Past

The birational classification of varieties is an interesting and ongoing problem in algebraic geometry. This talk aims to give an

overview of the progress made on this problem in the special case where the varieties considered are surfaces in projective space.

Linear operators preserving non-vanishing properties are an important

tool in e.g. combinatorics, the Lee-Yang program on phase transitions, complex analysis, matrix theory. We characterize all linear operators on spaces of multivariate polynomials preserving the property of being non-vanishing when the variables are in prescribed open circular domains, which solves the higher dimensional counterpart of a long-standing classification problem going back to P\'olya-Schur. This also leads to a self-contained theory of multivariate stable polynomials and a natural framework for dealing with Lee-Yang and Heilmann-Lieb type problems in a uniform manner. The talk is based on joint work with Petter Brändén.

In this talk I shall review analytical and numerical results on equilibrium configurations of rotating fluid bodies within Einstein's theory of gravitation.

I will discuss recent results concerning the uniqueness of Lagrangian particle trajectories associated to weak solutions of the Navier-Stokes equations. In two dimensions, for which the weak solutions are unique, I will present a mcuh simpler argument than that of Chemin & Lerner that guarantees the uniqueness of these trajectories (this is joint work with Masoumeh Dashti, Warwick). In three dimensions, given a particular weak solution, Foias, Guillopé, & Temam showed that one can construct at leaset one trajectory mapping that respects the volume-preserving nature of the underlying flow. I will show that under the additional assumption that $u\in L^{6/5}(0,T;L^\infty)$ this trajectory mapping is in fact unique (joint work with Witek Sadowski, Warsaw).

We consider one-dimensional Brownian motion conditioned (in a suitable

sense) to have a local time at every point and at every moment bounded by some fixed constant. Our main result shows that a phenomenon of entropic repulsion occurs: that is, this process is ballistic and has an asymptotic velocity approximately 4.5860... as high as required by the conditioning (the exact value of this constant involves the first zero of a Bessel function). I will also describe other conditionings of Brownian motion in which this principle of entropic repulsion manifests itself.

Joint work with Itai Benjamini.

Particle current is the net number of particles that pass an observer who moves with a deterministic velocity V. Its fluctuations in time-stationary interacting particle systems are nontrivial and draw serious attention. It has been known for a while that in most models diffusive scaling and the corresponding Central Limit Theorem hold for this quantity. However, such normal fluctuations disappear for a particular value of V, called the characteristic speed.

For this velocity value, the correct scaling of particle current fluctuations was shown to be t1/3 and the limit distribution was also identified by K. Johansson in 2000 and later by P. L. Ferrari and H. Spohn in 2006. These results use heavy combinatorial and analytic tools, and their application is limited to a few particular models, one of which is the totally asymmetric simple exclusion process (TASEP). I will explain a purely probabilistic, more robust approach that provides the t2/3-scaling of current variance, but not the limit distribution, in (non-totally) asymmetric simple exclusion (ASEP) and some other particle systems. I will also point out a key feature of the models which allows the proof of such universal behaviour.

Joint work with Júlia Komjáthy and Timo Seppälläinen)

Abstract: We discuss an extension of Hitchin's generalised geometry, based on the exceptional groups, that provides a unified geometrical description of supersymmetric flux backgrounds in eleven-dimensional supergravity. We focus on N=1 seven-dimensional compactifications. The background is characterised by an element phi, the analogue of the generalised complex structure, that lies in a particular orbit of the 912 representation of E7. As an application we show that the four-dimensional effective superpotential takes a universal form, that is, a homogeneous E7-invariant functional of phi.

Let E be an elliptic curve defined by a cubic equation with rational coefficients.
The Sato-Tate Conjecture is a statistical assertion about the variation of the number of points of E over finite fields. I review some of the main steps in my proof of this conjecture with Clozel, Shepherd-Barron, and Taylor, in the case when E has non-integral j-invariant. Emphasis will be placed on the steps involving moduli spaces of certain Calabi-Yau hypersurfaces with level structure.

If one admits a version of the stable trace formula that should soon be available, the same techniques imply that, when E and E' are two elliptic curves that are not isogenous, then the numbers of their points over finite fields are statistically independent. For reasons that have everything to do with the current limits to our understanding of the Langlands program, the analogous conjectures for three or more non-isogenous elliptic curves are entirely out of reach.

Fixed-point logics are a class of logics designed for formalising

recursive or inductive definitions. Being initially studied in

generalised recursion theory by Moschovakis and others, they have later

found numerous applications in computer science, in areas

such as database theory, finite model theory, and verification.

A common feature of most fixed-point logics is that they extend a basic

logical formalism such as first-order or modal logic by explicit

constructs to form fixed points of definable operators. The type of

fixed points that can be formed as well as the underlying logic

determine the expressive power and complexity of the resulting logics.

In this talk we will give a brief introduction to the various extensions

of first-order logic by fixed-point constructs and give some examples

for properties definable in the different logics. In the main part of

the talk we will concentrate on extensions of first-order

logic by least and inflationary fixed points. In particular, we

compare the expressive power and complexity of the resulting logics.

The main result will be to show that while the two logics have rather

different properties, they are equivalent in expressive power on the

class of all structures.

This talk will give a survey of results in continuous-time

contract theory, and discuss open problems and plans for further

research on this topic.

The general question is how a ``principal" (a company, investors ...)

should design a payoff for compensating an ``agent" (an executive, a

portfolio manager, ...) in order to induce the best possible

performance.

The following frameworks are standard in contract theory:

(i) the principal and the agent have same, full information;

(ii) the principal cannot monitor agent's actions

(iii) the principal does not know agent's type We will discuss all

three of these problems.

The mathematical tools used are those of stochastic control theory,

stochastic maximum principle and Forward Backward Stochastic

Differential Equations.

Among optimization problems, convex problems form a special subset with two important and useful properties: (1) the existence of a strongly related dual problem that provides certified bounds and (2) the possibility to find an optimal solution using polynomial-time algorithms. In the first part of this talk, we will outline how the framework of conic optimization, which formulates structured convex problems using convex cones, facilitates the exploitation of those two properties. In the second part of this talk, we will introduce a specific cone (called the power cone) that allows the formulation of a large class of convex problems (including linear, quadratic, entropy, sum-of-norm and geometric optimization).
For this class of problems, we present a primal-dual interior-point algorithm, which focuses on preserving the perfect symmetry between the primal and dual sides of the problem (arising from the self-duality of the power cone).

We study an equilibrium model for a defaultable bond in the asymmetric dynamic information setting. The market consists of noise traders, an insider and a risk neutral market maker. Under the assumption that the insider observes the firm value continuously in time we study the optimal strategies for the insider and the optimal pricing rules for the market maker. We show that there exists an equilibrium where the insider’s trades are inconspicuous. In this equilibrium the insider drives the total demand to a certain level at the default time. The solution follows from answering the following purely mathematical question which is of interest in its own: Suppose Z and B are two independent Brownian motions with B(0)=0 and Z(0) is a positive random variable. Let T be the first time that Z hits 0. Does there exists a semimartingale X such that

1) it is a solution to the SDE

dX(t) = dB(t) + g(t,X(t),Z(t))dt

with X(0) = 1, for some appropriate function g,

2) T is the first hitting time of 0 for X, and

3) X is a Brownian motion in its own filtration?

In St Anne's College. Confirmed Speakers:

• Alfio Quarteroni (EPFL) — Heterogeneous Domain Decomposition Methods
• Laure Saint-Raymond (Paris VI & ENS) — Weak compactness methods for singular penalization problems with boundary layers
• Bryce McLeod (Oxford) — A problem in dislocation theory
• Tom Bridges (Surrey) — Degenerate conservation laws, bifurcation of solitary waves and the concept of criticality in fluid mechanics
• Neshan Wickramasekera (Cambridge) — Frequency functions and singular set bounds for branched minimal graphs

The meeting is being held in the Mary Ogilvie Lecture Theatre, St Anne’s College and will start promptly at 9:30am with the last talk finishing at 4:30pm.

For the full programme and registration pages please see: http://www2.maths.ox.ac.uk/oxpde/meetings/

Abstract: We discuss recent techniques to compute one-loop amplitudes in QCD and show that all N-gluon one-loop helicity amplitudes can be computed numerically for arbitrary N with an algorithm which has a polynomial growth in N.

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.