Past

Mathematicians need to talk and writeabout their mathematics.  This includes undergraduates and MSc students, who may be writing a dissertation or project report, preparing a presentation on a summer research project, or preparing for a job interview.  We think that it can be helpful to think of this as a form of story telling, as this can lead to more effective communication.  For a story to be engaging you also need to know your audience.In this session, we'll discuss what we mean by telling a mathematical story, give you some top tips from our experience, and give you a chance to think about how you might put this into practice.  The session will be of relevance to all undergraduates and MSc students, not only those currently writing a dissertation or preparing an oral presentation.

There is great interest in the molecular characterisation of intestinal metaplasia, such as Barrett’s esophagus (BE), to understand the basic biology of metaplastic development from a tissue of origin. BE is asymptomatic, so it is not generally known how long a patient has lived with this precursor of esophageal adenocarcinoma (EAC) when initially diagnosed in the clinic. We previously constructed a BE clock model using patient-specific methylation data to estimate BE onset times using Bayesian inference techniques, and thus obtain the biological age of BE tissue (Curtius et al. 2016). We find such epigenetic drift to be widely evident in BE tissue (Luebeck et al. 2017) and the corresponding tissue ages show large inter-individual heterogeneity in two patient populations.               

From a basic biological mechanism standpoint, it is not fully understood how the Barrett’s tissue first forms in the human esophagus because this process is never observed in vivo, yet such information is critical to inform biomarkers of risk based on temporal features (e.g., growth rates, tissue age) reflecting the evolution toward cancer. We analysed multi-region samples from 17 BE patients to

1) measure the spatial heterogeneity in biological tissue ages, and 2) use these ages to calibrate mathematical models (agent-based and continuum) of the mechanisms for formation of the segment itself. Most importantly, we found that tissue must be regenerated nearer to the stomach, perhaps driven by wound healing caused by exposure to reflux, implying a gastric tissue of origin for the lesions observed in BE. Combining bioinformatics and mechanistic modelling allowed us to infer evolutionary processes that cannot be clinically observed and we believe there is great translational promise to develop such hybrid methods to better understand multiscale cancer data.

References:

Curtius K, Wong C, Hazelton WD, Kaz AM, Chak A, et al. (2016) A Molecular Clock Infers Heterogeneous Tissue Age Among Patients with Barrett's Esophagus. PLoS Comput Biol 12(5): e1004919

Luebeck EG, Curtius K, Hazelton WD, Made S, Yu M, et al. (2017) Identification of a key role of epigenetic drift in Barrett’s esophagus and esophageal adenocarcinoma. J Clin Epigenet 9:113

Data science has become a topic of great interest lately and has triggered new widescale research activities around efficientl first order methods for optimisation and Bayesian sampling. The National Physical Laboratory is addressing some of these challenges with particular focus on  robustness and confidence in the solution.  In this talk, I will present some problems and recent results concerning i. robust learning in the presence of outliers based on the Median of Means (MoM) principle and ii. stability of the solution in super-resolution (joint work with A. Thompson and B. Toader).

 The talk is about the normal modal logics of elementary classes defined by first-order formulas of the form
 'for all x_0 there exist x_1, ..., x_n phi(x_0, x_1, ... x_n)' with phi being a conjunction of binary atoms.
 I'll show that many properties of these logics, such as finite axiomatisability,
 elementarity,  axiomatisability by a set of canonical formulas or by a single generalised Sahlqvist formula,
 together with modal definability of the initial formula, either simultaneously hold or simultaneously do not hold.
 

The Asai (or twisted tensor) L-function attached to a Bianchi modular form is the 'restriction to the rationals' of the standard L-function. Introduced by Asai in 1977, subsequent study has linked its special values to the arithmetic of the corresponding form. In this talk, I will discuss joint work with David Loeffler in which we construct a p-adic Asai L-function -- that is, a measure on Z_p* that interpolates the critical values L^As(f,chi,1) -- for ordinary weight 2 Bianchi modular forms. We use a new method for constructing p-adic L-functions, using Kato's system of Siegel units to build a 'Betti analogue' of an Euler system, building on algebraicity results of Ghate. I will start by giving a brief introduction to p-adic L-functions and Bianchi modular forms, and if time permits, I will briefly mention another case where the method should apply, that of non-self-dual automorphic representations for GL(3).

A two-layer shear flow in the presence of surfactants is considered. The flow configuration comprises two superposed layers of viscous and immiscible fluids confined in a long horizontal channel, and characterised by different densities, viscosities and thicknesses. The surfactants can be insoluble, i.e. located at the interface between the two fluids only, or soluble in the lower fluid in the form of monomers (single molecules) or micelles (multi-molecule aggregates). A mathematical model is formulated, consisting of governing equations for the hydrodynamics and appropriate transport equations for the surfactant concentration at the interface, the concentration of monomers in the bulk fluid and the micelle concentration. A primary objective of this study is to investigate the effect of surfactants on the stability of the interface, and in particular surfactants in high concentrations and above the critical micelle concentration (CMC). Interfacial instabilities are induced due to the acting forces of gravity and inertia, as well as the action of Marangoni forces generated as a result of the dependence of surface tension on the interfacial surfactant concentration. The underlying physical mechanism responsible for the formation of interfacial waves will be discussed, together with the complex flow dynamics (typical nonlinear phenomena associated with interfacial flows include travelling waves, solitary pulses, quasi-periodic and chaotic dynamics).

I will describe a family of 2d TQFTs, due to Moore-Tachikawa, which take values in a category whose objects are Lie groups and whose morphisms are holomorphic symplectic varieties. They link many interesting aspects of geometry, such as moduli spaces of solutions to Nahm equations, hyperkähler reduction, and geometric invariant theory.

 Multiscale methods based on coupled partial differential equations defined on bulk and embedded manifolds are still poorly explored from the theoretical standpoint, although they are successfully used in applications, such as microcirculation and flow in perforated subsurface reservoirs. This work aims at shedding light on some theoretical aspects of a multiscale method consisting of coupled partial differential equations defined on one-dimensional domains embedded into three-dimensional ones. Mathematical issues arise because the dimensionality gap between the bulk and the inclusions is larger than one, named as the high dimensionality gap case. First, we show that such model derives from a system of full three-dimensional equations, by the application of a topological model reduction approach. Secondly, we rigorously analyze the problem, showing that the averaging operators applied for the model reduction introduce a regularization effect that resolves the issues due to the singularity of solutions and to the ill-posedness of restriction operators. Then, we discretize the problem by means of the finite element method and we analyze the approximation error. Finally, we exploit the structure of the model reduction technique to analyze the modeling error. This study confirms that for infinitesimally small inclusions, the modeling error vanishes.

This is a joint work with Federica Laurino, Department of Mathematics, Politecnico di Milano.

In this talk, we study the long time behaviour of a cloud of weakly interacting Brownian particles, conditionally on the observation of their initial and final configuration. In particular, we connect this problem, which may be regarded as a nonlinear version of the Schrödinger problem, to the study of the long time behaviour of Mean Field Games. Combining tools from optimal transport and stochastic control we prove convergence towards the equilibrium configuration and establish convergence rates. A key ingredient to derive these results is a new functional inequality, which generalises Talagrand’s inequality to the entropic transportation cost.

A stacking is a lift of an immersion of graphs $A\to B$ to an embedding of $A$ into the product of $B$ with the real line; their existence relates to orderability properties of groups. I will describe how Louder and Wilton used them to prove Wise's "$w$-cycles" conjecture: given a primitive word $w$ in a free group $F$, and a subgroup $H < F$, the number of conjugates of $H$ which intersect $<w>$ nontrivially is at most rank($H$). I will also discuss applications of the result to questions of coherence, and possible extensions of it.

The Grothendieck ring of varieties over a field is a simple idea that formalizes various cut-and-paste arguments in algebraic geometry. We will explain how this intuitive construction leads to nontrivial results, such as computing Euler characteristics, counting points of varieties over finite fields, and determining Hodge numbers. As an example, we will investigate cubic hypersurfaces, especially the varieties parametrizing lines on them. If time permits, we will discuss some of the stranger properties of the Grothendieck ring.

We study random (simple) graphs with given vertex degrees, in the sparse case where the average degree is bounded. Assume also that the second moment of the vertex degree is bounded. The standard method then is to use the configuration model to construct a random multigraph and condition it on
being simple.

This works well for results of the type that something holds with high probability, or that something converges in probability, but it does not immediately apply to convergence in distribution, for example asymptotic normality. (Although this has been done by special arguments in a couple of cases, by Janson and Luczak and by Riordan.) A typical example is the recent result by Barbour and Röllin on asymptotic normality of the size of the giant component of the multigraph (in the supercritical case); it is an obvious conjecture that the same results hold for the random simple graph.

We discuss two new approaches to this, both based on old methods. Both apply to the size of the giant component, using rather minor special arguments.

One approach uses the method of moments to obtain joint convergence of the variable of interest together with the numbers of loops and multiple edges
in the  multigraph.

The other approach uses switchings to modify the multigraph and construct a simple graph. This simple random graph will not have a uniform distribution,
but almost, and this is good enough.

The idea of adjusting prices in order to sell goods at the highest acceptable price, such as haggling in a market, is as old as money itself. We consider the problem of pricing multiple products on a network of resources, such as that faced by an airline selling tickets on its flight network. In this talk I will consider various optimization relaxations to the deterministic dynamic pricing problem on a network. This is joint work with Raphael Hauser.

In the 1990s Moy and Prasad revolutionized the representation theory of p-adic groups by showing how to use Bruhat-Tits theory to assign invariants to representations of p-adic groups. The tools they introduced resulted in rapid advancements in both representation theory and harmonic analysis -- areas of central importance in the Langlands program. A crucial ingredient for many results is an explicit construction of (types for) representations of p-adic groups. In this talk I will indicate why, survey what constructions are known (no knowledge about p-adic groups assumed) and present recent developments based on a refinement of Moy and Prasad's invariants.​

In recent years, there has been an increased interest in exploiting rank structure of matrices arising from the discretization of partial differential equations to develop fast direct solvers. In this talk, I will outline the fundamental ideas of this topic in the context of solving the integral equation formulation of the Helmholtz equation, known as the Lippmann-Schwinger equation, and will discuss some plans for future work to develop new, higher-order solvers. This is joint work with Gunnar Martinsson.

We propose a novel class of network models for temporal dyadic interaction data. Our objective is to capture important features often observed in social interactions: sparsity, degree heterogeneity, community structure and reciprocity. We use mutually-exciting Hawkes processes to model the interactions between each (directed) pair of individuals. The intensity of each process allows interactions to arise as responses to opposite interactions (reciprocity), or due to shared interests between individuals (community structure). For sparsity and degree heterogeneity, we build the non time dependent part of the intensity function on compound random measures following (Todeschini et al., 2016). We conduct experiments on real- world temporal interaction data and show that the proposed model outperforms competing approaches for link prediction, and leads to interpretable parameters.

Link to paper: https://papers.nips.cc/paper/7502-modelling-sparsity-heterogeneity-reci…

 I will give a short survey on classical inequalities for Laplace eigenvalues, tell about related history and questions. I will then discuss the so-called Korevaar method, and new results generalising to higher eigenvalues a number of classical inequalities known for the first Laplace eigenvalue only. 

In his famous book on partial differential relations Gromov formulates an exercise concerning local deformations of solutions to open partial differential relations. We will explain the content of this fundamental assertion and sketch a proof. 

In the sequel we will apply this to extend local deformations of closed $G_2$ structures, and to construct 
$C^{1,1}$-Riemannian metrics which are positively curved "almost everywhere" on arbitrary manifolds. 

This is joint work with Christian Bär (Potsdam).

We consider Brownian motion with Kato class measure-valued drift.   A small time large deviation principle and a Varadhan type asymptotics for the Brownian motion with singular drift are established. We also study the existence and uniqueness of the associated Dirichlet boundary value problems.

We first give a short introduction to analysis and stochastic processes on fractal state spaces and the typical difficulties involved.

We then discuss gradient operators and semilinear PDE. They are used to formulate the main result which establishes the hydrodynamic limit of the weakly asymmetric exclusion process on the Sierpinski gasket in the form of a law of large numbers for the particle density. We will explain some details and, if time permits, also sketch a corresponding large deviations principle for the symmetric case.