Past

I will explain the basics of deformation quantization of Poisson
algebras (an important tool in mathematical physics). Roughly, it is a
family of associative algebras deforming the original commutative
algebra. Following Fedosov, I will describe a classification of
quantizations of (algebraic) symplectic manifolds.
 

I will describe joint work with Manjul Bhargava (Princeton) and Tom Fisher (Cambridge) in which we determine the probability that random equation from certain families  has a solution either locally (over the reals or the p-adics), everywhere locally,  or globally. Three kinds of equation will be considered: quadratics in any number of variables, ternary cubics and hyperelliptic quartics.

In today's interconnected world, the dissemination of an idea, a trend, a rumor through social networks, as well as the propagation of information or cyber-viruses through digital networks are all common phenomena. They are conceptually similar to the spread of infectious diseases among hosts, as common to all these phenomena is the dissemination of a spreading agent on a networked system. A large body of research has been produced in recent years to characterize the spread of epidemics on static connectivity patterns in a wide range of biological and socio-technical systems. In particular, understanding the mechanisms and conditions for widespread dissemination represents a crucial step for its prevention and control (e.g. in the case of diseases) or for its enhancement (e.g. in the case of viral marketing). This task is however further hindered by the temporal nature characterizing the activation of the connections shaping the networked system, for which data has recently become available. As an example, in networks of proximity contacts among individuals, connections represent sequences of contacts that are active for given periods of time. The time variation of contacts in a networked system may fundamentally alter the properties of spreading processes occurring on it, with respect to static networks, and affect the condition at which epidemics become possible. In this talk I will present a novel theoretical framework adopting a multi-layer perspective for the analytical understanding of the interplay between temporal networks and spreading dynamics. The framework is tested on a set of time-varying network models and empirical networks.

In this talk, we discuss the development of fast iterative solvers for matrix systems arising from various constrained optimization problems. In particular, we seek to exploit the saddle point structure of these problems to construct powerful preconditioners for the resulting systems, using appropriate approximations of the (1,1)-block and Schur complement.

The problems we consider arise from two well-studied subject areas within computational optimization. Specifically, we investigate the
numerical solution of PDE-constrained optimization problems, and the interior point method (IPM) solution of linear/quadratic programming
problems. Indeed a particular focus in this talk is the interior point method solution of PDE-constrained optimization problems with
additional inequality constraints on the state and control variables.

We present a range of optimization problems which we seek to solve using our methodology, and examine the theoretical and practical
convergence properties of our iterative methods for these problems.
 

We investigate the problem of optimizing the shape and

location of actuators or sensors for evolution systems

driven by a partial differential equation, like for

instance a wave equation, a Schrödinger equation, or a

parabolic system, on an arbitrary domain Omega, in

arbitrary dimension, with boundary conditions if there

is a boundary, which can be of Dirichlet, Neumann,

mixed or Robin. This kind of problem is frequently

encountered in applications where one aims, for

instance, at maximizing the quality of reconstruction

of the solution, using only a partial observation. From

the mathematical point of view, using probabilistic

considerations we model this problem as the problem of

maximizing what we call a randomized observability

constant, over all possible subdomains of Omega having

a prescribed measure. The spectral analysis of this

problem reveals intimate connections with the theory of

quantum chaos. More precisely, if the domain Omega

satisfies some quantum ergodic assumptions then we

provide a solution to this problem.



These works are in collaboration with Emmanuel Trélat

(Univ. Paris 6) and Enrique Zuazua (BCAM Bilbao, Spain).

NOTE CHANGE OF TIME AND PLACE

It is known by results of Macintyre and Chatzidakis-Hrushovski that the theory ACFA of existentially closed difference fields is decidable. By developing techniques of difference algebraic geometry, we view quantifier elimination as an instance of a direct image theorem for Galois formulae on difference schemes. In a context where we restrict ourselves to directly presented difference schemes whose definition only involves algebraic correspondences, we develop a coarser yet effective procedure, resulting in a primitive recursive quantifier elimination. We shall discuss various algebraic applications of Galois stratification and connections to fields with Frobenius.

 Voevodsky asked what the topology of the universe is in a 
continuous interpretation of type theory, such as Johnstone's 
topological topos. We can actually give a model-independent answer: it 
is indiscrete. I will briefly introduce "intensional Martin-Loef type 
theory" (MLTT) and formulate and prove this in type theory (as opposed 
to as a meta-theorem about type theory). As an application or corollary, 
I will also deduce an analogue of Rice's Theorem for the universe: the 
universe (the large type of all small types) has no non-trivial 
extensional, decidable properties. Topologically this is the fact that 
it doesn't have any clopens other than the trivial ones.

A group is called residually finite if every non-trivial element can be homomorphically mapped to a finite group such that the image is again non-trivial. Residually finite groups are interesting because quite a lot of information about them can be reconstructed from their finite quotients. Baumslag showed that if G is a finitely generated residually finite group then Aut(G) is also residually finite. Using a similar method Grossman showed that if G is a finitely generated conjugacy separable group with "nice" automorphisms then Out(G) is residually finite. The graph product is a group theoretic construction naturally generalising free and direct products in the category of groups. We show that if G is a finite graph product of finitely generated residually finite groups then Out(G) is residually finite (modulo some technical conditions)

Soluble groups, and other classes of groups that can be built from simpler groups, are useful test cases for studying group properties. I will talk about techniques for building profinite groups from simpler ones, and how  to use these to investigate the cohomology of such groups and recover information about the group structure.

Given a Lagrangian submanifold invariant under a Hamiltonian loop, we partially compute the image of the loop's Seidel element under the closed-open string map into the Hochschild cohomology of the Lagrangian. This piece captures the homology class of the loop's orbits on the Lagrangian and can help to prove that the closed-open map is injective in some examples. As a corollary we prove that $\mathbb{RP}^n$ split-generates the Fukaya category of $\mathbb{CP}^n$ over a field of characteristic 2, and the same for real loci of some other toric  varieties.

Suppose that we have a tile $T$ in say $\mathbb{Z}^2$, meaning a finite subset of $\mathbb{Z}^2$. It may or may not be the case that $T$ tiles $\mathbb{Z}^2$, in the sense that $\mathbb{Z}^2$ can be partitioned into copies of $T$. But is there always some higher dimension $\mathbb{Z}^d$ that can be tiled with copies of $T$? We prove that this is the case: for any tile in $\mathbb{Z}^2$ (or in $\mathbb{Z}^n$, any $n$) there is a $d$ such that $\mathbb{Z}^d$ can be tiled with copies of it. This proves a conjecture of Chalcraft.

The first FFT-based algorithm for numerical homogenization from high-resolution images was proposed by Moulinec and Suquet in 1994 as an alternative to finite elements and twenty years later, it is still widely used in computational micromechanics of materials. The method is based on an iterative solution to an integral equation of the Lippmann-Schwinger type, whose kernel can be explicitly expressed in the Fourier domain. Only recently, it has been recognized that the algorithm has a variational structure arising from a Fourier-Galerkin method. In this talk, I will show how this insight can be used to significantly improve the performance of the original Moulinec-Suquet solver. In particular, I will focus on (i) influence of an iterative solver used to solve the system of linear algebraic equations, (ii) effects of numerical integration of the Galerkin weak form, and (iii) convergence of an a-posteriori bound on the solution during iterations.

A year ago I gave a talk raising questions about Faraday shielding which stimulated discussion with John Ockendon and others and led to a collaboration with Jon Chapman and Dave Hewett.  The problem is one of harmonic functions subject to constant-potential boundary conditions.  A year later, we are happy with the solution we have found, and the paper will appear in SIAM Review.  Though many assume as we originally did that Faraday shielding must be exponentially effective, and Feynman even argues this explicitly in his Lectures, we have found that in fact, the shielding is only linear.  Along the way to explaining this we make use of Mikhlin's numerical method of series expansion, homogenization by multiple scales analysis, conformal mapping, a phase transition, Brownian motion, some ideas recollected from high school about electrostatic induction, and a constrained quadratic optimization problem solvable via a block 2x2 KKT matrix.

We will focus on vortex gradient fields of unit-length. The associated stream function solves the eikonal equation, more precisely it is the distance function to a point. We will prove a kinetic formulation characterizing such vector fields in any dimension.
 

A permutation group is called sharply n-transitive if it acts 

freely and transitively on the set of ordered n-tuples of distinct 

points. The investigation of such permutation groups is a classical 

branch of group theory; it led Emile Mathieu to the discovery of the 

smallest finite simple sporadic groups in the 1860's. In this talk I 

will discuss the case where the permutation group is assumed to be a 

locally compact transformation group, and explain how this set-up is 

related to Gromov hyperbolicity and to arithmetic lattices in products 

of trees.

We consider the problem of minimising the commute or shuttle time for a diffusion between the endpoints of an interval. The control is the scale function for the diffusion. We show that the dynamic version of the problem has the same solution as the static version if we start at an end point and consider the much harder case where the starting point is in the interior.

tba

I shall demonstrate, under some mild assumptions, that the symmetry group of  extreme, Killing, supergravity horzions contains an sl(2, R) subalgebra.  The proof requires a generalization of the  Lichnerowicz theorem for non-metric connections. The techniques developed can also be applied in the classification
of AdS and Minkowski flux backgrounds.
 

Computational models of the heart have been primarily developed to simulate, analyse and understand experimental measurements. Increasingly biophysical models are being used to understand cardiac disease and pathologies in patients. This shift from laboratory to clinical contexts requires the development of new modelling frameworks to simulate pathological states that invalidate assumptions in existing modelling frameworks, work flows to integrate multiple data sets to constrain model parameters and an understanding of the clinical questions that models can answer. We report on the development and application of biophysical modelling frameworks representing the cardiac electrical and mechanical systems, which are currently being customised for modelling cardiac pathologies.

Our study addresses the question of how an arbitrage-free semimartingale model is affected when the knowledge about a random time is added. Precisely, we focus on the No-Unbounded-Profit-with-Bounded-Risk condition, which is also known in the literature as the first kind of no arbitrage. In the general semimartingale setting, we provide a sufficient condition on the random time and price process for which the no arbitrage is preserved under filtration enlargement. Moreover we study the condition on the random time for which the no arbitrage is preserved for any process. This talk is based on a joint work with Tahir Choulli, Jun Deng and Monique Jeanblanc.

We present some recent work - joint with Arno Fehm - in which we give an `existential Ax-Kochen-Ershov principle' for equicharacteristic henselian valued fields. More precisely, we show that the existential theory of such a valued field depends only on the existential theory of the residue field. In residue characteristic zero, this result is well-known and follows from the classical Ax-Kochen-Ershov Theorems. In arbitrary (but equal) characteristic, our proof uses F-V Kuhlmann's theory of tame fields. One corollary is an unconditional proof that the existential theory of F_q((t)) is decidable. We will explain how this relates to the earlier conditional proof of this result, due to Denef and Schoutens.
 

In this talk I shall discuss some classical results on isometric embedding of positively/nonegatively curved surfaces into $\mathbb{R}^3$. 

    The isometric embedding problem has played a crucial role in the development of geometric analysis and nonlinear PDE techniques--Nash invented his Nash-Moser techniques to prove the embeddability of general manifolds; later Gromov recast the problem into his ``h-Principle", which recently led to a major breakthrough by C. De Lellis et al. in the analysis of Euler/Navier-Stokes. Moreover, Nirenberg settled (positively) the Weyl Problem: given a smooth metric with strictly positive Gaussian curvature on a closed surface, does there exist a global isometric embedding into the Euclidean space $\mathbb{R}^3$? This work is proved by the continuity method and based on the regularity theory of the Monge-Ampere Equation, which led to Cheng-Yau's renowned works on the Minkowski Problem and the Calabi Conjecture. 

    Today we shall summarise Nirenberg's original proof for the Weyl problem. Also, we shall describe Hamilton's simplified proof using Nash-Moser Inverse Function Theorem, and Guan-Li's generalisation to the case of nonnegative Gaussian curvature. We shall also mention the status-quo of the related problems.

Thanks to the p-adic local Langlands correspondence for GL_2(Q_p), one "knows" all admissible unitary topologically irreducible representations of GL_2(Z_p). In this talk I will focus on some elementary properties of their restriction to GL_2(Z_p): for instance, to what extent does the restriction to GL_2(Z_p) allow one to recover the original representation, when is the restriction of finite length, etc.

Stability of the hp-Raviart-Thomas projection operator as a mapping H^1(K) -> H^1(K) on the unit cube K in R^3 has been addressed e.g. in [2], see also [1]. These results are suboptimal with respect to the polynomial degree. In this talk we present quasi-optimal stability estimates for the hp-Raviart-Thomas projection operator on the cube. The analysis involves elements of the polynomial approximation theory on an interval and the real method of Banach space interpolation.

(Joint work with Herbert Egger, TU Darmstadt)

[1] Mark Ainsworth and Katia Pinchedez. hp-approximation theory for BDFM and RT finite elements on quadrilaterals. SIAM J. Numer. Anal., 40(6):2047–2068 (electronic) (2003), 2002.

[2] Dominik Schötzau, Christoph Schwab, and Andrea Toselli. Mixed hp-DGFEM for incompressible flows. SIAM J. Numer. Anal., 40(6):2171–2194 (electronic) (2003), 2002.

In the presence of a translation space-like Killing field

the 3 + 1 vacuum Einstein equations reduce to the 2 + 1

Einstein equations with a scalar field. We work in

generalised wave coordinates. In this gauge Einstein

equations can be written as a system of quaslinear

quadratic wave equations. The main difficulty is due to

the weak decay of free solutions to the wave equation in 2

dimensions. To prove long time existence of solutions, we

have to rely on the particular structure of Einstein

equations in wave coordinates. We also have to carefully

choose the behaviour of our metric in the exterior region

to enforce convergence to Minkowski space-time at

time-like infinity.

A Kähler group is a group which is isomorphic to the fundamental group of a compact Kähler manifold. In 2008 Dimca and Suciu proved that the groups which are both Kähler and isomorphic to the fundamental group of a closed 3-manifold are precisely the finite subgroups of $O(4)$ which act freely on $S^3$. In this talk we will explain Kotschick's proof of this result. On the 3-manifold side the main tools that will be used are the first Betti number and Poincare Duality and on the Kähler group side we will make use of the Albanese map and some basic results about Kähler groups. All relevant notions will be explained in the talk.

In this talk we will introduce the (bounded) derived category of coherent sheaves on a smooth projective variety X, and explain how the geometry of X endows this category with a very rigid structure. In particular we will give an overview of a theorem of Orlov which states that any sufficiently ‘nice’ functor between such categories must be Fourier-Mukai.

Suppose we have a finite graph. We can view this as a resistor network where each edge has unit resistance. We can then calculate the resistance between any two vertices and ask questions like `which graph with $n$ vertices and $m$ edges minimises the average resistance between pairs of vertices?' There is a `obvious' solution; we show that this answer is not correct.

This problem was motivated by some questions about the design of statistical experiments (and has some surprising applications in chemistry) but this talk will not assume any statistical knowledge.

This is joint work with Robert Johnson.

When high-frequency acoustic or electromagnetic waves are incident upon an obstacle, the resulting scattered field is composed of rapidly oscillating waves. Conventional numerical methods for such problems use piecewise-polynomial approximation spaces which are not well-suited to capture the oscillatory solution. Hence these methods are prohibitively expensive in the high-frequency regime. Much work has been done in developing “hybrid numerical-asymptotic” (HNA) boundary element methods which utilise approximation spaces containing oscillatory functions carefully chosen to capture the high-frequency asymptotic behaviour of the solution. The computational cost of this approach is significantly smaller than that of conventional methods, and for many problems it is independent of the frequency. In this talk, I will outline the HNA method and discuss its extension to scattering by penetrable obstacles.​

I will present a systematic approach to two-dimensional N=(2,2) supersymmetric gauge theories in curved space, with a particular focus on the two-dimensional Omega deformation. I will explain how to compute Omega-deformed A-type topological correlation functions in purely field theoretic terms (i.e. without relying on a target-space picture), improving on previous techniques. The resulting general formula simplifies previous results in the Abelian (toric) case, while it leads to new results for non-Abelian GLSMs.

Abstract: We introduce a data-based approach to estimating key quantities which arise in the study of nonlinear control and random dynamical systems. Our approach hinges on the observation that much of the existing linear theory may be readily extended to nonlinear systems -with a reasonable expectation of success - once the nonlinear system has been mapped into a high or infinite dimensional Reproducing Kernel Hilbert Space. In particular, we develop computable, non-parametric estimators approximating controllability and observability energy/Lyapunov functions for nonlinear systems, and study the ellipsoids they induce. It is then shown that the controllability energy estimator provides a key means for approximating the invariant measure of an ergodic, stochastically forced nonlinear system. We also apply this approach to the problem of model reduction of nonlinear control systems.

In all cases the relevant quantities are estimated from simulated or observed data. These results collectively argue that there is a reasonable passage from linear dynamical systems theory to a data-based nonlinear dynamical systems theory through reproducing kernel Hilbert spaces. This is a joint work with J. Bouvrie (MIT).

We consider Euler equations on a fixed Lorentzian manifold. The fluid is initially supported on a compact domain and the boundary between the fluid and the vacuum is allowed to move. Imposing the so-called physical vacuum boundary condition, we will explain how to obtain a priori estimates for this problem. In particular, our functional framework allows us to track the regularity of the free boundary. This is joint work with S. Shkoller and J. Speck.

In 1937 Vinogradov showed that every sufficiently large odd number is the sum of three primes, using bounds on the sums of additive characters taken over the primes. He was improving, rather dramatically, on an earlier result of Schnirelmann, which showed that every sufficiently large integer is the sum of at most 37 000 primes. We discuss a natural analogue of this question in the multiplicative group (Z/pZ)* and find that, although the current unconditional character sum technology is too weak to use Vinogradov's approach, an idea from Schnirelmann's work still proves fruitful. We will use a result of Selberg-Delange, an application of a small sieve, and a few easy ideas from additive combinatorics. 

Let $G$ be a reductive group such as $SL_n$ over the field $k((t))$, where $k$ is an algebraic closure of a finite field, and let $W$ be the affine Weyl group of $G$.  The associated affine Deligne-Lusztig varieties $X_x(b)$ were introduced by Rapoport.  These are indexed by elements $x$ in $G$ and $b$ in $W$, and are related to many important concepts in algebraic geometry over fields of positive characteristic.  Basic questions about the varieties $X_x(b)$ which have remained largely open include when they are nonempty, and if nonempty, their dimension.  We use techniques inspired by geometric group theory and representation theory to address these questions in the case that $b$ is a translation.  Our approach is constructive and type-free, sheds new light on the reasons for existing results and conjectures, and reveals new patterns.  Since we work only in the standard apartment of the building for $G$, which is just the tessellation of Euclidean space induced by the action of the reflection group $W$, our results also hold over the p-adics.  This is joint work with Elizabeth Milicevic (Haverford) and Petra Schwer (Karlsruhe).

The lecture will introduce the notion of a folded 4-dimensional hyperkähler manifold, give examples and prove a local existence theorem from boundary data using twistor methods, following an idea of Biquard.