The Schur product is the commutative operation of entrywise
multiplication of two (possibly infinite) matrices. If we fix a matrix
A and require that the Schur product of A with the matrix of any
bounded operator is again the matrix of a bounded operator, then A is
said to be a Schur multiplier; Schur multiplication by A then turns
out to be a completely bounded map. The Schur multipliers were
characterised by Grothendieck in the 1950s. In a 2006 paper, Kissin
and Shulman study a noncommutative generalisation which they call
"operator multipliers", in which the theory of operator spaces plays
an important role. We will present joint work with Katja Juschenko,
Ivan Todorov and Ludmilla Turowska in which we determine the operator
multipliers which are completely compact (that is, they satisfy a
strengthening of the usual notion of compactness which is appropriate
for completely bounded maps).
This is the second of two seminars this afternoon describing a generalization of Donaldson-Thomas invariants, joint work of Yinan Song and Dominic Joyce. (Still work in progress.)
Behrend showed that conventional Donaldson-Thomas invariants can be written as the Euler characteristic of the moduli space of semistable sheaves weighted by a "microlocal obstruction function" \mu.
In previous work, the speaker defined Donaldson-Thomas type invariants "counting" coherent sheaves on a Calabi-Yau 3-fold using
Euler characteristics of sheaf moduli spaces, and more generally, of moduli spaces of "configurations" of sheaves. However, these invariants are not deformation-invariant.
We now combine these ideas, and insert Behrend's microlocal obstruction \mu into the speaker's previous definition to get new generalized Donaldson-Thomas invariants. Microlocal functions \mu have a multiplicative property implying that the new invariants transform according to the same multiplicative transformation law as the previous invariants under change of stability condition.
Then we show that the invariants counting pairs in the previous seminar are sums of products of the new generalized Donaldson-Thomas invariants. Since the pair invariants are deformation invariant, we can deduce by induction on rank that the new generalized Donaldson-Thomas invariants are unchanged under deformations of the underlying Calabi-Yau 3-fold.
For any simple graph G and any positive integer lambda, let
P(G,lambda) denote the number of mappings f from V(G) to
{1,2,..,lambda} such that f(u) not= f(v) for every two adjacent
vertices u and v in G. It can be shown that
P(G,lambda) = \sum_{A \subseteq E} (-1)^{|A|} lambda^{c(A)}
where E is the edge set of G and c(A) is the number of components
of the spanning subgraph of G with edge set A. Hence P(G,lambda)
is really a polynomial of lambda. Many results on the chromatic
polynomial of a graph have been discovered since it was introduced
by Birkhoff in 1912. However, there are still many unsolved
problems and this talk will introduce the progress of some
problems and also some new problems proposed recently.
This is the first of two seminars this afternoon describing a generalization of Donaldson-Thomas invariants, joint work of Yinan Song and Dominic Joyce. We shall define invariants "counting" semistable coherent sheaves on a Calabi-Yau 3-fold. Our invariants are invariant under deformations of the complex structure of the underlying Calabi-Yau 3-fold, and have known transformation law under change of stability condition.
This first seminar constructs an auxiliary invariant "counting" stable pairs (s,E), where E is a Gieseker semistable coherent sheaf with fixed Hilbert polynomial and s : O(-n) --> E for n >> 0 is a morphism of sheaves, and (s,E) satisfies a stability condition. Using Behrend-Fantechi's approach to obstruction theories and virtual classes we prove this auxiliary invariant is unchanged under deformation of the underlying Calabi-Yau 3-fold.
I will begin by talking briefly about the Lavrentiev phenomenon and its implications for computations. In short, if a minimization problem exhibits a Lavrentiev gap then `naive' numerical methods cannot be used to solve it. In the past, several regularization techniques have been used to overcome this difficulty. I will briefly mention them and discuss their strengths and weaknesses.
The main part of the talk will be concerned with a class of convex problems, and I will show that for this class, relatively simple numerical methods, namely (i) the Crouzeix--Raviart FEM and (ii) the P2-FEM with under-integration, can successfully overcome the Lavrentiev gap.
I will start by reviewing the current status of the stability
problem for black holes in general relativity. In the second part of the
talk I will focus on a particular (symmetry) class of five-dimensional
dynamical black holes recently introduced by Bizon et al as a model to
study gravitational collapse in vacuum. In this context I state a recent
result establishing the asymptotic stability of the five dimensional
Schwarzschild metric with respect to vacuum perturbations in the given
class.
We construct a kinetic SDE in the state variables (position,velocity), where the spatial dependency in the drift term of the velocity equation is a conditional expectation with respect to the position. Those systems are introduced in fluid mechanic by S. B. Pope and are used in the simulation of complex turbulent flows. Such simulation approach is known as Probability Density Function (PDF) method .
We construct a PDF method applied to a dynamical downscaling problem to generate fine scale wind : we consider a bounded domain D. A weather prediction model solves the wind field at the boundary of D (coarse resolution). In D, we adapt a Lagrangian model to the atmospheric flow description and we construct a particles algorithm to solve it (fine resolution).
In the second part of the talk, we give a (partial) construction of a Lagrangian SDE confined in a given domain and such that the corresponding Eulerian velocity at the boundary is given. This problem is related to stochastic impact problem and existence of trace at the boundary for the McKean-Vlasov equations with specular boundary condition
The Cameron-Martin theorem is a fundamental result in stochastic analysis. We will show that the Wiener measure on a geometrically and stochastically complete Riemannian manifold is quasi-invariant. This is a complete a complete generalization of the classical Cameron-Martin theorem for Euclidean space to Riemannian manifolds. We do not impose any curvature growth conditions.
We consider a financial contract that delivers a single cash flow given by the terminal value of a cumulative gains process.
The problem of modelling such an asset and associated derivatives is important, for example, in the determination of optimal insurance claims reserve policies, and in the pricing of reinsurance contracts. In the insurance setting, aggregate claims play the role of cumulative gains, and the terminal cash flow represents the totality of the claims payable for the given accounting period. A similar example arises when we consider the accumulation of losses in a credit portfolio, and value a contract that pays an amount equal to the totality of the losses over a given time interval. An expression for the value process of such an asset is derived as follows. We fix a probability space, together with a pricing measure, and model the terminal cash flow by a random variable; next, we model the cumulative gains process by the product of the terminal cash flow and an independent gamma bridge; finally, we take the filtration to be that generated by the cumulative gains process.
An explicit expression for the value process is obtained by taking the discounted expectation of the future cash flow, conditional on the relevant market information. The price of an Arrow–Debreu security on the cumulative gains process is determined, and is used to obtain a closed-form expression for the price of a European-style option on the value of the asset at the given intermediate time. The results obtained make use of remarkable properties of the gamma bridge process, and are applicable to a wide variety of financial products based on cumulative gains processes such as aggregate claims, credit portfolio losses, defined benefit pension schemes, emissions, and rainfall. (Co-authors: D. C. Brody, Imperial College London, and A.
Macrina, King's College London and ETH Zurich. Downloadable at
Let E be an elliptic curve over the rationals. To get an asymptotic to the number of primes p
joint work with R Gurjar
Dirichlet to Neumann maps and their generalizations are exceptionally useful tools in the study of eigenvalue problems for ODEs and PDEs. They also have real physical significance through their occurrence in electrical impedance tomography, with applications to medical imagine, landmine detection and non-destructive testing. This talk will review some of the basic properties of Dirichlet to Neumann maps, some new abstract results which make it easier to use them for a wide variety of models, and some analytical/numerical results which depend on them, including detection and elimination of spectral pollution.
We call a family of permutations A in Sn 'intersecting' if any two permutations in A agree in at least one position. Deza and Frankl observed that an intersecting family of permutations has size at most (n-1)!; Cameron and Ku proved that equality is attained only by families of the form {σ in Sn: σ(i)=j} for i, j in [n].
We will sketch a proof of the following `stability' result: an intersecting family of permutations which has size at least (1-1/e + o(1))(n-1)! must be contained in {σ in Sn: σ(i)=j} for some i,j in [n]. This proves a conjecture of Cameron and Ku.
In order to tackle this we first use some representation theory and an eigenvalue argument to prove a conjecture of Leader concerning cross-intersecting families of permutations: if n >= 4 and A,B is a pair of cross-intersecting families in Sn, then |A||B|
The limit shape of Young diagrams under the Plancherel measure was found by Vershik & Kerov (1977) and Logan & Shepp (1977). We obtain a central limit theorem for fluctuations of Young diagrams in the bulk of the partition 'spectrum'. More specifically, we prove that, under a suitable (logarithmic) normalization, the corresponding random process converges (in the FDD sense) to a Gaussian process with independent values. We also discuss the link with an earlier result by Kerov (1993) on the convergence to a generalized Gaussian process. The proof is based on the Poissonization of the Plancherel measure and an application of a general central limit theorem for determinantal point processes (joint work with Zhonggen Su).
The McKean stochastic game (MSG) is a two-player version of the perpetual American put option. The MSG consists of two agents and a certain payoff function of an underlying stochastic process. One agent (the seller) is looking for a strategy (stopping time) which minimises the expected pay-off, while the other agent (the buyer) tries to maximise this quantity.
For Brownian motion one can find the value of the MSG and the optimal stopping times by solving a free boundary value problem. For a Lévy process with jumps the corresponding free boundary problem is more difficult to solve directly and instead we use fluctuation theory to find the solution of the MSG driven by a Lévy process with no positive jumps. One interesting aspect is that the optimal stopping region for the minimiser "thickens" from a point to an interval in the presence of jumps. This talk is based on joint work with Andreas Kyprianou (University of Bath).
Trading a financial asset involves a sequence of decisions to buy or sell the asset over time. A traditional trading strategy is to buy low and sell high. However, in practice, identifying these low and high levels is extremely challenging and difficult. In this talk, I will present our ongoing research on characterization of these key levels when the underlying asset price is dictated by a mean-reversion model. Our objective is to buy and sell the asset sequentially in order to maximize the overall profit. Mathematically, this amounts to determining a sequence of stopping times. We establish the associated dynamic programming equations (quasi-variational
inequalities) and show that these differential equations can be converted to algebraic-like equations under certain conditions.
The two threshold (buy and sell) levels can be found by solving these algebraic-like equations. We provide sufficient conditions that guarantee the optimality of our trading strategy.
The lecture will describe two variants of thin film flows, one involving wetting and the other involving evaporation. First, describing the spreading of mostly wetting liquid droplets on surfaces decorated with assemblies of micron-size cylindrical posts arranged in regular arrays. A variety of deterministic final shapes of the spreading droplets are obtained, including octagons, squares, hexagons and cricles. Dynamic considerations provide a "shape" diagram and suggest rules for control. It is then shown how these ideas can be used to explore (and control) splashing and to create polygonal hydraulic jumps. Second, the evaporation of volatile liquid drops is considered. Using experiments and theory it is shown how the sense of the internal circulation depends on the ratio of the liquid and substrate conductivities. The internal motions control the deposition patterns and so may impact various printing processes. These ideas are then applied to colloid deposition porous media.
Recently there has been increasing interest in discrete analogues of classical operators in harmonic analysis. Often the difficulties one encounters in the discrete setting require completely new approaches; the most successful current approaches are motivated by ideas from classical analytic number theory. This talk will describe a menagerie of new results for discrete analogues of operators ranging from twisted singular Radon transforms to fractional integral operators both on R^n and on the Heisenberg group H^n. Although these are genuinely analytic results, key aspects of the methods come from number theory, and this talk will highlight the roles played by theta functions, Waring's problem, the Hypothesis K* of Hardy and Littlewood, and the circle method.