Past

In this talk we show how Teichmüller curves can be used to compute

quantum invariants of certain Pseudo-Anasov mapping tori. This involves

computing monodromy of the Hitchin connection along closed geodesics of

the Teichmüller curve using iterated integrals. We will mainly focus on

the well known Teichmüller curve generated by a pair of regular

pentagons. This is joint work with J. E. Andersen.

To any complex smooth variety Y with an action of a finite group G, Etingof associates a global Cherednik algebra. The usual rational Cherednik algebra corresponds to the case of Y= C^n and a finite Coxeter group G

Domain decomposition methods were introduced as techniques for solving partial differential equations based on a decomposition of the spatial domain of the problem into several subdomains. The initial equation restricted to the subdomains defines a sequence of new local problems. The main goal is to solve the initial equation via the solution of the local problems. This procedure induces a dimension reduction which is the major responsible of the success of such a method. Indeed, one of the principal motivations is the formulation of solvers which can be easily parallelized.

In this presentation we shall develop a domain decomposition algorithm to the minimization of functionals with total variation constraints. In this case the interesting solutions may be discontinuous, e.g., along curves in 2D. These discontinuities may cross the interfaces of the domain decomposition patches. Hence, the crucial difficulty is the correct treatment of interfaces, with the preservation of crossing discontinuities and the correct matching where the solution is continuous instead. I will present our domain decomposition strategy, including convergence results for the algorithm and numerical examples for its application in image inpainting and magnetic resonance imaging.

A number of pricing models for electricity and carbon credit pricing involve nonlinear dependencies between two, or more, of the processes involved; for example, the models developed by Schwarz and Howison. The consequences of these nonlinearities are not well understood.

In this talk I will discuss some much simpler models, namely options whose values are defined self-referentially, which have been looked at in order to better understand the effects of these non-linear dependencies.

Let F be a free group, and N a normal subgroup of F with derived subgroup N'. The Magnus embedding gives a way of seeing F/N' as a subgroup of a wreath product of a free abelian group over over F/N. The aim is to show that the Magnus embedding is a quasi-isometric embedding (hence "Q.I." in the title). For this I will use an alternative geometric definition of the embedding (hence "picture"), which I will show is equivalent to the definition which uses Fox calculus. Please note that we will assume no prior knowledge of calculus.

In our 2004 paper, Fritz Grunewald and I constructed the first
pairs of finitely presented, residually finite groups $u: P\to G$
such that $P$ is not isomorphic to $G$ but the map that $u$ induces on
profinite completions is an isomorphism. We were unable to determine if
there might exist finitely presented, residually finite groups $G$ that
with infinitely many non-isomorphic finitely presented subgroups $u_n:
P_n\to G$ such that $u_n$ induces a profinite isomorphism. I shall
discuss how two recent advances in geometric group theory can be used in
combination with classical work on Nielsen equivalence to settle this
question.

SEMINAR CANCELLED

Given a finite list of vectors X in $\R^d$, one can define the box spline $B_X$. Box splines are piecewise polynomial functions that are used in approximation theory. They are also interesting from a combinatorial point of view and many of their properties solely depend on the structure of the matroid defined by the list X. The support of the box spline is a certain polytope called zonotope Z(X). We will show that if the list X is totally unimodular, any real-valued function defined on the set of lattice points in the interior of Z(X) can be extended to a function on Z(X) of the form $p(D)B_X$ in a unique way, where p(D) is a differential operator that is contained in the so-called internal P-space. This was conjectured by Olga Holtz and Amos Ron. The talk will focus on combinatorial aspects and all objects mentioned above will be defined. (arXiv:1211.1187)

We analyze the fluctuation of the loss from default around its large portfolio limit in a class of reduced-form models of correlated default timing. We prove a weak convergence result for the fluctuation process and use it for developing a conditionally Gaussian approximation to the loss distribution. Numerical results illustrate the accuracy of the approximation.

This is joint work with Kostas Spiliopoulos (Boston University) and Justin Sirignano (Stanford).

The question of estimating the length of short conjugators in between
elements in a group could be described as an effective version of the
conjugacy problem. Given a finitely generated group $G$ with word metric
$d$, one can ask whether there is a function $f$ such that two elements
$u,v$ in $G$ are conjugate if and only if there exists a conjugator $g$ such
that $d(1,g) \leq f(d(1,u)+d(1,v))$. We investigate this problem in free
solvable groups, showing that f may be cubic. To do this we use the Magnus
embedding, which allows us to see a free solvable group as a subgroup of a
particular wreath product. This makes it helpful to understand conjugacy
length in wreath products as well as metric properties of the Magnus
embedding.

I will speak about the of weak (and the existence and uniqueness of strong solutions) to the stochastic
Landau-Lifshitz equations for multi (one)-dimensional spatial domains. I will also describe the corresponding Large Deviations principle and it's applications to a ferromagnetic wire. The talk is based on a joint works with B. Goldys and T. Jegaraj.

Motived by the desire to study geometry over the 'field with one element', in the past decade several authors have constructed extensions of scheme theory to geometries locally modelled on algebraic objects more general than rings. Semi-ring schemes exist in all of these theories, and it has been suggested that schemes over the semi-ring T of tropical numbers should describe the polyhedral objects of tropical geometry. We show that this is indeed the case by lifting Payne's tropicalization functor for subvarieties of toric varieties to the category of T-schemes. There are many applications such as tropical Hilbert schemes, tropical sheaf theory, and group actions and quotients in tropical geometry. This project is joint work with N. Giansiracusa (Berkeley).

I will report on joint work in progress with David Aldous, concerning a curious random metric space on the plane which can be constructed with the help of an improper Poisson line process.

Abstract: Sharp asymptotic lower bounds of the expected quadratic

variation of discretization error in stochastic integration are given.

The theory relies on inequalities for the kurtosis and skewness of a

general random variable which are themselves seemingly new.

Asymptotically efficient schemes which attain the lower bounds are

constructed explicitly. The result is directly applicable to practical

hedging problem in mathematical finance; it gives an asymptotically

optimal way to choose rebalancing dates and portofolios with respect

to transaction costs. The asymptotically efficient strategies in fact

reflect the structure of transaction costs. In particular a specific

biased rebalancing scheme is shown to be superior to unbiased schemes

if transaction costs follow a convex model. The problem is discussed

also in terms of the exponential utility maximization.

Turbidity currents - submarine flows of sediment - are capable of transporting particulate material over large distance. However direct observations of them are extremely rare and much is inferred from the deposits they leave behind, even though the characteristics of their source are often not known. The submarine flows of volcanic ash from the Soufriere Hills Volcano, Monsterrat provide a unique opportunity to study a particle-driven flow and the deposit it forms, because the details of the source are relatively well constrained and through ocean drilling, the deposit is well sampled.

We have formed simple mathematical models of this motion that capture ash transport and deposit. Our description brings out two dynamical features that strongly influence the motion and which have previously often been neglected, namely mixing between the particulate flow and the oceanic water and the distribution of sizes suspended by the flow. We show how, in even simple situations, these processes alter our views of how these currents propagate.

Notice that the time is 12:30, not 12:00!

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The following is joint work with Sylvia Serfaty and Cyrill Muratov.

We study the asymptotic behavior of the screened sharp interface

Ohta-Kawasaki energy in dimension 2 using the framework of Γ-convergence.

In that model, two phases appear, and they interact via a nonlocal Coulomb

type energy. We focus on the regime where one of the phases has very small

volume fraction, thus creating ``droplets" of that phase in a sea of the

other phase. We consider perturbations to the critical volume fraction

where droplets first appear, show the number of droplets increases

monotonically with respect to the perturbation factor, and describe their

arrangement in all regimes, whether their number is bounded or unbounded.

When their number is unbounded, the most interesting case we compute the

Γ limit of the `zeroth' order energy and yield averaged information for

almost minimizers, namely that the density of droplets should be uniform.

We then go to the next order, and derive a next order Γ-limit energy,

which is exactly the ``Coulombian renormalized energy W" introduced in the

work of Sandier/Serfaty, and obtained there as a limiting interaction

energy for vortices in Ginzburg-Landau. The derivation is based on their

abstract scheme, that serves to obtain lower bounds for 2-scale energies

and express them through some probabilities on patterns via the

multiparameter ergodic theorem. Without thus appealing at all to the

Euler-Lagrange equation, we establish here for all configurations which

have ``almost minimal energy," the asymptotic roundness and radius of the

droplets as done by Muratov, and the fact that they asymptotically shrink

to points whose arrangement should minimize the renormalized energy W, in

some averaged sense. This leads to expecting to see hexagonal lattices of

droplets.

The task is to estimate approach time (time-to-go (TTG)) of non-ballistic threats (e.g. missiles) using passive infrared imagery captured from a sensor on the target platform (e.g. a helicopter). The threat information available in a frame of data is angular position and signal amplitude.

A Kalman filter approach is presented that is applied to example amplitude data to estimate TTG. Angular information alone is not sufficient to allow analysis of missile guidance dynamics to provide a TTG estimate. Detection of the launch is required as is additional information in the form of a terrain database to determine initial range. Parameters that relate to missile dynamics might include proportional navigation constant and motor thrust. Differences between actual angular position observations and modelled values can beused to form an estimator for the parameter set and thence to the TTG.

The question posed here is, "how can signal amplitude information be employed to establish observability in a state-estimation-based model of the angular data to improve TTG estimate performance without any other source of range information?"