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.