Past

The talk is based on the joint papers [{\it Bourgain J., Korobkov

M.V. and Kristensen~J.}: Journal fur die reine und angewandte Mathematik

(Crelles

Journal).

DOI: 10.1515/crelle-2013-0002] \ and \

[{\it Korobkov~M.V., Pileckas~K. and Russo~R.}:

arXiv:1302.0731, 4 Feb 2013]

We establish Luzin $N$ and Morse--Sard

properties for functions from the Sobolev space

$W^{n,1}(\mathbb R^n)$. Using these results we prove

that almost all level sets are finite disjoint unions of

$C^1$-smooth compact manifolds of dimension

$n-1$. These results remain valid also within

the larger space of functions of bounded variation

$BV_n(\mathbb R^n)$.

As an application, we study the nonhomogeneous boundary value problem

for the Navier--Stokes equations of steady motion of a viscous

incompressible fluid in arbitrary bounded multiply connected

plane or axially-symmetric spatial domains. We prove that this

problem has a solution under the sole necessary condition of zero total

flux through the boundary.

The problem was formulated by Jean Leray 80 years ago.

The proof of the main result uses Bernoulli's law

for a weak solution to the Euler equations based on the above-mentioned

Morse-Sard property for Sobolev functions.

Convection in a porous medium plays an important role in many geophysical and industrial processes, and is of particular current interest due to its implications for the long-term security of geologically sequestered CO_2. I will discuss two different convective systems in porous media, with the aid of 2D direct numerical simulations: first, a Rayleigh-Benard cell at high Rayleigh number, which gives an accurate characterization both of the convective flux and of the remarkable dynamical structure of the flow; and second, the evolution and eventual `shut-down' of convection in a sealed porous domain with a source of buoyancy along only one boundary. The latter case is also studied using simple box models and laboratory experiments, and can be extended to consider convection across an interface that can move and deform, rather than across a rigid boundary. The relevance of these results in the context of CO_2 sequestration will be discussed.

We prove an analogue of the Tate isogeny conjecture and the
semi-simplicity conjecture for overconvergent crystalline Dieudonne modules
of abelian varieties defined over global function fields of characteristic
p, combining methods of de Jong and Faltings. As a corollary we deduce that
the monodromy groups of such overconvergent crystalline Dieudonne modules
are reductive, and after base change to the field of complex numbers they
are the same as the monodromy groups of Galois representations on the
corresponding l-adic Tate modules, for l different from p.

I will discuss some p-adic (and mod p) criteria ensuring that an elliptic curve over the rationals has algebraic and analytic rank one, as well as some applications.

In partially molten regions of Earth, rock and magma coexist as a two-phase aggregate in which the solid grains of rock form a viscously deformable matrix. Liquid magma resides within the permeable network of pores between grains. Deviatoric stress causes the distribution of contact area between solid grains to become anisotropic; this causes anisotropy of the matrix viscosity. The anisotropic viscosity tensor couples shear and volumetric components of stress/strain rate. This coupling, acting over a gradient in shear stress, causes segregation of liquid and solid. Liquid typically migrates toward higher shear stress, but under specific conditions, the opposite can occur. Furthermore, in a two-phase aggregate with a porosity-weakening viscosity, matrix shear causes porosity perturbations to grow into a banded structure. We show that viscous anisotropy reduces the angle between these emergent high-porosity features and the shear plane. This is consistent with lab experiments.

We shall introduce complex projective structures on a surface, and discuss a new result that relates grafting, which are certain geometric deformations of these structures, to the Teichmuller geodesic flow in the moduli space of Riemann surfaces. A consequence is that for any Fuchsian representation of a surface-group, the set of projective structures with that as holonomy, is dense in moduli space.

A Landau-Ginzburg B-model is a smooth scheme $X$, equipped with a global function $W$. From $(X,W)$ we can construct a category $D(X,W)$, which is called by various names, including ‘the category of B-branes’. In the case $W=0$ it is exactly the derived category $D(X)$, and in the case that $X$ is affine it is the category of matrix factorizations of $W$. There has been a lot of foundational work on this category in recent years, I’ll describe the most modern and flexible approach to its construction. I’ll then interpret Nick Addington’s thesis in this language. We’ll consider the case that $W$ is a quadratic form on a vector bundle, and the corresponding global version of Knorrer periodicity. We’ll see that interesting gerbe structures arise, related to the bundle of isotropic Grassmannians.

A Landau-Ginzburg B-model is a smooth scheme $X$, equipped with a global function $W$. From $(X,W)$ we can construct a category $D(X,W)$, which is called by various names, including ‘the category of B-branes’. In the case $W=0$ it is exactly the derived category $D(X)$, and in the case that $X$ is affine it is the category of matrix factorizations of $W$. There has been a lot of foundational work on this category in recent years, I’ll describe the most modern and flexible approach to its construction.

I’ll then interpret Nick Addington’s thesis in this language. We’ll consider the case that $W$ is a quadratic form on a vector bundle, and the corresponding global version of Knorrer periodicity. We’ll see that interesting gerbe structures arise, related to the bundle of isotropic Grassmannians.

In this talk I will focus on the method of barycentric interpolation, which ties up to the ideas that August Ferdinand Möbius published in his seminal work "Der barycentrische Calcül" in 1827. For univariate data, this gives a special kind of rational interpolant which is guaranteed to have no poles and favourable approximation properties.

I further discuss how to extend this idea to bivariate data, where it leads to the concept of generalized barycentric coordinates and to an efficient method for interpolating data given at the vertices of an arbitrary polygon.

We consider minimisers of integral functionals $F$ over a ‘constrained’ class of $W^{1,p}$-mappings from a bounded domain into a compact Riemannian manifold $M$, i.e. minimisers of $F$ subject to holonomic constraints. Integrands of the form $f(Du)$ and the general $f(x,u,Du)$ are considered under natural strict $p$-quasiconvexity and $p$-growth assumptions for any exponent $1 < p <+\infty$. Unlike the harmonic and $p$-harmonic map case, the quasiconvexity condition requires one to linearise the map at the level of the gradient. In a bid to give a direct proof of partial $C^{1,α}-regularity for such minimisers, we developing an appropriate notion of a tangential harmonic approximation together with a discussion on the difficulties in establishing Caccioppoli-type inequalities. The need in the latter problem to construct suitable competitors to the minimiser via the so-called Luckhaus Lemma presents difficulties quite separate to that of the unconstrained case. We will give a proof of this lemma together with a discussion on the implications for higher integrability.

To continue the day's questions of how complex groups can be I will be looking about some decision problems. I will prove that certain properties of finitely presented groups are undecidable. These properties are called Markov properties and include many nice properties one may want a group to have. I will also hopefully go into an algorithm of Whitehead on deciding if a set of n words generates F_n.

I shall talk about Subgroup Membership Problem for amalgamated products of finite rank free groups. I'm going to show how one can solve different versions of this problem in amalgams of free groups and give an estimate of the complexity of some algorithms involved.  This talk is based on a joint paper with A. J. Duncan.

In my talk I shall give a small survey on some algorithmic properties of amalgamated products of finite rank 
free groups. In particular, I'm going to concentrate on Membership Problem for this groups. Apart from being algorithmically interesting, amalgams of free groups admit a lot of interpretations. I shall show how to 
characterize this construction from the point of view of geometry and linguistic.  

I will explain some recent joint work with Georgios Dimitroglou Rizell in which we use moduli spaces of holomorphic discs with boundary on a monotone Lagrangian torus in ${\mathbb C}^n$ to prove that all such tori are smoothly isotopic when $n$ is odd and at least 5

The quantification and management of risk in financial markets
is at the center of modern financial mathematics. But until recently, risk
assessment models did not consider the effects of inter-connectedness of
financial agents and the way risk diversification impacts the stability of
markets. I will give an introduction to these problems and discuss the
implications of some mathematical models for dealing with them. 

Exact Lagrangian immersions are governed by an h-principle, whilst exact Lagrangian

embeddings are well-known to be constrained by strong rigidity theorems coming from

holomorphic curve theory. We consider exact Lagrangian immersions in Euclidean space with a

prescribed number of double points, and find that the borderline between flexibility and

rigidity is more delicate than had been imagined. The main result obtains constraints on such

immersions with exactly one double point which go beyond the usual setting of Morse or Floer

theory. This is joint work with Tobias Ekholm, and in part with Ekholm, Eliashberg and Murphy.

We relate the expected signature to the Fourier transform of n-point functions, first studied by O. Schramm, and subsequently
by J. Cardy and Simmon, D. Belyaev and J. Viklund. We also prove that the signatures determine the paths in the complement of a Chordal SLE null set. In the end, we will also discuss an idea on how to extend the uniqueness of signatures result by Hambly and Lyons (2006) to paths with finite 1<p<2variations.

Modelling reactive flows, diffusion, transport and mechanical interactions in media consisting of multiple phases, e.g. of a fluid and a solid phase in a porous medium, is giving rise to many open problems for multi-scale analysis and simulation. In this lecture, the following processes are studied:

diffusion, transport, and reaction of substances in the fluid and the solid phase,

mechanical interactions of the fluid and solid phase,

change of the mechanical properties of the solid phase by chemical reactions,

volume changes (“growth”) of the solid phase.

These processes occur for instance in soil and in porous materials, but also in biological membranes, tissues and in bones. The model equations consist of systems of nonlinear partial differential equations, with initial-boundary conditions and transmission conditions on fixed or free boundaries, mainly in complex domains. The coupling of processes on different scales is posing challenges to the mathematical analysis as well as to computing. In order to reduce the complexity, effective macroscopic equations have to be derived, including the relevant information from the micro scale.

In case of processes in tissues, a homogenization limit leads to an effective, mechanical system, containing a pressure gradient, which satisfies a generalized, time-dependent Darcy law, a Biot-law, where the chemical substances satisfy diffusion-transport-reaction equations and are influencing the mechanical parameters.

The interaction of the fluid and the material transported in a vessel with its flexible wall, incorporating material and changing its structure and mechanical behavior, is a process important e.g. in the vascular system (plague-formation) or in porous media.

The lecture is based on recent results obtained in cooperation with A. Mikelic, M. Neuss-Radu, F. Weller and Y. Yang.

Abstract: The aim of this lecture is to give a general introduction to the theory of interacting particle methods and an overview of its applications to numerical finance. We survey the main techniques and results on interacting particle systems and explain how they can be applied to deal with a variety of financial numerical problems such as: pricing complex path dependent European options, computing sensitivities, American option pricing or solving numerically partially observed control problems.

In this talk, we will introduce how to apply Green's function method to get  pointwise estimates for solutions of the Cauchy problem of nonlinear evolution equations with dissipative  structure. First of all, we introduce the pointwise estimates of the time-asymptotic shape of the solutions of the isentropic Navier-Stokes equations and exhibit the generalized Huygen's principle. Then, for other nonlinear dissipative evolution equations, we will  introduce some recent results and give brief explanations. Our approach is based on the detailed analysis of the Green's function of the linearized system and micro-local analysis, such as frequency decomposition and so on.

The Ross Recovery Theorem gives sufficient conditions under which the

market’s beliefs

can be recovered from risk-neutral probabilities. His approach places

mild restrictions on the form of the preferences of

the representative investor. We present an alternative approach which

has no restrictions beyond preferring more to less,

Instead, we restrict the form and risk-neutral dynamics of John Long’s

numeraire portfolio. We also replace Ross’ finite state Markov chain

with a diffusion with bounded state space. Finally, we present some

preliminary results for diffusions on unbounded state space.

In particular, our version of Ross recovery allows market beliefs to be

recovered from risk neutral probabilities in the classical Cox

Ingersoll Ross model for the short interest rate.

The original transport problem is to optimally move a pile of soil to an excavation.

Mathematically, given two measures of equal mass, we look for an optimal bijection that takes

one measure to the other one and also minimizes a given cost functional. Kantorovich relaxed

this problem by considering a measure whose marginals agree with given two measures instead of

a bijection. This generalization linearizes the problem. Hence, allows for an easy existence

result and enables one to identify its convex dual.

In robust hedging problems, we are also given two measures. Namely, the initial and the final

distributions of a stock process. We then construct an optimal connection. In general, however,

the cost functional depends on the whole path of this connection and not simply on the final value.

Hence, one needs to consider processes instead of simply the maps S. The probability distribution

of this process has prescribed marginals at final and initial times. Thus, it is in direct analogy

with the Kantorovich measure. But, financial considerations restrict the process to be a martingale

Interestingly, the dual also has a financial interpretation as a robust hedging (super-replication)

problem.

In this talk, we prove an analogue of Kantorovich duality: the minimal super-replication cost in

the robust setting is given as the supremum of the expectations of the contingent claim over all

martingale measures with a given marginal at the maturity.

This is joint work with Yan Dolinsky of Hebrew University.

Please note the change of venue!

Suppose there is a system where certain objects move through a network. The objects are detected only when they pass through a sparse set of points in the network. For example, the objects could be vehicles moving along a road network, and observed by a radar or other sensor as they pass through (or originate or terminate at) certain key points in the network, but which cannot be observed continuously and tracked as they travel from one point to another. Alternatively they could be data packets in a computer network. The detections only record the time at which an object passes by, and contain no information about identity that would trivially allow the movement of an individual object from one point to another to be deduced. It is desired to determine the statistics of the movement of the objects through the network. I.e. if an object passes through point A at a certain time it is desired to determine the probability density that the same object will pass through a point B at a certain later time.

The system might perhaps be represented by a graph, with a node at each point where detections are made. The detections at each node can be represented by a signal as a function of time, where the signal is a superposition of delta functions (one per detection). The statistics of the movement of objects between nodes must be deduced from the correlations between the signals at each node. The problem is complicated by the possibility that a given object might move between two nodes along several alternative routes (perhaps via other nodes or perhaps not), or might travel along the same route but with several alternative speeds.

What prior knowledge about the network, or constraints on the signals, are needed to make this problem solvable? Is it necessary to know the connections between the nodes or the pdfs for the transition time between nodes a priori, or can this be deduced? What conditions are needed on the information content of the signals? (I.e. if detections are very sparse on the time scale for passage through the network then the transition probabilities can be built up by considering each cascade of detections independently, while if detections are dense then it will presumably be necessary to assume that objects do not move through the network independently, but instead tend to form convoys that are apparent as a pattern of detections that persist for some distance on average). What limits are there on the noise in the signal or amount of unwanted signal, i.e. false detections, or objects which randomly fail to be detected at a particular node, or objects which are detected at one node but which do not pass through any other nodes? Is any special action needed to enforce causality, i.e. positive time delays for transitions between nodes?

One of the most subtle aspects of the correspondence between automorphic and Galois representations is the weight part of Serre conjectures, namely describing the weights of modular forms corresponding to mod p congruence class of Galois representations. We propose a direct geometric approach via studying the mod p cohomology groups of certain integral models of modular or Shimura curves, involving Deligne-Lusztig curves with the action of GL(2) over finite fields. This is a joint work with James Newton.

In this talk we will discuss the mathematical modelling of the performance of Lithium-ion batteries. A mathematical model based on a macro-homogeneous approach developed by John Neuman will be presented. The uniqueness and existence of solution of the corresponding problem will be also discussed.

Very-large scale data analytics are an alleged golden goose for efforts in parallel and distributed computing, and yet contemporary statistics remain somewhat of a dark art for the uninitiated. In this presentation, we are going to take a mathematical and algorithmic look beyond the veil of Big Data by studying the structure of the algorithms and data, and by analyzing the fit to existing and proposed computer systems and programming models. Towards highly scalable kernels, we will also discuss some of the promises and challenges of approximation algorithms using randomization, sampling, and decoupled processing, touching some contemporary topics in parallel numerics.

Following the recent paper of Ogasa, we attempt to construct Boy's surface using only paper and tape. If this is successful we hope to address such questions as:

Is that really Boy's surface?

Why should we care?

Do we have any more biscuits?

Consider non-negative integers assigned to the vertexes of an oriented graph. To this combinatorial data we associate a so-called quiver representation. We will study the geometry and the algebra of this representation, when the underlying un-oriented graph is of Dynkin type ADE.

A remarkable object we will consider is Kazarian's equivariant cohomology spectral sequence. The edge homomorphism of this spectral sequence defines the so-called quiver polynomials. These polynomials are generalizations of remarkable polynomials in algebraic combinatorics (Giambelli-Thom-Porteous, Schur, Schubert, their double, universal, and quantum versions). Quiver polynomials measure degeneracy loci of maps among vector bundles over a common base space. We will present interpolation, residue, and (conjectured) positivity properties of these polynomials.

The quiver polynomials are also encoded in the Cohomological Hall Algebra (COHA) associated with the oriented graph. This is a non-commutative algebra defined by Kontsevich and Soibelman in relation with Donaldson-Thomas invariants. The above mentioned spectral sequence has a structure identity expressing the fact that the sequence converges to explicit groups. We will show the role of this structure identity in understanding the structure of the COHA. The obtained identities are equivalent to Reineke's quantum dilogarithm identities associated to ADE quivers and certain stability conditions.

The recently completed auction for 4G mobile spectrum was the most importantcombinatorial auction ever held in the UK.  In general, combinatorial auctions allow bidders to place individual bids on packages of items,instead of separate bids on individual items, and this feature has theoretical advantages for bidders and sellers alike.  The accompanying challenges of implementation have been the subject of intense work over the last few years, with the result that the advantages of combinatorial auctions can now be realised in practice on a large scale.  Nowhere has this work been more prominent than in auctions for radio spectrum.  The UK's 4G auction is the most recent of these and the publication by Ofcom (the UK's telecommunications regulator) of the auction's full bidding activity creates a valuable case study of combinatorial auctions in action.

Consider non-negative integers assigned to the vertexes of an oriented graph. To this combinatorial data we associate a so-called quiver representation. We will study the geometry and the algebra of this representation, when the underlying un-oriented graph is of Dynkin type ADE.

A remarkable object we will consider is Kazarian's equivariant cohomology spectral sequence. The edge homomorphism of this spectral sequence defines the so-called quiver polynomials. These polynomials are generalizations of remarkable polynomials in algebraic combinatorics (Giambelli-Thom-Porteous, Schur, Schubert, their double, universal, and quantum versions). Quiver polynomials measure degeneracy loci of maps among vector bundles over a common base space. We will present interpolation, residue, and (conjectured) positivity properties of these polynomials.

The quiver polynomials are also encoded in the Cohomological Hall Algebra (COHA) associated with the oriented graph. This is a non-commutative algebra defined by Kontsevich and Soibelman in relation with Donaldson-Thomas invariants. The above mentioned spectral sequence has a structure identity expressing the fact that the sequence converges to explicit groups. We will show the role of this structure identity in understanding the structure of the COHA. The obtained identities are equivalent to Reineke's quantum dilogarithm identities associated to ADE quivers and certain stability conditions.

              Conventional decoherence (usually called 'Environmental

Decoherence') is supposed to be a result of correlations

established between some quantum system and the environment.

'Intrinsic decoherence' is hypothesized as being an essential

feature of Nature - its existence would entail a breakdown of

quantum mechanics. A specific mechanism of some interest is

'gravitational decoherence', whereby gravity causes intrinsic

decoherence.

I will begin by discussing what is now known about the mechanisms of

environmental decoherence, noting in particular that they can and do

involve decoherence without dissipation (ie., pure phase decoherence).

I will then briefly review the fundamental conflict between Quantum

Mechanics and General Relativity, and several arguments that suggest

how this might be resolved by the existence of some sort of 'gravitational

decoherence'.  I then outline a theory of gravitational decoherence

(the 'GR-Psi' theory) which attempts to give a quantitative discussion of

gravitational decoherence, and which makes predictions for

experiments.

The weak field regime of this theory (relevant to experimental

predictions) is discussed in detail, along with a more speculative

discussion of the strong field regime.

A time-invariant level surface is a (codimension one)

spatial surface on which, for every fixed time, the solution of an

evolution equation equals a constant (depending on the time). A

relevant and motivating case is that of the heat equation. The

occurrence of one or more time-invariant surfaces forces the solution

to have a certain degree of symmetry. In my talk, I shall present a

set of results on this theme and sketch the main ideas involved, that

intertwine a wide variety of old and new analytical and geometrical

techniques.

We prove that quasi-trees of spaces satisfying the axiomatisation given by Bestvina, Bromberg and Fujiwara are quasi-isometric to tree-graded spaces in the sense of Dru\c{t}u and Sapir. We then present a technique for obtaining `good' embeddings of such spaces into $\ell^p$ spaces, and show how results of Bestvina-Bromberg-Fujiwara and Mackay-Sisto allow us to better understand the metric geometry of such groups.

In many models of Applied Probability, the distributional limits of recursively defined quantities satisfy distributional identities that are reminiscent of equations of stability. Therefore, there is an interest in generalized concepts of equations of stability.

One extension of this concept is that of random variables ``stable by random weighted mean'' (this notion is due to Liu).

A random variable $X$ taking values in $\mathbb{R}^d$ is called ``stable by random weighted mean'' if it satisfies a recursive distributional equation of the following type:

\begin{equation} \tag{1} \label{eq:1}

X ~\stackrel{\mathcal{D}}{=}~ C + \sum_{j \geq 1} T_j X_j.

\end{equation}

Here, ``$\stackrel{\mathcal{D}}{=}$'' denotes equality of the corresponding distributions, $(C,T_1,T_2,\ldots)$ is a given sequence of real-valued random variables,

and $X_1, X_2, \ldots$ denotes a sequence of i.i.d.\;copies of the random variable $X$ that are independent of $(C,T_1,T_2,\ldots)$.

The distributions $P$ on $\mathbb{R}^d$ such that \eqref{eq:1} holds when $X$ has distribution $P$ are called fixed points of the smoothing transform

(associated with $(C,T_1,T_2,\ldots)$).

A particularly prominent instance of \eqref{eq:1} is the {\texttt Quicksort} equation, where $T_1 = 1-T_2 = U \sim \mathrm{Unif}(0,1)$, $T_j = 0$ for all $j \geq 3$ and $C = g(U)$ for some function $g$.

In this talk, I start with the {\texttt Quicksort} algorithm to motivate the study of \eqref{eq:1}.

Then, I consider the problem of characterizing the set of all solutions to \eqref{eq:1}

in a very general context.

Special emphasis is put on \emph{endogenous} solutions to \eqref{eq:1} since they play an important role in the given setting.