Past

In the first part of the talk I will introduce a notion of convexity ($\mathcal{X}$-convexity) which applies to any given family of vector fields: the main model which we have in mind is the case of vector fields satisfying the H\"ormander condition.

Then I will give a PDE-characterization for $\mathcal{X}$-convex functions using a viscosity inequality for the intrinsic Hessian and I will derive bounds for the intrinsic gradient and intrinsic local Lipschitz-continuity for this class of functions.\\

In the second part of the talk I will introduce a notion of subdifferential for any given family of vector fields (namely $\mathcal{X}$-subdifferential) and show that a non empty $\mathcal{X}$-subdifferential at any point characterizes the class of $\mathcal{X}$-convex functions.

As application I will prove a Jensen-type inequality for $\mathcal{X}$-convex functions in the case of Carnot-type vector fields. {\em (Joint work with Martino Bardi)}.

More perspectives on spectra.

We present recent results of Dani Wise which tie together many of the themes of this term's jGGT meetings: hyperbolic and relatively hyperbolic groups, (in particular limit groups), graphs of spaces, 3-manifolds and right-angled Artin groups.
Following this, we make an attempt at explaining some of the methods, beginning with special non-positively curved cube complexes.

It is known that the minimum number of generators d(G^n) of the n-th direct power G^n of a non-trivial finite group G tends to infinity with n. This prompts the question: in which ways can the sequence {d(G^n)} tend to infinity? This question was first asked by Wiegold who almost completely answered it for finitely generated groups during the 70's. The question can then be generalised to any algebraic structure and this is still an open problem currently being researched. I will talk about some of the results obtained so far and will try to explain some of the methods used to obtain them, both for groups and for the more general algebraic structure setting.

Fractional differential equations are becoming increasingly used as a modelling tool for processes associated with anomalous diffusion or spatial heterogeneity. However, the presence of a fractional differential operator causes memory (time fractional) or nonlocality (space fractional) issues that impose a number of computational constraints. In this talk we discuss efficient, scalable techniques for solving fractional-in-space reaction diffusion equations combining the finite element method with robust techniques for computing the fractional power of a matrix times a vector. We shall demonstrate the methods on a number examples which show the qualitative difference in solution profiles between standard and fractional diffusion models.

In the talk I plan to overview several constructions for finite dimensional represenations of DAHA: construction via quantization of Hilbert scheme of points in the plane (after Gordon, Stafford), construction via quantum Hamiltonian reduction (after Gan, Ginzburg), monodromic construction (after Calaque, Enriquez, Etingof). I will discuss the relations of the constructions to the conjectures from the first lecture.

We shall discuss how the algebra norm can be used to identify structure in groups. No prior familiarity with the area will be assumed.

NB: EXTRA SEMINAR THIS WEEK

Executives compensated with stock options generally receive grants periodically and so on

any given date, may have a portfolio of options of differing strikes and maturities on their

company’s stock. Non-transferability and trading restrictions in the company stock result in the executive facing unhedgeable risk. We employ exponential utility indifference pricing to analyse the optimal exercise thresholds for each option, option values and cost of the options to shareholders. Portfolio interaction effects mean that each of these differ, depending on the composition of the remainder of the portfolio. In particular, the cost to shareholders of an option portfolio is lowered relative to its cost computed on a per-option basis. The model can explain a number of empirical observations - which options are attractive to exercise first, how exercise changes following a new grant, and early exercise.

Joint work with Jia Sun and Elizabeth Whalley (WBS).

I shall start with an idea (somewhat heuristic) that I call `Thermal Holography' and use that to probe the thermal behaviour of quantum horizons, i.e., without using any classical geometry, but using ordinary statistical mechanics with Gaussian fluctuations. This approach leads to a criterion for thermal stability for thermally active horizons with an Isolated horizon as an equilibrium configuration, whose (microcanonical) entropy has been computed using Loop Quantum Gravity (I shall outline this computation). As fiducial checks, we briefly look at some very well-known classical black hole metrics for their thermal stability and recover known results. Finally, I shall speculate about a possible link between our stability criterion and the Chandrasekhar upper bound for the mass of stable neutron stars.

This talk will begin by recalling classical facts about the relationship between values of the Riemann zeta function at negative integers and the arithmetic of cyclotomic extensions of the rational numbers. We will then consider a generalisation of this theory due to Iwasawa, and along the way we shall define the p-adic Riemann zeta function. Time permitting, I will also say something about what zeta values at positive integers have to do with the fundamental group of the projective line minus three points

The concordance group of classical knots C was introduced

over 50 years ago by Fox and Milnor. It is a much-studied and elusive

object which among other things has been a valuable testing ground for

various new topological (and smooth 4-dimensional) invariants. In

this talk I will address the problem of embedding C in a larger group

corresponding to the inclusion of knots in links.

I will report joint work with Wolfhard Hansen, Tomasz Jakubowski, Sebastian Sydor and Karol Szczypkowski on perturbations of semigroups and integral kernels, ones which produce comparable semigroups and integral kernels.

By intersecting a small three-dimensional sphere which surrounds a singular point of a planar curve, with the curve, one obtains a link in three-dimensional space. In my talk I explain a conjectural formula for the  ranks Khovanov-Rozansky homology of the link which interpretsthe ranks in terms of topology of some natural stratification on the moduli space of torsion free sheaves on the curve. In particular I will present  a formula for the ranks of the Khovanov-Rozansky homology of the torus knots which generalizes Jones formula for HOMFLY invariants of the torus knots.  The later formula relates Khovanov-Rozansky homology to the represenation theory of Double Affine Hecke Algebras. The talk presents joint work with Gorsky, Shende and  Rasmussen.

Several stochastic simulation algorithms (SSAs) have been recently

proposed for modelling reaction-diffusion processes in cellular and molecular biology. In this talk, two commonly used SSAs will  be studied. The first SSA is an on-lattice model described by the  reaction-diffusion master equation. The second SSA is an off-lattice model based on the simulation of Brownian motion of individual  molecules and their reactive collisions. The connections between SSAs  and the deterministic models (based on reaction- diffusion PDEs) will  be presented. I will consider chemical reactions both at a surface  and in the bulk. I will show how the "microscopic" parameters should  be chosen to achieve the correct "macroscopic" reaction rate. This  choice is found to depend on which SSA is used. I will also present  multiscale algorithms which use models with a different level of  detail in different parts of the computational domain

In just the last year it has been realized that one can define supersymmetric gauge theories on non-trivial compact curved manifolds, coupled to a background R-symmetry gauge field, and moreover that expectation values of certain BPS operators reduce to finite matrix integrals via a form of localization. I will argue that a general approach to this topic is provided by the gauge/gravity correspondence. In particular, I will present several examples of supersymmetric gauge theories on different 1-parameter deformations of the three-sphere, which have a large N limit, together with their gravity duals (which are solutions to Einstein-Maxwell theory). The Euclidean gravitational partition function precisely matches a large N matrix model evaluation of the field theory partition function, as an exact \emph{function} of the deformation parameter.

 A common way to replace body tissue is via donors, but as the world population is ageing at an unprecedented rate there will be an even smaller supply to demand ratio for replacement parts than currently exists. Tissue engineering is a process in which damaged body tissue is repaired or replaced via the engineering of artificial tissues. We consider one type of this; a two-phase flow through a rotating high-aspect ratio vessel (HARV) bioreactor that contains a porous tissue construct. We extend the work of Cummings and Waters [2007], who considered a solid tissue construct, by considering flow through the porous construct described by a rotating form of Darcy's equations. By simplifying the equations and changing to bipolar variables, we can produce analytic results for the fluid flow through the system for a given construct trajectory. It is possible to calculate the trajectory numerically and couple this with the fluid flow to produce a full description of the flow behaviour. Finally, coupling with the numerical result for the tissue trajectory, we can also analytically calculate the particle paths for the flow which will lead to being able to calculate the spatial and temporal nutrient density.

Rising sea levels are frequently cited as one of the most pressing societal consequences of climate change. In order to understand the present day change in sea level we need to place it in the context of historical changes. The primary source of information on sea level change over the past 100-150 years is tide gauges. However, these tide gauges are a globally sparse set of point measurements located largely at the coast. "Global mean sea level" calculated from these tide gauges is therefore biased and is also more variable than than global mean sea level calculated from the past 19 years of satellite altimtery measurements.

The work presented here explores the use of simple statistical approaches which make use of reanalysis wind stress datasets and heat content reconstructions to model the sea level records. It is shown that these simple models have skill in reproducing variability at decadal time-scales. The results suggest that there are active regions of wind stress and heat content in the ocean which affect regional variability in sea level records that point to the atmospheric and oceanic processes which drive the variability. Acceleration seen in the longest continous sea level record at Brest is shown to be partially attributable to changes in wind stress over the past 140 years.

We give an exposition of some results from matroid theory which characterise the finite pregeometries arising from Hrushovski's predimension construction as the strict gammoids: a class of matroids studied in the early 1970's which arise from directed graphs. As a corollary, we observe that a finite pregeometry which satisfies Hrushovski's flatness condition arises from a predimension. We also discuss the isomorphism types of the pregeometries of countable, saturated strongly minimal structures in Hrushovski's 1993 paper and answer some open questions from there. This last part is joint work with Marco Ferreira, and extends results in his UEA PhD thesis.

In these two talks we will look at a recent paper by David Anderson and Des Higham: http://arxiv.org/pdf/1107.2181 This paper takes the Multilevel Monte Carlo method which I developed in 2006 for Brownian SDEs, and comes up with an elegant way of applying it to stochastic biochemical reaction networks.

change to previous speaker

• Review of the basic notions concerning bifurcation and asymptotic linearity.

• Review of differentiability in the sense of Gˆateaux, Fréchet, Hadamard.

• Examples which are Hadamard but not Fréchet differentiable.  The Dirichlet problem for a degenerate elliptic equation on a bounded domain. The stationary nonlinear Schrödinger equation on RN

Variational data assimilation techniques for optimal state estimation in very large environmental systems currently use approximate Gauss-Newton (GN) methods. The GN method solves a sequence of linear least squares problems subject to linearized system constraints. For very large systems, low resolution linear approximations to the model dynamics are used to improve the efficiency of the algorithm. We propose a new approach for deriving low order system approximations based on model reduction techniques from control theory which can be applied to unstable stochastic systems. We show how this technique can be combined with the GN method to retain the response of the dynamical system more accurately and improve the performance of the approximate GN method.

We consider a portfolio optimisation problem on infinite horizon when

the investment policy satisfies the drawdown constraint, which is the

wealth process of an investor is always above a threshold given as a

function of the past maximum of the wealth process. The preferences are

given by a utility function and investor aims to maximise an asymptotic

growth rate of her expected utility of wealth. This problem was firstly

considered by Grossman and Zhou [3] and solved for a Black-Scholes

market and linear drawdown constraint.

The main contribution of the paper is an equivalence result: the

constrained problem with utility U and drawdown function w has the same

value function as the unconstrained problem with utility UoF, where

function F is given explicitly in terms of w. This work was inspired by

ideas from [2], whose results are a special case of our work. We show

that the connection between constrained and unconstrained problems holds

for a much more general setup than their paper, i.e. a general

semimartingale market, larger class of utility functions and drawdown

function which is not necessarily linear. The paper greatly simplifies

previous approaches using the tools of Azema-Yor processes developed in

[1]. In fact we show that the optimal wealth process for constrained

problem can be found as an explicit Azema-Yor transformation of the

optimal wealth process for the unconstrained problem.

We further provide examples with explicit solution for complete and

incomplete markets.

[1] Carraro, L., Karoui, N. E., and Obloj, J. On Azema-Yor processes,

their optimal properties and the Bachelier-Drawdown equation, to appear in

Annals of Probability, 2011.

[2] Cvitanic, J., and Karatzas, I. On portfolio optimization under

drawdown constraints. IMA Volumes in Mathematics and Its Applications

65(3), 1994, 35-45

[3] Grossman, S. J., and Zhou, Z. Optimal investment strategies for

controlling drawdowns. Mathematical Finance 3(3), 1993, 241-276

An overview is given of some key issues and definitions in the Calculus of Variations, with a focus on lower semicontinuity and quasiconvexity. Some well known results and instructive counterexamples are also discussed. We then move to consider variational problems in the BV setting, and present a new lower semicontinuity result for quasiconvex integrals of subquadratic growth. The proof of this requires some interesting techniques, such as obtaining boundedness properties for an extension operator, and exploiting fine properties of Sobolev maps.

This is the first in a series of $\geq 2$ talks about Stable Homotopy Theory. We will motivate the definition of spectra by the Brown Representability Theorem, which allows us to interpret a spectrum as a generalized cohomology theory. Along the way we recall basic notions from homotopy theory, such as suspension, loop spaces and smash products.

The self-consistent analytic theory of the wind-driven sea can be developed due to the presence of small parameter, ratio of atmospheric and water densities. Because of low value of this parameter the sea is "weakly nonlinear" and the average steepness of sea surface is also relatively small. Nevertheless, the weakly nonlinear four-wave resonant interaction is the dominating process in the energy balance. The wind-driven sea can be described statistically in terms of the Hasselmann kinetic equation.

This equation has a rich family of Kolmogorov-type solutions perfectly describing "rear faces" of wave spectra right behind the spectral peak.

More short waves are described by steeper Phillips spectrum formed by ensemble of microbreakings. From the practical view-point the most important question is the spatial and temporal evolution of spectral peaks governed by self-similar solutions of the Hasselmann equation. This analytic theory is supported by numerous experimental data and computer

simulations.   

A dimer model on a surface with punctures is an embedded quiver such that its vertices correspond to the punctures and the arrows circle round the faces they cut out. To any dimer model Q we can associate two categories: A wrapped Fukaya category F(Q), and a category of matrix factorizations M(Q). In both categories the objects are arrows, which are interpreted as Lagrangian subvarieties in F(Q) and will give us certain matrix factorizations of a potential on the Jacobi algebra of the dimer in M(Q).

We show that there is a duality D on the set of all dimers such that for consistent dimers the category of matrix factorizations M(Q) is isomorphic to the Fukaya category of its dual,  F((DQ)). We also discuss the connection with classical mirror symmetry.

We say that a hypergraph is \emph{stable} if each sufficiently large subset of its vertices either spans many hyperedges or is very structured. Hypergraphs that arise naturally in many classical settings posses the above property. For example, the famous stability theorem of Erdos and Simonovits and the triangle removal lemma of Ruzsa and Szemeredi imply that the hypergraph on the vertex set $E(K_n)$ whose hyperedges are the edge sets of all triangles in $K_n$ is stable. In the talk, we will present the following general theorem: If $(H_n)_n$ is a sequence of stable hypergraphs satisfying certain technical conditions, then a typical (i.e., uniform random) $m$-element independent set of $H_n$ is very structured, provided that $m$ is sufficiently large. The above abstract theorem has many interesting corollaries, some of which we will discuss. Among other things, it implies sharp bounds on the number of sum-free sets in a large class of finite Abelian groups and gives an alternate proof of Szemeredi’s theorem on arithmetic progressions in random subsets of integers.

Joint work with Noga Alon, Jozsef Balogh, and Robert Morris.

The Market Selection Hypothesis is a principle which (informally) proposes that `less knowledgeable' agents are eventually eliminated from the market. This elimination may take the form of starvation (the proportion of output consumed drops to zero), or may take the form of going broke (the proportion of asset held drops to zero), and these are not the same thing. Starvation may result from several causes, diverse beliefs being only one.We firstly identify and exclude these other possible causes, and then

prove that starvation is equivalent to inferior belief, under suitable technical conditions. On the other hand, going broke cannot be characterized solely in terms of beliefs, as we show. We next present a remarkable example with two agents with different beliefs, in which one agent starves yet amasses all the capital, and the other goes broke yet consumes all the output -- the hungry miser and the happy bankrupt.

This example also serves to show that although an agent may starve, he may have long-term impact on the prices. This relates to the notion of price impact introduced by Kogan et al (2009), which we correct in the final section, and then use to characterize situations where asymptotically equivalent

pricing holds.

Attractive features of Lifshitz type theories are described with different

examples,

as the improvement of graphs convergence, the introduction of new

renormalizable

interactions, dynamical mass generation, asymptotic freedom, and other

features

related to more specific models. On the other hand, problems with the

expected

emergence of Lorentz symmetry in the IR are discussed, related to the

different

effective light cones seen by different particles when they interact.

We describe several bifurcation properties corresponding to various classes of nonlinear elliptic equations The purpose of this talk is two-fold. First, it points out different competition effects between the terms involved in the equations. Second, it provides several non standard phenomena that occur according to the structure of the differential operator.

A longstanding question in geometric group theory is the following. Suppose G is a hyperbolic group where all finitely generated subgroups of infinite index are free. Is G the fundamental group of a surface? This question is still open for some otherwise well understood classes of groups. In this talk, I will explain why the answer is affirmative for graphs of free groups with cyclic edge groups. I will also discuss the extent to which these techniques help with the harder problem of finding surface subgroups.