Past

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.

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.

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.

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.

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

TBA