Past

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 .

Speaker: Dr Aleksander Horawa (North Wing)
Title: Bitcoin, elliptic curves, and this building

Abstract:
We will discuss two motivations to work on Algebraic Number Theory: applications to cryptography, and fame and fortune. For the first, we will explain how Bitcoin and other companies use Elliptic Curves to digitally sign messages. For the latter, we will introduce two famous problems in Number Theory: Fermat's Last Theorem, worth a name on this building, and the Birch Swinnerton--Dyer conjecture, worth $1,000,000 according to some people in this building (Clay Mathematics Institute).
 

Speaker: Dr Jemima Tabeart (South Wing)
Title: Numerical linear algebra for weather forecasting

Abstract:
The quality of a weather forecast is strongly determined by the accuracy of the initial condition. Data assimilation methods allow us to combine prior forecast information with new measurements in order to obtain the best estimate of the true initial condition. However, many of these approaches require the solution an enormous least-squares problem. In this talk I will discuss some mathematical and computational challenges associated with data assimilation for numerical weather prediction, and show how structure-exploiting numerical linear algebra approaches can lead to theoretical and computational improvements.

Simulating diffusion computationally allows to predict the diffusivity of materials, understand diffusion mechanisms, and to tailor-make materials such as solid-state electrolytes with desired properties aiming at developing new batteries. By studying the geometry and topology of 3-periodic scalar fields (e.g. the potential of ions in the electrolyte), we develop a cost-efficient multi-scale model for diffusion in crystalline materials. This project is a typical example of a collaboration in the overlap of topology and materials science that started as a persistent homology project and turned into something else.

Nanopore sequencing is a method to infer the sequence of nucleotides in DNA or RNA molecules from small variations in ionic current during transit through a nanoscale pore. We will give an introduction to nanopore sequencing and some of its applications and then explore simple models of the signal generation process. These can provide insight to guide optimisation of the system and inform the design of more flexible neural network models, capable of extracting the rich contextual information required for accurate sequence inference.

The crepant resolutions of a singular threefold are related by a finite sequence of birational maps called flops. In the simplest cases, this network of flops is governed by simple combinatorics. I will begin the talk with an overview of flops and crepant resolutions. I will then move on to explain how their underlying combinatorial structure can be abstracted to define the notion of a cluster category.

Toward a characterization of modularity using shatter functions, we show that an o-minimal expansion of the  real ordered additive group $(\mathbb{R}; 0, +,<)$ does not define restricted multiplication if and only if the shatter function of every definable family is asymptotic to a polynomial. Our result implies that vc-density can only take integer values in $(\mathbb{R}; 0, +,<)$ confirming a special case of a conjecture by Chernikov. (Joint with Abdul Basit.)

Assuming the Riemann Hypothesis, Soundararajan established almost sharp upper bounds for all positive moments of the Riemann zeta-function. This result was later improved by Harper, who proved upper bounds of the right order of magnitude. I will describe some of the ideas in their proofs, and then discuss recent joint work with Alexandra Florea, where we consider negative moments of the Riemann zeta-function. For example, we can obtain asymptotic formulas for negative moments when the shift in the zeta function is large enough, confirming a conjecture of Gonek.  We also obtain an upper bound for the average of the generalised Mobius function.

In this talk, we first review the de Rham complex and the finite element exterior calculus, a cohomological framework for structure-preserving discretisation of PDEs. From de Rham complexes, we derive other complexes with applications in elasticity, geometry and general relativity. The derivation, inspired by the Bernstein-Gelfand-Gelfand (BGG) construction, also provides a general machinery to establish results for tensor-valued problems (e.g., elasticity) from de Rham complexes (e.g., electromagnetism and fluid mechanics). We discuss some applications and progress in this direction, including mechanics models and the construction of bounded homotopy operators (Poincaré integrals) and finite elements.

In this presentation, I will discuss the construction and results of numerical simulations quantifying flows around several species of soft corals. In the first project, the flows near the tentacles of xeniid soft corals are quantified for the first time. Their active pulsations are thought to enhance their symbionts' photosynthetic rates by up to an order of magnitude. These polyps are approximately 1 cm in diameter and pulse at frequencies between approximately 0.5 and 1 Hz. As a result, the frequency-based Reynolds number calculated using the tentacle length and pulse frequency is on the order of 10 and rapidly decays as with distance from the polyp. This introduces the question of how these corals minimize the reversibility of the flow and bring in new volumes of fluid during each pulse. We estimate the Péclet number of the bulk flow generated by the coral as being on the order of 100–1000 whereas the flow between the bristles of the tentacles is on the order of 10. This illustrates the importance of advective transport in removing oxygen waste. In the second project, the flows through the elaborate branching structures of gorgonian colonies are considered.  As water moves through the elaborate branches, it is slowed, and recirculation zones can form downstream of the colony. At the smaller scale, individual polyps that emerge from the branches expand their tentacles, further slowing the flow. At the smallest scale, the tentacles are covered in tiny pinnules where exchange occurs. We quantified the gap to diameter ratios for various gorgonians at the scale of the branches, the polyp tentacles and the pinnules. We then used computational fluid dynamics to determine the flow patterns at all three levels of branching. We quantified the leakiness between the branches, tentacles and pinnules over the biologically relevant range of Reynolds numbers and gap-to-diameter ratios, and found that the branches and tentacles can act as either leaky rakes or solid plates depending upon these dimensionless parameters. The pinnules, in contrast, mostly impede the flow. Using an agent-based modeling framework, we quantified plankton capture as a function of the gap-to diameter ratio of the branches and the Reynolds number. We found that the capture rate depends critically on both morphology and Reynolds number. 

The Lott-Sturm-Villani curvature-dimension condition CD(K,N) provides a synthetic notion for a metric measure space to have curvature bounded from below by K and dimension bounded from above by N. It was proved by Juillet that the CD(K,N) condition is not satisfied in a large class of sub-Riemannian manifolds, for every choice of the parameters K and N. In a joint work with Tommaso Rossi, we extended this result to the setting of almost-Riemannian manifolds and finally it was proved in full generality by Rizzi and Stefani. In this talk I present the ideas behind the different strategies, discussing in particular their possible adaptation to the sub-Finsler setting. Lastly I show how studying the validity of the CD condition in sub-Finsler Carnot groups could help in proving rectifiability of CD spaces.

Mike will briefly describe the scope and shape of science within the Met Office and of his career there. He will also outline the coordination of the development of the NEMO ocean model, which he leads, and work to ensure the marine systems at the Met Office work efficiently on modern High Performance Computers (HPCs).  In the second half of the talk, Mike will focus on two of his current scientific interests: accurate calculation of horizontal pressure forces in models with steeply sloping coordinates; and dynamical interpretations of meridional overturning circulations and ocean heat uptake.

Mathematicians frequently present their own work in a diachronic fashion, e.g. by comparing their "modern" methods to those supposedly of the "Ancients," or by situating their latest theories as an "abstract" counterpart to more "classical" ones. The construction of such contrasts entangle mathematical labour and cultural life writ large. Indeed, it involves on the part of mathematicians the shaping up of correspondences between their technical achievements and intellectual discussions taking place on a much broader stage, such as those surrounding the concept of modernity, its relation to an imagined ancient past, or the characterisation of scientific progress as an increase in abstraction. This talk will explore the creation and use of such mathematical diachronies, the focus being on the works of Felix Klein, Hieronymus Zeuthen, and Hermann Schubert.

Stable commutator length (scl) is a measure of homological complexity in groups that has attracted attention for its various connections with geometric topology and group theory. In this talk, I will introduce scl and discuss the (hard) problem of computing scl in surface groups. I will present some results concerning isometric embeddings of free groups for scl, and how they generalise to surface groups for the relative Gromov seminorm.

The joint spectral radius of a finite family S of matrices measures the rate of exponential growth of the maximal norm of an element from the product set S^n as n grows. This notion was introduced by Rota and Strang in the 60s. It arises naturally in a number of contexts in pure and applied mathematics. I will discuss its basic properties and focus on a formula of Berger and Wang and results of J. Bochi that extend to several matrices the classical for formula of Gelfand that relates the growth rate of the powers of a single matrix to its spectral radius. I give new proofs and derive explicit estimates with polynomial dependence on the dimension, refining these results. If time permits I will also discuss connections with the Tits alternative, the notion of joint spectrum, and a geometric version of these results regarding groups acting on non-positively curved spaces.

In 1977, Milnor formulated the following conjecture: every discrete group of affine transformations acting properly on the affine space is virtually solvable. We now know that this statement is false; the current goal is to gain a better understanding of the counterexamples to this conjecture. Every group that violates this conjecture "lives" in a certain algebraic affine group, which can be specified by giving a linear group and a representation thereof. Representations that give rise to counterexamples are said to be non-Milnor. We will talk about the progress made so far towards classification of these non-Milnor representations.

At the heart of all quasi-Newton methods is an update rule that enables us to gradually improve the Hessian approximation using the already available gradient evaluations. Theoretical results show that the global performance of optimization algorithms can be improved with higher-order derivatives. This motivates an investigation of generalizations of quasi-Newton update rules to obtain for example third derivatives (which are tensors) from Hessian evaluations. Our generalization is based on the observation that quasi-Newton updates are least-change updates satisfying the secant equation, with different methods using different norms to measure the size of the change. We present a full characterization for least-change updates in weighted Frobenius norms (satisfying an analogue of the secant equation) for derivatives of arbitrary order. Moreover, we establish convergence of the approximations to the true derivative under standard assumptions and explore the quality of the generated approximations in numerical experiments.

A regular bipartite tournament is an orientation of a complete balanced bipartite graph $K_{2n,2n}$ where every vertex has its in- and outdegree both equal to $n$. In 1981, Jackson conjectured that any regular bipartite tournament can be decomposed into Hamilton cycles. We prove this conjecture for sufficiently large $n$. Along the way, we also prove several further results, including a conjecture of Liebenau and Pehova on Hamilton decompositions of dense bipartite digraphs.

(jt work with David Cushing and George Stagg)

Prolog is a rather unusual programming language that was developed by Alain Colmerauer 50 years ago in one of the buildings on the way to the CIRM in Luminy. It is a declarative language that operates on a paradigm of first-order logic -- as distinct from imperative languages like C, GAP and Magma. Prolog operates by loading in a list of axioms as input, and then responds at the command line to queries that ask the language to achieve particular goals, given those axioms. It gained some notoriety through IBM’s implementation of ‘Watson’, which was a system designed to play the game show Jeopardy. Through a very efficiently implemented constraint logic programming module, it is also the worlds fastest sudoku solver. However, it has had barely any serious employment by pure mathematicians. So the aim of this talk is to advertise Prolog through an extended example: my co-authors and I used it to search for new simple Lie algebras over the field GF(2) and were able to classify a certain flavour of absolutely simple Lie algebra in dimensions 15 and 31, discovering a dozen or so new examples. With some further examples in dimension 63, we then extrapolated two previously undocumented infinite families of simple Lie algebras.

Sketch-and-precondition techniques are popular for solving large overdetermined least squares (LS) problems. This is when a right preconditioner is constructed from a randomized 'sketch' of the matrix. In this talk, we will see that the sketch-and-precondition technique is not numerically stable for ill-conditioned LS problems. We propose a modifciation: using an unpreconditioned iterative LS solver on the preconditioned matrix. Provided the condition number of A is smaller than the reciprocal of the unit round-off, we show that this modification ensures that the computed solution has a comparable backward error to the iterative LS solver applied to a well-conditioned matrix. Using smoothed analysis, we model floating-point rounding errors to provide a convincing argument that our modification is expected to compute a backward stable solution even for arbitrarily ill-conditioned LS problems.

Shortwave radiation penetrating beneath an ice-sheet surface can cause internal melting and the formation of a near-surface porous layer known as the weathering crust, a dynamic hydrological system that provides home to impurities and microbial life. We develop a mathematical model, incorporating thermodynamics and population dynamics, for the evolution of such layers. The model accounts for conservation of mass and energy, for internal and surface-absorbed radiation, and for logistic growth of a microbial species mediated by nutrients that are sourced from the melting ice. I will discuss one-dimensional steadily melting solutions of the model, which suggest a range of changes in behaviour of the weathering crust and its microbial community in response to climate change. In addition, time-dependent solutions of the model give insight into the formation and removal of the weathering crust in frequently changing weather conditions.

Factorization homology is an arguably abstract formalism which produces
well-behaved topological invariants out of certain "higher algebraic"
structures. In this talk, I'll explain how this formalism can be made
fairly concrete in the case where this input algebraic structure is a
braided tensor category. If the category at hand is semi-simple, this in
fact essentially recovers skein categories and skein algebras. I'll
present various applications of this formalism to quantum topology and
representation theory.
 

Given a finite graph, the arboreal gas is the measure on forests (subgraphs without cycles) in which each edge is weighted by a parameter β greater than 0. Equivalently this model is bond percolation conditioned to be a forest, the independent sets of the graphic matroid, or the q→0 limit of the random cluster representation of the q-state Potts model. Our results rely on the fact that this model is also the graphical representation of the nonlinear sigma model with target space the fermionic hyperbolic plane H^{0|2}, whose symmetry group is the supergroup OSp(1|2).

The main question we are interested in is whether the arboreal gas percolates, i.e., whether for a given β the forest has a connected component that includes a positive fraction of the total edges of the graph. We show that in two dimensions a Mermin-Wagner theorem associated with the OSp(1|2) symmetry of the nonlinear sigma model implies that the arboreal gas does not percolate for any β greater than 0. On the other hand, in three and higher dimensions, we show that percolation occurs for large β by proving that the OSp(1|2) symmetry of the non-linear sigma model is spontaneously broken. We also show that the broken symmetry is accompanied by massless fluctuations (Goldstone mode). This result is achieved by a renormalisation group analysis combined with Ward identities from the internal symmetry of the sigma model.

In this talk I'll discuss some applications of the mean curvature flow to self shrinkers in R^3 and R^4.

Sparse inversion and classification problems are ubiquitous in modern data science and imaging. They are often formulated as non-smooth minimisation problems. In sparse inversion, we minimise, e.g., the sum of a data fidelity term and an L1/LASSO regulariser. In classification, we consider, e.g., the sum of a data fidelity term and a non-smooth Ginzburg--Landau energy. Standard (sub)gradient descent methods have shown to be inefficient when approaching such problems. Splitting techniques are much more useful: here, the target function is partitioned into a sum of two subtarget functions -- each of which can be efficiently optimised. Splitting proceeds by performing optimisation steps alternately with respect to each of the two subtarget functions.

In this work, we study splitting from a stochastic continuous-time perspective. Indeed, we define a differential inclusion that follows one of the two subtarget function's negative subdifferential at each point in time. The choice of the subtarget function is controlled by a binary continuous-time Markov process. The resulting dynamical system is a stochastic approximation of the underlying subgradient flow. We investigate this stochastic approximation for an L1-regularised sparse inversion flow and for a discrete Allen-Cahn equation minimising a Ginzburg--Landau energy. In both cases, we study the longtime behaviour of the stochastic dynamical system and its ability to approximate the underlying subgradient flow at any accuracy. We illustrate our theoretical findings in a simple sparse estimation problem and also in low- and high-dimensional classification problems.

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).