Past

The second order sensitivity of a trading position, the so

called gamma, has a very real and intuitive meaning to the traders.

People think that convex payoffs must generate convex prices. Being long

or short of gamma is a strategy used to balance risks in options books.

While the simples models, like Black Scholes, are consistent with this

intuition other popular models used in the industry are not. I will give

examples of simple and popular models which do not always convert a

convex payoff into a convex price. I will also give the necessary and

sufficient conditions under which the convexity is propagated.

The new schedule will follow shortly

First of all, I will give an overview of what the phenomenon of homological stability is and why it's useful, with plenty of examples. I will then introduce configuration spaces -- of various different kinds -- and give an overview of what is known about their homological stability properties. A "configuration" here can be more than just a finite collection of points in a background space: in particular, the points may be equipped with a certain non-local structure (an "orientation"), or one can consider unlinked embedded copies of a fixed manifold instead of just points. If by some miracle time permits, I may also say something about homological stability with local coefficients, in general and in particular for configuration spaces.

 We consider the emergence of classical correlations in macroscopic quantum systems, and its connection to monogamy relations for violation of Bell-type inequalities. We work within the framework of Abramsky and Brandenburger [1], which provides a unified treatment of non-locality and contextuality in the general setting of no-signalling empirical models. General measurement scenarios are represented by simplicial complexes that capture the notion of compatibility of measurements. Monogamy and locality/noncontextuality of macroscopic correlations are revealed by our analysis as two sides of the same coin: macroscopic correlations are obtained by averaging along a symmetry (group action) on the simplicial complex of measurements, while monogamy relations are exactly the inequalities that are invariant with respect to that symmetry. Our results exhibit a structural reason for monogamy relations and consequently for the classicality of macroscopic correlations in the case of multipartite scenarios, shedding light on and generalising the results in [2,3].More specifically, we show that, however entangled the microscopic state of the system, and provided the number of particles in each site is large enough (with respect to the number of allowed measurements), only classical (local realistic) correlations will be observed macroscopically. The result depends only on the compatibility structure of the measurements (the simplicial complex), hence it applies generally to any no-signalling empirical model. The macroscopic correlations can be defined on the quotient of the simplicial complex by the symmetry that lumps together like microscopic measurements into macroscopic measurements. Given enough microscopic particles, the resulting complex satisfies a structural condition due to Vorob'ev [4] that is necessary and sufficient for any probabilistic model to be classical.  The generality of our scheme suggests a number of promising directions. In particular, they can be applied in more general scenarios to yield monogamy relations for contextuality inequalities and to study macroscopic non-contextuality.

[1] Samson Abramsky and Adam Brandenburger, The sheaf-theoretic structure of non-locality and contextuality, New Journal of Physics 13 (2011), no. 113036.
[2] MarcinPawłowski and Caslav Brukner, Monogamy of Bell’s inequality violations in nonsignaling theories, Phys. Rev. Lett. 102 (2009), no. 3, 030403.
[3] R. Ramanathan, T. Paterek, A. Kay, P. Kurzynski, and D. Kaszlikowski, Local realism of macroscopic correlations, Phys. Rev. Lett. 107 (2011), no. 6, 060405.
[4] N.N.Vorob’ev, Consistent families of measures and their extensions, Theory of Probability and its Applications VII (1962), no. 2, 147–163, (translated by N. Greenleaf, Russian original published in Teoriya Veroyatnostei i ee Primeneniya).

Several recent works by D. Tamarkin, D. Nadler, E. Zaslow make use of the microlocal theory of sheaves of M. Kashiwara and P. Schapira to obtain results in symplectic geometry. The link between sheaves on a manifold $M$ and the symplectic geometry of the cotangent bundle of $M$ is given by the microsupport of a sheaf, which is a conic co-isotropic subset of the cotangent bundle. In the above mentioned works properties of a given Lagrangian submanifold $\Lambda$ are deduced from the existence of a sheaf with microsupport $\Lambda$, which we call a quantization of $\Lambda$.

In the second talk we will introduce a stack on $\Lambda$ by localization of the category of sheaves on $M$. We deduce topological obstructions on $\Lambda$ for the existence of a quantization.

A number is called transcendental if it is not algebraic, that is it does not satisfy a polynomial equation with rational coefficients. It is easy to see that the algebraic numbers are countable, hence the transcendental numbers are uncountable. Despite this fact, it turns out to be very difficult to determine whether a given number is transcendental. In this talk I will discuss some famous examples and the theorems which allow one to construct many different transcendental numbers. I will also give an outline of some of the many open problems in the field.

I propose to present modeling and experimental data about the organization of telomeres (ends of the chromosomes): the 32 telomeres in Yeast form few local aggregates. We built a model of diffusion-aggregation-dissociation for a finite number of particles to estimate the number of cluster and the number of telomere/cluster and other quantities. We will further present based on eingenvalue expansion for the Fokker-Planck operator, asymptotic estimation for the mean time a piece of a polymer (DNA) finds a small target in the nucleus.

We discuss the problem to what extend a group action determines geometry of the space. 
More precisely, we show that for a large class of actions measurable isomorphisms must preserve 
the geometric structure as well. This is a joint work with Bader, Furman, and Weiss.

Several recent works by D. Tamarkin, D. Nadler, E. Zaslow make use of the microlocal theory of sheaves of M. Kashiwara and P. Schapira to obtain results in symplectic geometry. The link between sheaves on a manifold $M$ and the symplectic geometry of the cotangent bundle of $M$ is given by the microsupport of a sheaf, which is a conic co-isotropic subset of the cotangent bundle. In the above mentioned works properties of a given Lagrangian submanifold $\Lambda$ are deduced from the existence of a sheaf with microsupport $\Lambda$, which we call a quantization of $\Lambda$. In the first talk we will see that the graph of a Hamiltonian isotopy admits a canonical quantization and we deduce a new proof of Arnold's non-displaceability conjecture.

Several recent works by D. Tamarkin, D. Nadler, E. Zaslow make use of the microlocal theory of sheaves of M. Kashiwara and P. Schapira to obtain results in symplectic geometry. The link between sheaves on a manifold $M$ and the symplectic geometry of the cotangent bundle of $M$ is given by the microsupport of a sheaf, which is a conic co-isotropic subset of the cotangent bundle. In the above mentioned works properties of a given Lagrangian submanifold $\Lambda$ are deduced from the existence of a sheaf with microsupport $\Lambda$, which we call a quantization of $\Lambda$.

In the first talk we will see that the graph of a Hamiltonian isotopy admits a canonical quantization and we deduce a new proof of Arnold's non-displaceability conjecture.

I will describe how a search for the answer to an old question about the existence of monotone arithmetic progressions in permutations of positive integers led to the study of infinite words with bounded additive complexity. The additive complexity of a word on a finite subset of integers is defined as the function that, for a positive integer $n$, counts the maximum number of factors of length $n$, no two of which have the same sum.

Defect measures have successfully been used, in a variety of

contexts, as a tool to quantify the lack of compactness of bounded

sequences of square-integrable functions due to concentration and

oscillation effects. In this talk we shall present some results on the

structure of the set of possible defect measures arising from sequences

of solutions to the linear Schrödinger equation on a compact manifold.

This is motivated by questions related to understanding the effect of

geometry on dynamical aspects of the Schrödinger flow, such as

dispersive effects and unique continuation.

It turns out that the answer to these questions depends strongly on

global properties of the geodesic flow on the manifold under

consideration: this will be illustrated by discussing with a certain

detail the examples of the the sphere and the (flat) torus.

In 2005, Bonk and Kleiner showed that a hyperbolic group admits a

quasi-isometrically embedded copy of the hyperbolic plane if and only if the

group is not virtually free. This answered a question of Papasoglu. I will

discuss a generalisation of their result to certain relatively hyperbolic

groups (joint work with Alessandro Sisto). Key tools involved are new

existence results for quasi-circles, and a better understanding of the

geometry of boundaries of relatively hyperbolic groups.

(Joint work with P.E. Chaudru de Raynal and F. Delarue)

Several problems in financial mathematics, in particular that of contingent claim pricing, may be casted as decoupled Forward Backward Stochastic Differential Equations (FBSDEs). It is then of a practical interest to find efficient algorithms to solve these equations numerically with a reasonable complexity.

An efficient numerical approach using cubature on Wiener spaces was introduced by Crisan and Manoralakis [1]. The algorithm uses an approximation scheme requiring the calculation of conditional expectations, a task achieved through the cubature kernel approximation. This algorithm features several advantages, notably the fact that it is possible to solve with the same cubature tree several decoupled FBSDEs with different boundary conditions. There is, however, a drawback of this method: an exponential growth of the algorithm's complexity.

In this talk, we introduce a modification on the cubature method based on the recombination method of Litterer and Lyons [2] (as an alternative to Tree Branch Based Algorithm proposed in [1]). The main idea of the method is to modify the nodes and edges of the cubature trees in such a way as to preserve, up to a constant, the order of convergence of the expectation and conditional expectation approximations obtained via the cubature method, while at the same time controlling the complexity growth of the algorithm.

We have obtained estimations on the order of convergence and complexity growth of the algorithm under smoothness assumptions on the coefficients of the FBSDE and uniform ellipticity of the forward equation. We discuss that, just as in the case of the plain cubature method, the order of convergence of the algorithm may be degraded as an effect of solving FBSDEs with rougher boundary conditions. Finally, we illustrate the obtained estimations with some numerical tests.

References

[1] Crisan, D., and K. Manolarakis. “Solving Backward Stochastic Differential Equations Using the Cubature Method. Application to Nonlinear Pricing.” In Progress in Analysis and Its Applications, 389–397. World Sci. Publ., Hackensack, NJ, 2010.

[2] Litterer, C., and T. Lyons. “High Order Recombination and an Application to Cubature on Wiener Space.” The Annals of Applied Probability 22, no. 4 (August 2012): 1301–1327. doi:10.1214/11-AAP786.

In this talk I will consider the Burgers equation with a homogeneous Possion process as a forcing potential. In recent years, the randomly forced Burgers equation, with forcing that is ergodic in time, received a lot of attention, especially the almost sure existence of unique global solutions with given average velocity, that at each time only depend on the history up to that time. However, in all these results compactness in the space dimension of the forcing was essential. It was even conjectured that in the non-compact setting such unique global solutions would not exist. However, we have managed to use techniques developed for first and last passage percolation models to prove that in the case of Poisson forcing, these global solutions do exist almost surely, due to the existence of semi-infinite minimizers of the Lagrangian action. In this talk I will discuss this result and explain some of the techniques we have used.

This is joined work Yuri Bakhtin and Konstantin Khanin.

We discuss sparse portfolio optimization in continuous time.

Optimization objective is to maximize an expected utility as in the

classical Merton problem but with regularizing sparsity constraints.

Such constraints aim for asset allocations that contain only few assets or

that deviate only in few coordinates from a reference benchmark allocation.

With a focus on growth optimization, we show empirical results for various

portfolio selection strategies with and without sparsity constraints,

investigating different portfolios of stock indicies, several performance

measures and adaptive methods to select the regularization parameter.

Sparse optimal portfolios are less sensitive to estimation

errors and performance is superior to portfolios without sparsity

constraints in reality, where estimation risk and model uncertainty must

not be ignored.

We are interested in finding the Probability Density Function (PDF) of high dimensional chaotic systems such as a global atmospheric circulation model. The key difficulty stems from the so called “curse of dimensionality”. Representing anything numerically in a high dimensional space seems to be just too computationally expensive. Methods applied to dodge this problem include representing the PDF analytically or applying a (typically linear) transformation to a low dimensional space. For chaotic systems these approaches often seem extremely ad-hoc with the main motivation being that we don't know what else to do.

The Lorenz 95 system is one of the simplest systems we could come up with that is both chaotic and high dimensional. So it seems like a good candidate for initial investigation. We look at two attempts to approximate the PDF of this system to an arbitrary level of accuracy, firstly using a simple Monte-Carlo method and secondly using the Fokker-Planck equation. We also describe some of the (sometimes surprising) difficulties encountered along the way.

• Jean-Charles Seguis - Simulation in chemotaxis and comparison of cell models
• Laura Kimpton (née Gallimore) - A viscoelastic two-phase flow model of a crawling cell
• Benjamin Franz - Particles and PDEs and robots

The Outer Model Programme investigates L-like forcing  extensions of the universe, where we say that a model of Set Theory  is L-like if it satisfies properties of Goedel's constructible universe of sets L. I will introduce the Outer Model Programme, talk  about its history, motivations, recent results and applications. I  will be presenting joint work with Sy Friedman and Philipp Luecke.

I discuss models for the planar spreading of a viscous fluid between an elastic lid and an underlying rigid plane. These have application to the growth of magmatic intrusions, as well as to other industrial and biological processes; simple experiments of an inflated blister will be used for motivation. The height of the fluid layer is described by a sixth order non-linear diffusion equation, analogous to the fourth order equation that describes surface tension driven spreading. The dynamics depend sensitively on the conditions at the contact line, where the sheet is lifted from the substrate and where some form of regularization must be applied to the model. I will explore solutions with a pre-wetted film or a constant-pressure fluid lag, for flat and inclined planes, and compare with the analogous surface tension problems.

Toby Gee and I have proposed the definition of a "C-group", an extension of Langlands' notion of an L-group, and argue that for an arithmetic version of Langlands' philosophy such a notion is useful for controlling twists properly. I will give an introduction to this business, and some motivation. I'll start at the beginning by explaining what an L-group is a la Langlands, but if anyone is interested in doing some background preparation for the talk, they might want to find out for themselves what an L-group (a Langlands dual group) is e.g. by looking it up on Wikipedia!

This talk is a basic introduction to the wonderful world of Higgs bundles on a Riemann Surface, and their moduli space. We will only survey the basics of the theory focusing on the rich geometry of the moduli space of Higgs bundles, and the relation to moduli space of vector bundles. In the end we consider small applications of Higgs bundles. As this talk will be very basic we won't go into any new developments of the theory, but just mention the areas in which Higgs bundles are used today.

Subtitle:

And applications to problems involving pointwise constraints on the gradient of the state on non smooth polygonal domains

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In this talk we are concerned with an analysis of Moreau-Yosida regularization of pointwise state constrained optimal control problems. As recent analysis has already revealed the convergence of the primal variables is dominated by the reduction of the feasibility violation in the maximum norm.

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We will use a new method to derive convergence of the feasibility violation in the maximum norm giving improved the known convergence rates.

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Finally we will employ these techniques to analyze optimal control problems governed by elliptic PDEs with pointwise constraints on the gradient of the state on non smooth polygonal domains. For these problems, standard analysis, fails because the control to state mapping does not yield sufficient regularity for the states to be continuously differentiable on the closure of the domain. Nonetheless, these problems are well posed. In particular, the results of the first part will be used to derive convergence rates for the primal variables of the regularized problem.

We consider a problem of maximising lifetime utility of consumption subject to a drawdown constraint on undiscounted wealth

process. This problem was solved by Elie and Touzi in the case of zero interest rate. We apply methodology of Azema-Yor processes to connect

constrained and unconstrained wealth processes, which allows us to get the results for non-zero interest rate.

The theory of equations over groups goes back to the very beginning of group theory and is linked to many deep problems in mathematics, such as the Diophantine problem over rationals. In this talk, we shall survey some of the key results on equations over groups, give an outline of the Makanin-Razborov process (an algorithm for solving equations over free groups) and its connections to other results in group theory and low-dimensional topology.

We will present a regularity result for degenerate elliptic equations in nondivergence form.

In joint work with Charlie Smart, we extend the regularity theory of Caffarelli to equations with possibly unbounded ellipticity-- provided that the ellipticity satisfies an averaging condition. As an application we obtain a stochastic homogenization result for such equations (and a new estimate for the effective coefficients) as well as an invariance principle for random diffusions in random environments. The degenerate equations homogenize to uniformly elliptic equations, and we give an estimate of the ellipticity.

In the Greek mythology the hydra is a many-headed poisonous beast. When cutting one of its heads off, it will grow two more. Inspired by how Hercules defeated the hydra, Dison and Riley constructed a family of groups defined by two generators and one relator, which is an Engel  word: the hydra groups. I will talk about its remarkably wild subgroup distortion and its hyperbolic cousin. Very recent discussions of Baumslag and Mikhailov show that those groups are residually torsion-free nilpotent and they introduce generalised hydra groups.

When two drops come into contact they will rapidly merge and form a single drop. Here we address the coalescence of drops on a substrate, focussing on the initial dynamics just after contact. For very viscous drops we present similarity solutions for the bridge that connects the two drops, the size of which grows linearly with time. Both the dynamics and the self-similar bridge profiles are verified quantitatively by experiments. We then consider the coalescence of water drops, for which viscosity can be neglected and liquid inertia takes over. Once again, we find that experiments display a self-similar dynamics, but now the bridge size grows with a power-law $t^{2/3}$. We provide a scaling theory for this behavior, based on geometric arguments. The main result for both viscous and inertial drops is that the contact angle is important as it determines the geometry of coalescence -- yet, the contact line dynamics appears irrelevant for the early stages of coalescence.

A Lagrangian submanifold of a Calabi-Yau manifold is called positive if the real part of the holomorphic volume form restricted to it is positive. A Hamiltonian isotopy class of positive Lagrangian submanifolds admits a Riemannian metric with non-positive curvature. Its universal cover

admits a functional, with critical points special Lagrangians, that is strictly convex with respect to the metric. If time permits, I'll explain

how mirror symmetry relates the metric and functional to the infinite dimensional symplectic reduction picture of Atiyah, Bott, and Donaldson in

the context of the Kobayashi-Hitchin correspondence.

This talk will give an introduction to the recent paper by Arkani Hamed et. al. arxiv:1212:5605.  

Results of Bourgain and Kindler-Safra state that if $f$ is a Boolean function on $\{0,1\}^n$, and

the Fourier transform of $f$ is highly concentrated on low frequencies, then $f$ must be close

to a ‘junta’ (a function depending upon a small number of coordinates). This phenomenon is

known as ‘Fourier stability’, and has several interesting consequences in combinatorics,

theoretical computer science and social choice theory. We will describe some of these,

before turning to the analogous question for Boolean functions on the symmetric group. Here,

genuine stability does not occur; it is replaced by a weaker phenomenon, which we call

‘quasi-stability’. We use our 'quasi-stability' result to prove an isoperimetric inequality

for $S_n$ which is sharp for sets of size $(n-t)!$, when $n$ is large. Several open questions

remain. Joint work with Yuval Filmus (University of Toronto) and Ehud Friedgut (Weizmann

Institute).

Let ${T_1,...,T_l}$ be a collection of differential operators

with constant coefficients on the torus $\mathbb{T}^n$. Consider the

Banach space $X$ of functions $f$ on the torus for which all functions

$T_j f$, $j=1,...,l$, are continuous. The embeddability of $X$ into some

space $C(K)$ as a complemented subspace will be discussed. The main result

is as follows. Fix some pattern of mixed homogeneity and extract the

senior homogeneous parts (relative to the pattern chosen)

${\tau_1,...,\tau_l}$ from the initial operators ${T_1,...,T_l}$. If there

are two nonproportional operators among the $\tau_j$ (for at least one

homogeneity pattern), then $X$ is not isomorphic to a complemented

subspace of $C(K)$ for any compact space $K$.

The main ingredient of the proof is a new Sobolev-type embedding

theorem. It generalises the classical embedding of

${\stackrel{\circ}{W}}_1^1(\mathbb{R}^2)$ to $L^2(\mathbb{R}^2)$. The difference is that

now the integrability condition is imposed on certain linear combinations

of derivatives of different order of several functions rather than on the

first order derivatives of one function.

This is a joint work with D. Maksimov and D. Stolyarov.

Abstract: When a strict local martingale is projected onto a subfiltration to which it is not adapted, the local martingale property may be lost, and the finite variation part of the projection may have singular paths. This phenomenon has consequences for arbitrage theory in mathematical finance. In this paper it is shown that the loss of the local martingale property is related to a measure extension problem for the associated Föllmer measure. When a solution exists, the finite variation part of the projection can be interpreted as the compensator, under the extended measure, of the explosion time of the original local martingale. In a topological setting, this leads to intuitive conditions under which its paths are singular. The measure extension problem is then solved in a Brownian framework, allowing an explicit treatment of several interesting examples.

We study portfolio optimization and contingent claim valuation in markets where illiquidity may affect the transfer of wealth over time and between investment classes. In addition to classical frictionless markets and markets with transaction costs, our model covers nonlinear illiquidity effects that arise in limit order markets. We extend basic results on arbitrage bounds, attainable claims and optimal portfolios to illiquid markets and general swap contracts where both claims and premiums may have multiple payout dates. We establish the existence of optimal trading strategies and the lower semicontinuity of the optimal value of portfolio optimization under conditions that extend the no-arbitrage condition in the classical linear market model.

The operation of sub-cellular processes in living organisms often require the transfer of biopolymers across impermeable lipid membranes. The emergence of new experimental techniques for manipulation of single molecules at nanometer scales have made possible in vitro experiments that can directly probe such translocation processes in cells as well as in synthetic systems. Some of these ideas have spawned novel bio-technologies with many more likely to emerge in the near future. In this talk I would review some of these experiments and attempt to provide a quantitative understanding of the data in terms of physical laws, primarily mechanics and electrostatics.