Past

Pluripotential theory studies plurisubharmonic functions and complex Monge-Ampère equations on complex manifolds, and has played a key role in recent progress on Kähler-Einstein and constant scalar curvature Kähler metrics. This theory admits a non-Archimedean analogue over Berkovich spaces, that can be used to study K-stability. The purpose of this talk is to provide an introduction to this circle of ideas, and to discuss more specifically recent joint work with Mattias Jonsson studying Green's functions in this context.

Generative AI has seen tremendous successes recently, most notably the chatbot ChatGPT and the DALLE2 software creating realistic images and artwork from text descriptions. Underlying these and other generative AI systems are usually neural networks trained to produce text, images, audio, or video from text inputs. The aim of this talk is to develop an understanding of the fundamental capabilities of generative neural networks. Specifically and mathematically speaking, we consider the realization of high-dimensional random vectors from one-dimensional random variables through deep neural networks. The resulting random vectors follow prescribed conditional probability distributions, where the conditioning represents the text input of the generative system and its output can be text, images, audio, or video. It is shown that every d-dimensional probability distribution can be generated through deep ReLU networks out of a 1-dimensional uniform input distribution. What is more, this is possible without incurring a cost—in terms of approximation error as measured in Wasserstein-distance—relative to generating the d-dimensional target distribution from d independent random variables. This is enabled by a space-filling approach which realizes a Wasserstein-optimal transport map and elicits the importance of network depth in driving the Wasserstein distance between the target distribution and its neural network approximation to zero. Finally, we show that the number of bits needed to encode the corresponding generative networks equals the fundamental limit for encoding probability distributions (by any method) as dictated by quantization theory according to Graf and Luschgy. This result also characterizes the minimum amount of information that needs to be extracted from training data so as to be able to generate a desired output at a prescribed accuracy and establishes that generative ReLU networks can attain this minimum.

This is joint work with D. Perekrestenko and L. Eberhard


 

We propose new G2-holonomy manifolds, which geometrize the Gaiotto-Kim 4d N = 1 duality
domain walls of 5d N = 1 theories. These domain walls interpolate between different extended
Coulomb branch phases of a given 5d superconformal field theory. Our starting point is the
geometric realization of such a 5d superconformal field theory and its extended Coulomb
branch in terms of M-theory on a non-compact singular Calabi-Yau three-fold and its Kahler
cone. We construct the 7-manifold that realizes the domain wall in M-theory by fibering the
Calabi-Yau three-fold over a real line, whilst varying its Kahler parameters as prescribed by
the domain wall construction. In particular this requires the Calabi-Yau fiber to pass through
a canonical singularity at the locus of the domain wall. Due to the 4d N = 1 supersymmetry
that is preserved on the domain wall, we expect the resulting 7-manifold to have holonomy G2.
Indeed, for simple domain wall theories, this construction results in 7-manifolds, which are
known to admit torsion-free G2-holonomy metrics. We develop several generalizations to new
7-manifolds, which realize domain walls in 5d SQCD theories.

abstract-Oxford.pdf

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 work in progress with Cédric Pépin.

I will explain an ongoing program, joint with Kari Vilonen, that aims to study unitary representations of real reductive Lie groups using mixed Hodge modules on flag varieties. The program revolves around a conjecture of Schmid and Vilonen that natural filtrations coming from the geometry of flag varieties control the signatures of Hermitian forms on real group representations. This conjecture is expected to facilitate new progress on the decades-old problem of determining the set of unitary irreducible representations by placing it in a more conceptual context. Our results to date centre around the interaction of Hodge theory with the unitarity algorithm of Adams, van Leeuwen, Trapa, and Vogan, which calculates the signature of a canonical Hermitian form on an arbitrary representation by reducing to the case of tempered representations using deformations and wall crossing. Our results include a Hodge-theoretic proof of the ALTV wall crossing formula as a consequence of a more refined result and a verification of the Schmid-Vilonen conjecture for tempered representations.

Tbc

This course will serve as an introduction to the theory of gradient flows with an emphasis on the recent advances in metric spaces. More precisely, we will start with an overview of gradient flows from the Euclidean theory to its generalisation to metric spaces, in particular Wasserstein spaces. This also includes a short introduction to the Optimal Transport theory, with a focus on specific concepts and tools useful subsequently. We will then analyse the time-discretisation scheme à la Jordan--Kinderlehrer-Otto (JKO), also known as minimising movement, and discuss the role of convexity in proving stability, uniqueness, and long-time behaviour for the PDE under study. Finally, we will comment on recent advances, e.g., in the study of PDEs on graphs and/or particle approximation of diffusion equations.

PhD_course_Esposito_1.pdf

This course will serve as an introduction to the theory of gradient flows with an emphasis on the recent advances in metric spaces. More precisely, we will start with an overview of gradient flows from the Euclidean theory to its generalisation to metric spaces, in particular Wasserstein spaces. This also includes a short introduction to the Optimal Transport theory, with a focus on specific concepts and tools useful subsequently. We will then analyse the time-discretisation scheme à la Jordan--Kinderlehrer-Otto (JKO), also known as minimising movement, and discuss the role of convexity in proving stability, uniqueness, and long-time behaviour for the PDE under study. Finally, we will comment on recent advances, e.g., in the study of PDEs on graphs and/or particle approximation of diffusion equations.

PhD_course_Esposito_0.pdf

Hitchin's equations are a system of gauge theoretic equations on a Riemann surface that are of interest in many areas including representation theory, Teichmüller theory, and the geometric Langlands correspondence. The Hitchin moduli space carries a natural hyperkähler metric.  An intricate conjectural description of its asymptotic structure appears in the work of Gaiotto-Moore-Neitzke and there has been a lot of progress on this recently.  I will discuss some recent results using tools coming out of geometric analysis which are well-suited for verifying these extremely delicate conjectures. This strategy often stretches the limits of what can currently be done via geometric analysis, and simultaneously leads to new insights into these conjectures.

In their seminal 2012 paper, Elekes and Szabó found that a certain weak combinatorial non-expansion property of an algebraic relation suffices to trigger the group configuration theorem, showing that only (approximate subgroups of) algebraic groups can be responsible for it. I will discuss some more recent variations and elaborations on this result, focusing on the case of ternary relations on varieties of dimension >1.

PhD_course_Esposito.pdfThis course will serve as an introduction to the theory of gradient flows with an emphasis on the recent advances in metric spaces. More precisely, we will start with an overview of gradient flows from the Euclidean theory to its generalisation to metric spaces, in particular Wasserstein spaces. This also includes a short introduction to the Optimal Transport theory, with a focus on specific concepts and tools useful subsequently. We will then analyse the time-discretisation scheme à la Jordan--Kinderlehrer-Otto (JKO), also known as minimising movement, and discuss the role of convexity in proving stability, uniqueness, and long-time behaviour for the PDE under study. Finally, we will comment on recent advances, e.g., in the study of PDEs on graphs and/or particle approximation of diffusion equations.

There has been a great deal of interest in understanding the link between the axiomatic descriptions of conformal field theory given by vertex operator algebras and conformal nets. In recent work, we establish an equivalence between certain vertex algebras and conformally-symmetric quantum field theories in the sense of Wightman. In this talk I will give an overview of these results and discuss some of the difficulties that arise, the functional analytic properties of vertex algebras, and some of the ideas for future work in this area.

This is joint work with James Tener and Yoh Tanimoto.

This DPhil short course will serve as an introduction to the theory of gradient flows with an emphasis on the recent advances in metric spaces. More precisely, we will start with an overview of gradient flows from the Euclidean theory to its generalisation to metric spaces, in particular Wasserstein spaces. This also includes a short introduction to the Optimal Transport theory, with a focus on specific concepts and tools useful subsequently. We will then analyse the time-discretisation scheme à la Jordan--Kinderlehrer-Otto (JKO), also known as minimising movement, and discuss the role of convexity in proving stability, uniqueness, and long-time behaviour for the PDE under study. Finally, we will comment on recent advances, e.g., in the study of PDEs on graphs and/or particle approximation of diffusion equations.

Donaldson-Thomas theory associates integers (which are virtual counts of sheaves) to a Calabi-Yau threefold X. The simplest example is that of $\mathbb{C}^3$, when the Donaldson-Thomas (DT) invariant of sheaves of zero dimensional support and length d is $p(d)$, the number of plane partitions of $d$. The DT invariants have several refinements, for example a cohomological one, where instead of a DT invariant, one studies a graded vector space with Euler characteristic equal to the DT invariant. I will talk about two other refinements (categorical and K-theoretic) of DT invariants, focusing on the explicit case of $\mathbb{C}^3$. In particular, we show that the K-theoretic DT invariant for $d$ points on $\mathbb{C}^3$ also equals $p(d)$. This is joint work with Yukinobu Toda.

Dr. Keven Roy from RMS Moody's Analytics (who are currently hiring!) will share his experience of transitioning from academia to industry, discussing his fascinating work in modelling climate change and how his mathematical background has helped him succeed in industry. After the approximately 20-minute talk, we'll hold a Q&A session to discuss the importance of considering industry in your job search. Aimed primarily at PhD students and postdocs, this session will explore options beyond academia that can provide a fulfilling career (as well as good work-life balance, and financial compensation!).

Join us for a thought-provoking discussion at Fridays@4 to expand your career horizons and help you make informed decisions about your future. There will be lots of time for Q&A in this session, but if you have questions for Keven you can also send them in advance to Jess Crawshaw (session organiser) - @email.

Mapper and Ball Mapper are Topological Data Analysis tools used for exploring high dimensional point clouds and visualizing scalar–valued functions on those point clouds. Inspired by open questions in knot theory, new features are added to Ball Mapper that enable encoding of the structure, internal relations and symmetries of the point cloud. Moreover, the strengths of Mapper and Ball Mapper constructions are combined to create a tool for comparing high dimensional data descriptors of a single dataset. This new hybrid algorithm, Mapper on Ball Mapper, is applicable to high dimensional lens functions. As a proof of concept we include applications to knot and game  theory, as well as material science and cancer research. 

Permanent geological carbon storage will reduce greenhouse gas emissions and help mitigate climate change. Storage security is increased by CO₂ capillary trapping in cm-to-m scale, layered rock heterogeneities; features that are ubiquitous across storage sites worldwide. This talk will outline the challenges associated with modelling the impact of small-scale heterogeneity on large scale saturation distributions and trapping during geological CO₂ storage, including the difficulties in incorporating petrophysical and geological uncertainty into field-scale numerical models. Experimental results demonstrate the impact of cm-scale heterogeneity on pore-scale processes, which in turn influence large scale behaviour. Heterogeneity is shown to have a leading order impact on saturation distribution and storage capacity during geological CO₂ storage.

The overall goal of the Sherwood lab is to advance genomic and precision medicine applications through high-throughput, multi-disciplinary science. In a shortened talk this past autumn, I described our recent efforts using combined analysis of rare coding variants from the UK Biobank and genome-scale CRISPR-Cas9 knockout and activation screening to improve the identification of genes, coding variants, and non-coding variants whose alteration impacts serum LDL cholesterol (LDL-C) levels.

In this talk, I will discuss our emerging efforts to optimize and employ precision CRISPR techniques such as base editing and prime editing to better understand the impacts of coding and non-coding variation on serum LDL-C levels and coronary artery disease risk. This work involves the development of novel high-throughput screening platforms and computational analysis approaches that have wide applicability in dissecting complex human disease genetics.

In the theory of relative algebraic geometry, our affines are objects in the opposite category of commutative monoids in a symmetric monoidal category $\mathcal{C}$. This categorical approach simplifies many constructions and allows us to compare different geometries. Toën and Vezzosi's theory of homotopical algebraic geometry considers the case when $\mathcal{C}$ has a model structure and is endowed with a compatible symmetric monoidal structure. Derived algebraic geometry is recovered when we take $\mathcal{C}=\textbf{sMod}_k$, the category of simplicial modules over a simplicial commutative ring $k$.

In Kremnizer et al.'s version of derived analytic geometry, we consider geometry relative to the category $\textbf{sMod}_k$ where $k$ is now a simplicial commutative complete bornological ring. In this talk we discuss, from an algebraist's perspective, the main ideas behind the theory of relative algebraic geometry and discuss briefly how it provides us with a convenient framework to consider derived analytic geometry. 

Let $k,l>1$ be two multiplicatively independent integers. A subset $X$ of $\mathbb{N}^n$ is $k$-recognizable if the set of $k$-ary representations of $X$ is recognized by some finite automaton. Cobham's famous theorem states that a subset of the natural numbers is both $k$-recognizable and $l$-recognizable if and only if it is Presburger-definable (or equivalently: semilinear). We show the following strengthening. Let $X$ be $k$-recognizable, let $Y$ be $l$-recognizable such that both $X$ and $Y$ are not Presburger-definable. Then the first-order logical theory of $(\mathbb{N},+,X,Y)$ is undecidable. This is in contrast to a well-known theorem of Büchi that the first-order logical theory of $(\mathbb{N},+,X)$ is decidable. Our work strengthens and depends on earlier work of Villemaire and Bès. The essence of Cobham's theorem is that recognizability depends strongly on the choice of the base $k$. Our results strengthens this: two non-Presburger definable sets that are recognizable in multiplicatively independent bases, are not only distinct, but together computationally intractable over Presburger arithmetic. This is joint work with Christian Schulz.

This talk presents a general framework for modelling the dynamics of limit order books, built on the combination of two modelling ingredients: the order flow, modelled as a general spatial point process, and the market clearing, modelled via a deterministic ‘mass transport’ operator acting on distributions of buy and sell orders. At the mathematical level, this corresponds to a natural decomposition of the infinitesimal generator describing the  order book evolution into two operators: the generator of the order flow and the clearing operator. Our model provides a flexible framework for modelling and simulating order book dynamics and studying various scaling limits of discrete order book models. We show that our framework includes previous models as special cases and yields insights into the interplay between order flow and price dynamics. This talk is based on joint work with Rama Cont and Pierre Degond.

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.
 

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

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.

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.

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.

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.

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.

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. 
 

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.

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.)

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.