Past

The aim of this lecture is to give a general introduction to

the interacting particle system and applications in finance, especially

in the pricing of American options. We survey the main techniques and

results on Snell envelope, and provide a general framework to analyse

these numerical methods. New algorithms are introduced and analysed

theoretically and numerically.

In 1985 Moffatt suggested that stationary flows of the 3D Euler equations with non-trivial topology could be obtained as the time-asymptotic limits of certain solutions of the equations of magnetohydrodynamics. Heuristic arguments also suggest that the same is true of the system
\[ -\Delta u+\nabla p=(B\cdot\nabla)B\qquad\nabla\cdot u=0\qquad \]
\[ B_t-\eta\Delta B+(u\cdot\nabla)B=(B\cdot\nabla)u \] when $\eta=0$.

In this talk I will discuss well posedness of this coupled elliptic-parabolic equation in the two-dimensional case when $B(0)\in L^2$ and $\eta$ is positive.

Crucial to the analysis is a strengthened version of the 2D Ladyzhenskaya inequality: $\|f\|_{L^4}\le c\|f\|_{L^{2,\infty}}^{1/2}\|\nabla f\|_{L^2}^{1/2}$, where $L^{2,\infty}$ is the weak $L^2$ space. I will also discuss the problems that arise in the case $\eta=0$.

This is joint work with David McCormick and Jose Rodrigo.

This talk is not a detailed and precise exposition on DAG, but it is conceived more as a kind of advertisement on this theory and some of its interesting new features one should contemplate and try to understand, as they might reveal interesting new insights also on classical objects. We select some of the several motivations for introducing it (non-representability of moduli problem and non-naturality of the obstruction theory), and then we will go through the homotopy theory of simplicial commutative algebras and their cotangent complex. We will introduce the category of derived schemes and we will describe their relation with classical schemes. A good amount of time will be dedicated to examples.

This talk consists of three parts. As a motivation, we are first going to introduce von Neumann algebras associated with discrete groups and briefly describe their interplay with measurable group theory. Next, we are going to consider group von Neumann algebras of general locally compact groups and highlight crucial differences between the discrete and the non-discrete case. Finally, we present some recent results on group von Neumann algebras associated with certain locally compact HNN-extensions.

Concepts such as infinitesimal numbers and fluxions have been used by Leibnitz and Newton for the initial development of calculus. However, their non-rigorous nature has caused a lot of controversy and they have eventually been phased out by epsilon-delta definitions. In early 60s Abraham Robinson realised that methods of mathematical logic can be used to provide rigorous meaning to such concepts. This talk is a gentle introduction to some of Robinson's ideas.

We will first recall the known notion of commensurating actions

and its link to actions on CAT(0) cube complexes. We define a

group to have Property FW if every isometric action on a CAT(0)

cube complex has a fixed point. We conjecture that every

irreducible lattice in a semisimple Lie group of higher rank has

Property FW, and will give some instances beyond the trivial

case of Kazhdan groups.

The first application of Szemeredi's regularity method was the following celebrated Ramsey-Turan result proved by Szemeredi in 1972: any K_4-free graph on N vertices with independence number o(N) has at most (1/8 + o(1))N^2 edges. Four years later, Bollobas and Erdos gave a surprising geometric construction, utilizing the isodiametric inequality for the high dimensional sphere, of a K_4-free graph on N vertices with independence number o(N) and (1/8 - o(1)) N^2 edges. Starting with Bollobas and Erdos in 1976, several problems have been asked on estimating the minimum possible independence number in the critical window, when the number of edges is about N^2 / 8.

These problems have received considerable attention and remained one of the main open problems in this area.  More generally, it remains an important problem to determine if, for certain applications of the regularity method, alternative proofs exist which avoid using the regularity lemma and give better quantitative estimates.  In this work, we develop new regularity-free methods which give nearly best-possible bounds, solving the various open problems concerning this critical window.

Joint work with Jacob Fox and Yufei Zhao.

Many interesting examples of singular symplectic algebraic varieties and their symplectic resolutions are built by Hamiltonian reduction. There is a corresponding construction of "quantum Hamiltonian reduction" which is of substantial interest to representation theorists. It starts from a twisted-equivariant D-module, an analogue of an algebraic vector bundle (or coherent sheaf) on a moment map fiber, and produces an object on the quantum analogue of the symplectic resolution. In order to understand how far apart the quantisation of the singular symplectic variety and its symplectic resolution can be, one wants to know "what gets killed by quantum Hamiltonian reduction?" I will give a precise answer to this question in terms of effective combinatorics. The answer has consequences for exactness of direct images, and thus for t-structures, which I will also explain. The beautiful geometry behind the combinatorics is that of a stratification of a GIT-unstable locus called the "Kirwan-Ness stratification." The lecture will not assume familiarity with D-modules, nor with any previous talks by the speaker or McGerty in this series. The new results are joint work with McGerty.

Many interesting examples of singular symplectic algebraic varieties and their symplectic resolutions are built by Hamiltonian reduction. There is a corresponding construction of "quantum Hamiltonian reduction" which is of substantial interest to representation theorists. It starts from a twisted-equivariant D-module, an analogue of an algebraic vector bundle (or coherent sheaf) on a moment map fiber, and produces an object on the quantum analogue of the symplectic resolution. In order to understand how far apart the quantisation of the singular symplectic variety and its symplectic resolution can be, one wants to know "what gets killed by quantum Hamiltonian reduction?" I will give a precise answer to this question in terms of effective combinatorics. The answer has consequences for exactness of direct images, and thus for t-structures, which I will also explain. The beautiful geometry behind the combinatorics is that of a stratification of a GIT-unstable locus called the "Kirwan-Ness stratification." The lecture will not assume familiarity with D-modules, nor with any previous talks by the speaker or McGerty in this series. The new results are joint work with McGerty.

Consider the complete graph on n vertices with independent and identically distributed edge-weights having some absolutely continuous distribution. The minimum spanning tree (MST) is simply the spanning subtree of smallest weight. It is straightforward to construct the MST using one of several natural algorithms. Kruskal's algorithm builds the tree edge by edge starting from the globally lowest-weight edge and then adding other edges one by one in increasing order of weight, as long as they do not create any cycles. At each step of this process, the algorithm has generated a forest, which becomes connected on the final step. In this talk, I will explain how it is possible to exploit a connection between the forest generated by Kruskal's algorithm and the Erd\"os-R\'enyi random graph in order to prove that $M_n$, the MST of the complete graph, possesses a scaling limit as $n$ tends to infinity. In particular, if we think of $M_n$ as a metric space (using the graph distance), rescale edge-lengths by $n^{-1/3}$, and endow the vertices with the uniform measure, then $M_n$ converges in distribution in the sense of the Gromov-Hausdorff-Prokhorov distance to a certain random measured real tree.

This is joint work with Louigi Addario-Berry (McGill), Nicolas Broutin (INRIA Paris-Rocquencourt) and Grégory Miermont (ENS Lyon).

We detect communities on time-dependent correlation networks to study the geographical spread of disease. Using data on country-wide dengue fever, rubella, and H1N1 influenza occurrences spanning several years, we create multilayer similarity networks, with the provinces of a country as nodes and the correlations between the time series of case numbers giving weights to the edges.

We perform community detection on these temporal networks of disease outbreaks, looking for groups of provinces in which disease patterns change in similar ways. Optimizing multilayer modularity with a Newman-Girvan null model over a wide parameter range, we observe several partitions that corresponding roughly to relevant historical time points, such as large epidemics and introduction of new disease strains, as well as many strongly spatial partitions.

We develop a novel null model for community detection that takes into account spatial information, thereby allows to uncover additional structure that might otherwise be obscured by spatial proximity. The null model is based on a radiation model that was proposed recently for modelling human mobility, and we believe that it might be better at capturing disease spread than existing spatial null models based on gravity models for interaction between nodes.

The radiation null model performs better than the Newman-Girvan null model and similarly to the gravity model on benchmark spatial networks with distance-dependent links and a known community structure (both static and multislice networks), and it strongly outperforms both on flux-based benchmarks. When applied to the disease networks, the radiation null model uncovers novel, clear temporal partitions, that might shed light on disease patterns, the introduction of new strains, and provide epidemic warning signals.

In this paper we prove a weak necessary and sufficient maximum principle for Markov regime switching stochastic optimal control problems. Instead of insisting on the maximum condition of the Hamiltonian, we show that 0 belongs to the sum of Clarke's generalized gradient of the Hamiltonian and Clarke's normal cone of the control constraint set at the optimal control. Under a joint concavity condition on the Hamiltonian and a convexity condition on the terminal objective function, the necessary condition becomes sufficient. We give four examples to demonstrate the weak stochastic maximum principle.

“Flash sintering” is a process reported by R Raj and co-workers in which very rapid densification of a ceramic powder compact is achieved by the passage of an electrical current through the specimen. Full density can be achieved in a few seconds (sintering normally takes several hours) and at furnace temperatures several hundred Kelvin below the temperatures required with conventional sintering. The name of the process comes from a runaway power spike that is observed at the point of sintering. Although it is acknowledged by Raj that Joule heating plays a role in the process, he and his co-authors claim that this is of minor importance and that entirely new physical effects must also be involved. However, the existence and possible relevance of these other effects of the electric field/current remains controversial. The aim of this workshop is to introduce the subject and to stimulate discussion of how mathematics could shed light on some the factors that are difficult to measure and understand experimentally.

A mini-lecture series consisting of four 1 hour lectures.

The Schelling segregation model has been extensively studied, by researchers in fields as diverse as economics, physics and computer science. While the explicit concern when the model was first introduced back in 1969, was to model the kind for racial segregation observed in large American cities, the model is sufficiently abstract to apply to almost situation in which agents or nodes arrange themselves geographically according to a preference not to be of a minority type within their own neighbourhhood. Kirman and Vinkovik have established, for example, that Schelling's model is a finite difference version of a differential equation describing interparticle forces (and applied in the modelling of cluster formation). Despite the large literature relating to the model, however, it has largely resisted rigorous analysis -- it has not been possible to prove the segregation behaviour easily observed when running simulations. For the first time we have now been able to rigorously analyse the model, and have also established some rather surprising threshold behaviour.

This talk will require no specialist background knowledge.

I'll discuss questions about the structure of long sums of

Dirichlet characters --- that is, sums of length comparable to the modulus.

For example: How often do character sums get large? Where do character sums

get large? What do character sums "look like" when then get large? This will

include some combination of theorems and experimental data.

A program for Geometric Unity is presented to argue that the seemingly baroque features of the standard model of particle physics are in fact inexorable and geometrically natural when generalizations of the Yang-Mills and Dirac theories are unified with one of general relativity.

We investigate the effect of mass transfer on the evolution of a thin two-dimensional partially wetting drop. While the effects of viscous dissipation, capillarity, slip and uniform mass transfer are taken into account, the effects of inter alia gravity, surface tension gradients, vapour transport and heat transport are neglected in favour of mathematical tractability. Our matched asymptotic analysis reveals that the leading-order outer formulation and contact-line law that is selected in the small-slip limit depends delicately on both the sign and size of the mass transfer flux. We analyse the resulting evolution of the drop and report good agreement with numerical simulations.

"Among the first successes of the h-cobordism theorem was the classification of simply connected closed 5-manifolds. Dimension five is sufficiently large to be able to implement the tools of surgery theory, yet low enough to allow an explicit classification of the manifolds. These traits make dimension five interesting in terms of existence results of geometric structures, like Riemannian metrics of positive Ricci/nonnegative sectional/positive sectional curvature, Einstein metrics, contact structures, Sasakian structures, among others. The talk will be a limited survey of the five-dimensional symbiosis between topology and geometry"

In imaging science, efficient acquisition of images by few samples with the possibility to precisely recover the complete image is a topic of significant interest. The area of compressed sensing, which in particular advocates random sampling strategies, has had already a tremendous impact on both theory and applications. The necessary requirement for such techniques to be applicable is the sparsity of the original data within some transform domain. Recovery is then achieved by, for instance, $\ell_1$ minimization. Various applications however do not allow complete freedom in the choice of the samples. Take Magnet Resonance Imaging (MRI) for example, which only provides access to Fourier samples. For this particular application, empirical results still showed superior performance of compressed sensing techniques.

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In this talk, we focus on sparse sampling strategies under the constraint that only Fourier samples can be accessed. Since images -- and in particular images from MRI -- are governed by anisotropic features and shearlets do provide optimally sparse approximations of those, compactly supported shearlet systems will be our choice for the reconstruction procedure. Our sampling strategy then exploits a careful variable density sampling of the Fourier samples with $\ell_1$-analysis based reconstruction using shearlets. Our main result provides success guarantees and shows that this sampling and reconstruction strategy is optimal.

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This is joint work with Wang-Q Lim (Technische Universit\"at Berlin).

Inspired by some recents developments in the theory of small-strain elastoplasticity, we

both revisit and generalize the formulation of the quasistatic evolutionary problem in

perfect plasticity for heterogeneous materials recently given by Francfort and Giacomini.

We show that their definition of the plastic dissipation measure is equivalent to an

abstract one, where it is defined as the supremum of the dualities between the deviatoric

parts of admissible stress fields and the plastic strains. By means of this abstract

definition, a viscoplastic approximation and variational techniques from the theory of

rate-independent processes give the existence of an evolution statisfying an energy-

dissipation balance and consequently Hill's maximum plastic work principle for an

abstract and very large class of yield conditions.

 A classical question in the model theory of fields is to find out which fields are model complete in the language of rings. It turns out that all well-known examples of model complete fields are quite rigid when it comes to henselianity. We discuss some first results which indicate that in residue characteristic zero, definable henselian valuations prevent model completeness.

In this talk I will describe an attempt to construct a conformal field theory with target space a symmetric product of $R^D$ (referred to by physicists as orbifold sigma model). The construction uses branched covers of $S^2$ to lift the well studied formulation of a sigma model on $S^2$, in terms of vertex operator algebras, to higher genus surfaces. I will motivate and explain this construction.

I will give a gentle introduction to uniformly finite homology. The highlight application will be showing existence of aperiodic tilings of the hyperbolic plane.

The spectral presheaf of a nonabelian von Neumann algebra or C*-algebra

was introduced as a generalised phase space for a quantum system in the

so-called topos approach to quantum theory. Here, it will be shown that

the spectral presheaf has many features of a spectrum of a

noncommutative operator algebra (and that it can be defined for other

classes of algebras as well). The main idea is that the spectrum of a

nonabelian algebra may not be a set, but a presheaf or sheaf over the

base category of abelian subalgebras. In general, the spectral presheaf

has no points, i.e., no global sections. I will show that there is a

contravariant functor from unital C*-algebras to their spectral

presheaves, and that a C*-algebra is determined up to Jordan

*-isomorphisms by its spectral presheaf in many cases. Moreover, time

evolution of a quantum system can be described in terms of flows on the

spectral presheaf, and commutators show up in a natural way. I will

indicate how combining the Jordan and Lie algebra structures may lead to

a full reconstruction of nonabelian C*- or von Neumann algebra from its

spectral presheaf.

The ``k-commodity flow problem'' is: we are given k pairs of vertices of a graph, and we ask whether there are k flows in the graph, where the ith flow is between the ith pair of vertices, and has total value one, and for each edge, the sum of the absolute values of the flows along it is at most one. We may also require the flows to be 1/2-integral, or indeed 1/p-integral for some fixed p.

If the problem is feasible (that is, the desired flows exist) then it is still feasible after contracting any edge, so let us say a flow problem is ``critical'' if it is infeasible, but becomes feasible when we contract any edge. In many special cases, all critical instances have only two vertices, but if we ask for integral flows (that is, p = 1, essentially the edge-disjoint paths problem), then there arbitrarily large critical instances, even with k = 2. But it turns out that p = 1 is the only bad case; if p>1 then all critical instances have bounded size (depending on k, but independent of p), and the same is true if there is no integrality requirement at all.

The proof gives rise to a very simple algorithm for the k edge-disjoint paths problem in 4-edge-connected graphs.

Cutting-edge experiments in quantum communications are reaching regimes

where relativistic effects can no longer be neglected. For example, there

are advanced plans to use satellites to implement teleportation and quantum

cryptographic protocols. Relativistic effects can be expected at these

regimes: the Global Positioning System (GPS), which is a system of

satellites that is used for time dissemination and navigation, requires

relativistic corrections to determine time and positions accurately.

Therefore, it is timely to understand what are the effects of gravity and

motion on entanglement and other quantum properties exploited in quantum

information.

In this talk I will show that entanglement can be created or degraded by

gravity and non-uniform motion. While relativistic effects can degrade the

efficiency of teleportation between moving observers, the effects can also

be exploited in quantum information. I will show that the relativistic

motion of a quantum system can be used to perform quantum gates. Our

results, which will inform future space-based experiments, can be

demonstrated in table-top experiments using superconducting circuits.

This talk is regarding PDE systems from geophysical applications with multiple time scales, in which linear skew-self-adjoint operators of size 1/epsilon gives rise to highly oscillatory solutions. Analysis is performed in justifying the limiting dynamics as epsilon goes to zero; furthermore, the analysis yields estimates on the difference between the multiscale solution and the limiting solution. We will introduce a simple yet effective time-averaging technique which is especially useful in general domains where Fourier analysis is not applicable.

Random wavelet series were introduced in the mid 90s as simple and flexible models that allow to take into account observed statistics of wavelet coefficients in signal and image processing. One of their most interesting properties is that they supply random processes whose pointwise regularity jumps form point to point in a very erratic way, thus supplying examples of multifractal processes.

Interest in such models has been renewed recently under the spur of new applications coming from widely different fields; e.g.

-in functional analysis, they allow to derive the regularity properties of ``generic'' functions in a given function space (in the sense of

prevalence)

-they offer toy examples on which one can check the accuracy of numerical algorithms that allow to derive the multifractal parameters associated with signals and images.

We will give an overview of these properties, and we will focus on recent extensions whose sample paths are not locally bounded, and offer models for signals which share this property.

 In this talk I will introduce my joint work with Kreck on a classification of
certain 5-manifolds with fundamental group Z. This result can be interpreted as a
generalization of the classical Browder-Levine's fibering theorem to dimension 5.

The talk will survey some results about smooth and topological 4-manifolds obtained via surgery, and discuss some contrasting information provided by gauge theory about smooth finite group actions on 4-manifolds.

Free probability theory has been introduced by Voiculescu in the 80's for the study of the von Neumann algebras of the free groups. It consists in an algebraic setting of non commutative probability, where one encodes "non commutative random variables" in abstract (non commutative) algebras endowed with linear forms (which satisfies properties in order to play the role of the expectation). In this context, Voiculescu introduce the notion of freeness which is the analogue of the classical independence.

A decade later, he realized that a family of independent random matrices invariant in law by conjugation by unitary matrices are asymptotically free. This phenomenon is called asymptotic freeness. It had a deep impact in operator algebra and probability and has been generalized in many directions. A simple particular case of Voiculescu's theorem gives an estimate, for N large, of the spectrum of an N by N Hermitian matrix H_N = A_N + 1/\sqrt N X_N, where A_N is a given deterministic Hermitian matrix and X_N has independent gaussian standard sub-diagonal entries.

Nevertheless, it turns out that asymptotic freeness does not hold in certain situations, e.g. when the entries of X_N as above have heavy-tails. To infer the spectrum of a larger class of matrices, we go further into Voiculescu's approach and introduce the distributions of traffics and their free product. This notion of distribution is richer than Voiculescu's notion of distribution of non commutative random variables and it generalizes the notion of law of a random graph. The notion of freeness for traffics is an intriguing mixing between the classical independence and Voiculescu's notion of freeness. We prove an asymptotic freeness theorem in that context for independent random matrices invariant in law by conjugation by permutation matrices.

The purpose of this talk is to give an introductory presentation of these notions.

We discuss the superhedging problem under model uncertainty based on existence

and duality results for minimal supersolutions of backward stochastic differential equations.

The talk is based on joint works with Samuel Drapeau, Gregor Heyne and Reinhard Schmidt.

Nowadays there are a large number of satellite and airborne observations of the large ice sheet that covers Antarctica. These include maps of the surface elevation, ice thickness, surface velocity, the rate of snow accumulation, and the rate of change of surface elevation. Uncertainty in the possible rate of future sea level rise motivates using all of these observations and models of ice-sheet flow to project how the ice sheet will behave in future, but this is still a challenge. To make useful predictions, especially in the presence of potential dynamic instabilities, models will need accurate initial conditions, including flow velocity throughout the ice thickness. The ice sheet can be several kilometres thick, but most of the observations identify quantities at the upper surface of the ice sheet, not within its bulk. There is thus a question of how the subsurface flow can be inferred from surface observations. The key parameters that must be identified are the viscosity in the interior of the ice and the basal drag coefficient that relates the speed of sliding at the base of the ice sheet to the basal shear stress. Neither is characterised well by field or laboratory studies, but for incompressible flow governed by the Stokes equations they can be investigated by inverse methods analogous to those used in electric impedance tomography (which is governed by the Laplace equation). Similar methods can also be applied to recently developed 'hybrid' approximations to Stokes flow that are designed to model shallow ice sheets, fast-sliding ice streams, and floating ice shelves more efficiently. This talk will give a summary of progress towards model based projections of the size and shape of the Antarctic ice sheet that make use of the available satellite data. Some of the outstanding problems that will need to be tackled to improve the accuracy of these projections will also be discussed.