Past

We introduce Outer space, a contractible finite dimensional topological space on which the outer automorphism group of a free group acts 'nicely.' We will explain what 'nicely' is, and provide motivation with comparisons to symmetric spaces, analogous spaces associated to linear groups.

Following work of Bridgeland in the smooth case and Chen in the terminal singularities case, I will explain a proposal that extends the existence of flops for threefolds (and the required derived equivalences) to also cover canonical singularities.  Moreover this technique conjecturally says much more than just the existence of the flop, as it shows how the dual graph changes under the flop and also which curves in the flopped variety contract to points without contracting divisors.  This allows us to continue the Minimal Model Programme on the flopped variety in an easy way, thus producing many varieties birational to the given input.    

Due to the scaling properties of the Navier-Stokes equations,

homogeneous initial data may lead to forward self-similar solutions.

When the initial data is small enough, it is well known that the

formalism of mild solutions (through the Picard-Duhamel formula) give

such self-similar solutions. We shall discuss the issue of large initial

data, where we can only prove the existence of weak solutions; those

solutions may lack self-similarity, due to the fact that we have no

results about uniqueness for such weak solutions. We study some tools

which may be useful to get a better understanding of those weak solutions.

Suppose that $C$ and $C'$ are cubic forms in at least 19 variables over a

$p$-adic field $k$. A special case of a conjecture of Artin is that the

forms $C$ and $C'$ have a common zero over $k$. While the conjecture of

Artin is false in general, we try to argue that, in this case, it is

(almost) correct! This is still work in progress (joint with

Heath-Brown), so do not expect a full answer.

As a historical note, some cases of Artin's conjecture for certain

hypersurfaces are known. Moreover, Jahan analyzed the case of the

simultaneous vanishing of a cubic and a quadratic form. The approach

we follow is closely based on Jahan's approach, thus there might be

some overlap between his talk and this one. My talk will anyway be

self-contained, so I will repeat everything that I need that might

have already been said in Jahan's talk.

Abstract: There has in the last year been much progresson the universality problem for the spectra of a Wigner random matrices, i.e.Hermitian or symmetric random matrices with independent elements. I will givesome background on this problem and also discuss what can be said when we onlyassume a few moments of the matrix elements to be finite.

A copolymer is a chain of repetitive units (monomers) that

are almost identical, but they differ in their degree of

affinity for certain solvents. This difference leads to striking

phenomena when the polymer fluctuates

in a non-homogeneous medium, for example made up by two solvents

separated by an interface.

One may observe, for exmple, the localization of the polymer at the

interface between the two solvents.

Much of the literature on the subject focuses on the most basic model

based on the simple symmetric random walk on the integers, but

E. Bolthausen and F. den Hollander (AP 1997) pointed out

the convergence of the (rescaled) free energy of such a discrete model

toward

the free energy of a continuum model, based on Brownian motion,

in the limit of weak polymer-solvent coupling. This result is

remarkable because it strongly suggests

a universal feature for copolymer models. In this work we prove that

this is indeed the case. More precisely,

we determine the weak coupling limit for a general class of discrete

copolymer models, obtaining as limits

a one-parameter (alpha in (0,1)) family of continuum models, based on

alpha-stable regenerative sets.

This is a review of hep-th/0912.4912

McBurnie: “Sound propagation through bubbly liquids”.

Hewett: "Switching on a time-harmonic acoustic source".

When the human eye looks at a distant object, the lens is held in a state of tension by a set of fibres (known as zonules) that connect the lens to the ciliary body. To view a nearby object, the ciliary muscle (which is part of the ciliary body) contracts. This reduces the tension in the zonules, the lens assumes a thicker and more rounded shape and the optical power of the eye increases.

This process is known as accommodation.

With increased age, however, the accommodation mechanism becomes increasingly ineffective so that, from an age of about 50 years onwards, it effectively ceases to function. This condition is known as presbyopia. There is considerable interest in the ophthalmic community on developing a better understanding of the ageing processes that cause presbyopia. As well as being an interesting scientific question in its own right, it is hoped that this improved understanding will lead to improved surgical procedures (e.g. to re-start the accommodation process in elderly cataract patients).

Let R be a number field (or a recursive subring of anumber field) and consider the polynomial ring R[T].

We show that the set of polynomials with integercoefficients is diophantine (existentially definable) over R[T].

Applying a result by Denef, this implies that everyrecursively enumerable subset of R[T]^k is diophantine over R[T].

Some years ago Hall and Smith in a number of papers developed a theory governing the interaction of vortices and waves in shear flows. In recent years immense interest has been focused on so-called self-sustained processes in turbulent shear flows where the importance of waves interacting with streamwise vortex flows has been elucidated in a number of; see for example the work of Waleffe and colleagues, Kerswell, Gibson, etc. These processes have a striking resemblance to coherent structures observed in turbulent shear flow and for that reason they are often referred to as exact coherent structures. It is shown that the structures associated with the so-called 'lower branch' state, which has been shown to play a crucial role in these self-sustained process, is nothing but a Rayleigh wave vortex interaction with a wave system generating streamwise vortices inside a critical layer. The theory enables the reduction of the 3D Navier Stokes equations to a coupled system for a steady streamwise vortex and an inviscid wave system. The reduced system for the streamwise vortices must be solved with jump conditions in the shear across the critical layer and the position of that layer constitutes a nonlinear pde eigenvalue problem. Remarkable agreement between the asymptotic theory and numerical simulations is found thereby demonstrating the importance of vortex-wave interaction theory in the mathematical description of coherent structures in turbulent shear flows. The theory offers the possibility of drag reduction in turbulent shear flows by controlling the flow to the neighborhood of the lower branch state. The relevance of the work to more general shear flows is also discussed.

tba

In the last few years, there has been renewed interest in stochastic

finite element methods (SFEMs), which facilitate the approximation

of statistics of solutions to PDEs with random data. SFEMs based on

sampling, such as stochastic collocation schemes, lead to decoupled

problems requiring only deterministic solvers. SFEMs based on

Galerkin approximation satisfy an optimality condition but require

the solution of a single linear system of equations that couples

deterministic and stochastic degrees of freedom. This is regarded as

a serious bottleneck in computations and the difficulty is even more

pronounced when we attempt to solve systems of PDEs with

random data via stochastic mixed FEMs.

In this talk, we give an overview of solution strategies for the

saddle-point systems that arise when the mixed form of the Darcy

flow problem, with correlated random coefficients, is discretised

via stochastic Galerkin and stochastic collocation techniques. For

the stochastic Galerkin approach, the systems are orders of

magnitude larger than those arising for deterministic problems. We

report on fast solvers and preconditioners based on multigrid, which

have proved successful for deterministic problems. In particular, we

examine their robustness with respect to the random diffusion

coefficients, which can be either a linear or non-linear function of

a finite set of random parameters with a prescribed probability

distribution.

After reviewing the salient details from last week's seminar, I will construct an explicit example of a spectral curve, using co-Higgs bundles of rank 2. The role of the spectral curve in understanding the moduli space will be made clear by appealing to the Hitchin fibration, and from there inferences (some of them very concrete) can be made about the structure of the moduli space. I will make some conjectures about the higher-dimensional picture, and also try to show how spectral varieties might live in that picture.