*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.*

*To receive notifications of upcoming seminars, please subscribe to the weekly seminar email update.*

##### SMRI Seminar ‘Relative equilibria in the full N-body problem’

**Francisco Crespo**, Universidad del Bío-Bío

**1 December 2022**

*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. Watch the recording.

##### SMRI Seminar ‘Constructing highly regular expanders from hyperbolic Coxeter groups’

**Jeroen Schillewaert**, University of Auckland

**17 November 2022**

*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. Watch the recording.

##### SMRI Seminar ‘Some New Inequalities in Analysis and Geometry’

**Changfeng Gui**, **University of Texas at San Antonio**

**Changfeng Gui**

**17 November 2022**

*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. Watch the recording.

##### SMRI Seminar ‘Disjoint and sliding blocks estimators for heavy tailed time series’

**Rafał Kulik**, University of Ottawa

**Rafał Kulik**

**10 November 2022**

*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. Watch the recording.

##### SMRI Seminar ‘Discrete two-generator subgroups of PSL(2,Q_p)’

**Matthew Conder,** University of Auckland

**Matthew Conder,**

**10 November 2022**

*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. Watch the recording.

##### SMRI Seminar ‘What can the working mathematician expect from deep learning?’

Geordie Williamson, Sydney Mathematical Research Institute

**2 November 2022**

*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.)

##### SMRI Seminar ‘Categories, approximation, representation theory and algebraic geometry’

Bregje Pauwels, Sydney Mathematical Research Institute

**20 October 2022**

*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.

##### SMRI Seminar ‘Sequential Bayesian Learning’

**Jana de Wiljes**, University of Potsdam

**Jana de Wiljes**

**8 September 2022**

*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. Watch the recording.

##### SMRI Seminar ‘On the invariance of plurigenera’

**Henri Guenancia**, Paul Sabatier University

**Henri Guenancia**

**26 August 2022**

*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.

##### SMRI Seminar ‘Finding structures in observations: consistent(?) clustering analysis’

**Clara Grazian**, University of Sydney

**Clara Grazian**

**10 May 2022**

*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. Watch the recording.

**SMRI Seminar** ‘Canards, Cardiac Cycles, and Chimeras’

**Theodore Vo**, Monash** **University

**Theodore Vo**

**8 March 2022**

*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. Watch the recording.

**SMRI Seminar** ‘Stochastic Optimal Transport in Financial Mathematics’

**Ivan Guo**, Monash University

**Ivan Guo**

**22 February 2022**

*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. Watch the recording.