Past

In this talk, I will present several results on the Anderson Hamiltonian with white noise potential in dimension 1. This operator formally writes « - Laplacian + white noise ». It arises as the scaling limit of various discrete models and its explicit potential allows for a detailed description of its spectrum. We will discuss localization of its eigenfunctions as well as the behavior of the local statistics of its eigenvalues. Around large energies, we will see that the eigenfunctions are localized and follow a universal shape given by the exponential of a Brownian motion plus a drift, a behavior already observed by Rifkind and Virag in tridiagonal matrix models. Based on joint works with Cyril Labbé.

Shrinking Ricci solitons are Ricci flow solutions that self-similarly shrink under the flow. Their significance comes from the fact that finite-time Ricci flow singularities are typically modeled on gradient shrinking Ricci solitons. Here, we shall address a certain converse question, namely, “Given a complete, noncompact gradient shrinking Ricci soliton, does there exist a Ricci flow on a closed manifold that forms a finite-time singularity modeled on the given soliton?” We’ll discuss work that shows the answer is yes when the soliton is asymptotically conical. No symmetry or Kahler assumption is required, and so the proof involves an analysis of the Ricci flow as a nonlinear degenerate parabolic PDE system in its full complexity. We’ll also discuss applications to the (non-)uniqueness of weak Ricci flows through singularities.

Numerical optimization is an indispensable tool of modern data analysis, and there are many optimization problems where it is difficult or impossible to compute the full gradient of the objective function. The field of derivative free optimization (DFO) addresses these cases by using only function evaluations, and has wide-ranging applications from hyper-parameter tuning in machine learning to PDE-constrained optimization.

We present two projects that attempt to scale DFO techniques to higher dimensions.  The first method converges slowly but works in very high dimensions, while the second method converges quickly but doesn't scale quite as well with dimension.  The first-method is a family of algorithms called "stochastic subspace descent" that uses a few directional derivatives at every step (i.e. projections of the gradient onto a random subspace). In special cases it is related to Spall's SPSA, Gaussian smoothing of Nesterov, and block-coordinate descent. We provide convergence analysis and discuss Johnson-Lindenstrauss style concentration.  The second method uses conventional interpolation-based trust region methods which require large ill-conditioned linear algebra operations.  We use randomized linear algebra techniques to ameliorate the issues and scale to larger dimensions; we also use a matrix-free approach that reduces memory issues.  These projects are in collaboration with David Kozak, Luis Tenorio, Alireza Doostan, Kevin Doherty and Katya Scheinberg.

I will describe a procedure, known as holomorphic twist, to isolate protected quantities in supersymmetric quantum field theories. The resulting theories are holomorphic, interacting and have infinite dimensional symmetries, analogous to the holomorphic half of a 2D CFT. I will explain how to study quantum corrections to these symmetries and other  higher operations.
As a surprise, we find a novel UV manifestation of
confinement, dubbed "holomorphic confinement," in the example of pure
SU(N) super Yang-Mills.

Ilia Smilga
Margulis spacetimes and crooked planes

We are interested in the following problem: which groups can act 
properly on R^n by affine transformations, or in other terms, can occur 
as a symmetry group of a "regular affine tiling"? If we additionally 
require that they preserve a Euclidean metric (i.e. act by affine 
isometries), then these groups are well-known: they all contain a 
finite-index abelian subgroup. If we remove this requirement, a 
surprising result due to Margulis is that the free group can act 
properly on R^3. I shall explain how to construct such an action.

Charles Parker
Unexpected Behavior in Finite Elements for Linear Elasticity
One of the first problems that finite elements were designed to approximate is the small deformations of a linear elastic body; i.e. the 2D/3D version of Hooke's law for springs from elementary physics. However, for nearly incompressible materials, such as rubber, certain finite elements seemingly lose their approximation power. After briefly reviewing the equations of linear elasticity and the basics of finite element methods, we will spend most of the time looking at a few examples that highlight this unexpected behavior. We conclude with a theoretical result that (mostly) explains these findings.

The recent curation of large-scale databases with 3D surface scans of shapes has motivated the development of tools that better detect global patterns in morphological variation. Studies which focus on identifying differences between shapes have been limited to simple pairwise comparisons and rely on pre-specified landmarks (that are often known). In this talk, we present SINATRA: a machine learning pipeline for analyzing collections of shapes without requiring any correspondences. Our method takes in two classes of shapes and highlights the physical features that best describe the variation between them. We develop a rigorous simulation framework to assess our approach, which themselves are a novel contribution to 3D image and shape analyses. Lastly, as case studies with real data, we use SINATRA to (1) analyze mandibular molars from four different suborders of primates and (2) facilitate the visual identification of structural signatures differentiating between the trajectories of two protein ensembles resulting from molecular dynamics simulations. Together, these results highlight a promising future for interpretable machine learning to facilitate the non-trivial task of pattern recognition in evolutionary and structural biology with substantially increased resolution.

2:30 -3.00 Will Turner (CDT Student, Imperial College London)

Topologies on unparameterised path space

The signature of a path is a non-commutative exponential introduced by K.T. Chen in the 1950s, and appears as a central object in the theory of rough paths developed by T. Lyons in the 1990s. For continuous paths of bounded variation, the signature may be realised as a sequence of iterated integrals, which provides a succinct summary for multimodal, irregularly sampled, time-ordered data. The terms in the signature act as an analogue to monomials for finite dimensional data: linear functionals on the signature uniformly approximate any compactly supported continuous function on unparameterised path space (Levin, Lyons, Ni 2013). Selection of a suitable topology on the space of unparameterised paths is then key to the practical use of this approximation theory. We present new results on the properties of several candidate topologies for this space. If time permits, we will relate these results to two classical models: the fixed-time solution of a controlled differential equation, and the expected signature model of Levin, Lyons, and Ni. This is joint work with Thomas Cass.

3.05 -3.35 Ross Zhang (CDT Student, University of Oxford)

Random vortex dynamics via functional stochastic differential equations

The talk focuses on the representation of the three-dimensional (3D) Navier-Stokes equations by a random vortex system. This new system could give us new numerical schemes to efficiently approximate the 3D incompressible fluid flows by Monte Carlo simulations. Compared with the 2D Navier-Stokes equation, the difficulty of the 3D Navier-Stokes equation lies in the stretching of vorticity. To handle the stretching term, a system of stochastic differential equations is coupled with a functional ordinary differential equation in the 3D random vortex system. Two main tools are developed to derive the new system: the first is the investigation of pinned diffusion measure, which describes the conditional distribution of a time reversal diffusion, and the second is a forward-type Feynman Kac formula for nonlinear PDEs, which utilizes the pinned diffusion measure to delicately overcome the time reversal issue in PDE. Although the main focus of the research is the Navier-stokes equation, the tools developed in this research are quite general. They could be applied to other nonlinear PDEs as well, thereby providing respective numerical schemes.

3.40 - 4.25pm Dr Cris Salvi (Imperial College London)

Signature kernel methods

Kernel methods provide a rich and elegant framework for a variety of learning tasks including supervised learning, hypothesis testing, Bayesian inference, generative modelling and scientific computing. Sequentially ordered information often arrives in the form of complex streams taking values in non-trivial ambient spaces (e.g. a video is a sequence of images). In these situations, the design of appropriate kernels is a notably challenging task. In this talk, I will outline how rough path theory, a modern mathematical framework for describing complex evolving systems, allows to construct a family of characteristic kernels on pathspace known as signature kernels. I will then present how signature kernels can be used to develop a variety of algorithms such as two-sample hypothesis and (conditional) independence tests for stochastic processes, generative models for time series and numerical methods for path-dependent PDEs.

4.30 Refreshments

Building computationally capable biological systems has long been an aim of synthetic biology. The potential utility of bio-computing devices ranges from biosafety and environmental applications to diagnosis and personalised medicine. Here we present work on the design of bacterial computers which use spatial patterning to process information. A computer is composed of a number of bacterial colonies which, inspired by patterning in embryo development, communicate using diffusible morphogen-like signals. A computation is programmed into the overall physical arrangement of the system by arranging colonies such that the resulting diffusion field encodes the desired function, and the output is represented in the spatial pattern displayed by the colonies. We first mathematically demonstrate the simple digital logic capability of single bacterial colonies and show how additional structure is required to build complex functions. Secondly, inspired by electronic design automation, an algorithm for designing optimal spatial circuits computing two-level digital logic functions is presented, extending the capability of our system to complex digital functions without significantly increasing the biological complexity. We implement experimentally a proof-of-principle system using engineered Escherichia coli interpreting diffusion fields formed from droplets of an inducer molecule. Our approach will open up new ways to perform biological computation, with applications in synthetic biology, bioengineering and biosensing. Ultimately, these computational bacterial communities will help us explore information processing in natural biological systems.

The canonical dimension is a fundamental integer-valued invariant that is attached to mod p representations of p-adic Lie groups. I will explain why it is both an asymptotic measure of growth, and an algebraic quantity strongly related to Krull dimension. We will survey algebraic tools that can be applied in its calculation, and describe results spanning the last twenty years. I'll present a new theorem and suggest its possible significance for the mod p local Langlands programme. 

Movie Me would like offer dynamical pricing for movie tickets, considering consumer’s demand for the movie, showtime and lead time before the show begins, such that the overall quantity of tickets sold is maximized. We encourage all interested party to join us and especially those interested in data science, optimization and mathematical finance.

If $Q(x_0,x_1,x_2)$ is a quadratic form, how many solutions, of size at most $B$, does $Q=0$ have? How does this depend on $Q$? We apply the answers to the surface $y_0 Q_0 +y_1 Q_1 = 0$ in $P^1 x P^2$. (Joint work with Dan Loughran.)
 

This talk presents a new framework for Merton’s optimal investment problem which uses the theory of Meyer $\sigma$-fields to allow for signals that possibly warn the investor about impending jumps. With strategies no longer predictable, some care has to be taken to properly define wealth dynamics through stochastic integration. By means of dynamic programming, we solve the problem explicitly for power utilities. In a case study with Gaussian jumps, we find, for instance, that an investor may prefer to disinvest even after a mildly positive signal. Our setting also allows us to investigate whether, given the chance, it is better to improve signal quality or quantity and how much extra value can be generated from either choice.
This talk is based on joint work with Peter Bank.

Physics-informed neural networks (PINNs) [2] are a solution method for solving boundary value problems based on differential equations (PDEs). The key idea of PINNs is to incorporate the residual of the PDE as well as boundary conditions into the loss function of the neural network. This provides a simple and mesh-free approach for solving problems relating to PDEs. However, a key limitation of PINNs is their lack of accuracy and efficiency when solving problems with larger domains and more complex, multi-scale solutions. 

In a more recent approach, Finite Basis Physics-Informed Neural Networks (FBPINNs) [1], the authors use ideas from domain decomposition to accelerate the learning process of PINNs and improve their accuracy in this setting. In this talk, we show how Schwarz-like additive, multiplicative, and hybrid iteration methods for training FBPINNs can be developed. Furthermore, we will present numerical experiments on the influence on convergence and accuracy of these different variants. 

This is joint work with Alexander Heinlein (Delft) and Benjamin Moseley (Oxford).

References 
1.    [1]  B. Moseley, A. Markham, and T. Nissen-Meyer. Finite basis physics- informed neural networks (FBPINNs): a scalable domain decomposition approach for solving differential equations. arXiv:2107.07871, 2021. 
2.    [2]  M. Raissi, P. Perdikaris, and G. E. Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378:686–707, 2019.

We will be joined by Charlotte Turner-Smith to discuss issues surrounding harassment and mental health, and how the department is helping to tackle these.

Many microorganisms must navigate strange biological environments whose physics are unique and counter-intuitive, with wide-ranging consequences for evolutionary biology and human health. Mucus, for instance, behaves like both a fluid and an elastic solid. This can affect locomotion dramatically, which can be highly beneficial (e.g. for mammalian spermatozoa swimming through cervical fluid) or extremely problematic (e.g. the Lyme disease spirochete B. burgdorferi swimming through the extracellular matrix of human skin). Mathematical modeling and numerical simulations continue to provide new fundamental insights about the biological world in and around us and point toward new possibilities in biomedical engineering. These complex fluid phenomena can either enhance or retard a microorganism's swimming speed, and can even change the direction of swimming, depending on the body geometry and the properties of the fluid. We will discuss analytical and numerical insights into swimming through model viscoelastic (Oldroyd-B) and liquid-crystalline (Ericksen-Leslie) fluids, with a special focus on the important and in some cases dominant roles played by the presence of nearby boundaries.

We review a few instances in which the first $\ell^2$ Betti number of a group is a profinite invariant and we discuss some applications and open problems.

From each point of a Poisson point process start growing a balloon at rate 1. When two balloons touch, they pop and disappear. Will balloons reach the origin infinitely often or not? We answer this question for various underlying spaces. En route we find a new(ish) 0-1 law, and generalize bounds on independent sets that are factors of IID on trees. Joint work with Omer Angel and Gourab Ray.

The classification by K-theory and traces of the category of simple, separable, nuclear, Z-stable C*-algebras satisfying the UCT is an extraordinary feat of mathematics. What's more, it provides powerful machinery for the analysis of the internal structure of these regular C*-algebras. In this talk, I will explain one such application of classification: In the subclass of classifiable C*-algebras consisting of those for which the simplex of tracial states is nonempty, with extremal boundary that is compact and has the structure of a connected topological manifold, automorphisms can be shown to be generically tracially chaotic. Using similar ideas, I will also show how certain stably projectionless C*-algebras can be described as crossed products.

In 1983 Thomassen conjectured that every graph of sufficiently large average degree has a subgraph of large girth and large average degree. While this conjecture remains open, recent evidence suggests that something stronger might be true; perhaps the subgraph can be made induced when a clique and biclique are forbidden. We overview our proof for removing 4-cycles from $K_{t,t}$-free bipartite graphs. Moreover, we discuss consequences to tau-boundedness, which is an analog of chi-boundedness.

The Calogero-Painlevé systems were introduced in 2001 by K. Takasaki as a natural generalization of the classical Painlevé equations to the case of the several Painlevé “particles” coupled via the Calogero type interactions. In 2014, I. Rumanov discovered a remarkable fact that a particular case of the Calogero– Painlevé II equation describes the Tracy-Widom distribution function for the general $\beta$-ensembles with the even values of parameter $\beta$. in 2017 work of M. Bertola, M. Cafasso , and V. Rubtsov, it was proven that all Calogero-Painlevé systems are Lax integrable, and hence their solutions admit a Riemann-Hilbert representation. This important observation has opened the door to rigorous asymptotic analysis of the Calogero-Painlevé equations which in turn yields the possibility of rigorous evaluation of the asymptotic behavior of the Tracy-Widom distributions for the values of $\beta$ beyond the classical $\beta =1, 2, 4$. In the talk these recent developments will be outlined with a special focus on the Calogero-Painlevé system corresponding to $\beta = 6$. This is a joint work with Andrei Prokhorov.

I will discuss some new invariants of 2-complexes. They are inspired by recent developments in the theory of one-relator groups, but also have the potential to unify the theories of many well-studied families including small-cancellation presentation complexes, CAT(0) 2-complexes and 3-manifold spines, in addition to the motivating examples of one-relator presentation complexes. The fundamental result is that these invariants are the extrema of explicit linear-programming problems, and in particular are rational, computable and realised. The definitions suggest a conjectural “map” of 2-complexes, which I will attempt to describe.
 

Spectral methods offer a tractable, global framework for clustering in graphs via eigenvector computations on graph matrices. Hypergraph data, in which entities interact on edges of arbitrary size, poses challenges for matrix representations and therefore for spectral clustering. We study spectral clustering for arbitrary hypergraphs based on the hypergraph nonbacktracking operator. After reviewing the definition of this operator and its basic properties, we prove a theorem of Ihara-Bass type which allows eigenpair computations to take place on a smaller matrix, often enabling faster computation. We then propose an alternating algorithm for inference in a hypergraph stochastic blockmodel via linearized belief-propagation which involves a spectral clustering step, again using nonbacktracking operators. We provide proofs related to this algorithm that both formalize and extend several previous results. We pose several conjectures about the limits of spectral methods and detectability in hypergraph stochastic blockmodels in general, supporting these with in-expectation analysis of the eigeinpairs of our studied operators. We perform experiments with real and synthetic data that demonstrate the benefits of hypergraph methods over graph-based ones when interactions of different sizes carry different information about cluster structure.

Joint work with Nicole Eikmeier (Grinnell) and Jamie Haddock (Harvey Mudd).