Past

Multibody interactions can reveal higher-order dynamical effects that are not captured by traditional two-body network models. We derive and analyze models for consensus dynamics on hypergraphs, where nodes interact in groups rather than in pairs. Our work reveals that multibody dynamical effects that go beyond rescaled pairwise interactions can appear only if the interaction function is nonlinear, regardless of the underlying multibody structure. As a practical application, we introduce a specific nonlinear function to model three-body consensus, which incorporates reinforcing group effects such as peer pressure. Unlike consensus processes on networks, we find that the resulting dynamics can cause shifts away from the average system state. The nature of these shifts depends on a complex interplay between the distribution of the initial states, the underlying structure, and the form of the interaction function. By considering modular hypergraphs, we discover state-dependent, asymmetric dynamics between polarized clusters where multibody interactions make one cluster dominate the other.

Building on these results, we generalise the model allowing for interactions within hyper edges of any cardinality and explore in detail the role of involvement and stubbornness on polarisation.

What is fairness, and to what extent is it practically achievable? I'll talk about a simple mathematical model under which one might hope to understand such questions. Red and blue points occur as independent homogeneous Poisson processes of equal intensity in Euclidean space, and we try to match them to each other. We would like to minimize the sum of a some function (say, a power, $\gamma$) of the distances between matched pairs. This does not make sense, because the sum is infinite, so instead we satisfy ourselves with minimizing *locally*. If the points are interpreted as agents who would like to be matched as close as possible, the parameter $\gamma$ encodes a measure of fairness - large $\gamma$ means that we try to avoid occasional very bad outcomes (long edges), even if that means inconvenience to others - small $\gamma$ means everyone is in it for themselves.
    In dimension 1 we have a reasonably complete picture, with a phase transition at $\gamma=1$. For $\gamma<1$ there is a unique minimal matching, while for $\gamma>1$ there are multiple matchings but no stationary solution. In higher dimensions, even existence is not clear in all cases.

In this talk I will explain how theories with local symmetries are treated in perturbative Algebraic Quantum Field Theory (pAQFT). The main mathematical tool used here is the Batalin Vilkovisky (BV) formalism. I will show how the perturbative Master Ward Identity can be applied in this formalism to make sense of the renormalised Quantum Master Equation. I will also comment on perspectives for a non-perturbative formulation.

The risk and return profiles of a broad class of dynamic trading strategies, including pairs trading and other statistical arbitrage strategies, may be characterized in terms of excursions of the market price of a portfolio away from a reference level. We propose a mathematical framework for the risk analysis of such strategies, based on a description in terms of price excursions, first in a pathwise setting, without probabilistic assumptions, then in a Markovian setting.

We introduce the notion of δ-excursion, defined as a path which deviates by δ from a reference level before returning to this level. We show that every continuous path has a unique decomposition into δ-excursions, which is useful for scenario analysis of dynamic trading strategies, leading to simple expressions for the number of trades, realized profit, maximum loss and drawdown. As δ is decreased to zero, properties of this decomposition relate to the local time of the path. When the underlying asset follows a Markov process, we combine these results with Ito's excursion theory to obtain a tractable decomposition of the process as a concatenation of independent δ-excursions, whose distribution is described in terms of Ito's excursion measure. We provide analytical results for linear diffusions and give new examples of stochastic processes for flexible and tractable modeling of excursions. Finally, we describe a non-parametric scenario simulation method for generating paths whose excursion properties match those observed in empirical data.

Joint work with Anna Ananova and Rama Cont: https://ssrn.com/abstract=3723980

Given an extension L/K of number fields, we say that the Hasse norm principle (HNP) holds if every non-zero element of K which is a norm everywhere locally is in fact a global norm from L. If L/K is cyclic, the original Hasse norm theorem states that the HNP holds. More generally, there is a cohomological description (due to Tate) of the obstruction to the HNP for Galois extensions. In this talk, I will present work (joint with Rachel Newton) developing explicit methods to study this principle for non-Galois extensions. As a key application, I will describe how these methods can be used to characterize the HNP for extensions whose normal closure has Galois group A_n or S_n. I will additionally discuss some recent generalizations of these methods to study the Hasse principle and weak approximation for multinorm equations as well as consequences in the statistics of these local-global principles.

Operads are tools to encode operations satisfying algebro-homotopic relations. They have proved to be extremely useful tools, for instance for detecting spaces that are iterated loop spaces. However, in many natural examples, composition of operations is only associative up to homotopy and operads are too strict to captured these phenomena. This leads to the notion of infinity operads. While they are a well-established tool, there are few examples of infinity operads in the literature that are not the nerve of an actual operad. I will introduce new topological operad of bracketed trees that can be used to identify and construct natural examples of infinity operads. The key example for this talk will be the normalised cacti model for genus 0 surfaces.

Glueing surfaces along their boundaries defines composition laws that have been used to construct topological field theories and to compute the homology of the moduli space of Riemann surfaces. Normalised cacti are a graphical model for the moduli space of genus 0 oriented surfaces. They are endowed with a composition that corresponds to glueing surfaces along their boundaries, but this composition is not associative. By using the operad of bracketed trees, I will show that this operation is associative up to all higher homotopies and hence that normalised cacti form an infinity operad.

Given a nested pair X and Y of complex projective varieties, there is a single positive integer e which measures the singularity type of X inside Y. This is called the Hilbert-Samuel multiplicity of Y along X, and it appears in the formulations of several standard intersection-theoretic constructions including Segre classes, Euler obstructions, and various other multiplicities. The standard method for computing e requires knowledge of the equations which define X and Y, followed by a (super-exponential) Grobner basis computation. In this talk we will connect the HS multiplicity to complex links, which are fundamental invariants of (complex analytic) Whitney stratified spaces. Thanks to this connection, the enormous computational burden of extracting e from polynomial equations reduces to a simple exercise in clustering point clouds. In fact, one doesn't even need the polynomials which define X and Y: it suffices to work with dense point samples. This is joint work with Martin Helmer.

In this talk I will present an optical theorem for perturbative CFTs, which directly computes the double discontinuity of CFT correlators in terms of the discontinuities of correlators at lower loops or lower points, in analogy to the optical theoreom for scattering amplitudes. I will then discuss the application of this theorem to high-energy scattering of type IIb strings in AdS at one loop and finite 't Hooft coupling. Tidal excitations are taken into account and shown to be efficiently described by an AdS vertex function. The result is related to the 1987 flat space result of Amati, Ciafaloni and Veneziano via the flat space limit in impact parameter space.

In this new session a speaker tells us about how their area of mathematics can be used in different applications.

In this talk, Yuji Nakatsukasa tells us about how random matrix theory can be used in numerical linear algebra. 

Abstract

Randomized SVD is a topic in numerical linear algebra that draws heavily from random matrix theory. It has become an extremely successful approach for efficiently computing a low-rank approximation of matrices. In particular the paper by Halko, Martinsson, and Tropp (SIREV 2011) contains extensive analysis, and has made it a very popular method. The classical Nystrom method is much faster, but only applicable to positive semidefinite matrices. This work studies a generalization of Nystrom's method applicable to general matrices, and shows that (i) it has near-optimal approximation quality comparable to competing methods, (ii) the computational cost is the near-optimal O(mnlog n+r^3) for a rank-r approximation of dense mxn matrices, and (iii) crucially, it can be implemented in a numerically stable fashion despite the presence of an ill-conditioned pseudoinverse. Numerical experiments illustrate that generalized Nystrom can significantly outperform state-of-the-art methods. In this talk I will highlight the crucial role played by a classical result in random matrix theory, namely the Marchenko-Pastur law, and also briefly mention its other applications in least-squares problems and compressed sensing.

I will lead an informal discussion centered on discrete data that need to be specified when reducing 6d relative theories on an internal manifold M and how they determine symmetries of the resulting theory T[M].

In this talk, I will present the theory of combinatorial multivector fields for finite topological spaces, the main subject of my thesis. The idea of combinatorial vector fields came from Forman and emerged naturally from discrete Morse theory. Lately, Mrozek generalized it to the multivector fields theory for Lefschetz complexes. In our work, we simplified and extended it to the finite topological spaces settings. We developed a combinatorial counterpart for dynamical objects, such as isolated invariant sets, isolating neighbourhoods, Conley index, limit sets, and Morse decomposition. We proved the additivity property of the Conley index and the Morse inequalities. Furthermore, we applied persistence homology to study the evolution and the stability of Morse decomposition. In the last part of the talk, I will show numerical results and potential future directions from a data-analysis perspective. 

 “In this talk, I shall present the past research track passing through quantum mechanical studies of small molecules to biomolecules, to proteome-wide big data analyses and computational genomics. Next, the ongoing research in our group will be presented that builds upon the expertise on different levels of information processing in life (genome, transcriptome, proteins, small molecules), to develop self-consistent “first principles” models in biology with a wide spectrum of usage. The immediate benefits and the targeted processes will be described covering different layers of the central dogma of biology, multigenic diseases and disease driver/passenger mutation predictions."

A Real representation of a $C_2$-graded group $H < G$ ($H$ an index two subgroup) is a complex representation of $H$ with an action of the other coset $G \backslash H$ (“odd" elements) satisfying appropriate algebraic coherence conditions. In this talk I will present three such Real representation theories. In these, each odd element acts as an antilinear operator, a bilinear form or a sesquilinear form (equivalently a linear map to $V$ from the conjugate, the dual, or the conjugate dual of $V$) respectively. I will describe how these theories are related, how representations in each are classified, and how the first generalises the classical representation theory of $H$ over the real numbers - retaining much of its beauty and subtlety.

The classical fractional crystallisation scenario for magma ocean solidification on the Moon suggests that its crust formed by flotation of light anorthite minerals on top of a liquid ocean, which has been used to explain the anorthositic composition of the lunar crust. However, this model points to rapid crustal formation over tens of million years and struggles to predict the age range of primitive ferroan anorthosites from 4.5 and 4.3 Ga. 

Here I will present a new paradigm of slushy magma ocean crystallisation in which crystals are suspended throughout the magma ocean, and the lunar crust forms by magmatic processes over several hundreds of thousand years.

We will then focus on the effects of the particular characteristics of this primary crust on the transport and eruption of magma on the Moon.

We present a partial differential equation framework for deep residual neural networks and for the associated learning problem. This is done by carrying out the continuum limits of neural networks with respect to width and depth. We study the wellposedness, the large time solution behavior, and the characterization of the steady states of the forward problem. Several useful time-uniform estimates and stability/instability conditions are presented. We state and prove optimality conditions for the inverse deep learning problem, using standard variational calculus, the Hamilton-Jacobi-Bellmann equation and the Pontryagin maximum principle. This serves to establish a mathematical foundation for investigating the algorithmic and theoretical connections between neural networks, PDE theory, variational analysis, optimal control, and deep learning.

This is based on joint work with Hailiang Liu.

Agent-based models are an intuitive, interpretable way to model markets and give us a powerful mechanism to analyze counterfactual scenarios that might rarely occur in historical market data. However, building realistic agent-based models is challenging and requires that we (a) ensure that agent behaviors are realistic, and (b) calibrate the agent composition using real data. In this talk, we will present our work to build realistic agent-based models using a multi-agent reinforcement learning approach. Firstly, we show that we can learn a range of realistic behaviors for heterogeneous agents using a shared policy conditioned on agent parameters and analyze the game-theoretic implications of this approach. Secondly, we propose a new calibration algorithm (CALSHEQ) which can estimate the agent composition for which calibration targets are approximately matched, while simultaneously learning the shared policy for the agents. Our contributions make the building of realistic agent-based models more efficient and scalable.

Amy Kent

Multiscale Mathematical Models for Tendon Tissue Engineering

Tendon tissue engineering aims to grow functional tendon in vitro. In bioreactor chambers, cells growing on a solid scaffold are fed with nutrient-rich media and stimulated by mechanical loads. The Nuffield Department of Orthopaedics, Rheumatology and Musculoskeletal Sciences is developing a Humanoid Robotic Bioreactor, where cells grow on a flexible fibrous scaffold actuated by a robotic shoulder. Tendon cells modulate their behaviour in response to shear stresses - experimentally, it is desirable to design robotic loading regimes that mimic physiological loads. The shear stresses are generated by flowing cell media; this flow induces deformation of the scaffold which in turn modulates the flow. Here, we capture this fluid-structure interaction using a homogenised model of fluid flow and scaffold deformation in a simplified bioreactor geometry. The homogenised model admits analytical solutions for a broad class of forces representing robotic loading. Given the solution to the microscale problem, we can determine microscale shear stresses at any point in the domain. In this presentation, we will outline the model derivation and discuss the experimental implications of model predictions.

=======================

Michael Negus

High-Speed Droplet Impact Onto Deformable Substrates: Analysis And Simulations

The impact of a high-speed droplet onto a substrate is a highly non-linear, multiscale phenomenon and poses a formidable challenge to model. In addition, when the substrate is deformable, such as a spring-suspended plate or an elastic sheet, the fluid-structure interaction introduces an additional layer of complexity. We present two modeling approaches for droplet impact onto deformable substrates: matched asymptotics and direct numerical simulations. In the former, we use Wagner's theory of impact to derive analytical expressions which approximate the behaviour during the early stages of the impact. In the latter, we use the open source volume-of-fluid code Basilisk to conduct direct numerical simulations designed to both validate the analytical framework and provide insight into the later times of impact. Through both methods, we are able to observe how the properties of the substrate, such as elasticity, affect the behaviour of the flow. We conclude by showing how these methods are complementary, as a combination of both can lead to a thorough understanding of the droplet impact across timescales.

=======================

Edwina Yeo

Modelling of Magnetically Targeted Stem Cell Delivery

Targeting delivery of stem cells to the site of an injury is a key challenge in regenerative medicine. One possible approach is to inject cells implanted withmagnetic nanoparticles into the blood stream. Cells can then be targeted to the delivery site by an external magnetic field. At the injury site, it is of criticalimportance that the cells do not form an aggregate which could significantly occlude the vessel.We develop a model for the transport of magnetically tagged cells in blood under the action of an external magnetic field. We consider a system of blood and stem cells in a single vessel.  We exploit the small aspect ratio of the vessel to examine the system asymptotically. We consider the system for a range of magnetic field strengths and varying strengths of the diffusion coefficient of the stem cells. We explore the different regimes of the model and determine the optimal conditions for the effective delivery of stem cells while minimising vessel occlusion.

=======================

Helen Zha

Mathematical model of a valve-controlled, gravity driven bioreactor for platelet production

Hospitals sometimes experience shortages of donor blood platelet supplies, motivating research into~\textit{in vitro}~production of platelets. We model a novel platelet bioreactor described in Shepherd et al [1]. The bioreactor consists of an upper channel, a lower channel, and a cell-seeded porous collagen scaffold situated between the two. Flow is driven by gravity, and controlled by valves on the four inlets and outlets. The bioreactor is long relative to its width, a feature which we exploit to derive a lubrication reduction of unsteady Stokes flow coupled to Darcy. As the shear stress experienced by cells influences platelet production, we use our model to quantify the effect of varying pressure head and valve dynamics on shear stress.

[1] Shepherd, J.H., Howard, D., Waller, A.K., Foster, H.R., Mueller, A., Moreau, T., Evans, A.L., Arumugam, M., Chalon, G.B., Vriend, E. and Davidenko, N., 2018. Structurally graduated collagen scaffolds applied to the ex vivo generation of platelets from human pluripotent stem cell-derived megakaryocytes: enhancing production and purity. Biomaterials.

Scientific machine learning is a burgeoning discipline for mixing machine learning into scientific simulation. Use cases of this field include automated discovery of physical equations and accelerating physical simulators. However, making the analyses of this field automated will require building a set of tools that handle stiff and ill-conditioned models without requiring user tuning. The purpose of this talk is to demonstrate how the methods and tools of scientific machine learning can be consolidated to give a single high performance and robust software stack. We will start by describing universal differential equations, a flexible mathematical object which is able to represent methodologies for equation discovery, 100-dimensional differential equation solvers, and discretizations of physics-informed neural networks. Then we will showcase how adjoint sensitivity analysis on the universal differential equation solving process gives rise to efficient and stiffly robust training methodologies for a large variety of scientific machine learning problems. With this understanding of differentiable programming we will describe how the Julia SciML Software Organization is utilizing this foundation to provide high performance tools for deploying battery powered airplanes, improving the energy efficiency of buildings, allow for navigation via the Earth's magnetic field, and more.

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please send email to @email.

We analyze the steady motion of a viscous incompressible fluid in a two- and three-dimensional channel containing an obstacle through the Navier-Stokes equations under different types of boundary conditions. In the 2D case we take constant non-homogeneous Dirichlet boundary data in a (virtual) square containing the obstacle, and emphasize the connection between the appearance of lift and the unique solvability of Navier-Stokes equations. In the 3D case we consider mixed boundary conditions: the inflow is given by a fairly general datum and the flow is assumed to satisfy a constant traction boundary condition on the outlet. In the absence of external forcing, explicit bounds on the inflow velocity guaranteeing existence and uniqueness of such steady motion are provided after estimating some Sobolev embedding constants and constructing a suitable solenoidal extension of the inlet velocity. In the 3D case, this solenoidal extension is built through the Bogovskii operator and explicit bounds on its Dirichlet norm (in terms of the geometric parameters of the obstacle) are found by solving a variational problem involving the infinity-Laplacian.

The talk accounts for results obtained in collaboration with Filippo Gazzola and Ilaria Fragalà (both at Politecnico di Milano).

I will introduce left-orderable groups and discuss constructions and examples of such groups. I will then motivate studying left-orders under the framework of formal languages and discuss some recent results.

The Burali-Forti paradox suggests that the transfinite cardinals “go on forever,” surpassing any conceivable bound one might try to place on them. The traditional Zermelo-Frankel axioms for set theory fall into a hierarchy of axiomatic systems formulated by reasserting this intuition in increasingly elaborate ways: the large cardinal hierarchy. Or so the story goes. A serious problem for this already naive account of large cardinal set theory is the Kunen inconsistency theorem, which seems to impose an upper bound on the extent of the large cardinal hierarchy itself. If one drops the Axiom of Choice, Kunen’s proof breaks down and a new hierarchy of choiceless large cardinal axioms emerges. These axioms, if consistent, represent a challenge for those “maximalist” foundational stances that take for granted both large cardinal axioms and the Axiom of Choice. This talk concerns some recent advances in our understanding of the weakest of the choiceless large cardinal axioms and the prospect, as yet unrealized, of establishing their consistency and reconciling them with the Axiom of Choice.