Past

5d Superconformal Field Theories (SCFTs) are intrinsically strongly-coupled UV fixed points, whose realization hinges on string theoretic methods: they can be constructed by compactifying M-theory on local Calabi-Yau threefold singularities or alternatively from the world-volume of 5-brane-webs in type IIB string theory. There is a correspondence between 5-brane-webs and toric Calabi-Yau threefolds, however this breaks down when multiple 5-branes are allowed to end on a single 7-brane. In this talk, we extend this connection and provide a geometric realization of brane configurations including 7-branes. Along the way, we also review techniques developed in the past few years to describe the Higgs branch of these 5d SCFTs, including magnetic quivers and Hasse diagram for symplectic singularities. 

This session will introduce the opportunities for entrepreneurship and generating commercial impact available to researchers and students across MPLS. Representatives from the Maths Institute and across the university will discuss training and resources to help you begin enterprising and develop your ideas. We will hear from Paul Gass and Dawn Gordon about the support that can be provided by Oxford University Innovation, discussing commercialisation of research findings, consultancy, utilising your expertise and the protection and licensing of Intellectual Property. 

Please see below slides from the talk:

20230217 Short Seminar - Maths Fri@4_FINAL- Dept (1)_0.pdf

Introductory talk Maths 2023.pdf

Leadership and Innovation Presentation.pdf

There are several ways of defining the persistence diagram, but the definition using the Möbius inversion formula (for posets) offers the greatest amount of flexibility. There are now many variations of the so called Generalized Persistence Diagrams by many people.  In this talk, I will focus on the approach I am developing. I will cover the state-of-the-art and where I see this work going.

The Turing pattern is a key concept in the modern study of reaction-diffusion systems, with Turing patterns proposed a possible explanation for the spatial structure observed in myriad physical, chemical, and biological systems. Real-world systems are not always so clean as idealized Turing systems, and in this talk we will take up the case of more messy reaction-diffusion systems involving explicit space or time dependence in diffusion or reaction terms. Turing systems of this nature arise in several applications, such as when a Turing system is studied on a growing substrate, is subjected to a temperature gradient, or is immersed within a fluid flow. The analysis of these messy Turing systems is not as straightforward as in the idealized case, with the explicit space or time dependence greatly complicating or even preventing most standard routes of analysis. Motivated by patterning in phenomena involving explicit space or time dependence, and by the interesting mathematical challenges inherent in the study of such systems, in this talk we consider the following questions:

* Is it possible to obtain generalizations of the Turing instability conditions for non-autonomous, spatially heterogeneous reaction-diffusion systems?
* What can one say about predicting nascent patterns in these systems?
* What role does explicit space or time dependence play in selecting fully developed patterns in these systems?
* Is it possible to exploit this space or time dependence in order to manipulate the form of emergent patterns?

We will also highlight some of the applications opened up by the analysis of heterogeneous Turing systems.

We will define a notion of a semi-retraction between two first-order structures introduced by Scow. We show how a semi-retraction encodes Ramsey problems of finitely-generated substructes of one structure into the other under the most general conditions. We will compare semi-retractions to a category-theoretic notion of pre-adjunction recently popularized by Masulovic. We will accompany the results with examples and questions. This is a joint work with Lynn Scow.

The Uniswap v3 ecosystem is built upon liquidity pools, where pairs of tokens are exchanged subject to a fee. We propose a systematic workflow to extract a meaningful but tractable sub-universe out of the current > 6,000 pools. We filter by imposing minimum levels on individual pool features, e.g. liquidity locked and agents’ activity, but also maximising the interconnection between the chosen pools to support broader dynamics. Then, we investigate liquidity consumption behaviour on the most relevant pools for Jan-June 2022. We propose to describe each liquidity taker by a transaction graph, which is a complete graph where nodes are transactions on pools and edges have weights from the time elapsed between pairs of transactions. Each graph is embedded into a vector by our own variant of the NLP rooted graph2vec algorithm. Thus, we are able to investigate the structural equivalence of liquidity takers behaviour and extract seven clusters with interpretable features. Finally, we introduce an ideal crypto law inspired from the ideal gas law of thermodynamics. Our model tests a relationship between variables that govern the mechanisms of each pool, i.e. liquidity provision, consumption, and price variation. If the law is satisfied, we say the pool has high cryptoness and demonstrate that it constitutes a better venue for the activity of market participants. Our metric could be employed by regulators and practitioners for developing pool health monitoring tools and establishing minimum levels of requirements.

Conditional on finiteness of relevant Shafarevich--Tate groups, Harpaz and Skorobogatov established the Hasse principle for Kummer varieties associated to a 2-covering of a principally polarised abelian variety A, under certain large image assumptions on the Galois action on A[2]. However, their method stops short of treating the case where the image is the full symplectic group, due to the possible failure of the Shafarevich--Tate group to have square order in this setting. I will explain work in progress which overcomes this obstruction by combining second descent ideas of Harpaz with new results on the 2-parity conjecture. 

The neutron transport equation is a linear Boltzmann-type PDE that models radiative transfer processes, and fission nuclear reactions. The computation of the largest eigenvalue of this Boltzmann operator is crucial in nuclear safety studies but it has classically been formulated only at a discretized level, so the predictive capabilities of such computations are fairly limited. In this talk, I will give an overview of the modeling for this equation, as well as recent analysis that leads to an infinite dimensional formulation of the eigenvalue problem. We leverage this point of view to build a numerical scheme that comes with a rigorous, a posteriori estimation of the error between the exact, infinite-dimensional solution, and the computed one.

Andrea Giudici: Multiple shapes from one elastomer sheet

Active soft materials, such as Liquid Crystal Elastomers (LCEs), possess a unique property: the ability to change shape in response to thermal or optical stimuli. This makes them attractive for various applications, including bioengineering, biomimetics, and soft robotics. The classic example of a shape change in LCEs is the transformation of a flat sheet into a complex curved surface through the imprinting of a spatially varying deformation field. Despite its effectiveness, this approach has one important limitation: once the deformation field is imprinted in the material, it cannot be amended, hindering the ability to achieve multiple target shapes.

In this talk, I present a solution to this challenge and discuss how modulating the degree of actuation using light intensity offers a route towards programming multiple shapes. Moreover, I introduce a theoretical framework that allows us to sculpt any surface of revolution using light.

Edwina Yeo: Modelling the onset of arterial blood clotting

Arterial blood clot formation (thrombosis) is the leading cause of both stroke and heart attack. The blood protein Von Willebrand Factor (VWF) is critical in facilitating arterial thrombosis. At pathologically high shear rates the protein unfolds and rapidly captures platelets from the flow.

I will present two pieces of modelling to predict the location of clot formation in a diseased artery. Firstly a continuum model to describe the mechanosensitive protein VWF and secondly a model for platelet transport and deposition to VWF. We interface this model with in vitro data of thrombosis in a long, thin rectangular microfluidic geometry. Using a reduced model, the unknown model parameters which determine platelet deposition can be calibrated.

Although Green's theorem, currently considered one of the cornerstones of multivariate calculus, was published in 1828, its widespread introduction into calculus textbooks can be traced back to the first decades of the twentieth century, when vector calculus emerged as a slightly autonomous discipline. In addition, its contemporary version (and its demonstration), currently found in several calculus textbooks, is the result of some adaptations during its almost 200 years of life. Comparing some books and articles from this long period, I would like to discuss in this lecture the didactic adaptations, the editorial strategies and visual representations that shaped the theorem in its current form.

Although tensor products and their symmetrisation have appeared in mathematical literature since at least the mid-nineteenth century, they rarely appear in the function-theoretic operator theory literature. In this talk, I will introduce the symmetric and antisymmetric tensor products from an operator theoretic point of view. I will present results concerning some of the most fundamental operator-theoretic questions in this area, such as finding the norm and spectrum of the symmetric tensor products of operators. I will then work through some examples of symmetric tensor products of familiar operators, such as the unilateral shift, the adjoint of the shift, and diagonal operators.

The talk is based on joint work with Uri Bader. We prove that arithmetic lattices in a semisimple Lie group G satisfy a higher-degree version of property T below the rank of G. The proof relies on functional analysis and the polynomiality of higher Dehn functions of arithmetic lattices below the rank and avoids any automorphic machinery. If time permits, we describe applications to the cohomology and stability of arithmetic groups (the latter being joint work with Alex Lubotzky and Shmuel Weinberger).

Let $p \ge 3$ be a prime integer. The density of a non-empty solution set of a system of affine equations $L_i(x) = b_i$, $i=1,\dots,k$ on a vector space over the field $\mathbb{F}_p$ is determined by the dimension of the linear subspace $\langle L_1,\dots,L_k \rangle$, and tends to $0$ with the dimension of that subspace. In particular, if the solution set is dense, then the system of equations contains at most boundedly many pairwise distinct linear forms. In the more general setting of systems of affine conditions $L_i(x) \in E_i$ for some strict subsets $E_i$ of $\mathbb{F}_p$ and where the solution set and its density are taken inside $S^n$ for some non-empty subset $S$ of $\mathbb{F}_p$ (such as $\{0,1\}$), we can however usually find subsets of $S^n$ with density at least $1/2$ but such that the linear subspace $\langle L_1,\dots,L_k \rangle$ has arbitrarily high dimension. We shall nonetheless establish an approximate boundedness result: if the solution set of a system of affine conditions is dense, then it contains the solution set of a system of boundedly many affine conditions and which has approximately the same density as the original solution set. Using a recent generalisation by Gowers and the speaker of a result of Green and Tao on the equidistribution of high-rank polynomials on finite prime fields we shall furthermore prove a weaker analogous result for polynomials of small degree.

Based on joint work with Timothy Gowers (College de France and University of Cambridge).

***CANCELLED*** In this talk, I will discuss recent results related to the mathematical justification of PDEs which model multi-phase flows at the macroscopic level from mesoscopic descriptions with jump conditions at interfaces. We will also present interesting and difficult open problems.

The notion of a pseudocharacter was introduced by A.Wiles for GL_2 and generalized by R.Taylor to GL_n. It is a tool that allows us to deal with the
deformation theory of a residually reducible Galois representation when the usual techniques fail. G.Chenevier gave an alternative theory of "determinants" extending that of pseudocharacters to arbitrary rings. In this talk we will discuss some aspects of this theory and introduce a similar definition in the case of the symplectic group, which is the subject of a forthcoming work joint with J.Quast. 

I will describe the proof of the following classification result for manifolds with positive scalar curvature. Let M be a closed manifold of dimension $n=4$ or $5$ that is "sufficiently connected", i.e. its second fundamental group is trivial (if $n=4$) or second and third fundamental groups are trivial (if $n=5$). Then a finite covering of $M$ is homotopy equivalent to a sphere or a connect sum of $S^{n-1} \times S^1$. The proof uses techniques from minimal surfaces, metric geometry, geometric group theory. This is a joint work with Otis Chodosh and Chao Li.
 

Using rough path techniques, we provide a priori estimates for the output of Deep Residual Neural Networks in terms of both the input data and the (trained) network weights. As trained network weights are typically very rough when seen as functions of the layer, we propose to derive stability bounds in terms of the total p-variation of trained weights for any p∈[1,3]. Unlike the C1-theory underlying the neural ODE literature, our estimates remain bounded even in the limiting case of weights behaving like Brownian motions, as suggested in [Cohen-Cont-Rossier-Xu, "Scaling Properties of Deep Residual Networks”, 2021]. Mathematically, we interpret residual neural network as solutions to (rough) difference equations, and analyse them based on recent results of discrete time signatures and rough path theory. Based on joint work with C. Bayer and P. K. Friz.
 

Glueing construction has been used extensively to construct solutions to nonlinear geometric PDEs. In this talk, I will focus on the glueing construction of solutions to Lagrangian mean curvature flow. Specifically, I will explain the construction of Lagrangian translating solitons by glueing a small special Lagrangian 'Lawlor neck' into the intersection point of suitably rotated Lagrangian Grim Reaper cylinders. I will also discuss an ongoing joint project with Chung-Jun Tsai and Albert Wood, where we investigate the construction of solutions to Lagrangian mean curvature flow with infinite time singularities.

In her recent work, Mina Aganagic proposed novel perspectives on computing knot homologies associated with any simple Lie algebra. One of her proposals relies on counting intersection points between Lagrangians in Landau-Ginsburg models on symmetric powers of Riemann surfaces. In my talk, I am going to present a concrete algebraic algorithm for finding such intersection points, turning the proposal into an actual calculational tool. I am going to illustrate the construction on the example of the sl_2 invariant for the Hopf link. I am also going to comment on the extension of the story to homological invariants associated to gl(m|n) super Lie algebras, solving this long-standing problem. The talk is based on our work in progress with Mina Aganagic and Elise LePage.

What is curiosity? Is it an emotion? A behavior? A cognitive process? Curiosity seems to be an abstract concept—like love, perhaps, or justice—far from the realm of those bits of nature that mathematics can possibly address. However, contrary to intuition, it turns out that the leading theories of curiosity are surprisingly amenable to formalization in the mathematics of network science. In this talk, I will unpack some of those theories, and show how they can be formalized in the mathematics of networks. Then, I will describe relevant data from human behavior and linguistic corpora, and ask which theories that data supports. Throughout, I will make a case for the position that individual and collective curiosity are both network building processes, providing a connective counterpoint to the common acquisitional account of curiosity in humans.

What is curiosity? Is it an emotion? A behavior? A cognitive process? Curiosity seems to be an abstract concept—like love, perhaps, or justice—far from the realm of those bits of nature that mathematics can possibly address. However, contrary to intuition, it turns out that the leading theories of curiosity are surprisingly amenable to formalization in the mathematics of network science. In this talk, I will unpack some of those theories, and show how they can be formalized in the mathematics of networks. Then, I will describe relevant data from human behavior and linguistic corpora, and ask which theories that data supports. Throughout, I will make a case for the position that individual and collective curiosity are both network building processes, providing a connective counterpoint to the common acquisitional account of curiosity in humans.