Past

In this talk we will review some recent results on the long-time/large-scale, weak-friction asymptotics for the one dimensional Langevin equation with a periodic potential. First we show that the Freidlin-Wentzell and central limit theorem (homogenization) limits commute. We also show that, in the combined small friction, long-time/large-scale limit the particle position converges weakly to a Brownian motion with a singular diffusion coefficient which we compute explicitly. Furthermore we prove that the same result is valid for a whole one parameter family of space/time rescalings. We also present a new numerical method for calculating the diffusion coefficient and we use it to study the multidimensional problem and the problem of Brownian motion in a tilted periodic potential.

Hedge fund managers receive a large fraction of their funds' gains, in addition to the small fraction of funds' assets typical of mutual funds. The additional fee is paid only when the fund exceeds its previous maximum - the high-water mark. The most common scheme is 20 percent of the fund profits + 2 percent of assets.

To understand the incentives implied by these fees, we solve the portfolio choice problem of a manager with Constant Relative Risk Aversion and a Long Horizon, who maximizes the utility from future fees.

With constant investment opportunities, and in the absence of fixed fees, the optimal portfolio is constant. It coincides with the portfolio of an investor with a different risk aversion, which depends on the manager's risk aversion and on the size of the fees. This portfolio is also related to that of an investor facing drawdown constraints. The combination of both fees leads to a more complex solution.

The model involves a stochastic differential equation involving the running maximum of the solution, which is related to perturbed Brownian Motions. The solution of the control problem employs a verification theorem which relies on asymptotic properties of positive local martingales.

Joint work with Jan Obloj.

A theorem of Kuyk says that every Abelian extension of a

Hilbertian field is Hilbertian.

We conjecture that for an Abelian variety $A$ defined over

a Hilbertian field $K$

every extension $L$ of $K$ in $K(A_\tor)$ is Hilbertian.

We prove our conjecture when $K$ is a number field.

The proofs applies a result of Serre about $l$-torsion of

Abelian varieties, information about $l$-adic analytic

groups, and Haran's diamond theorem.

An elastic sheet will buckle out of the plane when subjected to an in-plane compression. In the simplest systems the typical lengthscale of the buckled structure is that of the system itself but with additional physics (e.g. an elastic substrate) repeated buckles with a well-defined wavelength may be seen. We discuss two examples in which neither of these scenarios is realized: instead a small number of localized structures are observed with a size different to that of the system itself. The first example is a heavy sheet on a rigid floor - a ruck in a rug. We study the static properties of these rucks and also how they propagate when one end of the rug is moved quickly. The second example involves a thin film adhered to a much softer substrate. Here delamination blisters are formed with a well-defined size, which we characterize in terms of the material properties of the system. We then discuss the possible application of these model systems to real world problems ranging from the propagation of slip pulses in earthquakes to the manufacture of flexible electronic devices."

Quasicontinuum methods are a prototypical class of atomistic-to-continuum coupling methods. For example, we may wish to model a lattice defect (a vacancy or a dislocation) by an atomistic model, but the elastic far field by a continuum model. If the continuum model is consistent with the atomistic model (e.g., the Cauchy--Born model) then the main question is how the interface treatment affects the method.

In this talk I will introduce three of the main ideas how to treat the interface. I will explain their strengths and weaknesses by formulating the simplest possible non-trivial model problem and then simply analyzing the two classical concerns of numerical analysis: consistency and stability.

A parabolic bundle on a marked curve is a vector bundle with extra structure (a flag) in each of the fibres over the marked points, together with data corresponding to a choice of stability condition Parabolic bundles are natural generalisations of vector bundles when the base comes with a marking (for example, they partially generalise the Narasimhan-Seshadri correspondence between representations of the fundamental group and semistable vector bundles), but they also play an important role in the study of pure sheaves on nodal curves (which are needed to compactify moduli of vector bundles on stable curves). Consider the following moduli problem: pairs $(C,E)$ of smooth marked curves $C$

and semistable parabolic bundles $E\rightarrow C$. I will sketch a construction of projective moduli spaces which compactify the above moduli problem over the space of stable curves. I'll discuss further questions of interest, including strategies for understanding the cohomology of these moduli spaces, generalisations of the construction to higher-dimensional base schemes, and possible connections with Torelli theorems for parabolic vector bundles on marked curves.

We begin by proving the abc theorem for polynomial rings and looking at a couple of its consequences. We then move on to the abc conjecture and its equivalence with the generalized Szpiro conjecture, via Frey polynomials. We look at a couple of consequences of the abc conjecture, and finally consider function fields, where we introduce the abc theorem in that case.

In the Said Business School

As the shockwaves of the financial crisis of 2008 propagate throughout the global economy, the "blame game" has begun in earnest, with some fingers pointing to the complexity of certain financial securities, and the mathematical models used to manage them. In this talk, I will review the evidence for and against this view, and argue that a broader perspective will show a much different picture.Blaming quantitative analysis for the financial crisis is akin to blaming F = MA for a fallen mountain climber's death. A more productive line of inquiry is to look deeper into the underlying causes of financial crisis, which ultimately leads to the conclusion that bubbles, crashes, and market dislocation are unavoidable consequences of hardwired human behavior coupled with free enterprise and modern capitalism. However, even though crises cannot be legislated away, there are many ways to reduce their disruptive effects, and I will conclude with a set of proposals for regulatory reform.

A polynomial $f$ is said to be a Brieskorn-Pham polynomial if

$ f = x_1^{p_1} + ... + x_n^{p_n}$

for positive integers $p_1,\ldots, p_n$. In the talk, I will discuss my joint work with Masahiro Futaki on the equivalence between triangulated category of matrix factorizations of $f$ graded with a certain abelian group $L$ and the Fukaya-Seidel category of an exact symplectic Lefschetz fibration obtained by Morsifying $f$.

I'll define the category of B-branes in a LG model, and show that for affine models the Hochschild homology of this category is equal to the physically-predicted closed state space. I'll also explain why this is a step towards proving that LG B-models define TCFTs.

Some results of R.Harvey and B.Lawson on the Dirichlet problem for a class of fully nonlinear elliptic equations will be presented.

No background is required; the talk will be expository.

This is the first (of two) talks which will be given concerning the Birch--Swinnerton-Dyer Conjecture.

If a distribution, say F, has all moments finite, then either F is unique (M-determinate) in the sense that F is the only distribution with these moments, or F is non-unique (M-indeterminate).  In the latter case we suggest a method for constructing a Stieltjes class consisting of infinitely many distributions different from F and all having the same moments as F.  We present some shocking examples involving distributions such as N, LogN, Exp and explain what and why.  We analyse conditions which are sufficient for F to be M-determinate or M-indeterminate.  Then we deal with recent problems from the following areas:

(A)  Non-linear (Box-Cox) transformations of random data.

(B) Distributional properties of functionals of stochastic processes.

(C) Random sums of random variables.

If time permits, some open questions will be outlined.  The talk will be addressed to colleagues, including doctoral and master students, working or having interests in the area of probability/stochastic processes/statistics and their applications. 

I will discuss the so-called Lasso method for signal recovery for high-dimensional data and show applications in computational biology, machine learning and image analysis.

This talk will be based on a joint work with Professor Terry Lyons and Mr Gechun Liang (OMI). I will explain a new approach to define and to solve a class of backward dynamic systems including the well known examples of non-linear backward SDE. The new approach does not require any kind of martingale representation or any specific restriction on the probability base in question, and therefore can be applied to a much wider class of backward systems.

I will briefly describe a differential geometric construction of Hitchin's projectively flat connection in the Verlinde bundle, over Teichm\"uller space, formed by the Hilbert spaces arising from geometric quantization of the moduli space of flat connections on a Riemann surface. We will work on a general symplectic manifold sharing certain properties with the moduli space. Toeplitz operators enter the picture when quantizing classical observables, but they are also closely connected with the notion of deformation quantization. Furthermore, through an intimate relationship between Toeplitz operators, the Hitchin connection manifests itself in the world of deformation quantization as a connection on formal functions. As we shall see, this formal Hitchin connection can be used to construct a deformation quantization, which is independent of the Kähler polarization used for quantization. In the presence of a symmetry group, this deformation quantization can (under certain cohomological conditions) be constructed invariantly. The talk presents joint work with J. E. Andersen.

For incompressible Navier-Stokes equations in a bounded domain, I will

first present a formula for the pressure that involves the commutator

of the Laplacian and Leray-Helmholtz projection operators. This

commutator and hence the pressure is strictly dominated by the viscous

term at leading order. This leads to a well-posed and computationally

congenial unconstrained formulation for the Navier-Stokes equations.

Based on this pressure formulation, we will present a new

understanding and design principle for third-order stable projection

methods. Finally, we will discuss the delicate inf-sup stability issue

for these classes of methods. This is joint work with Bob Pego and Jie Liu.