Past

An important moral truth about the mapping class group of a closed orientable surface is the following: a generic mapping class has no power fixing a finite family of simple closed curves on the surface. Such "generic" elements are called pseudo-Anosov. In this talk I will discuss one instantiation of this principle, namely that the probability of a simple random walk on the mapping class group returning a non-pseudo Anosov element decays exponentially quickly.

A finitely generated group has the Haagerup property if it admits a proper isometric action on a Hilbert space. It was a long open question whether Haagerup property is a quasi-isometry invariant. The negative answer was recently given by Mathieu Carette, who constructed two quasi-isometric generalized Baumslag-Solitar groups, one with the Haagerup property, the other not. Elaborating on these examples, we proved (jointly with S. Arnt and T. Pillon) that the equivariant Hilbert compression is not a quasi-isometry invariant. The talk will be devoted to describing Carette's examples.

Given a model dynamical system, a model of any measuring apparatus relating states to observations, and a prior assessment of uncertainty, the probability density of subsequent system states, conditioned upon the history of the observations, is of some practical interest.

When observations are made at discrete times, it is known that the evolving probability density is a solution of the Bayesian filtering equations. This talk will describe the difficulties in approximating the evolving probability density using a Gaussian mixture (i.e. a sum of Gaussian densities). In general this leads to a sequence of optimisation problems and related high-dimensional integrals. There are other problems too, related to the necessity of using a small number of densities in the mixture, the requirement to maintain sparsity of any matrices and the need to compute first and, somewhat disturbingly, second derivatives of the misfit between predictions and observations. Adjoint methods, Taylor expansions, Gaussian random fields and Newton’s method can be combined to, possibly, provide a solution. The approach is essentially a combination of filtering methods and '4-D Var’ methods and some recent progress will be described.

Erdos, Faudree, Gould, Gyarfas, Rousseau and Schelp, conjectured that
whenever the edges of a complete graph are coloured using three colours
there always exists a set of at most three vertices which have at least
two-thirds of their neighbours in one of the colours.  We will describe a
proof of this conjecture. This is joint work with Rahil Baber

Everybody has heard of the Faraday cage effect, in which a wire mesh does a good job of blocking electric fields and electromagnetic waves. For example, the screen on the front of your microwave oven keeps the microwaves from getting out, while light with its smaller wavelength escapes so you can see your burrito.  Surely the mathematics of such a famous and useful phenomenon has been long ago worked out and written up in the physics books, right?

Well, maybe.   Dave Hewett and I have communicated with dozens of mathematicians, physicists, and engineers on this subject so far, and we've turned up amazingly little.   Everybody has a view of why the Faraday cage mathematics is obvious, and most of their views are different.  Feynman discusses the matter in his Lectures on Physics, but so far as we can tell, he gets it wrong. 

For the static case at least (the Laplace equation), Hewett and I have made good progress with numerical explorations based on  Mikhlin's method backed up by a theorem.   The effect seems to much weaker than we had imagined -- are we missing something?  For time-harmonic waves (the Helmholtz equation), our simulations lead to further puzzles.  We need advice!  Where in the world is the literature on this problem? 

Monotonicity formulas play a pervasive role in the study of variational inequalities and free boundary problems. In this talk we will describe a new approach to a classical problem, namely the thin obstacle (or Signorini) problem, based on monotonicity properties for a family of so-called frequency functions.

Let $F \in \mathbb{Z}[x_1,\ldots,x_n]$. Suppose $F(\mathbf{x})=0$ has infinitely many integer solutions $\mathbf{x} \in \mathbb{Z}^n$. Roughly how common should be expect the solutions to be? I will tell you what your naive first guess ought to be, give a one-line reason why, and discuss the reasons why this first guess might be wrong.

I then will apply these ideas to explain the intriguing parallels between the handling of the Brauer-Manin obstruction by Heath-Brown/Skorobogotov [doi:10.1007/BF02392841] on the one hand and Wei/Xu [arXiv:1211.2286] on the other, despite the very different methods involved in each case.

I will present a formula that relates a Higgs bundle on an algebraic curve and Gromov-Witten invariants. I will start with the simplest example, which is a rank 2 bundle over the projective line with a meromorphic Higgs field. The corresponding quantum curve is the Airy differential equation, and the Gromov-Witten invariants are the intersection numbers on the moduli space of pointed stable curves. The formula connecting them is exactly the path that Airy took, i.e., from wave mechanics to geometric optics, or what we call the WKB method, after the birth of quantum mechanics. In general, we start with a Higgs bundle. Then we apply a generalization of the topological recursion, originally found by physicists Eynard and Orantin in matrix models, to this context. In this way we construct a quantization of the spectral curve of the Higgs bundle. 

Morrey's lower semicontinuity theorem for quasiconvex integrands is a classical result that establishes the existence of minimisers to variational problems by the Direct Method, provided the integrand satisfies "standard" growth conditions (i.e. when the growth and coercivity exponents match). This theorem has more recently been refined to consider convergence in Sobolev Spaces below the growth exponent of the integrand: such results can be used to show existence of solutions to a "Relaxed minimisation problem" when we have "non-standard'" growth conditions.

When the integrand satisfies linear coercivity conditions, it is much more useful to consider the space of functions of Bounded Variation, which has better compactness properties than $W^{1,1}$. We review the key results in the standard growth case, before giving an overview of recent results that we have obtained in the non-standard case. We find that new techniques and ideas are required in this setting, which in fact provide us with some interesting (and perhaps unexpected) corollaries on the general nature of quasiconvex functions. 

In this talk I will describe a multiple frequency approach to the boundary control of Helmholtz and Maxwell equations. We give boundary conditions and a finite number of frequencies such that the corresponding solutions satisfy certain non-zero constraints inside the domain. The suitable boundary conditions and frequencies are explicitly constructed and do not depend on the coefficients, in contrast to the illuminations given as traces of complex geometric optics solutions. This theory finds applications in several hybrid imaging modalities. Some examples will be discussed.

The first half of this talk will be an introduction to the wonderful world of Higgs bundles. The last half concerns Fourier--Mukai transforms, and we will discuss how to merge the two concepts by constructing a Fourier--Mukai transform for Higgs bundles. Finally we will discuss some properties of this transform. We will along the way discuss why you would want to transform Higgs bundles.

One of the things sustainability applications have in common with industrial applications is their close connection with decision-making and policy. We will discuss how a decision-support viewpoint may inspire new mathematical questions. For example, the concept of resilience (of ecosystems, food systems, communities, economies, etc) is often described as the capacity of a system to withstand disturbance and retain its functional characteristics. This has several familiar mathematical interpretations, probing the interaction between transient dynamics and noise. How does a focus on resilience change the modeling, dynamical and policy questions we ask? I look forward to your ideas and discussion.

Let $G=SL(2,\R)\ltimes R^2$ and $\Gamma=SL(2,Z)\ltimes Z^2$. Building on recent work of Strombergsson we prove a rate of equidistribution for the orbits of a certain 1-dimensional unipotent flow of $\Gamma\G$, which projects to a closed horocycle in the unit tangent bundle to the modular surface. We use this to answer a question of Elkies and McMullen by making effective the convergence of the gap distribution of $\sqrt n$ mod 1.

We present a new model of financial markets that studies the evolution of wealth

among investment strategies. An investment strategy can be generated by maximizing utility

given some expectations or by behavioral rules. The only requirement is that any investment strategy

is adapted to the information filtration. The model has the mathematical structure of a random dynamical system.

We solve the model by characterizing evolutionary properties of investment strategies (survival, evolutionary stability, dominance).

It turns out that only a fundamental strategy investing according to expected relative dividends satisfies these evolutionary criteria.

Thermoacoustic oscillations occur in combustion chambers when heat release oscillations lock into pressure oscillations. They were first observed in lamps in the 18th century, in rockets in the 1930s, and are now one of the most serious problems facing gas turbine manufacturers.

This theoretical and numerical study concerns an infinite-rate chemistry diffusion flame in a tube, which is a simple model for a flame in a combustion chamber. The problem is linearized around the non-oscillating state in order to derive the direct and adjoint equations governing the evolution of infinitesimal oscillations.

The direct equations are used to predict the frequency, growth rate, and mode shape of the most unstable thermoacoustic oscillations. The adjoint equations are then used to calculate how the frequency and growth rate change in response to (i) changes to the base state such as the flame shape or the composition of the fuel (ii) generic passive feedback mechanisms that could be added to the device. This information can be used to stabilize the system, which is verified by subsequent experiments.

This analysis reveals that, as expected from a simple model, the phase delay between velocity and heat-release fluctuations is the key parameter in determining the sensitivities. It also reveals that this thermo-acoustic system is exceedingly sensitive to changes in the base state. This analysis can be extended to more accurate models and is a promising new tool for the analysis and control of thermo-acoustic oscillations.

For any infinite group with a distinguished family of normal subgroups of finite index -- congruence subgroups-- one can ask whether every finite index subgroup contains a congruence subgroup. A classical example of this is the positive solution for $SL(n,\mathbb{Z})$ where $n\geq 3$, by Mennicke and Bass, Lazard and Serre. \\

Groups acting on infinite rooted trees are a natural setting in which to ask this question. In particular, branch groups have a sufficiently nice subgroup structure to yield interesting results in this area. In the talk, I will introduce this family of groups and the congruence subgroup problem in this context and will present some recent results.

A virtual endomorphism of a group $G$ is a homomorphism $f : H \rightarrow G$ where $H$

is a subgroup of $G$ of fi…nite index $m$: A recursive construction using $f$ produces a

so called state-closed (or, self-similar in dynamical terms) representation of $G$ on

a 1-rooted regular $m$-ary tree. The kernel of this representation is the $f$-core $(H)$;

i.e., the maximal subgroup $K$ of $H$ which is both normal in G and is f-invariant.

Examples of state-closed groups are the Grigorchuk 2-group and the Gupta-

Sidki $p$-groups in their natural representations on rooted trees. The affine group

$Z^n \rtimes GL(n;Z)$ as well as the free group $F_3$ in three generators admit state-closed

representations. Yet another example is the free nilpotent group $G = F (c; d)$ of

class c, freely generated by $x_i (1\leq i \leq d)$: let $H = \langle x_i^n | \

(1 \leq i \leq d) \rangle$ where $n$ is a

fi…xed integer greater than 1 and $f$ the extension of the map $x^n_i

\rightarrow x_i$ $(1 \leq i \leq d)$.

We will discuss state-closed representations of general abelian groups and of

…nitely generated torsion-free nilpotent groups.

Counting curves with given topological properties in a variety is a very old question. Example questions are: How many conics pass through five points in a plane, how many lines are there on a Calabi-Yau 3-fold? There are by now several ways to count curves and the numbers coming from different curve counting theories may be different. We would then like to have methods to compare these numbers. I will present such a general method and show how it works in the case of stable maps and stable quasi-maps.

Fix positive integers p and q with 2 \leq q \leq {p \choose 2}. An edge-colouring of the complete graph K_n is said to be a (p, q)-colouring if every K_p receives at least q different colours. The function f(n, p, q) is the minimum number of colours that are needed for K_n to have a (p,q)-colouring. This function was introduced by Erdos and Shelah about 40 years ago, but Erdos and Gyarfas were the first to study the function in a systematic way. They proved that f(n, p, p) is polynomial in n and asked to determine the maximum q, depending on p, for which f(n,p,q) is subpolynomial in n. We prove that the answer is p-1.

We also discuss some related questions.

Joint work with Jacob Fox, Choongbum Lee and Benny Sudakov.

In 1890, G. H. Bryan demonstrated that when a ringing wine glass rotates, the shape of the vibration pattern precesses, and this effect is the basis for a family of high-precision gyroscopes. Mathematically, the precession can be described in terms of a symmetry-breaking perturbation due to gyroscopic effects of a geometrically degenerate pair of vibration modes.  Unfortunately, current attempts to miniaturize these gyroscope designs are subject to fabrication imperfections that also break the device symmetry. In this talk, we describe how these devices work and our approach to accurate and efficient simulations of both ideal device designs and designs subject to fabrication imperfections.

Quantum field theory (QFT) originated in physics in the context of

elementary particles. Although, over the years, surprising and profound

connections to very diverse branches of mathematics have been discovered,

QFT does not have, as yet, found a universally accepted "standard"

mathematical formulation. In this talk, I shall outline an approach to QFT

that emphasizes its underlying algebraic structure. Concretely, this is

represented by a concept called "Operator Product Expansion". I explain the

properties of such expansions, how they can be constructed in concrete QFT

models, and the emergent relationship between "perturbation theory" on the

physics side and

"Hochschild cohomology" on the physics side. This talk is based on joint

work

with Ch. Kopper and J. Holland from Ecole Polytechnique, Paris.

The conformal approach to scattering theory goes back to the 1960's

and 1980's, essentially with the works of Penrose, Lax-Phillips and

Friedlander. It is Friedlander who put together the ideas of Penrose

and Lax-Phillips and presented the first conformal scattering theory

in 1980. Later on, in the 1990's, Baez-Segal-Zhou explored Friedlander's

method and developed several conformal scattering theories. Their

constructions, just like Friedlander's, are on static spacetimes. The

idea of replacing spectral analysis by conformal geometry is however

the door open to the extension of scattering theories to general non

stationary situations, which are completely inaccessible to spectral

methods. A first work in collaboration with Lionel Mason explained

these ideas and applied them to non stationary spacetimes without

singularity. The first results for nonlinear equations on such

backgrounds was then obtained by Jeremie Joudioux. The purpose is now

to extend these theories to general black holes. A first crucial step,

recently completed, is a conformal scattering construction on

Schwarzschild's spacetime. This talk will present the history of the

ideas, the principle of the constructions and the main ingredients

that allow the extension of the results to black hole geometries.

In this presentation, we focus on the decay rate of the expected signature of a stopped Brownian motion; more specifically we consider two types of the stopping time: the first one is the Brownian motion up to the first exit time from a bounded domain $\Gamma$, denoted by $\tau_{\Gamma}$, and the other one is the Brownian motion up to $min(t, \tau_{\Gamma\})$. For the first case, we use the Sobolev theorem to show that its expected signature is geometrically bounded while for the second case we use the result in paper (Integrability and tail estimates for Gaussian rough differential equation by Thomas Cass, Christian Litterer and Terry Lyons) to show that each term of the expected signature has the decay rate like 1/ \sqrt((n/p)!) where p>2. The result for the second case can imply that its expected signature determines the law of the signature according to the paper (Unitary representations of geometric rough paths by Ilya Chevyrev)

In order to study the group of (outer) automorphisms of

any group G by geometric methods one needs a well-behaved "outer

space" with an interesting action of Out(G). If G is free abelian, the

classic symmetric space SL(n,R)/SO(n) serves this role, and if G is

free non-abelian an appropriate outer space was introduced in the

1980's. I will recall these constructions and then introduce joint

work with Ruth Charney on constructing an outer space for any

right-angled Artin group.

By solving the Homogeneous Monge-Ampere equation on the deformation to the normal cone of a complex submanifold of a Kahler manifold, we get a canonical tubular neighbourhood adapted to both the holomorphic and the symplectic structure. If time permits I will describe an application, namely an optimal regularity result for certain naturally defined plurisubharmonic envelopes.

Simple probabilistic models for sequential data (text, music...), e.g., hidden Markov models, cannot capture some structures such as
long-term dependencies or combinations of simultaneous patterns and probabilistic rules hidden in the data. On the other hand, models such as
recurrent neural networks can in principle handle any structure but are notoriously hard to learn given training data. By analyzing the structure of
neural networks from the viewpoint of Riemannian geometry and information theory, we build better learning algorithms, which perform well on difficult
toy examples at a small computational cost, and provide added robustness.

Dynamics in nature often proceed in the form of reactive events, aka activated processes. The system under study spends very long periods of time in various metastable states; only very rarely does it transition from one such state to another. Understanding the dynamics of such events requires us to study the ensemble of transition paths between the different metastable states. Transition path theory (TPT) is a general mathematical framework developed for this purpose. It is also the foundation for developing modern numerical algorithms such as the string method for finding the transition pathways or milestoning to calculate the reaction rate, and it can also be used in the context of Markov State Models (MSMs). In this talk, I will review the basic ingredients of the transition path theory and discuss connections with transition state theory (TST) as well as approaches to metastability based on potential theory and large deviation theory. I will also discuss how the string method arises in order to find approximate solutions in the framework of the transition path theory, the connections between milestoning and TPT, and the way the theory help building MSMs. The concepts and methods will be illustrated using examples from molecular dynamics, material science and atmosphere/ocean sciences.

We will describe a generalization of the algebra of differential operators, which gives a

geometric approach to quantization of cotangent field theories. This construction is compatible

with "integration" thus giving a local-to-global construction of volume forms on derived mapping

spaces using a version of non-abelian duality. These volume forms give interesting invariants of

varieties such as the Todd genus, the Witten genus and the B-model operations on Hodge

cohomology.

The observations of the first traces of cosmic structure in the

Cosmic Microwave Background are in excellent agreement with the

predictions of Inflation. However as we shall see, that account

is not fully satisfactory, as it does not address the transition

from an homogeneous and isotropic early stage to a latter one

lacking those symmetries. We will argue that new physics along the

lines of the dynamical quantum state reduction theories is needed

to account for such transition and, motivated by Penrose's ideas

suggest that quantum gravity might be the place from where

this new physics emerges. Moreover we will show that observations

can be used to constrain the various phenomenological proposals

made in this regard.

Microbial biofilms grow on most rock and stone surfaces and may play critical roles in weathering. With climate change and improving air quality in many cities in Europe biofilms are growing rapidly on many historic stone buildings and posing practical problems for heritage conservation. With many new field and lab techniques available it is now possible to identify the microbes present and start to clarify their roles. We now need help modelling microbial biofilm growth and impacts in order to provide better advice for conservators.

The question of how to derive useful bounds on

arbitrage-free prices of exotic options given only prices of liquidly

traded products like European call und put options has received much

interest in recent years. It also led to new insights about classic

problems in probability theory like the Skorokhod embedding problem. I

will take this as a starting point and show how this progress can be

used to give new results on general Monte-Carlo schemes.

This topic was the subject of an OCIAM workshop on 8th March 2013

given by Nick Hall Taylor . The presentation will start with a review

of the physical problem and experimental evidence. A mathematical

model leading to a hydrodynamic free boundary problem has been derived

and some mathematical and computational results will be described.

Finally we will assess the results so far and list a number of

interesting open problems.

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After the workshop and during coffee at 11:30, we will also give a preview of the

upcoming problems at the Malaysian Study Group (Mar. 17-21). Problem

descriptions can be found here:

www.maths.ox.ac.uk/~trinh/2014_studygroup_problems.pdf.

Several number theorists have stressed that the proofs of FLT focus on small concrete arithmetically defined groups rings and modules, so the steps can be checked by direct calculation in any given case. The talk looks at this in relation both to Hilbert's idea of contentual (inhaltlich) mathematics, and to formal provability in Peano arithmetic and other stronger and weaker axioms.

We study lower- and dual upper-bounds for Bermudan options in a MonteCarlo/MC setting and provide four contributions. 1) We introduce a local least-squares MC method, based on maximizing the Bermudan price and which provides a lower-bound, which "also" minimizes (not the dual upper-bound itself, but) the gap between these two bounds; where both bounds are specified recursively. 2) We confirm that this method is near optimal, for both lower- and upper-bounds, by pricing Bermudan max-call options subject to an up-and-out barrier; state-of-the-art methods including Longstaff-Schwartz produce a large gap of 100--200 basis points/bps (Desai et al. (2012)), which we reduce to just 5--15 bps (using the same linear basis of functions). 3) For dual upper-bounds based on continuation values (more biased but less time intensive), it works best to reestimate the continuation value in the continuation region only. And 4) the difference between the Bermudan option Delta and the intrinsic value slope at the exercise boundary gives the sensitivity to suboptimal exercise (up to a 2nd-order Taylor approximation). The up-and-out feature flattens the Bermudan price, lowering the Bermudan Delta well below one when the call-payoff slope is equal to one, which implies that optimal exercise "really" matters.

I plan to give a non technical introduction (i.e. no prerequisites required apart basic differential geometry) to some analytic aspects of the theory of harmonic maps between Riemannian manifolds, motivate it by briefly discussing some relations to other areas of geometry (like minimal submanifolds, string topology, symplectic geometry, stochastic geometry...), and finish by talking about the heat flow approach to the existence theory of harmonic maps with some open problems related to my research.

I will discuss a novel approach to estimating communication costs of an algorithm (also known as its I/O complexity), which is based on small-set expansion for computational graphs. Various applications and implications will be discussed as well, mostly having to do with linear algebra algorithms. This includes, in particular, first known (and tight) bounds on communication complexity of fast matrix multiplication.

Joint work with Grey Ballard, James Demmel, Benjamin Lipshitz and Oded Schwartz.