodlo Y, Safaeesirat A, Ghanbarnejad F. The Big Bang of an epidemic: a metapopulation approach to identify the spatiotemporal origin of contagious diseases and their universal spreading pattern.
Sci Rep 15, 1 (2025).
SMRI Seminar ‘There and back again: groupoids and $C^*$-algebras‘
Aidan Sims, University of New South Wales
Thursday 25th September 2025
Abstract: Whatever \(C^* \)-algebras are, what they do is “linearise’” topological and dynamical data a la representation theory. The problem is that while every \(C^* \)-algebra can in principle be realised as a collection of linear operators on Hilbert space, this has, in general, the same sort of descriptive power as the observation that every group can, in principle, be realised as a group of permutations—and for the same reason: the set of which it is a group of permutations is…itself. Groupoids provide a means of “seeing” a coordinate system in a \(C^* \) -algebra. I’ll try to outline how, and why this might be useful. Watch on YouTube.
SMRI Seminar ‘Introduction to Optimal Transport and Free Boundary Problems’
Jiakun Liu, The University of Sydney
Thursday 21st August 2025
Abstract: In this talk, we will introduce the optimal transportation with some interesting applications in various fields including a reconstruction problem in cosmology; a brief proof of isoperimetric and Brunn-Minkowski inequalities in geometry; and an application in image recognition relating to a transport between hypercubes. We will also discuss our recent research on the regularity of free boundaries in optimal transportation. Watch on YouTube.
SMRI Seminar ‘Navigating the space of surfaces’
Lynn Heller, Beijing Institute of Mathematical Sciences and Applications
Thursday 7th August 2025
Abstract: In this talk, I will explore the space of surfaces satisfying certain variational principles. Through a series of examples, I will illustrate their geometric and analytic properties, and highlight surprising connections to number theory that arises in their study. Watch on YouTube.
SMRI Seminar ‘From surfaces to 4-manifolds: a leisurely journey in the topology of manifolds’
Diarmuid Crowley, University of Melbourne
Thursday 19th June 2025
Abstract: Let F be a closed connected orientable surface (2-manifold). Here are two foundational facts in the study of surfaces:
1) F_1 and F_2 are homeomorphic if and only if they have the same Euler characteristic;
2) Every self-homotopy equivalence of F is homotopic to a homeomorphism.
In this talk will I review these facts and then move to dimension 4, where the analogous statements for 4-manifolds are far from true. Indeed, in finding weak analogues of (1), i.e. necessary and sufficient conditions for two 4-manifolds within some class to be homeomorphic, it is often helpful to have counter-examples to the strict analogue of (2); i.e. self-homotopy equivalences of 4-manifolds which are not homotopic to homeomorphisms. The last part of this talk will contain a preliminary report on new examples of such “exotic self-homotopy equivalences”, which have shown up in joint work with Daniel Kasprowski, Mark Powell, and Arunima Ray.
SMRI Seminar ‘Graph Product Structure: Theory and Applications’
David Wood, Monash University
Thursday 29th May 2025
Abstract: Graph product structure theory describes graphs in complicated classes as subgraphs of products of simple tree-like graphs. There has been an explosion of interest in this field since 2019, when the speaker and his colleagues proved that every planar graph is contained in the product of a tree-like graph and a path.
This result opened up a new research direction, and has been the key tool in solving a number of longstanding open problems. This talk will introduce graph product structure theory, describing the main results and several of the applications. Only elementary graph theory will be assumed.
SMRI Seminar ‘Around the Danzer Problem and the construction of dense forests’
Faustin Adiceam, University Paris-Est Créteil
Thursday 8th May 2025
Abstract: A 1965 problem due to Danzer asks whether there exists a set with finite density in Euclidean space (i.e. « not containing too many points ») intersecting any convex body of volume one. A suitable weakening of the volume constraint leads one to the (much more recent) problem of constructing dense forests. Progress towards these problems have so far involved a very wide range of areas in mathematics (including number theory, ergodic theory, geometry and harmonic analysis). After surveying some of the known results related to the Danzer Problem and to the construction of dense forests, the talk will present some new constructions.
SMRI Seminar ‘Can language models learn arithmetic?’
François Charton, Research Engineer at FAIR, Meta
Thursday 17 April 2025
Abstract: Language models have become surprisingly good at many tasks, from text summarization to image generation, and speech recognition. Yet, they are still embarrassingly weak on basic arithmetic operations, like integer multiplication. I present recent results demonstrating that language models (transformers) can indeed learn complex calculations, and sometimes capture some of the underlying mathematics. This research demonstrates the importance of the distribution of training examples in deep learning.
SMRI Seminar ‘Positive geometry and scattering amplitudes’
Thomas Lam, University of Michigan
Thursday 10 April 2025
Abstract: Positive geometry is a new field developing at the intersection of combinatorial geometry in mathematics and scattering amplitudes in physics. Scattering amplitudes are functions that predict the outcome of particle physics experiments. The goal of positive geometry is to recover these functions using a mixture of combinatorics and geometry. I will give an introduction to the subject.
SMRI Seminar ‘Perverse sheaves and representations of p-adic groups’
Charlotte Chan, University of Michigan
Thursday 3 April 2025
Abstract: One of the first basic ideas we all learn is that a continuous function is determined by its values on a dense open subset. In representation theory, this allows us to recognize a representation of a Lie group from an especially well-behaved locus—that of regular semisimple elements. But what if we want to study representations of matrix groups over finite fields? Lusztig’s revolutionary idea in the 1980s was that intermediate extension—the algebro-geometric version of the analytic notion of limit—applies in representation theory in discrete settings. I will explain this picture and describe a recent construction (joint with R. Bezrukavnikov) of perverse sheaves that give rise to positive-depth supercuspidal representations of p-adic groups. In the simplest nontrivial case, this resolves a conjecture of Lusztig.
SMRI Seminar ‘Singularities in fluid: Self-similar analysis, computer assisted proofs and neural networks’
Tristan Buckmaster, Courant Institute of Mathematical Sciences, New York University
Thursday 27 March 2025
Abstract: In this presentation, I will provide an overview of how techniques involving self-similar analysis, computer assisted proofs and neural networks can be employed to investigate singularity formation in the context of fluids.
SMRI Seminar ‘Coupled Chaotic Maps and Self-Consistent Transfer Operators’
Matteo Tanzi, King’s College London
Thursday 6th March 2025
Abstract: At the end of the 1980s, globally coupled maps (GCMs) emerged as high-dimensional models for complex systems. These models feature simple equations where several variables are coupled symmetrically all-to-all, and display a rich variety of behaviors, including synchronization, phase ordering, and turbulence. Rigorous mathematical studies of the dynamics of GCMs have primarily focused on their mean-field limit—that is, the behavior of the system’s average state as the number of maps approaches infinity. This limit is governed by a nonlinear operator known as the self-consistent transfer operator, which dictates the evolution of the mean field. In this talk, I will provide a brief overview of the origin of the study of self-consistent transfer operators and discuss some recent progress in the field focusing on coupled chaotic maps.
SMRI Seminar ‘Hausdorff dimension of the Apollonian gasket’
Caroline Wormell, The University of Sydney
Thursday 27 February 2025
Abstract: The Apollonian gasket is a well-studied circle packing, created by iteratively filling a region with mutually tangent circles. Important properties of the packing, including the distribution of the circle radii, are universal and governed by its Hausdorff dimension. No closed form is currently known for the Hausdorff dimension, and its computation is a special case of a more general and hard problem: effective, rigorous estimates of dimension of a limit set generated by non-uniform contractions. In this talk, I will talk about an efficient method for solving this problem. With this method we can not only compute the dimension of the gasket to a lot of decimal places, but also rigorously validate this computation as a mathematical theorem. Our method is not particularly specialised to the Apollonian gasket, and could generalise easily to other “difficult” parabolic fractals. Based on joint work with Polina Vytnova.