Past

The theory of equations over groups goes back to the very beginning of group theory and is linked to many deep problems in mathematics, such as the Diophantine problem over rationals. In this talk, we shall survey some of the key results on equations over groups, give an outline of the Makanin-Razborov process (an algorithm for solving equations over free groups) and its connections to other results in group theory and low-dimensional topology.

We will present a regularity result for degenerate elliptic equations in nondivergence form.

In joint work with Charlie Smart, we extend the regularity theory of Caffarelli to equations with possibly unbounded ellipticity-- provided that the ellipticity satisfies an averaging condition. As an application we obtain a stochastic homogenization result for such equations (and a new estimate for the effective coefficients) as well as an invariance principle for random diffusions in random environments. The degenerate equations homogenize to uniformly elliptic equations, and we give an estimate of the ellipticity.

In the Greek mythology the hydra is a many-headed poisonous beast. When cutting one of its heads off, it will grow two more. Inspired by how Hercules defeated the hydra, Dison and Riley constructed a family of groups defined by two generators and one relator, which is an Engel  word: the hydra groups. I will talk about its remarkably wild subgroup distortion and its hyperbolic cousin. Very recent discussions of Baumslag and Mikhailov show that those groups are residually torsion-free nilpotent and they introduce generalised hydra groups.

When two drops come into contact they will rapidly merge and form a single drop. Here we address the coalescence of drops on a substrate, focussing on the initial dynamics just after contact. For very viscous drops we present similarity solutions for the bridge that connects the two drops, the size of which grows linearly with time. Both the dynamics and the self-similar bridge profiles are verified quantitatively by experiments. We then consider the coalescence of water drops, for which viscosity can be neglected and liquid inertia takes over. Once again, we find that experiments display a self-similar dynamics, but now the bridge size grows with a power-law $t^{2/3}$. We provide a scaling theory for this behavior, based on geometric arguments. The main result for both viscous and inertial drops is that the contact angle is important as it determines the geometry of coalescence -- yet, the contact line dynamics appears irrelevant for the early stages of coalescence.

A Lagrangian submanifold of a Calabi-Yau manifold is called positive if the real part of the holomorphic volume form restricted to it is positive. A Hamiltonian isotopy class of positive Lagrangian submanifolds admits a Riemannian metric with non-positive curvature. Its universal cover

admits a functional, with critical points special Lagrangians, that is strictly convex with respect to the metric. If time permits, I'll explain

how mirror symmetry relates the metric and functional to the infinite dimensional symplectic reduction picture of Atiyah, Bott, and Donaldson in

the context of the Kobayashi-Hitchin correspondence.

This talk will give an introduction to the recent paper by Arkani Hamed et. al. arxiv:1212:5605.  

Results of Bourgain and Kindler-Safra state that if $f$ is a Boolean function on $\{0,1\}^n$, and

the Fourier transform of $f$ is highly concentrated on low frequencies, then $f$ must be close

to a ‘junta’ (a function depending upon a small number of coordinates). This phenomenon is

known as ‘Fourier stability’, and has several interesting consequences in combinatorics,

theoretical computer science and social choice theory. We will describe some of these,

before turning to the analogous question for Boolean functions on the symmetric group. Here,

genuine stability does not occur; it is replaced by a weaker phenomenon, which we call

‘quasi-stability’. We use our 'quasi-stability' result to prove an isoperimetric inequality

for $S_n$ which is sharp for sets of size $(n-t)!$, when $n$ is large. Several open questions

remain. Joint work with Yuval Filmus (University of Toronto) and Ehud Friedgut (Weizmann

Institute).

Let ${T_1,...,T_l}$ be a collection of differential operators

with constant coefficients on the torus $\mathbb{T}^n$. Consider the

Banach space $X$ of functions $f$ on the torus for which all functions

$T_j f$, $j=1,...,l$, are continuous. The embeddability of $X$ into some

space $C(K)$ as a complemented subspace will be discussed. The main result

is as follows. Fix some pattern of mixed homogeneity and extract the

senior homogeneous parts (relative to the pattern chosen)

${\tau_1,...,\tau_l}$ from the initial operators ${T_1,...,T_l}$. If there

are two nonproportional operators among the $\tau_j$ (for at least one

homogeneity pattern), then $X$ is not isomorphic to a complemented

subspace of $C(K)$ for any compact space $K$.

The main ingredient of the proof is a new Sobolev-type embedding

theorem. It generalises the classical embedding of

${\stackrel{\circ}{W}}_1^1(\mathbb{R}^2)$ to $L^2(\mathbb{R}^2)$. The difference is that

now the integrability condition is imposed on certain linear combinations

of derivatives of different order of several functions rather than on the

first order derivatives of one function.

This is a joint work with D. Maksimov and D. Stolyarov.

Abstract: When a strict local martingale is projected onto a subfiltration to which it is not adapted, the local martingale property may be lost, and the finite variation part of the projection may have singular paths. This phenomenon has consequences for arbitrage theory in mathematical finance. In this paper it is shown that the loss of the local martingale property is related to a measure extension problem for the associated Föllmer measure. When a solution exists, the finite variation part of the projection can be interpreted as the compensator, under the extended measure, of the explosion time of the original local martingale. In a topological setting, this leads to intuitive conditions under which its paths are singular. The measure extension problem is then solved in a Brownian framework, allowing an explicit treatment of several interesting examples.

We study portfolio optimization and contingent claim valuation in markets where illiquidity may affect the transfer of wealth over time and between investment classes. In addition to classical frictionless markets and markets with transaction costs, our model covers nonlinear illiquidity effects that arise in limit order markets. We extend basic results on arbitrage bounds, attainable claims and optimal portfolios to illiquid markets and general swap contracts where both claims and premiums may have multiple payout dates. We establish the existence of optimal trading strategies and the lower semicontinuity of the optimal value of portfolio optimization under conditions that extend the no-arbitrage condition in the classical linear market model.

The operation of sub-cellular processes in living organisms often require the transfer of biopolymers across impermeable lipid membranes. The emergence of new experimental techniques for manipulation of single molecules at nanometer scales have made possible in vitro experiments that can directly probe such translocation processes in cells as well as in synthetic systems. Some of these ideas have spawned novel bio-technologies with many more likely to emerge in the near future. In this talk I would review some of these experiments and attempt to provide a quantitative understanding of the data in terms of physical laws, primarily mechanics and electrostatics.

When a rubber membrane tube is inflated, a localized bulge will initiate when the internal pressure reaches a certain value known as the initiation pressure. As inflation continues, the bulge will grow in diameter until it reaches a maximum size, after which the bulge will spread in both directions. This simple phenomenon has previously been studied both experimentally, numerically, and analytically, but surprisingly it is only recently that the character of the initiation pressure has been fully understood. In this talk, I shall first show how the entire inflation process can be described analytically, and then apply the ideas to the mathematical modelling of aneurysm initiation in human arteries.

A well known theorem of Coleman states that an overconvergent modular eigenform of weight k>1 and slope less than k-1 is classical. This theorem was later reproved and generalized using a geometric method very different from Coleman's cohomological approach. In this talk I will explain how one might go about generalizing the cohomological method to some higher-dimensional Shimura varieties.

The aim of this talk is to introduce the notion of a stack, by considering in some detail the example of the the stack of vector bundles on a curve. One of the key areas of modern geometry is the study of moduli problems and associated moduli spaces, if they exist. For example, can we find a `fine moduli space' which parameterises isomorphism classes of vector bundles on a smooth curve and contains information about how such vector bundles vary in families? Quite often such a space doesn't exist in the category where we posed the original moduli problem, but we can enlarge our category and construct a `stack' which in a reasonable sense gives us the key properties of a fine moduli space we were looking for. This talk will be quite sketchy and won't even properly define a stack, but we hope to at least give some feel of how these objects are defined and why one might want to consider them.

Domain decomposition methods have been developed in various contexts, and with very different goals in mind. I will start my presentation with the historical inventions of the Schwarz method, the Schur methods and Waveform Relaxation. I will show for a simple model problem how all these domain decomposition methods function, give precise results for the model problem, and also explain the most general convergence results available currently for these methods. I will conclude with the parareal algorithm as a new variant for parallelization of evolution problems in the time direction.

General Arrow-Debreu equilibrium can be determined for expected utility maximisers by explicit solutions for individual players. When the expected

utilities are distorted by probability weighting functions, players cannot find explicit optimal decisions. Zhou and Xia studied the existence of equilibrium when the probability weighting functions are the same for all individual players. In this paper, we investigate the same problem but with heterogeneous probability weighting function.

I will discuss the motion of screw dislocations in an elastic body under antiplane shear. In this setting, dislocations are viewed as points in a two-dimensional domain where the strain field fails to be a gradient. The motion is determined by the Peach-Koehler force and the slip-planes in the material. This leads to a system of discontinuous ODE, where the vector field depends on the solution to an elliptic PDE with Neumann data. We show short-time existence of solutions; we also have uniqueness for a restricted class of domains. In general, global solutions do not exist because of collisions.

 Following Prestel and Ziegler, we will explore what it means for a field
to be t-henselian, i.e. elementarily equivalent (in the language of
rings) to some non-trivially henselian valued field. We will discuss
well-known as well as some new properties of t-henselian fields.

***** PLEASE NOTE THIS SEMINAR WILL TAKE PLACE ON THURSDAY 31ST JANUARY *****

We will discuss (very) recent work by Hensel; Przytycki and Webb, who describe unicorn paths in the arc graph and show that they form 1-slim triangles and are invariant under taking subpaths. We deduce that all arc graphs are 7-hyperbolic. Considering the same paths in the arc and curve graph, this also shows that all curve graphs are 17-hyperbolic, including closed surfaces.

The eponymous result is due to Bridson and Vogtmann, and was proven in their paper "Automorphisms of Automorphism Groups of Free Groups" (Journal of Algebra 229). While I'll remind you all the basic definitions, it would be very helpful to be already somewhat familiar with the outer space.

I will look at some tools for proving expansion in the Cayley graphs of finite quotients of a given infinite group, with particular emphasis on Bourgain-Gamburd’s work on expansion in Zariski-dense subgroups of SL_2(Z), and speculate to what extent such expansion may be said to be “uniform”.

I will introduce the notion of Kurosh rank for subgroups of 
free products. This rank satisfies the Howson property, i.e. the 
intersection of two subgroups of finite Kurosh rank has finite Kurosh rank.
I will present a version of the Strengthened Hanna Neumann inequality in 
the case of free products of right-orderable groups. Joint work with  A. 
Martino and I. Schwabrow.

A self-avoiding walk on a lattice is a walk that never visits the same vertex twice.  Self-avoiding walks (SAW) have attracted interest for decades, first in statistical physics, where they are considered as polymer models, and then in combinatorics and in probability theory (the first mathematical contributions are probably due to John Hammersley, from Oxford, in the early sixties). However, their properties remain poorly understood in low dimension, despite the existence of remarkable conjectures.

About two years ago, Duminil-Copin and Smirnov proved an "old" and remarkable conjecture of Nienhuis (1982), according to which the number of SAWs of length n on the honeycomb (hexagonal) lattice grows like mu^n, with mu=sqrt(2 +sqrt(2)).

This beautiful result has woken up the hope to prove other simple looking conjectures involving these objects. I will thus present the proof of a younger conjecture (1995) by Batchelor and Yung, which deals with SAWs confined to a half-plane and interacting with its boundary.

(joint work with N. Beaton, J. de Gier, H. Duminil-Copin and A. Guttmann)

The talk will discuss generalizations of the classical Hadamard jump  condition to general locally Lipschitz maps, and applications to
polycrystals. This is joint work with Carsten Carstensen.

By a "coarse median" we mean a ternary operation on a path metric space, satisfying certain conditions which generalise those of a median algebra. It can be interpreted as a kind of non-positive curvature condition, and is applicable, for example to finitely generated groups. It is a consequence of work of Behrstock and Minsky, for example, that the mapping class group of a surface satisfies this condition. We aim to give some examples, results and applications concerning this notion.

Abstract: We study measures on half planar maps that satisfy a natural domain Markov property. I will discuss their classification and some of their geometric properties. Joint work with Gourab Ray.

A multi-dimensional extension of the structural default model with firms' values driven by diffusion processes with Marshall-Olkin-inspired

correlation structure is presented. Semi-analytical methods for solving

the forward calibration problem and backward pricing problem in three

dimensions are developed. The model is used to analyze bilateral counter- party risk for credit default swaps and evaluate the corresponding credit and debt value adjustments.

In many places, the Antarctic and Greenland ice sheets are fringed by tongues of ice floating on the ocean, called ice shelves. Recent observations and modelling suggest that melting and disintegration of the floating ice shelves can impact ice sheet flow, and hence have consequences for sea level rise. Of particular interest are observations of channels and undulations in the ice shelf base, for which the conditions for genesis remain unclear. To build insight into the potential for melting-driven instability of the ice shelf base, this talk will consider a free boundary problem with melting at the ice-ocean interface coupled to a buoyant plume of meltwater confined below a stationary ice shelf. An asymptotic model of turbulent heat transfer in the meltwater plume reveals that melting rates depend on ice-shelf basal slope, with potentially shocking consequences for the evolving ice-shelf geometry