Past

Cohomological Hall algebras and vertex algebras are two structures whose origins are (at least in part) from physics. I will explain what these objects are, how the latter was related to moduli stacks by Joyce, and a theorem relating these two structures. The main tool is torus localisation, a method for "turning geometry into combinatorics", or rather a new formulation of it which works in the singular setting.

We consider the problem of parameter estimation for a McKean stochastic differential equation, and the associated system of weakly interacting particles. The problem is motivated by many applications in areas such as neuroscience, social sciences (opinion dynamics, cooperative behaviours), financial mathematics, statistical physics. We will first survey some model properties related to propagation of chaos and ergodicity and then move on to discuss the problem of parameter estimation both in offline and on-line settings. In the on-line case, we propose an online estimator, which evolves according to a continuous-time stochastic gradient descent algorithm on the asymptotic log-likelihood of the interacting particle system. The talk will present our convergence results and then show some numerical results for two examples, a linear mean field model and a stochastic opinion dynamics model. This is joint work with Louis Sharrock, Panos Parpas and Greg Pavliotis. Preprint: https://arxiv.org/abs/2106.13751

The analytic structure of scattering amplitudes exhibit striking
properties that are not at all evident from the first principles of
Quantum Field Theory. These are often rich and powerful enough to be
considered as their defining features, and this makes the problem of
finding a set of universal rules a compelling one. I will review the
recently mounting evidence for the relevance of tropical Grassmannians
in this respect, including implications on symbol alphabets and
adjacency conditions

This session will take place live in L1 only and not online on Teams. 

Are you interested in sharing your love of Maths with the next generation of mathematicians, but you don’t know where to start? In this session we will discuss some basic principles and top tips for creating a workshop for students aged 14–16, and get you started on developing your own. There will also be the opportunity to work on this further afterwards and potentially deliver your session as part of the Oxfordshire Maths Masterclasses (for local school students) in Hilary Term. Bring along your favourite bit of maths and a willingness to have a go.

In this talk I will review the “holomorphic modular bootstrap,” i.e. the classification of rational conformal field theories via an analysis of the modular differential equations satisfied by their characters. By making use of the representation theory of PSL(2, Zn), we describe a method to classify allowed central charges and weights (c, hi) for theories with any number of characters d. This allows us to avoid various bottlenecks encountered previously in the literature, and leads to a classification of consistent characters up to d = 5 whose modular differential equations are uniquely fixed in terms of (c, hi). In the process, we identify the full set of constraints on the allowed values of the Wronskian index for fixed d ≤ 5.

In this talk, I'll present inequalities bounding the number of critical cells in a filtered cell complex on the one hand, and the entries of the Betti tables of the multi-parameter persistence modules of such filtrations on the other hand. Using the Mayer-Vietoris spectral sequence we first obtain strong and weak Morse inequalities involving the above quantities, and then we improve the weak inequalities achieving a sharp lower bound for the number of critical cells. Furthermore, we prove a sharp upper bound for the minimal number of critical cells, expressed again in terms of the entries of Betti tables. This is joint work with Andrea Guidolin (KTH, Stockholm). The full paper is posted online as arxiv:2108.11427.

The Specht modules are of fundamental importance to the representation theory of the symmetric group, and their 0th cohomology is understood through entirely combinatorial methods due to Gordon James. Over fields of odd characteristic, Hemmer proposed a similar combinatorial approach to calculating their 1st degree cohomology, or extensions by the trivial module. This combinatorial approach motivates the definition of universal $p$-ary designs, which we shall classify. We then explore the consequences of this classification to problem of determining extensions of Specht modules. In particular, we classify all extensions of Specht modules indexed by two-part partitions by the trivial module and shall see some far-reaching conditions on when the first cohomology of a Specht module is trivial.

This session is particularly aimed at fourth-year and OMMS students who are completing a dissertation this year. The talk will be given by Dr Richard Earl who chairs Projects Committee. For many of you this will be the first time you have written such an extended piece on mathematics. The talk will include advice on planning a timetable, managing the  workload, presenting mathematics, structuring the dissertation and creating a narrative, providing references and avoiding plagiarism.

TBC

Uncertainty Quantification through Markov Chain Monte Carlo (MCMC) can be prohibitively expensive for target probability densities with expensive likelihood functions, for instance when the evaluation it involves solving a Partial Differential Equation (PDE), as is the case in a wide range of engineering applications. Multilevel Delayed Acceptance (MLDA) with an Adaptive Error Model (AEM) is a novel approach, which alleviates this problem by exploiting a hierarchy of models, with increasing complexity and cost, and correcting the inexpensive models on-the-fly. The method has been integrated within the open-source probabilistic programming package PyMC3 and is available in the latest development version.

In this talk I will talk about the problems with the Multilevel Markov Chain Monte Carlo (Dodwell et al. 2015). In so we will prove detailed balance for Adaptive Multilevel Delayed Acceptance, as well as showing that multilevel variance reduction can be achieved without bias, not possible in the original MLMCMC framework.

I will talk about our implementation in the latest version of pymc3, and demonstrate how for classical inverse problem benchmarks the AMLDA sampler offers huge computational savings (> factor of 100 fold speed up).

Finally I will talk heuristically about new / future research, in which we seek to develop parallel strategies for this inherently sequential sampler, as well as point to interesting applied application areas in which the method is proving particular effective.

--

This talk will be in person.

The complexity of biological systems necessitates that we develop mathematical models to further our understanding of these systems. Mathematical models of these systems are generally based on heterogeneous sets of experimental data, resulting in a seemingly heterogeneous collection of models that ostensibly represent the same system. To understand the system, and to reveal underlying design principles, we therefore need to understand how the different models are related to each other with a view to obtaining a unified mathematical description. This goal is complicated by the number of distinct mathematical formalisms that may be employed to represent the same system, making direct comparison of the models very difficult. In this talk I will discuss two general methodologies, namely comparison by distance and comparison by equivalence, that allow us to compare model structures in a systematic way by representing models as labelled simplicial complexes. The distance can be obtained either directly from the simplicial complexes, or from the persistence intervals obtained by employing persistent homology with a flat filtration. Model equivalence is used to determine the conceptual similarity of models and can be automated by using group actions on the simplicial complexes. We apply our methodology for model comparison to demonstrate a particular equivalence between a positional-information model and a Turing-pattern model from developmental biology, which constitutes a novel observation for two classes of models that were previously regarded as unrelated. We also discuss an alternative framework for model comparison by representing models as groups, which allows for the application of group-theoretic techniques within our model comparison methodology.

In this talk, we describe some recent work on applying tools from category theory in finite model theory, descriptive complexity, constraint satisfaction, and combinatorics.

The motivations for this work come from Computer Science, but there may be something of interest for model theorists and other logicians.

The basic setting involves studying the category of relational structures via a resource-indexed family of adjunctions with some process category - which unfolds relational structures into treelike forms, allowing natural resource parameters to be assigned to these unfoldings.

One basic instance of this scheme allows us to recover, in a purely structural, syntax-free way:

- the Ehrenfeucht-Fraisse game

- the quantifier rank fragments of first-order logic

- the equivalences on structures induced by (i) the quantifier rank fragments, (ii) the restriction to the existential-positive part, and (iii) the extension with counting quantifiers

- the combinatorial parameter of tree-depth (Nesetril and Ossona de Mendez).

Another instance recovers the k-pebble game, the finite-variable fragments, the corresponding equivalences, and the combinatorial parameter of treewidth.

Other instances cover modal, guarded and hybrid fragments, generalized quantifiers, and a wide range of combinatorial parameters.

This whole scheme has been axiomatized in a very general setting, of arboreal categories and arboreal covers.

Beyond this basic level, a landscape is beginning to emerge, in which structural features of the resource categories, adjunctions and comonads are reflected in degrees of logical and computational tractability of the corresponding languages.

Examples include semantic characterisation and preservation theorems, Lovasz-type results on  isomorphisms, and classification of constraint satisfaction problems.

A big mapping class group is the mapping class group (MCG) of a surface of infinite type. Although several aspects of big MCGs remain mysterious, their geometric definition allows some simple, interesting arguments. In this talk, we will use big MCGs as an excuse to survey some (more or less) classical results in geometric group theory: we will present a quick introduction to infinite type surfaces, highlight differences between standard and large MCGs, and use Higman’s embedding theorem to deduce that there exists a big MCG that contains every finitely presented group as a subgroup.

In recent years it has been fruitful to model the physical world in a categorical framework. In this talk I will give an outline of this process theoretic view with a particular focus on its applications to quantum mechanics and quantum computing. I will discuss how abstract categorical structure captures certain quantum protocols, such as teleportation, unearthing the topological nature of them, and how we can use algebraic structures internal to a category to develop a framework for circuit-based quantum computing in the form of the ZX-calculus.

I will report on recent progress concerning eigenvalues of Schrödinger operators with complex potentials. We are interested in the magnitude and distribution of eigenvalues, and we seek bounds that only depend on an L^p norm of the potential.

These questions are well understood for real potentials, but completely new phenomena arise for complex potentials. I will explain how techniques from harmonic analysis, particularly those related to Fourier restriction theory, can be used to prove upper and lower bounds. We will also discuss some open problems. The talk is based on recent joint work with Sabine Bögli (Durham).

The question asked in the title is addressed from two points of view: First, we show that providing enough (term to be explained) spectral data, suffices to reconstruct uniquely generic (term to be explained) matrices. The method is well defined but requires somewhat cumbersome computations. Second, restricting the attention to banded matrices with band-width much smaller than the dimension, one can provide more spectral data than the number of unknown matrix elements. We make use of this redundancy to reconstruct generic banded matrices in a much more straight-forward fashion where the “cumbersome computations” can be skipped over. Explicit criteria for a matrix to be in the non-generic set are provided.

The additive Schwarz method with vertex-centered patches and a low-order coarse space gives a p-robust solver for FEM discretizations of symmetric and coercive problems. However, for very high polynomial degree it is not feasible to assemble or factorize the matrices for each patch. In this work we introduce a direct solver for separable patch problems that scales to very high polynomial degree on tensor product cells. The solver constructs a tensor product basis that diagonalizes the blocks in the stiffness matrix for the internal degrees of freedom of each individual cell. As a result, the non-zero structure of the cell matrices is that of the graph connecting internal degrees of freedom to their projection onto the facets. In the new basis, the patch problem is as sparse as a low-order finite difference discretization, while having a sparser Cholesky factorization. We can thus afford to assemble and factorize the matrices for the vertex-patch problems, even for very high polynomial degree. In the non-separable case, the method can be applied as a preconditioner by approximating the problem with a separable surrogate. We apply this approach as a relaxation for the displacement block of mixed formulations of incompressible linear elasticity.

Viscous contact problems describe the time evolution of fluid flows in contact with a surface from which they can detach. These type of problems arise in glaciology when, for example, modelling the evolution of the grounding line of a marine ice sheet or the formation of a subglacial cavity. Such problems are generally modelled as a time dependent viscous Stokes flow with a free boundary and contact boundary conditions. Although these applications are of great importance in glaciology, a systematic study of the numerical approximation of viscous contact problems has not been carried out yet. In this talk, I will present some of the challenges that arise when approximating these problems and some of the ideas we have come up with for overcoming them.

We study a family of evolving directed random graphs that includes the directed preferential model and the directed uniform attachment model. The directed preferential model is of particular interest since it is known to produce scale-free graphs with regularly varying in-degree distribution. We start by describing the local weak limits for our family of random graphs in terms of randomly stopped continuous-time branching processes, and then use these limits to establish the asymptotic behavior of the corresponding PageRank distribution. We show that the limiting PageRank distribution decays as a power-law in both models, which is surprising for the uniform attachment model where the in-degree distribution has exponential tails. And even for the preferential attachment model, where the power-law hypothesis suggests that PageRank should follow a power-law, our result shows that the two tail indexes are different, with the PageRank distribution having a heavier tail than the in-degree distribution.

We are confronted with signals defined on the nodes of a graph in many applications.  Think for instance of a sensor network measuring temperature; or a social network, in which each person (node) has an opinion about a specific issue.  Graph signal processing (GSP) tries to device appropriate tools to process such data by generalizing classical methods from signal processing of time-series and images -- such as smoothing, filtering and interpolation -- to signals defined on graphs.  Typically, this involves leveraging the structure of the graph as encoded in the spectral properties of the graph Laplacian.

In other applications such as traffic network analysis, however, the signals of interest are naturally defined on the edges of a graph, rather than on the nodes. After a very brief recap of the central ideas of GSP, we examine why the standard tools from GSP may not be suitable for the analysis of such edge signals.  More specifically, we discuss how the underlying notion of a 'smooth signal' inherited from (the typically considered variants of) the graph Laplacian are not suitable when dealing with edge signals that encode flows.  To overcome this limitation we devise signal processing tools based on the Hodge-Laplacian and the associated discrete Hodge Theory for simplicial (and cellular) complexes.  We discuss applications of these ideas for signal smoothing, semi-supervised and active learning for edge-flows on discrete (or discretized) spaces.

I shall present joint work with Maxim Kontsevich describing an interesting
domain of complex metrics on a smooth manifold. It is a complexification of
the space of ordinary Riemannian metrics, and has the Lorentzian metrics
(but not metrics of other signatures) on its boundary. Use of the domain
leads to a modified axiom system for QFT which illuminates not only the
special role of Lorentz signature, but also of features such as local
commutativity, unitarity, and global hyperbolicity.