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 below and in the SMRI Seminar YouTube playlist.
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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.