Past

"We introduce some type of generalized Poisson formula which is equivalent 
to Langlands' automorphic transfer from an arbitrary reductive group over a 
global field to a general linear group."

The martingale optimal transportation problem is motivated by

model-independent bounds for the pricing and hedging exotic options in

financial mathematics.

In the simplest one-period model, the dual formulation of the robust

superhedging cost differs from the standard optimal transport problem by

the presence of a martingale constraint on the set of coupling measures.

The one-dimensional Brenier theorem has a natural extension. However, in

the present martingale version, the optimal coupling measure is

concentrated on a pair of graphs which can be obtained in explicit form.

These explicit extremal probability measures are also characterized as

the unique left and right monotone martingale transference plans, and

induce an optimal solution of the kantorovitch dual, which coincides

with our original robust hedging problem.

By iterating the above construction over n steps, we define a Markov

process whose distribution is optimal for the n-periods martingale

transport problem corresponding to a convenient class of cost functions.

Similarly, the optimal solution of the corresponding robust hedging

problem is deduced in explicit form. Finally, by sending the time step

to zero, this leads to a continuous-time version of the one-dimensional

Brenier theorem in the present martingale context, thus providing a new

remarkable example of Peacock, i.e. Processus Croissant pour l'Ordre

Convexe. Here again, the corresponding robust hedging strategy is

obtained in explicit form.

*****     PLEASE NOTE THAT THIS WILL TAKE PLACE ON FRIDAY 7TH JUNE     *****

Collective behaviours of coupled linear or nonlinear resonators have been of interest to engineers as well as mathematician for a long time. In this presentation, using the example of coupled resonant nano-sensors (which leads to a Linear pencil with a Jacobian matrix), I will show how previously feared and often avoided coupling between nano-devices along with their weak nonlinear behaviour can be used with inverse eigenvalue analysis to design multiple-input-single-output nano-sensors. We are using these matrices in designing micro/Nano electromechanical systems, particularly resonant sensors capable for measuring very small mass for use as environmental as well as biomedical monitors. With improvement in fabrication technology, we can design and build several such sensors on one substrate. However, this leads to challenges in interfacing them as well as introduces undesired parasitic coupling. More importantly, increased nonlinearity is being observed as these sensors reduce in size. However, this also presents an opportunity to experimentally study chains or matrices of coupled linear and/or nonlinear structures to develop new sensing modalities as well as to experimentally verify theoretically or numerically predicted results. The challenge for us is now to identify sensing modalities with chain of linear or nonlinear resonators coupled either linearly or nonlinearly. We are currently exploring chains of Duffing resonators, van der Pol oscillators as well as FPU type lattices.

Some real resource allocation problems are so large and complex that optimization would computationally infeasible, even with complete information about all the relevant values. For example, the proposal in the US to use television broadcasters' bids to determine which stations go off air to make room for wireless broadband is characterized by hundreds of thousands of integer constraints. We use game theory and auction theory to characterize a class of simple, strategy-proof auctions for such problems and show their equivalence to a class of "clock auctions," which make the optimal bidding strategy obvious to all bidders. We adapt the results of optimal auction theory to reduce expected procurement costs and prove that the procurement cost of each clock auction is the same as that of the full information equilibrium of its related paid-as-bid (sealed-bid) auction.

An externally definable set of a first order structure $M$ is a set of the form $X\cap M^n$ for a set $X$ that is parametrically definable in some elementary extension of $M$. By a theorem of Shelah, these sets form again a first order structure if $M$ is NIP. If $M$ is a real closed field, externally definable sets can be described as some sort of limit sets (to be explained in the talk), in the best case as Hausdorff limits of definable families. It is conjectured that the Shelah structure on a real closed field is generated by expanding the field with convex subsets of the line. This is known to be true in the archimedean case by van den Dries (generalised by Marker and Steinhorn). I will report on recent progress around this question, mainly its confirmation on real closed fields that are close to being maximally valued with archimedean residue field. The main tool is an algebraic characterisation of definable types in real closed valued fields. I also intend to give counterexamples to a localized version of the conjecture. This is joint work with Francoise Delon.

I will show how to associate a quaternion algebra to a hyperbolic 3-manifold. I will then go through some examples and applications of this theory

The coastal ocean contains a diversity of physical and biological

processes, often occurring at vastly different scales. In this talk,

we will outline some of these processes and their mathematical

description. We will then discuss how finite element methods are used

in coastal ocean modeling and recent research into

improvements to these algorithms. We will also highlight some of the

successes of these methods in simulating complex events, such as

hurricane storm surges. Finally, we will outline several interesting

challenges which are ripe for future research.

Consider in the phase space of classical mechanics a Radon measure that is a probability density carried by the graph of a Lipschitz continuous (or even less regular) vector field. We study the structure of the push-forward of such a measure by a Hamiltonian flow. In particular, we provide an estimate on the number of folds in the support of the transported measure that is the image of the initial graph by the flow. We also study in detail the type of singularities in the projection of the transported measure in configuration space (averaging out the momentum variable). We study the conditions under which this projected measure can have atoms, and give an example in which the projected measure is singular with respect to the Lebesgue measure and diffuse. We discuss applications of our results to the classical limit of the Schrödinger equation. Finally we present various examples and counterexamples showing that our results are sharp.

In this talk we study numerical approximations of continuous solutions to a nonlocal $p$-Laplacian type diffusion equation, \[ u_t (t, x) = \int_\Omega J(x − y)|u(t, y) − u(t, x)|^{p-2} (u(t, y) − u(t, x)) dy. \]

First, we find that a semidiscretization in space of this problem gives rise to an ODE system whose solutions converge uniformly to the continuous one, as the mesh size goes to zero. Moreover, the semidiscrete approximation shares some properties with the continuous problem: it preserves the total mass and the solution converges to the mean value of the initial condition, as $t$ goes to infinity.

Next, we discretize also the time variable and present a totally discrete method which also enjoys the above mentioned properties.

In addition, we investigate the limit as $p$ goes to infinity in these approximations and obtain a discrete model for the evolution of a sandpile.

Finally, we present some numerical experiments that illustrate our results.

This is a joint work with J. D. Rossi.

 We consider two decision problems for linear recurrence sequences (LRS) 
over the integers, namely the Positivity Problem (are all terms of a given 
LRS positive?) and the Ultimate Positivity Problem (are all but finitely 
many terms of a given LRS positive?). We show decidability of both 
problems for LRS of order 5 or less, and for simple LRS (i.e. whose 
characteristic polynomial has no repeated roots) of order 9 or less. Our 
results rely on on tools from Diophantine approximation, including Baker's 
Theorem on linear forms in logarithms of algebraic numbers. By way of 
hardness, we show that extending the decidability of either problem to LRS 
of order 6 would entail major breakthroughs on Diophantine approximation 
of transcendental numbers.

This is joint with work with Joel Ouaknine and Matt Daws.

The Einstein equation from general relativity is a

quasilinear hyperbolic, geometric PDE (when viewed in an appropriate

coordinate system) for a manifold. A particularly interesting set of

known, exact solutions describe black holes. The wave and Maxwell

equations on these manifolds are models for perturbations of the known

solutions and have attracted a significant amount of attention in the

last decade. Key estimates are conservation of energy and Morawetz (or

integrated local energy) estimates. These can be proved using both

Fourier analytic methods and more geometric methods. The main focus of

the talk will be on decay estimates for solutions of the Maxwell

equation outside a slowly rotating Kerr black hole.

I will talk about random walks on groups and define the Poisson boundary of such. Studying it gives criteria for amenability or growth. I will outline how this can be used and describe recent related results and still open questions.

Saturated fusion systems are both a convenient language in which to formulate p-local finite simple group theory and interesting structures in their own right. In this talk, we will start by explaining what is meant by a 'tree of fusion systems' and give conditions on such an object for there to exist a saturated completion. We then describe how this theory can be used to understand a class of fusion systems first considered by Bob Oliver, which are determined by modular representations of finite groups. If time permits, we will discuss joint work with David Craven towards a complete classification of such fusion systems. The talk is aimed at a general mathematical audience with some background in algebra.

Let g be the complex semisimple Lie algebra associated to a complex semisimple algebraic group G, b a Borel subalgebra of g, h the Cartan sublagebra contained in b and N the unipotent subgroup corresponding to the nilradical n of b. Extremal projection operators are projection operators onto the subspaces of n-invariants in certain g-modules the action of n on which is locally nilpotent. Zhelobenko also introduced a family of operators which are analogues to extremal projection operators. These operators are called now Zhelobenko operators.
I shall show that the explicit formula for the extremal projection operator for g obtained by Asherova, Smirnov and Tolstoy and similar formulas for Zhelobenko operators are related to the existence of a birational equivalence (N, h) -> b given by the restriction of the adjoint action. Simple geometric proofs of  formulas for the ``classical'' counterparts of the extremal projection operator and of Zhelobenko operators are also obtained.

Convergence of the Bayes posterior measure is considered in canonical statistical settings (like density estimation or nonparametric regression) where observations sit on a geometrical object such as a compact manifold, or more generally on a compact metric space verifying some conditions.

A natural geometric prior based on randomly rescaled solutions of the heat equation is considered. Upper and lower bound posterior contraction rates are derived.

A-infinity algebras arise whenever one has a multiplication which is "associative up to homotopy". There is an important theory of minimal models which involves studying differential graded algebras via A-infinity structures on their homology algebras. However, this only works well over a ground field. Recently Sagave introduced the more general notion of a derived A-infinity algebra in order to extend the theory of minimal models to a general commutative ground ring.

Operads provide a very nice way of saying what A-infinity algebras are - they are described by a kind of free resolution of a strictly associative structure. I will explain the analogous result for derived A_infinity algebras - these are obtained in the same manner from a strictly associative structure with an extra differential.

This is joint work with Muriel Livernet and Constanze Roitzheim.

Jump processes, and Lévy processes in particular, are notoriously difficult to simulate. The task becomes even harder if the process is stopped when it crosses a certain boundary, which happens in applications to barrier option pricing or structural credit risk models. In this talk, I will present novel adaptive discretization

schemes for the simulation of stopped Lévy processes, which are several orders of magnitude faster than the traditional approaches based on uniform discretization, and provide an explicit control of the bias. The schemes are based on sharp asymptotic estimates for the exit probability and work by recursively adding discretization dates in the parts of the trajectory which are close to the boundary, until a specified error tolerance is met.

This is a joint work with Jose Figueroa-Lopez (Purdue).

In an equity market with stable capital distribution, a capitalization-weighted index of small stocks tends to outperform a capitalization-weighted index of large stocks.} This is a somewhat careful statement of the so-called "size effect", which has been documented empirically and for which several explanations have been advanced over the years. We review the analysis of this phenomenon by Fernholz (2001) who showed that, in the presence of (a suitably defined) stability for the capital structure, this phenomenon can be attributed entirely to portfolio rebalancing effects, and will occur regardless of whether or not small stocks are riskier than their larger brethren. Collision local times play a critical role in this analysis, as they capture the turnover at the various ranks on the capitalization ladder.

We shall provide a rather complete study of this phenomenon in the context of a simple model with stable capital distribution, the so-called ``Atlas model" studied in Banner et al.(2005).

This is a Joint work with Adrian Banner, Robert Fernholz, Vasileios Papathanakos and Phillip Whitman.

Landslides are generally associated with a trigger, such as an earthquake, a rapid snowmelt or a large storm. The trigger event can generate a single landslide or many thousands. This paper examines: (i) The frequency-area statistics of several triggered landslide event inventories, which are characterized by a three-parameter inverse-gamma probability distribution (exponential for small landslide areas, power-law for medium and large areas). (ii) The use of proxies (newspapers) for compiling long-time series of landslide activity in a given region, done in the context of the Emilia-Romagna region, northern Italy. (iii) A stochastic model developed to evaluate the probability of landslides intersecting a simple road network during a landslide triggering event.

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.

Please see http://www.cs.ox.ac.uk/seminars/CompBioPublicSeminars/

In order to reduce cost, the MOD are attempting to reduce the number of array types fitted to their assets. There is also a requirement for the arrays to increase their frequency coverage. A wide bandwidth capability is thus needed from a single array. The need for high sensitivity and comparatively high frequencies of operation has led to the view that 1 3 composites are suitable hydrophones for this purpose. These hydrophones are used widely in ultra-sonics, but are not generally used down to the frequency of the new arrays.

Experimental work using a single hydrophone (small in terms of wavelengths) has shown that the sensitivity drops significantly as the frequency approaches the bottom of the required band, and then recovers as the frequency reduces further. Complex computer modelling appears to suggest the loss in sensitivity is due to a "lateral mode" where the hydrophone "breathes" in and out. In order to engineer a solution, the mechanics of the cause of this problem and the associated parameters of the materials need to be identified (e.g. is changing the 1 3 filler material the best option?). In order to achieve this understanding, a mathematical model of the 1 3 composite hydrophone (ceramic pegs and filler) is required that can be used to explain why the hydrophone changes from the simple compression and expansion in the direction of travel of the wave front to a lateral "breathing" mode.

More details available from @email

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

Non-trivial henselian valuations are often so closely related to the arithmetic of the underlying field that they are encoded in it, i.e., that their valuation ring is first-order definable in the language of rings. In this talk, we will give a complete classification of all henselian valued fields of residue characteristic 0 that allow a (0-)definable henselian valuation. This requires new tools from the model theory of ordered abelian groups (joint work with Franziska Jahnke).

The study of human mobility patterns can provide important information for city planning or predicting epidemic spreading, has recently been achieved with various methods available nowadays such as tracking banknotes, airline transportation, official migration data from governments, etc. However, the dearth of data makes it much more difficult to study human mobility patterns from the past. In the present study, we show that Korean family books (called "jokbo") can be used to estimate migration patterns for the past 500 years. We

apply two generative models of human mobility, which are conventional gravity-like models and radiation models, to quantify how relevant the geographical information is to human marriage records in the data. Based on the different migration distances of family names, we show the almost dichotomous distinction between "ergodic" (spread in the almost entire country) and (localized) "non-ergodic" family names, which is a characteristic of Korean family names in contrast to Czech family names. Moreover, the majority of family names are ergodic throughout the long history of Korea, suggesting that they are stable not only in terms of relative fractions but also geographically.

Solutions to translation invariant linear forms in dense sets  (for example: k-term arithmetic progressions), have been studied extensively in additive combinatorics and number theory. Finding solutions to translation invariant quadratic forms is a natural generalization and at the same time a simple instance of the hard general problem of solving diophantine equations in unstructured sets. In this talk I will explain how to modify the  classical circle method approach to obtain quantitative results  for quadratic forms with at least 17 variables.

FEAST is a new general purpose eigenvalue algorithm that takes its inspiration from the density-matrix representation and contour integration technique in quantum mechanics [Phys. Rev. B 79, 115112, (2009)], and it can be understood as a subspace iteration algorithm using approximate spectral projection [http://arxiv.org/abs/1302.0432 (2013)]. The algorithm combines simplicity and efficiency and offers many important capabilities for achieving high performance, robustness, accuracy, and multi-level parallelism on modern computing platforms. FEAST is also the name of a comprehensive numerical library package which currently (v2.1) focuses on solving the symmetric eigenvalue problems on both shared-memory architectures (i.e. FEAST-SMP version -- also integrated into Intel MKL since Feb 2013) and distributed architectures (i.e. FEAST-MPI version) including three levels of parallelism MPI-MPI-OpenMP.

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In this presentation, we aim at expanding the current capabilies of the FEAST eigenvalue algorithm and developing an unified numerical approach for solving linear, non-linear, symmetric and non-symmetric eigenvalue problems. The resulting algorithms retain many of the properties of the symmetric FEAST including the multi-level parallelism. It will also be outlined that the development strategy roadmap for FEAST is closely tied together with the needs to address the variety of eigenvalue problems arising in computational nanosciences. Consequently, FEAST will also be presented beyond the "black-box" solver as a fundamental modeling framework for electronic structure calculations.

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Three problems will be presented and discussed: (i) a highly efficient and robust FEAST-based alternative to traditional self-consistent field

(SCF) procedure for solving the non-linear eigenvector problem (J. Chem. Phys. 138, p194101 (2013)]); (ii) a fundamental and practical solution of the exact muffin-tin problem for performing both accurate and scalable all-electron electronic structure calculations using FEAST on parallel architectures [Comp. Phys. Comm. 183, p2370 (2012)]; (iii) a FEAST-spectral-based time-domain propagation techniques for performing real-time TDDFT simulations. In order to illustrate the efficiency of the FEAST framework, numerical examples are provided for various molecules and carbon-based materials using our in-house all-electron real-space FEM implementation and both the DFT/Kohn-Sham/LDA and TDDFT/ALDA approaches.

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.