Past

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.

The term cancer covers a multitude of bodily diseases, broadly categorised by having cells which do not behave normally. Cancer cells can arise from any type of cell in the body; cancers can grow in or around any tissue or organ making the disease highly complex. My research is focused on understanding the specific mechanisms that occur in the tumour microenvironment via mathematical and computational modelling. In this talk I shall present a 3D individual-based force-based model for tumour growth and development in which we simulate the behaviour of, and spatio-temporal interactions between, cells, extracellular matrix fibres and blood vessels. Each agent is fully realised, for example, cells are described as viscoelastic sphere with radius and centre given within the off-lattice model. Interactions are primarily governed by mechanical forces between elements. However, as well as he mechanical interactions we also consider chemical interactions, by coupling the code to a finite element solver to model the diffusion of oxygen from blood vessels to cells, as well as intercellular aspects such as cell phenotypes. 

Quantum toroidal algebras $U_{q}(\mathfrak{g}_{\mathrm{tor}})$ are certain Drinfeld quantum affinizations of quantum groups associated to affine Lie algebras, and can therefore be thought of as `double affine quantum groups'.

In particular, they contain (and are generated by) a horizontal and vertical copy of the affine quantum group. 

Utilising an extended double affine braid group action, Miki obtained in type $A$ an automorphism of $U_{q}(\mathfrak{g}_{\mathrm{tor}})$ which exchanges these subalgebras. This has since played a crucial role in the investigation of its structure and representation theory.

In this talk I shall present my recent work -- which extends the braid group action to all types and generalises Miki's automorphism to the ADE case -- as well as potential directions for future work in this area.

Given a finite set A of integer lattice points in d dimensions, let NA denote the N-fold iterated sumset (i.e. the set comprising all sums of N elements from A). In 1992 Khovanskii observed that there is a fixed polynomial P(N), depending on A, such that the size of the sumset NA equals P(N) exactly (once N is sufficiently large, that is). In addition to this 'size stability', there is a related 'structural stability' property for the sumset NA, which Granville and Shakan recently showed also holds for sufficiently large N. But what does 'sufficiently large' mean in practice? In this talk I will discuss some perspectives on these questions, and explain joint work with Granville and Shakan which proves the first explicit bounds for all sets A. I will also discuss current work with Granville, which gives a tight bound 'up to logarithmic factors' for one of these properties. 

Skeletal joints are often modelled as two adjacent layers of poroviscoelastic cartilage that are permitted to slide past each other.  The talk will begin by outlining a mathematical model that may be used, focusing on two unusual features of the model: (i) the solid component of the poroviscoelastic body has a charged surface that ionises the fluid within the pores, generating a swelling pressure; and (ii) appropriate conditions are required at the interface between the two adjacent layers of cartilage.  The remainder of the talk will then address various theoretical and practical issues in computing a finite element solution of the governing equations.

We investigate the long-time behaviour of solutions to a nonlocal partial differential equation on smooth Riemannian manifolds of bounded sectional curvature. The equation models self-collective behaviour with intrinsic interactions that are modeled by an interaction potential. Without the diffusion term, we consider attractive interaction potentials and establish sufficient conditions for a consensus state to form asymptotically. In addition, we quantify the approach to consensus, by deriving a convergence rate for the diameter of the solution’s support. With the diffusion term, the attractive interaction and the diffusion compete. We provide the conditions of the attractive interaction for each part to win.

By definition, planktonic organisms drift with the water flows. But these millimetric organisms are not necessarily passive; many can swim and can sense the surrounding flow through mechanosensory hairs. But how useful can be flow sensing in a turbulent environment? To address this question, we show two examples of smart planktonic behavior: (1) we develop a model where plantkters choose a swimming direction as a function of the velocity gradient to "surf on turbulence" and move efficiently upwards; (2) we show how a plankter measuring the velocity gradient may track the position of a swimming target from its hydrodynamic signature. 

Ernst Haeckel, Kunstformen der Natur (1904), Copepoda 

Given a mathematical object, what can you compute about it? In some settings, you cannot say very much. Given an arbitrary group presentation, for example, there is no procedure to decide whether the group is trivial. In 3-manifolds, however, algorithms are a fruitful and active area of study (and some of them are even implementable!). One of the main tools in this area is normal surface theory, which allows us to describe interesting surfaces in a 3-manifold with respect to an arbitrary triangulation. I will discuss some results in this area, particularly around Seifert fibered spaces.

We will discuss the pros and cons of targets vs quotas in increasing diversity in Mathematics.

By Jewett-Krieger theorems minimal dynamical systems on the Cantor set are topological analogous of ergodic systems on probability Lebesgue spaces. In this analogy and to study a Cantor minimal system, indicator functions of clopen sets (as continuous integer or real valued functions) are considered while they are mod out by the subgroup of all co-boundary functions. That is how dimension group which is an operator algebraic object appears in dynamical systems. In this talk, I try to explain a bit more about dimension groups from dynamical point of view and how it relates to topological factoring and spectrum of Cantor minimal systems.

Bounded cohomology is a functional-analytic analogue of ordinary cohomology that has become a fundamental tool in many fields, from rigidity theory to the geometry of manifolds. However it is infamously hard of compute, and the lack of very basic examples makes the overall picture still hard to grasp. I will report on recent progress in this direction, and draw attention to some natural questions that remain open.

Random greedy algorithms became ubiquitous in Combinatorics after Rödl's nibble (semi-random method), which was repeatedly refined for various applications, such as iterative graph colouring algorithms (Molloy-Reed) and lower bounds for the Ramsey number $R(3,t)$ via the triangle-free process (Bohman-Keevash / Fiz Pontiveros-Griffiths-Morris). More recently, when combined with absorption, they have played a key role in many existence and approximate counting results for combinatorial structures, following a paradigm established by my proofs of the Existence of Designs and Wilson's Conjecture on the number of Steiner Triple Systems. Here absorption (converting approximate solutions to exact solutions) is generally the most challenging task, which has spurred the development of many new ideas, including my Randomised Algebraic Construction method, the Kühn-Osthus Iterative Absorption method and Montgomery's Addition Structures (for attacking the Ryser-Brualdi-Stein Conjecture). The design and analysis of a suitable guiding mechanism for the random process can also come with major challenges, such as in the recent proof of Erdős' Conjecture on Steiner Triple Systems of high girth (Kwan-Sah-Sawhney-Simkin). This talk will survey some of this background and also mention some recent results on the Queens Problem (Bowtell-Keevash / Luria-Simkin / Simkin) and the Existence of Subspace Designs (Keevash-Sah-Sawhney). I may also mention recent solutions of the Talagrand / Kahn-Kalai Threshold Conjectures (Frankston-Kahn-Narayanan-Park / Park-Pham) and thresholds for Steiner Triple Systems / Latin Squares (Keevash / Jain-Pham), where the key to my proof is constructing a suitable spread measure via a guided random process.

Perverse sheaves are an indispensable tool in representation theory. Their stalks often encode important representation theoretic information such as composition multiplicities or canonical bases. For the nilpotent cone, there is an algorithm that computes these stalks, known as the Lusztig-Shoji algorithm. In this talk, we discuss how this algorithm can be modified to compute stalks of perverse sheaves on more general varieties. As an application, we obtain a new algorithm for computing canonical bases in certain quantum groups as well as composition multiplicities for standard modules of the affine Hecke algebra of $\mathrm{GL}_n$.

In mathematical modelling, data and solutions are often represented as measurable functions, and their quality is being captured by their membership to a certain function space. One of the core questions arising in applications of this approach is the comparison of the quality of the data and that of the solution. A particular attention is being paid to optimality of the results obtained. A delicate choice of scales of suitable function spaces is required in order to balance the expressivity (the ability to capture fine mathematical properties of the model) and the accessibility (the level of its technical difficulty) for a practical use. We will give an overview of the research area which grew out of these questions and survey recent results obtained in this direction as well as challenging open questions. We will describe a development of a powerful method based on the so-called reduction principles and demonstrate its use on specific problems including the continuity of Sobolev embeddings or boundedness of pivotal integral operators such as the Hardy - Littlewood maximal operator and the Laplace transform.

We ask to what extend the SL(2,C)-character variety of the
fundamental group of the complement of a knot in S^3 determines the
knot. Our methods use results from group theory, classical 3-manifold
topology, but also geometric input in two ways: the geometrisation
theorem for 3-manifolds, and instanton gauge theory. In particular this
is connected to SU(2)-character varieties of two-component links, a
topic where much less is known than in the case of knots. This is joint
work with Michel Boileau, Teruaki Kitano, and Steven Sivek.

We consider an investor who is dynamically informed about the future evolution of one of the independent Brownian motions driving a stock's price fluctuations. The resulting rough semimartingale dynamics allow for strong arbitrage, but with linear temporary price impact the resulting optimal investment problem with exponential utility turns out to be well posed. The dynamically revealed Brownian path segment makes the problem infinite-dimensional, but by considering its convex-analytic dual problem, we show that it still can be solved explicitly and we give some financial-economic insights into the optimal investment strategy and the properties of maximum expected utility. (This is joint work with Yan Dolinsky, Hebrew University of Jerusalem).

It is well understood by works of Fukaya and Teleman that the Fukaya category of a symplectic reduction at a regular value of the moment map can be computed before taking the quotient as an equivariant Fukaya category. Informed by mirror calculations,  we will give a new geometric interpretation of the equivariant Fukaya category corresponding to a singular value of the moment map where the equivariance is traded with wrapping.

Joint work in progress with Ed Segal.

1:30-2:15 Deborah Miori, CDT student, University of Oxford

DeFi: Data-Driven Characterisation of Uniswap v3 Ecosystem & an Ideal Crypto Law for Liquidity Pools

Uniswap is a Constant Product Market Maker built around liquidity pools, where pairs of tokens are exchanged subject to a fee that is proportional to the size of transactions. At the time of writing, there exist more than 6,000 pools associated with Uniswap v3, implying that empirical investigations on the full ecosystem can easily become computationally expensive. Thus, we propose a systematic workflow to extract and analyse a meaningful but computationally tractable sub-universe of liquidity pools.

Leveraging on the 34 pools found relevant for the six-months time window January-June 2022, we then investigate the related liquidity consumption behaviour of market participants. We propose to represent each liquidity taker by a suitably constructed transaction graph, which is a fully connected network where nodes are the liquidity taker’s executed transactions, and edges contain weights encoding the time elapsed between any two transactions. We extend the NLP-inspired graph2vec algorithm to the weighted undirected setting, and employ it to obtain an embedding of the set of graphs. This embedding allows us to extract seven clusters of liquidity takers, with equivalent behavioural patters and interpretable trading preferences.

We conclude our work by testing for relationships between the characteristic mechanisms of each pool, i.e. liquidity provision, consumption, and price variation. We introduce a related ideal crypto law, inspired from the ideal gas law of thermodynamics, and demonstrate that pools adhering to this law are healthier trading venues in terms of sensitivity of liquidity and agents’ activity. Regulators and practitioners could benefit from our model by developing related pool health monitoring tools.

2:15-3:00 Žan Žurič, CDT student, Imperial College London

A Random Neural Network Approach to Pricing SPDEs for Rough Volatility

We propose a novel machine learning-based scheme for solving partial differential equations (PDEs) and backward stochastic partial differential equations (BSPDE) stemming from option pricing equations of Markovian and non-Markovian models respectively. The use of the so-called random weighted neural networks (RWNN) allows us to formulate the optimisation problem as linear regression, thus immensely speeding up the training process. Furthermore, we analyse the convergence of the RWNN scheme and are able to specify error estimates in terms of the number of hidden nodes. The performance of the scheme is tested on Black-Scholes and rBergomi models and shown to have superior training times with accuracy comparable to existing deep learning approaches.

A 2d SCFT given as a non-linear sigma model of a Ricci-flat Kahler target 

space is not a rational CFT in general; only special points in the moduli 

space of the target-space metric, the 2d SCFTs are rational. 

Gukov-Vafa's paper in 2002 hinted at a possibility that such special points 

may be characterized by the property "complex multiplication" of the target space, 

which has its origin in number theory. We revisit the idea, refine the Conjecture, 

and prove it in the case the target space is T^4. 
 

This presentation is based on arXiv:2205.10299 and 2212.13028 .