Past

I describe recent unique continuation results for linear wave equations obtained jointly with Spyros Alexakis and Arick Shao. They state, informally speaking, that solutions to the linear wave equation on asymptotically flat spacetimes are completely determined, in a neighbourhood of infinity, from their radiation towards infinity, understood in a suitable sense. We find that the mass of the spacetime plays a decisive role in the analysis.

We consider spacetimes arising from perturbations of the interior of Kerr

black holes. These spacetimes have a null boundary in the future such that

the metric extends continuously beyond. However, the Christoffel symbols

may fail to be square integrable in a neighborhood of any point on the

boundary. This is joint work with M. Dafermos

We present a mechanism of shock formation for a class of quasilinear wave equations. The solutions are stable and no symmetry assumption is assumed. The proof is based on the energy estimates and on the study of Lorentzian geometry defined by the solution.

 When given an explicit solution to an evolutionary partial differential equation, it is natural to ask whether the solution is stable, and if yes, what is the mechanism for stability and whether this mechanism survives under perturbations of the equation itself. Many familiar linear equations enjoy some notion of stability for the zero solution: solutions of the heat equation dissipate and decay uniformly and exponentially to zero, solutions of the Schrödinger equations disperse at a polynomial rate in time depending on spatial dimension, while solutions of the wave equation enjoy radiative decay (in the presence of at least two spatial dimensions) also at polynomial rates.

For this set of short course sessions, we will focus on the wave equation and its nonlinear perturbations. As mentioned above, the stability mechanism for the linear wave equation is that of radiative decay. Radiative decay depends on the number of spatial dimensions, and hence so does the stability of the zero solution for nonlinear wave equations. By the mid-1980s it was well understood that the stability mechanism survives generally (for “smooth nonlinearities”) when the spatial dimension is at least four, but for lower dimensions (two and three specifically; in dimension one there is no linear stability mechanism to start with) obstructions can arise when the nonlinearities are “stronger” than can be controlled by radiative decay. This led to the discovery of the null condition as a structural condition on the nonlinearities preventing the aforementioned obstructions. But what happens when the null condition is violated? This development spanning a quarter of a century, from F. John’s qualitative analysis of the spherically symmetric case, though S. Alinhac’s sharp control of the asymptotic lifespan, and culminating in D. Christodoulou’s full description of the null geometry, is the subject of this short course.

(1) We will start by reviewing the radiative decay mechanism for wave equations, and indicate the nonlinear stability results for high spatial dimensions. We then turn our attention to the case of three spatial dimensions: after a quick discussion of the null condition for quasilinear wave equations, we sketch, at the semilinear level, what happens when the null condition fails (in particular the asymptotic approximation of the solution by a Riccati equation).

(2) The semilinear picture is built up using a version of the method of characteristics associated with the standard wave operator. Turning to the quasilinear problem we will hence need to understand the characteristic geometry for a variable coefficient wave operator. This leads us to introduce the optical/acoustical function and its associated null structure equations.

(3) From this modern geometric perspective we next discuss, in some detail, the blow-up results obtained in the mid-1980s by F. John for quasilinear wave equations assuming radial symmetry.

(4) Finally, we indicate the main difficulties in extending the analysis to the non-radially-symmetric case, and how they can be resolved à la the recent tour de force of D. Christodoulou. While some knowledge of Lorentzian geometry and dynamics of wave equations will be helpful, this short course should be accessible to also graduate students with training in partial differential equations.

Imperial College London, United Kingdom E-mail address: @email

École Polytechnique Fédérale de Lausanne, Switzerland E-mail address: @email

Question: Is it a realistic proposition for a mathematician to use his/her skills to make a living from sports betting? The introduction of betting exchanges have fundamentally changed the potential profitability of gambling, and a professional mathematician's arsenal of numerical and theoretical weapons ought to give them a huge natural advantage over most "punters", so what might be realistically possible and what potential risks are involved? This talk will give some idea of the sort of plan that might be required to realise this ambition, and what might be further required to attain the aim of sustainable gambling profitability.

We show that for every $n\ge 2$ there exists a torsion-free one-ended word-hyperbolic group $G$ of rank $n$ admitting generating $n$-tuples $(a_1,\ldots ,a_n)$ and $(b_1,\ldots ,b_n)$ such that the $(2n-1)$-tuples $$(a_1,\ldots ,a_n, \underbrace{1,\ldots ,1}_{n-1 \text{ times}})\hbox{ and }(b_1,\ldots, b_n, \underbrace{1,\ldots ,1}_{n-1 \text{ times}} )$$ are not Nielsen-equivalent in $G$. The group $G$ is produced via a probabilistic construction (joint work with Ilya Kapovich).

TQFTs have received widespread attention in recent years. In mathematics

for example due to Lurie's proof of the cobordism hypothesis. In physics

they are used as toy models to understand structure, especially

boundaries and defects.

I will present a lattice construction of 2d Spin TFT. This mostly

motivated as both a toy model and stepping stone for a mathematical

construction of rational conformal field theories with fermions.

I will first describe a combinatorial model for spin surfaces that

consists of a triangulation and a finte set of extra data. This model is

then used to construct TFT correlators as morphisms in a symmetric

monoidal category, given a Frobenius algebra as input. The result is

shown to be independent of the triangulation used, and one obtains thus

a 2dTFT.

All results and constructions can be generalised to framed surfaces in a

relatively straightforward way.

 This lecture will cover two recent works on the mock modular
forms of Ramanujan.

I. Solution of Ramanujan's original conjectures about these functions.
(Joint work with Folsom and Rhoades)

II. A new theorem that mock modular forms are "generating functions" for
central L-values and derivatives of quadratic twist L-functions.
(Joint work with Alfes, Griffin, Rolen).

Worst-case portfolio optimization has been introduced in Korn and Wilmott

(2002) and is based on distinguishing between random stock price

fluctuations and market crashes which are subject to Knightian

uncertainty. Due to the absence of full probabilistic information, a

worst-case portfolio problem is considered that will be solved completely.

The corresponding optimal strategy is of a multi-part type and makes an

investor indifferent between the occurrence of the worst possible crash

and no crash at all.

We will consider various generalizations of this setting and - as a very

recent result - will in particular answer the question "Is it good to save

for bad times or should one consume more as long as one is still rich?"

Stick-slip behavior is a distinguishing characteristic of the flow of Whillans Ice Stream. Distinct from stick-slip on northern hemisphere glaciers, which is generally attributed to supraglacial melt, the behavior is thought be be controlled by fast processes at the bed and by tidally-induced stress. Modelling approaches to studying this phenomenon typically consider ice to be an elastically-deforming solid (e.g. Winberry et al, 2008; Sergienko et al, 2009). However, there remains a question of whether irreversible, i.e. viscous, deformation is important to the stick-slip process; and furthermore whether the details of stick-slip oscillations are important to ice stream evolution on longer time scales (years to decades).

To address this question I use two viscoelastic models of varying complexity. The first is a modification to the simple block-and-slider models traditionally used to examine earthquake processes on a very simplistic fashion. Results show that the role of viscosity in stick-slip depends on the dominant stress balance. These results are then considered in the context of a continuum description of a viscoelastic ice stream with a rate-weakening base capable of exhibiting stick-slip behavior. With the continuum model we examine the spatial and temporal aspects of stick-slip, their dependence on viscous effects, and how this behavior impacts the mean flow. Different models for the evolution of basal shear stress are examined in the experiments, with qualitatively similar results. A surprising outcome is that tidal effects, while greatly affecting the spectrum of the stick-slip cycle, may have relatively little effect on the mean flow.

In this talk, we will review the classical Bass-Serre theory of groups acting on trees and introduce its real version, Rips' theory. If time permits, I will briefly discuss some higher dimensional spaces that are currently being investigated, namely cubings and real cubings.

The plan: start with an introduction to several random matrix ensembles and discuss asymptotic properties of the eigenvalues of the matrices, the last one being the so-called "Normal Matrix Model", and the connection described in the title will be explained. If all goes well I will end with an explanation of asymptotic computations for a new normal matrix model example, which demonstrates a form of universality.

(NOTE CHANGE OF VENUE TO L2)

In this talk, two concepts are brought together: Algebras with triangular decomposition (as studied by Bazlov & Berenstein) and pointed Hopf algebra. The latter are Hopf algebras for which all simple comodules are one-dimensional (there has been recent progress on classifying all finite-dimensional examples of these by Andruskiewitsch & Schneider and others). Quantum groups share both of these features, and we can obtain possibly new classes of deformations as well as a characterization of them.

We propose a strategy which allows computing eigenvalue enclosures for the Maxwell operator by means of the finite element method. The origins of this strategy can be traced back to over 20 years ago. One of its main features lies in the fact that it can be implemented on any type of regular mesh (structured or otherwise) and any type of elements (nodal or otherwise). In the first part of the talk we formulate a general framework which is free from spectral pollution and allows estimation of eigenfunctions.

We then prove the convergence of the method, which implies precise convergence rates for nodal finite elements. Various numerical experiments on benchmark geometries, with and without symmetries, are reported.

We study the behavior of atomistic models under uniaxial tension and investigate the system for critical fracture loads. We rigorously prove that in the discrete-to- continuum limit the minimal energy satisfies a particular cleavage law with quadratic response to small boundary displacements followed by a sharp constant cut-off beyond some critical value. Moreover, we show that the minimal energy is attained by homogeneous elastic configurations in the subcritical case and that beyond critical loading cleavage along specific crystallographic hyperplanes is energetically favorable. We present examples of mass spring models with full nearest and next-to-nearest pair interactions and provide the limiting minimal energy and minimal configurations.

Hrushovski-Martin-Rideau have proved rationality of Poincare series counting 
numbers of equivalence classes of a definable equivalence relation on the p-adic field (in connection to a problem on counting representations of groups). For this they have proved 
uniform p-adic elimination of imaginaries. Their work implies that these Poincare series are 
motivic. I will talk about their work.