Past

Let $E/k$ be an elliptic curve over a number field and $K/k$ a Galois extension with group $G$. What can we say about $E(K)$ as a Galois module? Not just what complex representations appear, but its structure as a $\mathbb{Z}[G]$-module. We will look at some examples with small $G$.

One of the key unsolved challenges at the interface of physical and life sciences is to formulate comprehensive computational modeling of cells of higher organisms that is based on microscopic molecular principles of chemistry and physics. Towards addressing this problem, we have developed a unique reactive mechanochemical force-field and software, called MEDYAN (Mechanochemical Dynamics of Active Networks: http://medyan.org).  MEDYAN integrates dynamics of multiple mutually interacting phases: 1) a spatially resolved solution phase is treated using a reaction-diffusion master equation; 2) a polymeric gel phase is both chemically reactive and also undergoes complex mechanical deformations; 3) flexible membrane boundaries interact mechanically and chemically with both solution and gel phases.  In this talk, I will first outline our recent progress in simulating multi-micron scale cytosolic/cytoskeletal dynamics at 1000 seconds timescale, and also highlight the outstanding challenges in bringing about the capability for routine molecular modeling of eukaryotic cells. I will also report on MEDYAN’s applications, in particular, on developing a theory of contractility of actomyosin networks and also characterizing dissipation in cytoskeletal dynamics. With regard to the latter, we devised a new algorithm for quantifying dissipation in cytoskeletal dynamics, finding that simulation trajectories of entropy production provide deep insights into structural evolution and self-organization of actin networks, uncovering earthquake-like processes of gradual stress accumulation followed by sudden rupture and subsequent network remodeling.
 

A fine compactified Jacobian is a proper open substack of the moduli space of simple sheaves. We will see that fine compactified Jacobians correspond to a certain combinatorial datum, essentially obtained by taking multidegrees of all elements of the compactified Jacobian. This picture generalizes to flat families of curves. We will discuss a classification result in the case when the family is the universal family over the moduli space of curves. This is a joint work with Jesse Kass.

We'll discuss the large charge expansion in CFTs with supersymmetry, focussing on 1908.10306 by Grassi, Komargodski and Tizzano.

The development of effective solvers for high frequency wave propagation problems, such as those described by the Helmholtz equation, presents significant challenges. One promising class of solvers for such problems are parallel domain decomposition methods, however, an appropriate coarse space is typically required in order to obtain robust behaviour (scalable with respect to the number of domains, weakly dependant on the wave number but also on the heterogeneity of the physical parameters). In this talk we introduce a coarse space based on generalised eigenproblems in the overlap (GenEO) for the Helmholtz equation. Numerical results within FreeFEM demonstrate convergence that is effectively independent of the wave number and contrast in the heterogeneous coefficient as well as good performance for minimal overlap.

Let A be a Tannakian category. Any exact tensor functor defined on A is either zero, or faithful. In this talk, I want to draw attention to a derived analogue of this statement (in characteristic zero) due to Jack Hall and David Rydh, and discuss some remarkable consequences for certain classification problems in algebraic geometry.

The Anderson Hamiltonian is used to model particles moving in
disordered media, it can be thought of as a Schrödiger operator with an
extremely irregular random potential. Using the recently developed theory of
"Paracontrolled Distributions" we are able to define the Anderson
Hamiltonian as a self-adjoint non-positive operator on the 2- and
3-dimensional torus and give an explicit description of its domain.
Then we use these results to solve some semi-linear PDEs whose linear part
is given by the Anderson Hamiltonian, more precisely the multiplicative
stochastic NLS and nonlinear Wave equation.
This is joint work with M. Gubinelli and B. Ugurcan.

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.