Past

I will talk about some current work with Jesse Jaasaari and Steve Lester where we investigate the analogue of the Quantum Unique Ergodicity (QUE) conjecture in the weight aspect for Siegel cusp forms of degree 2 and full level. Assuming the Generalized Riemann Hypothesis (GRH) we establish QUE for Saito–Kurokawa lifts as the weight tends to infinity. As an application, we prove the equidistribution of zero divisors.

Polynomial splines over simplicial meshes in R^n (triangulations in 2D, tetrahedral meshes in 3D, and so on) sometimes have extra orders of smoothness at a vertex. This property is known as supersmoothness, and plays a role both in the construction of macroelements and in the finite element method.
Supersmoothness depends both on the number of simplices that meet at the vertex and their geometric configuration.

In this talk we review what is known about supersmoothness of polynomial splines and then discuss the more general setting of splines whose individual pieces are any infinitely smooth functions.

This is joint work with Kaibo Hu.

TBA

Topological data analysis (TDA) comprises a set of techniques of computational topology that has had enormous growth in the last decade, with applications to a wide variety of fields, such as images,  biological data, meteorology, materials science, time-dependent data, economics, etc. In this talk, we will first have a walk through a typical pipeline in TDA, to move later to its adaptation to a specific context: the topological characterization of the spatial distribution of regions in a segmented image

Geometric (even combinatorial) group theory suffers from the unfortunate situation that many obvious questions about group presentations (ex: is this a presentation of the trivial group? is this word the identity in that group?) cannot be answered. Not only "we don't know how to tell" but "we know that we cannot know how to tell" - this is called undecidability. This talk will serve as an introduction (for non-experts, since I am also such) to the area of group theoretic decision problems: I'll aim to cover some problems, some solutions (or half-solutions) and some of the general sources of undecidability, as well as featuring some of my (least?) favourite pathological groups.

The Hilbert transform H is a basic example of a Fourier multiplier, and Riesz proved that H is a bounded operator on Lp(T) for all p between 1 and infinity.  We study Hilbert transform type Fourier multipliers on group algebras and their boundedness on corresponding non-commutative L_p spaces. The pioneering work in this direction is due to Mei and Ricard who proved L_p-boundedness of Hilbert transforms on free group von Neumann algebras using a Cotlar identity. In this talk, we introduce a generalised Cotlar identity and a new geometric form of Hilbert transform for groups acting on tree-like structures. This class of groups includes amalgamated free products, HNN extensions, left orderable groups and many others.  This is joint work with Adrián González and Javier Parcet.

I will discuss statistical inference problems on edge-correlated stochastic block models. We determine the information-theoretic threshold for exact recovery of the latent vertex correspondence between two correlated block models, a task known as graph matching. As an application, we show how one can exactly recover the latent communities using multiple correlated graphs in parameter regimes where it is information-theoretically impossible to do so using just a single graph. Furthermore, we obtain the precise threshold for exact community recovery using multiple correlated graphs, which captures the interplay between the community recovery and graph matching tasks. This is based on joint work with Julia Gaudio and Anirudh Sridhar.

I will explain that all affine isometric actions of higher rank simple Lie groups and their lattices on arbitrary uniformly convex Banach spaces have a fixed point. This vastly generalises a recent breakthrough of Oppenheim. Combined with earlier work of Lafforgue and of Liao on strong Banach property (T) for non-Archimedean higher rank simple groups, this confirms a long-standing conjecture of Bader, Furman, Gelander and Monod. As a consequence, we deduce that box space expanders constructed from higher rank lattices are superexpanders. This is joint work with Mikael de la Salle.

Coupled differential equations generally present an important algebraic structure.
For example in the incompressible Navier-Stokes equations, the velocity is affected only by the selenoidal part of the applied force.
This structure can be translated naturally by the notion of complex.
One idea is then to exploit this complex structure at the discrete level in the creation of numerical methods.

The goal of the presentation is to expose the notion of complex by motivating its uses. 
We will present in more detail the creation of a scheme for the Navier-Stokes equations and study its properties.
 

In 1977, Erdős and Hajnal made the conjecture that, for every graph $H$, there exists $c>0$ such that every $H$-free graph $G$ has a clique or stable set of size at least $|G|^c$; and they proved that this is true with $|G|^c$ replaced by $2^{c\sqrt{\log |G|}}$. Until now, there has been no improvement on this result (for general $H$). We recently proved a strengthening: that for every graph $H$, there exists $c>0$ such that every $H$-free graph $G$ with $|G|\ge 2$ has a clique or stable set of size at least $2^{c\sqrt{\log |G| \log\log|G|}}$. This talk will outline the proof. Joint work with Matija Bucić, Tung Nguyen and Alex Scott.

The recent advancements in additive manufacturing have enabled the creation of lattice structures with intricate small-scale details. This has led to the need for new techniques in the field of topology optimization that can handle a vast number of design variables. Despite the efforts to develop multi-scale topology optimization techniques, their high computational cost has limited their application. To overcome this challenge, a new technique called dehomogenization has shown promising results in terms of performance and computational efficiency for optimizing compliance problems.

In this talk, we extend the application of the dehomogenization method to stress minimization problems, which are crucial in structural design. The method involves homogenizing the macroscopic response of a proposed family of microstructures. Next, the macroscopic structure is optimized using gradient-based methods while orienting the cells according to the principal stress components. The final step involves dehomogenization of the structure. The proposed methodology also considers singularities in the orientation field by incorporating singular functions in the dehomogenization process. The validity of the methodology is demonstrated through several numerical examples.

Understanding the stability of ecological communities is a matter of increasing importance in the context of global environmental change. Yet it has proved to be a challenging task. Different metrics are used to assess the stability of ecological systems, and the choice of one metric over another may result in conflicting conclusions. While the need to consider this multitude of stability metrics has been clearly stated in the ecological literature for decades, little is known about how different stability metrics relate to each other. I’ll present results of dynamical simulations of ecological communities investigating the correlations between frequently used stability metrics, and I will discuss how these results may contribute to make progress in the quantification of stability in theory and in practice.

Zoom Link: https://zoom.us/j/93174968155?pwd=TUJ3WVl1UGNMV0FxQTJQMFY0cjJNdz09

Meeting ID: 931 7496 8155

Passcode: 502784

Kazhdan-Lusztig polynomials are remarkable polynomials associated to pairs of elements in a Coxeter group W. They describe the base change between the standard and Kazhdan-Lusztig bases for the corresponding Hecke algebra. They were discovered by Kazhdan and Lusztig in 1979 and have found applications throughout representation theory and geometry. In 1987, Deodhar introduced the parabolic Kazhdan-Lusztig polynomials associated to a Coxeter group W and a standard parabolic subgroup P. These describe the base change between the standard and Kazhdan-Lusztig bases for the anti-spherical module for the Hecke algebra. (We recover the original definition of Kazhdan and Lusztig by taking the trivial parabolic subgroup).

(Anti-spherical) Hecke categories first rose to mathematical celebrity as the centrepiece of the proof of the (parabolic) Kazhdan-Lusztig positivity conjecture. The Hecke category categorifies the Hecke algebra and the anti-spherical Hecke category categorifies the anti-spherical module. More precisely, it was shown by Elias-Williamson (and Libedinsky-Williamson) that the (parabolic) Kazhdan-Lusztig polynomials are precisely the graded decomposition numbers for the (anti-spherical) Hecke categories over fields of characteristic zero, hence proving positivity of their coefficients.
The (anti-spherical) Hecke categories can be defined over any field. Their graded decomposition numbers over fields of positive characteristic p, the so-called (parabolic) p-Kazhdan-Lusztig polynomials, have been shown to have deep connections with the modular representation theory of reductive groups and symmetric groups. However, these polynomials are notoriously difficult to compute.
Unlike in the case of the ordinary (parabolic) Kazhdan-Lusztig polynomials, there is not even a recursive algorithm to compute them in general.
In this talk, I will discuss the representation of the anti-spherical Hecke categories for (W,P) a Hermitian symmetric pair, over an arbitrary field. In particular, I will explain why the decomposition numbers are characteristic free in this case.
This is joint work with C. Bowman, A. Hazi and E. Norton.

We model a tokamak fusion reaction, combining Maxwell's equations with the Navier-Stokes equations, the heat equation and the Seebeck effect giving a model of thermoelectric magnetohydrodynamics (TEMHD). At leading order, we showed that the free surface must be flat, that the pressure is constant, and that the temperature decouples from the governing equations relating the fluid velocity and magnetic field. We also find that the fluid flow is driven entirely by the temperature gradient normal to the free surface. Using singular perturbation methods we obtained velocity profiles which exhibit so-called Hartmann layers and thicker side layers. The role of the aspect ratio has been seldom considered in classical MHD duct flow literature as a varying parameter. Here, we show it's importance and derive a relationship between the aspect ratio and Hartmann number that maximises flow rate of fluid down the duct.

Spacetimes formed from the cartesian product of Minkowski space and a flat torus play an important role as toy models for theories of supergravity and string theory. In this talk I will discuss an upcoming work with Huneau and Stingo showing the nonlinear stability of such a Kaluza-Klein spacetime. The result is also connected to a claim of Witten.

It was recently shown by Aidekon and Da Silva how to construct a growth fragmentation process from a planar Brownian excursion. I will explain how this same growth fragmentation process arises in another setting: when one decorates a certain “critical Liouville quantum gravity random surface” with a conformal loop ensemble of parameter 4. This talk is based on joint work with Juhan Aru, Nina Holden and Xin Sun. 
 

Despite their straightforward definition, the homeomorphism groups of surfaces are far from straightforward. Basic algebraic and dynamical problems are wide open for these groups, which is a far cry from the closely related and much better understood mapping class groups of surfaces. With Jonathan Bowden and Sebastian Hensel, we introduced the fine curve graph as a tool to study homeomorphism groups. Like its mapping class group counterpart, it is Gromov hyperbolic, and can shed light on algebraic properties such as scl, via geometric group theoretic techniques. This brings us to the enticing question of how much of Thurston's theory (e.g. Nielsen--Thurston classification, invariant foliations, etc.) for mapping class groups carries over to the homeomorphism groups. We will describe new phenomena which are not encountered in the mapping class group setting, and meet some new connections with topological dynamics, which is joint work with Bowden, Hensel, Kathryn Mann and Emmanuel Militon. I will survey what's known, describe some of the new and interesting problems that arise with this theory, and give an idea of what's next.

We consider the varifolds associated to phase transitions whose first variation of Allen-Cahn energy is $L^p$ integrable with respect to the energy measure. We can see that the Dirichlet and potential part of the energy are almost equidistributed. After passing to the phase field limit, one can obtain an integer rectifiable varifold with bounded $L^p$ mean curvature. This is joint work with Huy Nguyen.

The data revolution has changed the landscape of computational mathematics and has increased the demand for new numerical linear algebra tools to handle the vast amount of data. One crucial task is data compression to capture the inherent structure of data efficiently. Tensor-based approaches have gained significant traction in this setting by exploiting multilinear relationships in multiway data. In this talk, we will describe a matrix-mimetic tensor algebra that offers provably optimal compressed representations of high-dimensional data. We will compare this tensor-algebraic approach to other popular tensor decomposition techniques and show that our approach offers both theoretical and numerical advantages.

I will present results from arXiv:2202.13924, where we studied the difference between integrable and chaotic motion in quantum theory as manifested by the complexity of the corresponding evolution operators. The notion of complexity of interest to us will be Nielsen’s complexity applied to the time-dependent evolution operator of the quantum systems. I will review Nielsen’s complexity, discuss the difficulties associated with this definition and introduce a simplified approach which appears to retain non-trivial information about the integrable properties of the dynamical systems.

Let F be a p-adic local field and let G be a connected split reductive group over F. Let H be the pro-p Iwahori-Hecke algebra of the p-adic group G(F), with coefficients in an algebraically closed field k of characteristic p. The module theory over H (or a certain derived version thereof) is of considerable interest in the so-called mod p local Langlands program for G(F), whose aim is to relate the smooth modular representation theory of G(F) to modular representations of the absolute Galois group of F. In this talk, we explain a possible construction of a certain moduli space for those Galois representations into the Langlands dual group of G over k which are semisimple. We then relate this space to the geometry of H. This is a work in progress with Cédric Pépin.

Chair: Ian Hewitt (Associate HoD (People))

Panel:
James Sparks (Head of Department)
Helen Byrne (winner of MPLS Outstanding Supervisor Awards for 2022)
Ali Goodall (Head of Faculty Services and HR)
Matija Tapuskovic (EPSRC Postdoctoral Research Fellow and JRF at Corpus Christi)

Scientific discussions with colleagues, at conferences and seminars, during supervisions and collaborations, are a crucial part of our research process. How can we ensure our academic discussions are fruitful, respectful, and a positive experience for everyone involved? What factors and power dynamics can impact our conversations? How can we make sure everyone’s voice is heard and respected? This panel discussion will probe these questions and encourage us all to reflect on how we approach our academic discussions.

Using persistent homology to guide optimization has emerged as a novel application of topological data analysis. Existing methods treat persistence calculation as a black box and backpropagate gradients only onto the simplices involved in particular pairs. We show how the cycles and chains used in the persistence calculation can be used to prescribe gradients to larger subsets of the domain. In particular, we show that in a special case, which serves as a building block for general losses, the problem can be solved exactly in linear time. We present empirical experiments that show the practical benefits of our algorithm: the number of steps required for the optimization is reduced by an order of magnitude. (Joint work with Arnur Nigmetov.)