Past

Partition Lie algebras are generalisations of rational differential graded Lie algebras which, by a recent result of Mathew and myself, govern the formal deformation theory of algebro-geometric objects in finite and mixed characteristic. In this talk, we will take a closer look at these new gadgets and discuss some of their applications in algebra and topology

Supergravity backgrounds are an essential ingredient in string theory or field theories via AdS/CFT. The simplest example of a 4d, N=1 background is the product of four-dimensional Minkowski space with a seven-dimensional manifold with G_2 holonomy in M-theory. For more complicated backgrounds where we allow non-zero fluxes, the supersymmetry conditions can be rephrased in terms of G-structure data. The geometry of these backgrounds is often complicated and their general features are not well understood.

In this talk, I will define the analogue of G_2 geometry for generic 4d, N=1 backgrounds with flux in both type II and eleven-dimensional supergravity. The geometry is characterised by a G-structure in 'exceptional generalised geometry' that includes G_2 structures and Hitchin's generalised geometry as subcases. Supersymmetry is then equivalent to integrability of the structures, which appears as an involutivity condition and a moment map for diffeomorphisms and gauge transformations. I will show how this works in a few simple examples and discuss how this can be used to understand general properties of supersymmetric backgrounds.

 

A panel discussion on non-academic careers for mathematicians with PhDs, featuring Katia Babbar (AI Wealth Technologies & QuantBright), Jara Imbers (Risk Management Solutions) and Tom Hawes (Smith Institute).
 

Cardiac fibrosis plays a significant role in the disruption of healthy electrical signalling in the heart, creating structural heterogeneities that induce and stabilise arrhythmia.  However, a proper understanding of the consequences of cardiac fibrosis must take into account the complex and highly variable patterns of its spatial localisation in the heart, which significantly affects the extent and manner of its impacts on cardiac wave propagation. In this work we present a methodology for the algorithmic generation of fibrotic patterns via Perlin noise, a technique for computationally efficient generation of textures in computer graphics.

Our approach works directly from image data to create populations of pattern realisations that all resemble the target image under a set of metrics. Our technique thus serves as a type of data enrichment, enabling analysis of how variability in the precise placement of fibrotic structures modulates their electrophysiological impact. We demonstrate our method, and the types of analysis it can enable, using a widely referenced histological image of four different types of microfibrotic structure. Our generator and Bayesian tuning method prove flexible enough to successfully capture each of these very distinct patterns.

We demonstrate the importance of this tool, by presenting 2D simulations overlayed on the generated images that highlight the effects of microscopic variability on the electrophysiological impact of fibrosis. Finally, we discuss the application of our methodology to the increasingly available imaging data of fibrotic patterning on a more macroscopic scale, and indeed to other areas of science underpinned by image based modelling and simulation.    

Dimension reduction is an overarching theme in data science: we enjoy finding informative patterns, features or substructures in large, complex data sets. Within the field of network science, an important problem of this nature is to identify core-periphery structure. Given a network, our task is to assign each node to either the core or periphery. Core nodes should be strongly connected across the whole network whereas peripheral nodes should be strongly connected only to core nodes. More generally, we may wish to assign a non-negative value to each node, with a larger value indicating greater "coreness." This type of problem is related to, but distinct from, commumnity detection (finding clusters) and centrality assignment (finding key players), and it arises naturally in the study of networks in social science and finance. We derive and analyse a new iterative algorithm for detecting network core-periphery structure.

Using techniques in nonlinear Perron-Frobenius theory we prove global convergence to the unique solution of a relaxed version of a natural discrete optimization problem. On sparse networks, the cost of each iteration scales linearly with the number of nodes, making the algorithm feasible for large-scale problems. We give an alternative interpretation of the algorithm from the perspective of maximum likelihood reordering of a new logistic core--periphery random graph model. This viewpoint also gives a new basis for quantitatively judging a core--periphery detection algorithm. We illustrate the algorithm on a range of synthetic and real networks, and show that it offers advantages over the current state-of-the-art.

This is joint work with Francesco Tudisco (Strathclyde)

In Formula 1 engineers strive to produce the fastest car possible for their drivers. A lap simulation provides an objective evaluation of the performance of the car and the subsequent lap time achieved. Using this information, engineers aim to test new car concepts, determine performance limitations or compromises, and identify the sensitivity of performance to car setup parameters.

The latest state of the art lap simulation techniques use optimal control approaches. Optimisation methods are employed to derive the optimal control inputs of the car that achieve the fastest lap time within the constraints of the system. The resulting state trajectories define the complete behaviour of the car. Such approaches aim to create more robust, realistic and powerful simulation output compared to traditional methods.

In this talk we discuss our latest work in this area. A dynamic vehicle model is used within a free-trajectory solver based on direct optimal control methods. We discuss the reasons behind our design choices, our progress to date, and the issues we have faced during development. Further, we look at the short and long term aims of our project and how we wish to develop our mathematical methods in the future.

I will discuss the arithmetic significance of Fourier expansions of modular forms at cusps. I will talk about joint work with F. Brunault, where we determine the number field generated by Fourier coefficients of newforms at a cusp. Then I will discuss joint work with A. Saha and K. Česnavičius where we find denominator bounds for Fourier expansions at cusps and apply these bounds to a conjecture on the Manin constants of elliptic curves.

The problem of model uncertainty in financial mathematics has received considerable attention in the last years. In this talk I will follow a non-parametric point of view, and argue that an insightful approach to model uncertainty should not be based on the familiar Wasserstein distances. I will then provide evidence supporting the better suitability of the recent notion of adapted Wasserstein distances (also known as Nested Distances in the literature). Unlike their more familiar counterparts, these transport metrics take the role of information/filtrations explicitly into account. Based on joint work with M. Beiglböck, D. Bartl and M. Eder.

Likely instabilities in stochastic hyperelastic solids

L. Angela Mihai

School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4AG, UK

E-mail: @email.uk

Nonlinear elasticity has been an active topic of fundamental and applied research for several decades. However, despite numerous developments and considerable attention it has received, there are important issues that remain unresolved, and many aspects still elude us. In particular, the quantification of uncertainties in material parameters and responses resulting from incomplete information remain largely unexplored. Nowadays, it is becoming increasingly apparent that deterministic approaches, which are based on average data values, can greatly underestimate, or overestimate, mechanical properties of many materials. Thus, stochastic representations, accounting for data dispersion, are needed to improve assessment and predictions. In this talk, I will consider stochastic hyperelastic material models described by a strain-energy density where the parameters are characterised by probability distributions. These models, which are constructed through a Bayesian identification procedure, rely on the maximum entropy principle and enable the propagation of uncertainties from input data to output quantities of interest. Similar modelling approaches can be developed for other mechanical systems. To demonstrate the effect of probabilistic model parameters on large strain elastic responses, specific case studies include the classic problem of the Rivlin cube, the radial oscillatory motion of cylindrical and spherical shells, and the cavitation and finite amplitude oscillations of spheres.

One of the major steps in the adaptive finite element methods (AFEM) is the adaptive selection of the next partition. The process is usually governed by a strategy based on carefully chosen local error indicators and aims at convergence results with optimal rates. One can formally relate the refinement of the partitions with growing an oriented graph or a tree. Then each node of the tree/graph corresponds to a cell of a partition and the approximation of a function on adaptive partitions can be expressed trough the local errors related to the cell, i.e., the node. The total approximation error is then calculated as the sum of the errors on the leaves (the terminal nodes) of the tree/graph and the problem of finding an optimal error for a given budget of nodes is known as tree approximation. Establishing a near-best tree approximation result is a key ingredient in proving optimal convergence rates for AFEM.

The classical tree approximation problems are usually related to the so-called h-adaptive approximation in which the improvements a due to reducing the size of the cells in the partition. This talk will consider also an extension of this framework to hp-adaptive approximation allowing different polynomial spaces to be used for the local approximations at different cells while maintaining the near-optimality in terms of the combined number of degrees of freedom used in the approximation.

The problem of conformity of the resulting partition will be discussed as well. Typically in AFEM, certain elements of the current partition are marked and subdivided together with some additional ones to maintain desired properties of the partition like conformity. This strategy is often described as “mark → subdivide → complete”. The process is very well understood for triangulations received via newest vertex bisection procedure. In particular, it is proven that the number of elements in the final partition is limited by constant times the number of marked cells. This hints at the possibility to design a marking procedure that is limited only to cells of the partition whose subdivision will result in a conforming partition and therefore no completion step would be necessary. This talk will present such a strategy together with theoretical results about its near-optimal performance.

It is nowadays well understood that the multidimensional isentropic Euler system is desperately ill–posed. Even certain smooth initial data give rise to infinitely many solutions and all available selection criteria fail to ensure both global existence and uniqueness. We propose a different approach to well–posedness of this system based on ideas from the theory of Markov semigroups: we show the existence of a Borel measurable solution semiflow. To this end, we introduce a notion of dissipative solution which is understood as time dependent trajectories of the basic state variables - the mass density, the linear momentum, and the energy - in a suitable phase space. The underlying system of PDEs is satisfied in a generalized sense. The solution semiflow enjoys the standard semigroup property and the solutions coincide with the strong solutions as long as the latter exist. Moreover, they minimize the energy (maximize the energy dissipation) among all dissipative solutions.

Leighton's Theorem states that if two finite graphs have a common universal cover then they have a common finite cover. I will present a new proof of this using groupoids, and then talk about two generalisations of the theorem that can also be tackled with this groupoid approach: one gives us control over the local structure of the common finite cover, and the other deals with graphs of spaces.

Hilbert's fifth problem asks informally what is the difference between Lie groups and topological groups. In 1950s this problem was solved by Andrew Gleason, Deane Montgomery, Leo Zippin and Hidehiko Yamabe concluding that every locally compact topological group is "essentially" a Lie group. In this talk we will show the complete proof of this theorem.

 Massless scattering amplitudes in four-dimensional Minkowski spacetime can be Mellin transformed to correlation functions on the celestial sphere at null infinity called celestial amplitudes. We study various properties of massless four-point scalar and gluon celestial amplitudes such as conformal partial wave decomposition, crossing relations and optical theorem. As a byproduct, we derive the analog of the single and double soft limits for all gluon celestial amplitudes.

Black holes are predicted by Einstein's theory of general relativity, and now we have ample observational evidence for their existence. However theoretically there are many unanswered questions about how black holes come into being. In this talk, with tools from hyperbolic PDE, quasilinear elliptic equations and geometric analysis, we will prove that, through a nonlinear focusing effect, initially low-amplitude and diffused gravitational waves can give birth to a trapped (black hole) region in our universe. This result extends the 2008 Christodoulou’s monumental work and it also proves a conjecture of Ashtekar on black-hole thermodynamics

The moduli space of smooth hypersurfaces in projective space was constructed by Mumford in the 60’s using his newly developed classical (a.k.a. reductive) Geometric Invariant Theory.  I wish to generalise this construction to hypersurfaces in weighted projective space (or more generally orbifold toric varieties). The automorphism group of a toric variety is in general non-reductive and I will use new results in non-reductive GIT, developed by F. Kirwan et al., to construct a moduli space of quasismooth hypersurfaces in certain weighted projective spaces. I will give geometric characterisations of notions of stability arising from non-reductive GIT.

Optimisation in 1D is far simpler than multidimensional optimisation and this is largely due to the notion of a bracket. A bracket is a trio of points such that the middle point is the one with the smallest objective function value (of the three). The existence of a bracket is sufficient to guarantee that a continuous function has a local minimum within the bracket. The most stable 1D optimisation methods, such as Golden Section or Brent's Method, make use of this fact. The mentality behind these methods is to maintain a bracket at all times, all the while finding smaller brackets until the local minimum can be guaranteed to lie within a sufficiently small range. For smooth functions, Brent's method in particular converges quickly with a minimum of function evaluations required. However, when applied to a piece-wise smooth functions, it achieves its realistic worst case convergence rate. In this presentation, I will present a new method which uses ideas from Brent and Golden Section, while being designed to converge quickly for piece-wise smooth functions.

Let $p$ be an odd prime and $n$ a natural number. We determine the irreducible constituents of the permutation module induced by the action of the symmetric group $S_n$ on the cosets of a Sylow $p$-subgroup $P_n$. In the course of this work, we also prove a symmetric group analogue of a well-known result of Navarro for $p$-solvable groups on a conjugacy action of $N_G(P)$. Before describing some consequences of these results, we will give an overview of the background and recent related results in the area.

A stationary Navier-Stokes-Brinkman model coupled to a system of advection-diffusion equations serves as a model for so-called double-diffusive viscous flow in porous mediain which both heat and a solute within the fluid phase are subject to transport and diffusion. The solvability analysis of these governing equations results as a combination of compactness arguments and fixed-point theory. In addition an H(div)-conforming discretisation is formulated by a modification of existing methods for Brinkman flows. The well-posedness ofthe discrete Galerkin formulation is also discussed, and convergence properties are derived rigorously. Computational tests confirm the predicted rates of error decay and illustrate the applicability of the methods for the simulation of bacterial bioconvection and thermohaline circulation problems.

String compactifications are essential for connecting string theory to low energy particle physics and cosmology. Moduli stabilisation gives rise to effective Lagrangians that capture the low-energy degrees of freedom. Much recent interest has been on swampland consistency conditions on such effective
field theories - which low energy Lagrangians can arise from quantum gravity? Furthermore, given that moduli stabilisation scenarios often exist in AdS space, we can also ask: what do swampland conditions mean in the context of AdS/CFT? I describe work on developing a holographic understanding of moduli stabilisation and swampland consistency conditions. I focus in particular on the Large Volume Scenario, which is especially appealing from a holographic perspective as in the large volume limit all its interactions can be expressed solely in terms of the AdS radius, with no free dimensionless parameters.

Coupling dynamics of the states of the nodes of a network to the dynamics of the network topology leads to generic absorbing and fragmentation transitions. The coevolving voter model is a typical system that exhibits such transitions at some critical rewiring. We study the robustness of these transitions under two distinct ways of introducing noise. Noise affecting all the nodes destroys the absorbing-fragmentation transition, giving rise in finite-size systems to two regimes: bimodal magnetization and dynamic fragmentation. Noise targeting a fraction of nodes preserves the transitions but introduces shattered fragmentation with its characteristic fraction of isolated nodes and one or two giant components. Both the lack of absorbing state for homogeneous noise and the shift in the absorbing transition to higher rewiring for targeted noise are supported by analytical approximations.

Paper Link:

https://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.032803

A two-dimensional, minimally Supersymmetric Quantum Field Theory is "nullhomotopic" if it can be deformed to one with spontaneous supersymmetry breaking, including along deformations that are allowed to "flow up" along RG flow lines. SQFTs modulo nullhomotopic SQFTs form a graded abelian group $SQFT_\bullet$. There are many SQFTs with nonzero index; these are definitely not nullhomotopic, and indeed represent nontorision classes in $SQFT_\bullet$. But relations to topological modular forms suggests that $SQFT_\bullet$ also has rich torsion. Based on an analysis of mock modularity and holomorphic anomalies, I will describe explicitly a "secondary invariant" of SQFTs and use it to show that a certain element of $SQFT_3$ has exact order $24$. This work is joint with D. Gaiotto and E. Witten.