Past

The idea of isotropic localization is to substitute an algebro-geometric object (motive)
by its “local” versions, parametrized by finitely generated extensions of the ground field k. In the case of the so-called “flexible” ground field, the complexity of the respective “isotropic motivic categories” is similar to that of their topological counterpart. At the same time, new features appear: the isotropic motivic cohomology of a point encode Milnor’s cohomological operations, while isotropic Chow motives (hypothetically) coincide with Chow motives modulo numerical equivalence (with finite coefficients). Extended versions of the isotropic category permit to access numerical Chow motives with rational coefficients providing a new approach to the old questions related to them. The same localization can be applied to the stable homotopic category of Morel- Voevodsky producing “isotropic” versions of the topological world. The respective isotropic stable homotopy groups of spheres exhibit interesting features.

Complex spatial patterns such as superlattice patterns and quasipatterns occur in a variety of physical systems ranging from vibrated fluid layers to crystallising soft matter. Reduced order models that describe such systems are usually PDEs. Close to a phase transition, modal expansion along with perturbation methods can be applied to convert the PDEs to normal form equations in the form of coupled ODEs. I use equivariant bifurcation theory along with homotopy methods (developed in computational algebraic geometry) to obtain all solutions of the normal form equations in a non-iterative method. I want to talk about how this approach allows us to ask new questions about the physical systems of interest and what extensions to this method might be possible. This forms a step in my long-term interest to explore how to better ‘complete’ a bifurcation diagram!

We discuss sketching techniques for sparse Linear Least Squares (LLS) problems, that perform a randomised dimensionality reduction for more efficient and scalable solutions. We give theoretical bounds for the accuracy of the sketched solution/residual when hashing matrices are used for sketching, quantifying carefully the trade-off between the coherence of the original, un-sketched matrix and the sparsity of the hashing matrix. We then use these bounds to quantify the success of our algorithm that employs a sparse factorisation of the sketched matrix as a preconditioner for the original LLS, before applying LSQR. We extensively compare our algorithm to state-of-the-art direct and iterative solvers for large-scale and sparse LLS, with encouraging results.

Feynman integrals govern the perturbative expansion in quantum field theories. As periods, these integrals generate representations of a motivic Galois group. I will explain this idea and illustrate the 'coaction principle', a mechanism that constrains which periods can appear at any loop order.
 

Many complex systems can be represented as networks. However, it is often not possible or even desirable to observe the entire network structure. For example, in social networks, it is often difficult to obtain samples of large networks due to commercial sensitivity or privacy concerns relating to the data. However, it may be possible to provide a coarse grained picture of the graph given knowledge of the distribution of different demographics (e.g age, income, location, etc…) in a population and their propensities for forming ties between each other.

I will explore the degree to which it is possible to influence Ising systems, which are commonly used to model social influence, on unobserved graphs. Using both synthetic networks (stochastic blockmodels) and case studies of real world social networks, I will demonstrate how simple models which rely only on a coarse grained description of the system or knowledge of only the underlying external fields can perform comparably to more expensive optimization algorithms.

Given a morphism of schemes of characteristic p, we can construct a morphism from the exterior algebra of Kahler differentials to the cohomology of De Rham complex, which is an isomorphism when the original morphism is smooth.

We present a programme towards a combinatorial language for higher (stratified) Morse-Cerf theory. Our starting point will be the interpretation of a Morse function as a constructible bundle (of manifolds) over R^1. Generalising this, we describe a surprising combinatorial classification of constructible bundles on flag foliated R^n (the latter structure of a "flag foliation” is needed for us to capture the notions of "singularities of higher Morse-Cerf functions" independently of differentiable structure). We remark that flag foliations can also be seen to provide a notion of directed topology and in this sense higher Morse-Cerf singularities are closely related to coherences in higher category theory. The main result we will present is the algorithmic decidability of existence of mutual refinements of constructible bundles. Using this result, we discuss how "combinatorial stratified higher Morse-Cerf theory" opens up novel paths to the computational treatment of interesting questions in manifold topology.

Abstract: Gaussian multiplicative chaos (GMC) has attracted a lot of attention in recent years due to its applications in many areas such as Liouville CFT and random matrix theory, but despite its importance not much has been known about its distributional properties. In this talk I shall explain the study of the tail probability of subcritical GMC and establish a precise formula for the leading order asymptotics, resolving a conjecture of Rhodes and Vargas.

The sequence of so-called Signature moments describes the laws of many stochastic processes in analogy with how the sequence of moments describes the laws of vector-valued random variables. However, even for vector-valued random variables, the sequence of cumulants is much better suited for many tasks than the sequence of moments. This motivates the study of so-called Signature cumulants. To do so, an elementary combinatorial approach is developed and used to show that in the same way that cumulants relate to the lattice of partitions, Signature cumulants relate to the lattice of so-called "ordered partitions". This is used to give a new characterisation of independence of multivariate stochastic processes.

A local SL(2,Z) transformation on the Type IIB brane configuration gives rise to an interesting class of 3d superconformal field theories, known as the S-fold SCFTs.  One of the interesting features of such a theory is that, in general, it does not admit a conventional Lagrangian description. Nevertheless, it can be described by a quiver diagram with a link being a superconformal field theory, known as the T(U(N)) theory. In this talk, we discuss various properties of the S-fold theories, including their supersymmetric indices, supersymmetry enhancement in the infrared, as well as several interesting dualities.
 

We discuss a recent generalization of the classical 3-dimensional light bulb theorem to 4-dimensions. We connect this with fundamental questions about knotting of surfaces in 4-dimensional manifolds as well as new directions regarding knotting of 3-balls in 4-manifolds.

Of the canonical stratified shear flow instabilities (Kelvin–Helmholtz, Holmboe-wave and Taylor–Caulfield), the Taylor–Caulfield instability (TCI) has received relatively little attention, and forms the focus of the presentation. A diagnostic of the linear instability dynamics is developed that exploits the net pseudomomentum to distinguish TCI from the other two instabilities for any given flow profile. Next, the nonlinear dynamics of TCI is shown across its range of unstable horizontal wavenumbers and bulk Richardson numbers. At small bulk Richardson numbers, a cascade of billow structures of sequentially smaller size may form. For large bulk Richardson numbers, the primary nonlinear travelling waves formed by the linear instability break down via a small-scale, Kelvin– Helmholtz-like roll-up mechanism with an associated large amount of mixing. In all cases, secondary parasitic nonlinear Holmboe waves appear at late times for high Prandtl number. Finally, a nonlinear diagnostic is proposed to distinguish between the saturated states of the three canonical instabilities based on their distinctive density–streamfunction and generalised vorticity–streamfunction relations.

In this interactive workshop, we'll discuss what mathematicians are looking for in written solutions.  How can you set out your ideas clearly, and what are the standard mathematical conventions?  Please bring a pen or pencil! 

This session is likely to be most relevant for first-year undergraduates, but all are welcome.

We aim to establish and experimentally test mathematical models of embryogenesis. While the foundation of this research is based on models of isolated developmental events, the ultimate challenge is to formulate and understand dynamical systems encompassing multiple stages of development and multiple levels of regulation. These range from specific chemical reactions in single cells to coordinated dynamics of multiple cells during morphogenesis. Examples of our dynamical systems models of embryogenesis – from the events in the Drosophila egg to the early stages of gastrulation – will be presented. Each of these will demonstrate what had been learned from model analysis and model-driven experiments, and what further research directions are guided by these models.

The challenge is to produce a reduced order model which predicts the maximum temperature rise of a thermally conducting object subjected to a power deposition profile supplied by an external code. The target conducting object is basically cuboidal but with one or more shaped faces and may have complex internal cooling structures, the deposition profile may be time dependent and exhibit hot spots and sharp edged shadows among other features. An additional feature is the importance of radiation which makes the problem nonlinear, and investigation of control strategies is also of interest. Overall there appears to be a sequence of problems of degree of difficulty sufficient to tax the most gifted student, starting with a line profile on a cuboid (quasi-2D) with linearised radiation term, and moving towards increased difficulty.

The homotopy bordism category hCob_d has as objects closed (d-1)-manifolds and as morphisms diffeomorphism classes of d-dimensional bordisms. This is a simplified version of the topologically enriched bordism category Cob_d whose classifying space B(Cob_d) been completely determined by Galatius-Madsen-Tillmann-Weiss in 2006. In comparison, little is known about the classifying space B(hCob_d).

In the first part of the talk I will give an introduction to bordism categories and their classifying spaces. In the second part I will identify B(hCob_1) showing, in particular, that the rational cohomology ring of hCob_1 is polynomial on classes \kappa_i in degrees 2i+2 for all i>=1. The seemingly simpler category hCob_1 hence has a more complicated classifying space than Cob_1.

Guided by the analogy with certain moments of the Bessel function that appear as Feynman integrals, Broadhurst and Roberts recently studied a family of L-functions built up by assembling symmetric power moments of Kloosterman sums over finite fields. I will prove that these L-functions arise from potentially automorphic motives over the field of rational numbers, and hence admit a meromorphic continuation to the complex plane that satisfies the expected functional equation. If time permits, I will identify the periods of the corresponding motives with the Bessel moments and make a few comments about the special values of the L-functions. This is a joint work with Claude Sabbah and Jeng-Daw Yu.

Thin film flows of nematic liquid crystal will be considered, using the Leslie-Ericksen formulation for nematics. Our model can account for variations in substrate anchoring, which may exert a strong influence on patterns that arise in the flow. A number of simulations will be presented using an "in house" code, developed to run on a GPU. Current modeling directions involving flow over interlaced electrodes, so-called "dielectrowetting", will be discussed.

Can we get rigorous answers when computing with real and complex numbers? There are now many applications where this is possible thanks to a combination of tools from computer algebra and traditional numerical computing. I will give an overview of such methods in the context of two projects I'm developing. The first project, Arb, is a library for arbitrary-precision ball arithmetic, a form of interval arithmetic enabling numerical computations with rigorous error bounds. The second project, Fungrim, is a database of knowledge about mathematical functions represented in symbolic form. It is intended to function both as a traditional reference work and as a software library to support symbolic-numeric methods for problems involving transcendental functions. I will explain a few central algorithmic ideas and explain the research goals of these projects.