14:15
14:15
14:15
A CY Manifold with 3 Generations and Small Hodge Numbers
Abstract
14:15
The Mean-Variance Hedging and Exponential Utility in a Bond Market With Jumps
Abstract
We construct a market of bonds with jumps driven by a general marked point
process as well as by an Rn-valued Wiener process, in which there exists at least one equivalent
martingale measure Q0. In this market we consider the mean-variance hedging of a contingent
claim H 2 L2(FT0) based on the self-financing portfolios on the given maturities T1, · · · , Tn
with T0 T. We introduce the concept of variance-optimal martingale
(VOM) and describe the VOM by a backward semimartingale equation (BSE). We derive an
explicit solution of the optimal strategy and the optimal cost of the mean-variance hedging by
the solutions of two BSEs.
The setting of this problem is a bit unrealistic as we restrict the available bonds to those
with a a pregiven finite number of maturities. So we extend the model to a bond market with
jumps and a continuum of maturities and strategies which are Radon measure valued processes.
To describe the market we consider the cylindrical and normalized martingales introduced by
Mikulevicius et al.. In this market we the consider the exp-utility problem and derive some
results on dynamic indifference valuation.
The talk bases on recent common work with Dewen Xiong.
14:00
Modelling interactions in spatially-structured systems to understand the division of labour in nitrification
17:00
16:30
Faraday waves in Bose-Einstein condensates
Abstract
Traditional Faraday waves appear in a layer of liquid that is shaken vertically. These patterns can take the form of horizontal stripes, close-packed hexagons, or even squares or quasiperiodic patterns. Faraday waves are commonly observed as fine stripes on the surface of wine in a wineglass that is ringing like a bell when periodically forced.
Motivated by recent experiments on Faraday waves in Bose-Einstein condensates we investigate both analytically and numerically the dynamics of cigar-shaped Bose-condensed gases subject to periodic modulation of the strength of the transverse confinement's trap.
We offer a fully analytical explanation of the observed parametric resonance yielding the pattern periodicity versus the driving frequency. These results, corroborated by numerical simulations, match extremely well with the experimental observations.
(COW seminar) Moduli of irreducible symplectic manifolds
Abstract
I shall describe joint work with Gritsenko and Hulek in which we study the moduli spaces of polarised holomorphic symplectic manifolds via their periods. There are strong similarities with moduli spaces of K3 surfaces, but also some important differences, notably that global Torelli fails. I shall explain (conjecturally) why and show how the techniques used to obtain general type results for K3 moduli can be modified to give similar, and quite strong, results in this case. Mainly I shall concentrate on the case of deformations of Hilbert schemes of K3 surfaces.
Sparsity, $\ell_1$ Minimization, and the Geometric Separation Problem
Abstract
During the last two years, sparsity has become a key concept in various areas
of applied mathematics, computer science, and electrical engineering. Sparsity
methodologies explore the fundamental fact that many types of data/signals can
be represented by only a few non-vanishing coefficients when choosing a suitable
basis or, more generally, a frame. If signals possess such a sparse representation,
they can in general be recovered from few measurements using $\ell_1$ minimization
techniques.
One application of this novel methodology is the geometric separation of data,
which is composed of two (or more) geometrically distinct constituents -- for
instance, pointlike and curvelike structures in astronomical imaging of galaxies.
Although it seems impossible to extract those components -- as there are two
unknowns for every datum -- suggestive empirical results using sparsity
considerations have already been obtained.
In this talk we will first give an introduction into the concept of sparse
representations and sparse recovery. Then we will develop a very general
theoretical approach to the problem of geometric separation based on these
methodologies by introducing novel ideas such as geometric clustering of
coefficients. Finally, we will apply our results to the situation of separation
of pointlike and curvelike structures in astronomical imaging of galaxies,
where a deliberately overcomplete representation made of wavelets (suited
to pointlike structures) and curvelets/shearlets (suited to curvelike
structures) will be chosen. The decomposition principle is to minimize the
$\ell_1$ norm of the frame coefficients. Our theoretical results, which
are based on microlocal analysis considerations, show that at all sufficiently
fine scales, nearly-perfect separation is indeed achieved.
This is joint work with David Donoho (Stanford University).
(COW seminar) Reid's recipe and derived categories
Abstract
We give a three dimensional generalization of the classical McKay correspondence construction by Gonzales-Sprinberg and Verdier. This boils down to computing for the Bridgeland-King-Reid derived category equivalence the images of twists of the point sheaf at the origin of C^3 by irreducible representations of G. For abelian G the answer turns out to be closely linked to a piece of toric combinatorics known as Reid's recipe.
MARKET MODELS FOR EUROPEAN OPTIONS: DYNAMIC LOCAL VOLATILITY AND DYNAMIC LOCAL LE´VY MEASURE
Abstract
Most financial models introduced for the purpose of pricing and hedging derivatives concentrate
on the dynamics of the underlying stocks, or underlying instruments on which the derivatives
are written. However, as certain types of derivatives became liquid, it appeared reasonable to model
their prices directly and use these market models to price or hedge exotic derivatives. This framework
was originally advocated by Heath, Jarrow and Morton for the Treasury bond markets.
We discuss the characterization of arbitrage free dynamic stochastic models for the markets with
infinite number of European Call options as the liquid derivatives. Subject to our assumptions on the
presence of jumps in the underlying, the option prices are represented either through local volatility or
through local L´evy measure. Each of the latter ones is then given dynamics through an Itˆo stochastic
process in infinite dimensional space. The main thrust of our work is to characterize absence of arbitrage
in this framework and address the issue of construction of the arbitrage-free models.
10:10
Analytic Factorisation of Integers? A review of classical methods and a proposal for a new paradigm
A Tour of Normal Functions and Algebraic Cycles
Abstract
Associated to a pencil of algebraic curves with singular fibres is a bundle of Jacobians (which are abelian varieties off the discriminant locus of the family and semiabelian varieties over it). Normal functions, which are holomorphic sections of such a Jacobian bundle, were introduced by Poincare and used by Lefschetz to prove the Hodge Conjecture (HC) on algebraic surfaces. By a recent result of Griffiths and Green, an appropriate generalization of these normal functions remains at the center of efforts to establish the HC more generally and understand its implications. (Furthermore, the nature of the zero-loci of these normal functions is related to the Bloch-Beilinson conjectures on filtrations on Chow groups.)
Abel-Jacobi maps give the connection between algebraic cycles and normal functions. In this talk, we shall discuss the limits and singularities of Abel-Jacobi maps for cycles on degenerating families of algebraic varieties. These two features are strongly connected with the issue of graphing admissible normal functions in a Neron model, properly generalizing Poincare's notion of normal functions. Some of these issues will be passed over rather lightly; our main intention is to give some simple examples of limits of AJ maps and stress their connection with higher algebraic K-theory.
A very new theme in homological mirror symmetry concerns what the mirror of a normal function should be; in work of Morrison and Walcher, the mirror is related to counting holomorphic disks in a CY 3-fold bounding on a Lagrangian. Along slightly different lines, we shall briefly describe a surprising application of "higher" normal functions to growth of enumerative (Gromov-Witten) invariants in the context of local mirror symmetry.
Prim's algorithm and self-organized criticality, in the complete graph
Abstract
Let $G=(V,E)$ be a graph with weights $\{w_e : e \in E\}$, and assume all weights are distinct. If $G$ is finite, then the well-known Prim's algorithm constructs its minimum spanning tree in the following manner. Starting from a single vertex $v$, add the smallest weight edge connecting $v$ to any other vertex. More generally, at each step add the smallest weight edge joining some vertex that has already been "explored" (connected by an edge) to some unexplored vertex.
If $G$ is infinite, however, Prim's algorithm does not necessarily construct a spanning tree (consider, for example, the case when the underlying graph is the two-dimensional lattice ${\mathbb Z}^2$, all weights on horizontal edges are strictly less than $1/2$ and all weights on vertical edges are strictly greater than $1/2$).
The behavior of Prim's algorithm for *random* edge weights is an interesting and challenging object of study, even
when the underlying graph is extremely simple. This line of research was initiated by McDiarmid, Johnson and Stone (1996), in the case when the underlying graph is the complete graph $K_n$. Recently Angel et. al. (2006) have studied Prim's algorithm on regular trees with uniform random edge weights. We study Prim's algorithm on $K_n$ and on its infinitary analogue Aldous' Poisson-weighted infinite tree. Along the way, we uncover two new descriptions of the Poisson IIC, the critical Poisson Galton-Watson tree conditioned to survive forever.
Joint work with Simon Griffiths and Ross Kang.
14:15
Non-Expanding Horizons, Shear-Free Congruences and H-spaces
Adaptive evolution and concentrations in parabolic PDE
Abstract
Living systems are subject to constant evolution through the two processes of mutations and selection, a principle discovered by C. Darwin. In a very simple, general and idealized description, their environment can be considered as a nutrient shared by all the population. This alllows certain individuals, characterized by a 'phenotypical trait', to expand faster because they are better adapted to use the environment. This leads to select the 'best fitted trait' in the population (singular point of the system). On the other hand, the new-born individuals undergo small variation of the trait under the effect of genetic mutations. In these circumstances, is it possible to describe the dynamical evolution of the current trait?
We will give a mathematical model of such dynamics, based on parabolic equations, and show that an asymptotic method allows us to formalize precisely the concepts of monomorphic or polymorphic population. Then, we can describe the evolution of the 'fittest trait' and eventually to compute various forms of branching points which represent the cohabitation of two different populations.
The concepts are based on the asymptotic analysis of the above mentioned parabolic equations once appropriately rescaled. This leads to concentrations of the solutions and the difficulty is to evaluate the weight and position of the moving Dirac masses that desribe the population. We will show that a new type of Hamilton-Jacobi equation, with constraints, naturally describes this asymptotic. Some additional theoretical questions as uniqueness for the limiting H.-J. equation will also be addressed.
This work is based on collaborations with O. Diekmann, P.-E. Jabin, S. Mischler, S. Cuadrado, J. Carrillo, S. Genieys, M. Gauduchon, S. Mirahimmi and G. Barles.The Mordell-Weil group of elliptic curves over function fields
15:45
15:45