Past

In this talk I will explain a category theoretic perspective on geometry.  Starting with a category of local objects (of and algebraic nature), and a (Grothendieck) 
topology on it, one can define global objects such as schemes and stacks. Examples of this  approach are algebraic, analytic, differential geometries and also more exotic geometries  such as analytic and differential geometry over the integers and analytic geometry over  the field with one element. In this approach the notion of a point is not primary but is  derived from the local to global structure. The Zariski and Huber spectra are recovered  in this way, and we also get new spectra which might be of interest in model theory.

We will answer the following question: given a finite simplicial complex X acted on by a finite group G, which object stores the minimal amount of information about the symmetries of X in such a way that we can reconstruct both X and the group action? The natural first guess would be the quotient X/G, which remembers one representative from each orbit. However, it does not tell us the size of each orbit or how to glue together simplices to recover X. Our desired object is, in fact, a complex of groups. We will understand two processes: compression and reconstruction and see primarily through an example how to answer our initial question.

In 2010 Coecke, Sadrzadeh, and Clark formulated a new model of natural language which operates by combining the syntactics of grammar and the semantics of individual words to produce a unified ''meaning'' of sentences. This they did by using category theory to understand the component parts of language and to amalgamate the components together to form what they called a ''distributional compositional categorical model of meaning''. In this talk I shall introduce the model of Coecke et. al., and use it to compare the meaning of sentences in Irish and in English (and thus ascertain when a sentence is the translation of another sentence) using a cosine similarity score.

The Irish language is a member of the Gaelic family of languages, originating in Ireland and is the official language of the Republic of Ireland.

I will give an overview of the localization technique: a powerful dimension-reduction tool for proving geometric and functional inequalities.  Having its roots in a  pioneering work of Payne-Weinberger in the 60ies about sharp Poincare’-Wirtinger inequality on Convex Bodies in Rn, recently such a technique found new applications for a range of sharp geometric and functional inequalities in spaces with Ricci curvature bounded below.

This talk will be a survey on the applications of random matrix theory in neuroscience. We will explain why and how we use random matrices to model networks of neurons in the brain. We are mainly interested in the study of neuronal dynamics, and we will present results that cover two parallel directions taken by the field of theoretical neuroscience. First, we will talk about the critical point of transitioning to chaos in cases of random matrices that aim to be more "biologically plausible". And secondly, we will see how a deterministic and a random matrix (corresponding to learned structure and noise in a neuronal network) can interact in a dynamical system.

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.

As early as the 1920's Marshall suggested that firms co-locate in cities to reduce the costs of moving goods, people, and ideas. These 'forces of agglomeration' have given rise, for example, to the high tech clusters of San Francisco and Boston, and the automobile cluster in Detroit. Yet, despite its importance for city planners and industrial policy-makers, until recently there has been little success in estimating the relative importance of each Marshallian channel to the location decisions of firms.
Here we explore a burgeoning literature that aims to exploit the co-location patterns of industries in cities in order to disentangle the relationship between industry co-agglomeration and customer/supplier, labour and idea sharing. Building on previous approaches that focus on across- and between-industry estimates, we propose a network-based method to estimate the relative importance of each Marshallian channel at a meso scale. Specifically, we use a community detection technique to construct a hierarchical decomposition of the full set of industries into clusters based on co-agglomeration patterns, and show that these industry clusters exhibit distinct patterns in terms of their relative reliance on individual Marshallian channels.

The second part is to use industry relatedness, which we measure via a similar metric to co-location, to better understand the association of industrial emissions to city-industry agglomeration. Specifically, we see that industrial emissions (which are the largest source of greenhouse emissions in the US) are highly tied to certain industries, and furthermore that communities in the industry relatedness network tend to explain the tendency of particular industry clusters to produce emissions. This is important, because it limits cities' abilities to move to a greener industry basket as some cities may be more or less constrained to highly polluting industry clusters, while others have more potential for diversification away from polluting industries.

The aim of this talk is to give an overview about the structure theory of finite dimensional RCD metric measure spaces. I will first focus on rectifiability, existence, uniqueness and constancy of the dimension of tangents up to negligible sets.
Then I will motivate why boundaries of sets of finite perimeter are natural codimension one objects to look at in this framework and present some recent structure results obtained in their study.
This is based on joint works with Luigi Ambrosio, Elia Bruè and Enrico Pasqualetto.
 

The mapping class group of a surface is a group of homeomorphisms of that surface, and these groups have been very well studied in the last 50 years. The talk will be focused on a way to understand such a group by looking at the subsurfaces of the corresponding surface; this is the so-called "Masur-Minsky hierarchy machinery". We'll finish with a non-technical discussion of hierarchically hyperbolic groups, which are a popular area of current research, and of which mapping class groups are important motivating examples. No prior knowledge of the objects involved will be assumed.

We discuss asymptotics of Toeplitz determinants with Fisher--Hartwig singularities, and give an overview of past and more recent results.
Applications include the study of asymptotics of certain statistics of the characteristic polynomial of the Circular Unitary Ensemble (CUE) of random matrices. In particular recent results in the study of Toeplitz determinants allow for a proof of a conjecture by Fyodorov and Keating on moments of averages of the characteristic polynomial of the CUE.
 

Stability conditions on triangulated categories were introduced by Bridgeland, based on ideas from string theory. Conjecturally, they control existence of solutions to the deformed Hermitian Yang-Mills equation and the special Lagrangian equation (on the A-side and B-side of mirror symmetry, respectively). I will focus on the symplectic side and sketch a program which replaces special Lagrangians by "spectral networks", certain graphs enhanced with algebraic data. Based on joint work in progress with Katzarkov, Konstevich, Pandit, and Simpson.

In this talk, I will present a strong (or pathwise) approximation of standard Brownian motion by a class of orthogonal polynomials. Most notably, the coefficients obtained from this expansion are independent Gaussian random variables. This will enable us to generate approximate Brownian paths by matching certain polynomial moments. To conclude the talk, I will discuss related works and applications to numerical methods for SDEs.
 

The notion of colour Lie algebra, introduced by Ree (1960), generalises notions of Lie algebra and Lie superalgebra. From an orthogonal representation V of a quadratic colour Lie algebra g, we give various ways of constructing a colour Lie algebra g’ whose bracket extends the bracket of g and the action of g on V. A first possibility is to consider g’=g⊕V and requires the cancellation of an invariant studied by Kostant (1999). Another construction is possible when the representation is ``special’’ and in this case the extension is of the form g’=g⊕sl(2,k)⊕V⊗k^2. Covariants are associated to special representations and satisfy to particular identities generalising properties studied by Mathews (1911) on binary cubics. The 7-dimensional fundamental representation of a Lie algebra of type G_2 and the 8-dimensional spinor representation of a Lie algebra of type so(7) are examples of special representations.

In this work, we propose a class of numerical schemes for solving semilinear Hamilton-Jacobi-Bellman-Isaacs (HJBI) boundary value problems which arise naturally from exit time problems of diffusion processes with controlled drift. We exploit policy iteration to reduce the semilinear problem into a sequence of linear Dirichlet problems, which are subsequently approximated by a multilayer feedforward neural network ansatz. We establish that the numerical solutions converge globally in the H^2 -norm, and further demonstrate that this convergence is superlinear, by interpreting the algorithm as an inexact Newton iteration for the HJBI equation. Moreover, we construct the optimal feedback controls from the numerical value functions and deduce convergence. The numerical schemes and convergence results are then extended to oblique derivative boundary conditions. Numerical experiments on the stochastic Zermelo navigation problem and the perpetual American option pricing problems are presented to illustrate the theoretical results and to demonstrate the effectiveness of the method.
 

Introducing cheap function proxies for quickly producing approximate random numbers, we show convergence of modified numerical schemes, and coupling between approximation and discretisation errors. We bound the cumulative roundoff error introduced by floating-point calculations, valid for 16-bit half-precision (FP16). We combine approximate distributions and reduced-precisions into a nested simulation framework (via multilevel Monte Carlo), demonstrating performance improvements achieved without losing accuracy. These simulations predominantly perform most of their calculations in very low precisions. We will highlight the motivations and design choices appropriate for SVE and FP16 capable hardware, and present numerical results on Arm, Intel, and NVIDIA based hardware.

In this talk, I will describe new mathematical structures in the low-energy  expansion of one-loop string amplitudes. The insertion of external states on the open- and closed-string worldsheets requires integration over punctures on a cylinder boundary and a torus, respectively. Suitable bases of such integrals will be shown to obey simple first-order differential equations in the modular parameter of the surface. These differential equations will be exploited  to perform the integrals order by order in the inverse string tension, similar to modern strategies for dimensionally regulated Feynman integrals. Our method manifests the appearance of iterated integrals over holomorphic  Eisenstein series in the low-energy expansion. Moreover, infinite families of Laplace equations can be generated for the modular forms in closed-string  low-energy expansions.
 

In many applications we are confronted with the following scenario: we observe snapshots of data describing the state of a system at particular times, and based on these observations we want to infer the (dynamical) interactions between the entities we observe. However, often the number of samples we can obtain from such a process are far too few to identify the network exactly. Can we still reliable infer some aspects of the underlying system?
Motivated by this question we consider the following alternative system identification problem: instead of trying to infer the exact network, we aim to recover a (low-dimensional) statistical model of the network based on the observed signals on the nodes.  More concretely, here we focus on observations that consist of snapshots of a diffusive process that evolves over the unknown network. We model the (unobserved) network as generated from an independent draw from a latent stochastic block model (SBM), and our goal is to infer both the partition of the nodes into blocks, as well as the parameters of this SBM. We present simple spectral algorithms that provably solve the partition and parameter inference problems with high-accuracy.

The goal of the talk is to present a proof of the following statement:
Let (K,v) be an algebraic extension of (Q_p,v_p) whose completion is perfectoid. We show that K is relatively decidable to its tilt K^♭, i.e. if K^♭ is decidable in the language of valued fields, then so is K. 
In the first part [of the talk], I will try to cover the necessary background needed from model theory and the theory of perfectoid fields.

Localization technique permits to reduce full dimensional problems to possibly easier lower dimensional ones. During the last years a new approach to localization has been obtained using the powerful tools of optimal transport. Following this approach, we obtain quantitative versions of two relevant geometric inequalities  in comparison geometry as Levy-Gromov isoperimetric inequality (joint with F. Maggi and A. Mondino) and the spectral gap inequality (joint with A. Mondino and D. Semola). Both results are also valid in the more general setting of metric measure spaces verifying the so-called curvature dimension condition.

The tunnel number of a graph embedded in a 3-dimensional manifold is the fewest number of arcs needed so that the union of the graph with the arcs has handlebody exterior. The behavior of tunnel number with respect to connected sum of knots can vary dramatically, depending on the knots involved. However, a classical theorem of Scharlemann and Schultens says that the tunnel number of a composite knot is at least the number of factors. For theta graphs, trivalent vertex sum is the operation which most closely resembles the connected sum of knots. The analogous theorem of Scharlemann and Schultens no longer holds, however. I will provide a sharp lower bound for the tunnel number of composite theta graphs, using recent work on a new knot invariant which is additive under connected sum and trivalent vertex sum. This is joint work with Maggy Tomova.

The classic Fatou lemma states that the lower limit of expectations is greater or equal than the expectation of the lower limit for a sequence of nonnegative random variables. This talk describes several generalizations of this fact including generalizations to converging sequences of probability measures. The three types of convergence of probability measures are considered in this talk: weak convergence, setwise convergence, and convergence in total variation. The talk also describes the Uniform Fatou Lemma (UFL) for sequences of probabilities converging in total variation. The UFL states the necessary and sufficient conditions for the validity of the stronger inequality than the inequality in Fatou's lemma. We shall also discuss applications of these results to sequential optimization problems with completely and partially observable state spaces. In particular, the UFL is useful for proving weak continuity of transition probabilities for posterior state distributions of stochastic sequences with incomplete state observations known under the name of Partially Observable Markov Decision Processes. These transition probabilities are implicitly defined by Bayes' formula, and general method for proving their continuity properties have not been available for long time. This talk is based on joint papers with Pavlo Kasyanov, Yan Liang, Michael Zgurovsky, and Nina Zadoianchuk.

 Gaussian processes are well studied object in statistics and mathematics. In Machine Learning, we think of Gaussian processes as prior distributions over functions, which map from the index set to the realised path. To make Gaussian processes a practical tool for machine learning, we have developed tools around variational inference that allow for approximate computation in GPs leveraging the same hardware and software stacks that support deep learning. In this talk I'll give an overview of variational inference in GPs, show some successes of the method, and outline some exciting direction of potential future work.

I will explore the geometrical structure of Higgs branches of quantum field theories with 8 supercharges in 3, 4, 5 and 6 dimensions. They are hyperkahler singularities, and as such they can be described by a Hasse diagram built from a family of elementary transitions. This corresponds physically to the partial Higgs mechanism. Using brane systems and recently introduced notions of magnetic quivers and quiver subtraction, we formalise the rules to obtain the Hasse diagrams.

Viruses encapsulate their genetic material into protein containers that act akin to molecular Trojan horses, protecting viral genomes between rounds of infection and facilitating their release into the host cell environment. In the majority of viruses, including major human pathogens, these containers have icosahedral symmetry. Mathematical techniques from group, graph and tiling theory can therefore be used to better understand how viruses form, evolve and infect their hosts, and point the way to novel antiviral solutions.

In this talk, I will present an overarching theory of virus architecture, that contains the seminal Caspar Klug theory as a special case and solves long-standing open problems in structural virology. Combining insights into virus structure with a range of different mathematical modelling techniques, such as Gillespie algorithms, I will show how viral life cycles can be better understood through the lens of viral geometry. In particular, I will discuss a recent model for genome release from the viral capsid. I will also demonstrate the instrumental role of the Hamiltonian path concept in the discovery of a virus assembly mechanism that occurs in many human pathogens, such as Picornaviruses – a family that includes the common cold virus– and Hepatitis B and C virus. I will use multi-scale models of a viral infection and implicit fitness landscapes in order to demonstrate that therapeutic interventions directed against this mechanism have advantages over conventional forms of anti-viral therapy. The talk will finish with a discussion of how the new mathematical and mechanistic insights can be exploited in bio-nanotechnology for applications in vaccination and gene therapy.