Past

We study the problem of maximizing expected utility from terminal wealth in a semi-static market composed of derivative securities, which we assume can be traded only at time zero, and of stocks, which can be
traded continuously in time and are modeled as locally-bounded semi-martingales.

Using a general utility function defined on the positive real line, we first study existence and uniqueness of the solution, and then we consider the dependence of the outputs of the utility maximization problem on the price of the derivatives, investigating not only stability but also differentiability, monotonicity, convexity and limiting properties.

Let ${\cal E}$ be a family of elliptic curves over a base variety defined over $\mathbb C$. An additive extension ${\cal G}$ of ${\cal E}$ is a family of algebraic groups which fits into an exact sequence of group schemes $0\rightarrow {\mathbb G}_{\rm a}\rightarrow {\cal G}\rightarrow {\cal E}\rightarrow 0$. We can define the special subvarieties of ${\cal G}$ to be families of algebraic groups over the same base contained in ${\cal G}$. The relative Manin-Mumford conjecture suggests that the intersection of a curve in ${\cal G}$ with the special subvarieties of dimension 0 is contained in a finite union of special subvarieties.

To prove this we can assume that the family ${\cal E}$ is the Legendre family and then follow the strategy employed by Masser-Zannier for their proof of the relative Manin-Mumford conjecture for the fibred product of two legendre families. This has applications to classical problems such as the theory of elementary integration and Pell's equation in polynomials.

I will introduce simple homotopy theory and then discuss relations between some conjectures in 2 dimensional simple homotopy theory and the 3 and 4 dimensional Poincaré conjectures.

In order:

1. Michael Dallaston, "Modelling channelization under ice shelves"

2. Jeevanjyoti Chakraborty, "Growth, elasticity, and diffusion in 
lithium-ion batteries"

3. Roberta Minussi, "Lattice Boltzmann modelling of the generation and 
propagation of action potential in neurons"

Fix a prime $p$. In this talk, we will discuss the $p$-adic properties of the *coefficients* of the characteristic power series of $U_{p}$ acting on spaces of overconvergent $p$-adic modular forms. These coefficients are, by a theorem of Coleman, power series in the weight variable over $Z_{p}$.  Our first goal will be to show that in tame level one, the simplest case, every coefficient is non-zero mod $p$ and then to give some idea of the (finitely many) roots of each coefficient. The second goal will be to explain how it the previous result fails in higher levels, along with possible salvages. This will include revisiting the tame level one case. The progress we've made has applications, and lends understanding, to recent work being made elsewhere on the geometric structure of the eigencurve "near its boundary". This is joint work with Rob Pollack.

Gaussian quadrature rules are of theoretical and practical interest because of their role in numerical integration and interpolation. For general weighting functions, their computation can be performed with the Golub-Welsch algorithm or one of its refinements. However, for the specific case of Gauss-Legendre quadrature, computation methods based on asymptotic series representations of the Legendre polynomials have recently been proposed. 
For large quadrature rules, these methods provide superior accuracy and speed at the cost of generality. We provide an overview of the progress that was made with these asymptotic methods, focusing on the ideas and asymptotic formulas that led to them. 
Finally, the limited generality will be discussed with Gauss-Jacobi quadrature rules as a prominent possibility for extension.

The Nottingham Group of a finite field is an object of great interest in profinite group theory, owing to its extreme structural properties and the relative ease with which explicit computations can be made within it. In this talk I shall explore both of these themes, before describing some new work on efficient short-word approximation in the Nottingham Group, based on the profinite Solovay-Kitaev procedure. Time permitting, I shall give an application to the dynamics of compositions of random power series.

A regular map is a highly symmetric embedding of a finite graph into a closed surface. I will describe a programme to study such embeddings for a rather large class of graphs: namely, the class of orbital graphs of finite simple groups.

Given two probability distributions $P_R$ and $P_B$ on the positive reals with finite means, colour the real line alternately with red and blue intervals so that the lengths of the red intervals have distribution $P_R$, the lengths of the blue intervals have distribution $P_B$, and distinct intervals have independent lengths. Now iteratively update this colouring of the line by coalescing intervals: change the colour of any interval that is surrounded by longer intervals so that these three consecutive intervals subsequently form a single monochromatic interval. Say that a colour (either red or blue) `wins' if every point of the line is eventually of that colour. I will attempt to answer the following question: under what natural conditions on the distributions is one of the colours almost surely guaranteed to win?

This talk concerns the numerical solution of elliptic partial differential equations posed on general smooth surfaces by the Closest Point Method. Based on the closest point representation of the surface, we formulate an embedding equation in a narrow band surrounding the surface, then discretize it using standard finite differences and interpolation schemes. Numerical convergence of the method will be discussed. In order to solve the resulting large sparse linear systems, we propose a specific geometric multigrid method which makes use of the closest point representation of the surface.
 

We discuss the iterative solution of a finite element discretisation of the magma dynamics equations.  These equations share features of the Stokes equations, however, Elman-Silvester-Wathen (ESW) preconditioners for the magma dynamics equations are not optimal. By introducing a new field, the compaction pressure, into the magma dynamics equations, we have developed a new three-field preconditioner which is optimal in terms of problem size and less sensitive to physical parameters compared to the ESW preconditioners.

In this talk I am going to discuss our recent and on-going work on an integral representation of tree-level S-matrices for massless particles. Starting from the formula for gravity amplitudes, I will introduce three operations acting on the integrand that produce compact and closed formulas for amplitudes in various other theories of massless bosons. In particular these includes Yang-Mills coupled to gravity, (Dirac)-Born-Infeld, U(N) non-linear sigma model, and Galileon theory. The main references are arXiv:1409.8256, arXiv:1412.3479.

We discuss the local minimality of certain configurations for a nonlocal isoperimetric problem used to model microphase separation in diblock copolymer melts. We show that critical configurations  with positive second variation are local minimizers of the nonlocal area functional and, in fact, satisfy a quantitative isoperimetric inequality with respect to sets  that are  $L^1$-close.  As an application, we address the global and local minimality of certain lamellar configurations.

By Thurston's geometrisation theorem, the complement of any knot admits a unique hyperbolic structure, provided that the knot is not the unknot, a torus knot or a satellite knot. However, this is purely an existence result, and does not give any information about important geometric quantities, such as volume, cusp volume or the length and location of short geodesics. In my talk, I will explain how some of this information may be computed easily, in the case of alternating knots. The arguments involve a detailed analysis of the geometry of certain subsurfaces.

Hitchin's existence theorem asserts that a stable Higgs bundle of rank two carries a unitary connection satisfying Hitchin's self-duality equation. In this talk we discuss a new proof, via gluing methods, for
elements in the ends of the Higgs bundle moduli space and identify a dense open subset of the boundary of the compactification of this moduli space.
 

This is a report on an ongoing project to construct Calabi-Yau manifolds for which the Hodge numbers $(h^{11}, h^{21})$ are both relatively small. These manifolds are, in a sense, the simplest Calabi-Yau manifolds. I will report on joint work with Volker Braun, Andrei Constantin, Rhys Davies, Challenger Mishra and others.

This paper introduces a mean field game framework for optimal execution with continuous trading. We generalize the classical optimal liquidation problem to a setting where, in addition to the major agent who is liquidating a large portion of shares, there are a number of minor agents (high-frequency traders (HFTs)) who detect and trade along with the liquidator. Cross interaction between the minor and major agents occur through the impact that each trader has on the drift of the fundamental price. As in the classical approach, here, each agent is exposed to both temporary and permanent price impact and they attempt to balance their impact against price uncertainty. In all, this gives rise to a stochastic dynamic game with mean field couplings in the fundamental price. We obtain a set of decentralized strategies using a mean field stochastic control approach and explicitly solve for an epsilon-optimal control up to the solution of a deterministic fixed point problem. As well, we present some numerical results which illustrate how the liquidating agents trading strategy is altered in the presence of the HFTs, and how the HFTs trade to profit from the liquidating agents trading.

[ This is joint work with Mojtaba Nourin, Department of Statistical Sciences, U. Toronto ]