Past

The periodic orbits of a discrete dynamical system can be described as

permutations. We derive the composition law for such permutations. When

the composition law is given in matrix form the composition of

different periodic orbits becomes remarkably simple. Composition of

orbits in bifurcation diagrams and decomposition law of composed orbits

follow directly from that matrix representation.

This talk surveys the well known relationship between half-flat SU(3) structures on 6-manifolds M and metrics with holonomy in G_2 on Mx(a,b), focusing on the case in which M=S3xS3 with solutions invariant by SO(4).

We introduce a deformation process of universal enveloping algebras of Borcherds-Kac-Moody algebras, which generalises quantum groups' one and yields a large class of new algebras called coloured Borcherds-Kac-Moody algebras. The direction of deformation is specified by the choice of a collection of numbers. For example, the natural numbers lead to classical enveloping algebras, while the quantum numbers lead to quantum groups. We prove, in the finite type case, that every coloured BKM algebra have representations which deform representations of semisimple Lie algebras and whose characters are given by the Weyl formula. We prove, in the finite type case, that representations of two isogenic coloured BKM algebras can be interpolated by representations of a third coloured BKM algebra. In particular, we solve conjectures of Frenkel-Hernandez about the Langland duality between representations of quantum groups. We also establish a Langlands duality between representations of classical BKM algebras, extending results of Littelmann and McGerty, and we interpret this duality in terms of quantum interpolation.

We describe our current efforts to develop finite volume

schemes for solving PDEs on logically Cartesian locally adapted

surfaces meshes. Our methods require an underlying smooth or

piecewise smooth grid transformation from a Cartesian computational

space to 3d surface meshes, but does not rely on analytic metric terms

to obtain second order accuracy. Our hyperbolic solvers are based on

Clawpack (R. J. LeVeque) and the parabolic solvers are based on a

diamond-cell approach (Y. Coudi\`ere, T. Gallou\"et, R. Herbin et

al). If time permits, I will also discuss Discrete Duality Finite

Volume methods for solving elliptic PDEs on surfaces.

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To do local adaption and time subcycling in regions requiring high

spatial resolution, we are developing ForestClaw, a hybrid adaptive

mesh refinement (AMR) code in which non-overlapping fixed-size

Cartesian grids are stored as leaves in a forest of quad- or

oct-trees. The tree-based code p4est (C. Burstedde) manages the

multi-block connectivity and is highly scalable in realistic

applications.

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I will present results from reaction-diffusion systems on surface

meshes, and test problems from the atmospheric sciences community.

Our starting point is a recent characterisation of one-dimensional, time-homogeneous diffusion in terms of its distribution at an exponential time. The structure of this characterisation leads naturally to the idea of measuring `how far' a diffusion is away from being a martingale diffusion in terms of expected local time at the starting point. This work in progress has a connection to finance and to

a Skorokhod embedding.

A subgroup $H$ of a group $G$ is said to be engulfed if there is a
finite-index subgroup $K$ other than $G$ itself such that $H<K$, or
equivalently if $H$ is not dense in the profinite topology on $G$.  In
this talk I will present a variety of methods for showing that a
subgroup of a discrete group is engulfed, and demonstrate how these
methods can be used to study finite-sheeted covering spaces of
topological spaces.

Several shape optimization problems, e.g. in image processing, biology, or discrete geometry, involve the Willmore functional, which is for a surface the integrated squared mean curvature. Due to its singularity, minimizing this functional under constraints is a delicate issue. More precisely, it is difficult to characterize precisely the structure of the minimizers and to provide an explicit

formulation of their energy. In a joint work with Giacomo Nardi (Paris-Dauphine), we have studied an "integrated" version of the Willmore functional, i.e. a version defined for functions and not only for sets. In this talk, I will describe the tools, based on Young measures and varifolds, that we have introduced to address the relaxation issue. I will also discuss some connections with the phase-field numerical approximation of the Willmore flow, that we have investigated with Elie Bretin (Lyon) and Edouard Oudet (Grenoble).

Let G be a permutation group acting on a set Omega. For g in G, let pi(g) denote the partition of Omega given by the orbits of g. The set of all partitions of Omega is naturally ordered by refinement and admits lattice operations of meet and join. My talk concerns the groups G such that the partitions pi(g) for g in G form a sublattice. This condition is highly restrictive, but there are still many interesting examples. These include centralisers in the symmetric group Sym(Omega) and a class of profinite abelian groups which act on each of their orbits as a subgroup of the Prüfer group. I will also describe a classification of the primitive permutation groups of finite degree whose set of orbit partitions is closed under taking joins, but not necessarily meets.

This talk is on joint work with John R. Britnell (Imperial College).

Beautiful results of Deligne-Illusie, Sabbah, and Ogus-Vologodsky show that certain modifications of the de Rham complex (either the usual one, or twisted versions of it that appear in the study of the cyclic homology of categories of matrix factorizations) are formal in positive characteristic. These are the crucial steps in proving algebraic analogues of the Hodge theorem (again, either in the ordinary setting or in the presence of a twisting). I will present these results along with a new approach to understanding them using derived intersection theory. This is joint work with Dima Arinkin and Marton Hablicsek.

Rainbow connectivity is a new concept for measuring the connectivity of a graph which was introduced in 2008 by Chartrand, Johns, McKeon and Zhang. In a graph G with a given edge colouring, a rainbow path is a path all of whose edges have distinct colours. The minimum number of colours required to colour the edges of G so that every pair of vertices is joined by at least one rainbow path is called the rainbow connection number rc(G) of the graph G.

For any graph G, rc(G) >= diam(G). We will discuss rainbow connectivity in the random graph setting and present the result that for random graphs, rainbow connectivity 2 happens essentially at the same time as diameter 2. In fact, in the random graph process, with high probability the hitting times of diameter 2 and of rainbow connection number 2 coincide

Multi-layered cylinders, or 'multitubes', are ubiquitous throughout the biological world, from microscopic axons to plant stems. Whilst these structures share an underlying common geometry, each one fulfils a different key role in its relevant environment. For example plant stems provide a transport network for nutrients within the organism, whilst the tongue of a chameleon is used for prey capture. This talk will be concerned with the mechanical stability of multitubes. How do the material properties, applied tractions and geometry of elastic rods and tubes influence their critical buckling pressure and mode of buckling? We will discuss the phenomenon of differential growth, an important factor in the mechanical behaviour of such systems and introduce a mathematical framework, which can be used to model differential growth in soft tissues and predict the onset of buckling. We will also present a small number of applications for this research.

A nice bed-time story to end the term. It is often said that ideas like the group law or isogenies on elliptic curves were 'known to Fermat' or are 'found
in Diophantus', but this is rarely properly explained. I will discuss the first work on rational points on curves from the point of view of modern number
theory, asking if it really did anticipate the methods we use today.

Consider a diagram of the unknot with c crossings. There is a

sequence of Reidemeister

moves taking this to the trivial diagram. But how many moves are required?

In my talk, I will give

an overview of my recent proof that there is there is an upper bound on the

number of moves, which

is a polynomial function of c.

The lecture will discuss some applications of topology to a number of interesting physical systems:

1. Classifications of Phases, 2. Classifications of one-dimensional textures in Nematics and Superfluid HE-3,

3. Classification of defects, 4. Phase transition in Liquid membranes.

The solution of these problems leads to interesting mathematics but the talk will also include some historical remarks.

We consider the fractional Laplacian perturbed by the gradient operator b(x)\nabla for various classes of vector fields b. We construct end estimate the corresponding semigroup.

We derive an exact implied volatility expansion for any model whose European call price can be expanded analytically around a Black-Scholes call price. Two examples of our framework are provided (i) exponential Levy models and (ii) CEV-like models with local stochastic volatility and local stochastic jump-intensity.

The University of Oxford’s modelling4all software is a wonderful tool to simulate early medieval populations and their cemeteries in order to evaluate the influence of palaeodemographic variables, such as mortality, fertility, catastrophic events and disease on settlement dispersal. In my DPhil project I will study archaeological sites in Anglo-Saxon England and the German south-west in a comparative approach. The two regions have interesting similarities in their early medieval settlement pattern and include some of the first sites where both cemeteries and settlements were completely excavated.

An important discovery in bioarchaeology is that an excavated cemetery is not a straightforward representation of the living population. Preservation issues and the limitations of age and sex estimation methods using skeletal material must be considered. But also the statistical procedures to calculate the palaeodemographic characteristics of archaeological populations are procrustean. Agent-based models can help archaeologists to virtually bridge the chasm between the excavated dead populations and their living counterparts in which we are really interested in.

This approach leads very far away from the archaeologist’s methods and ways of thinking and the major challenge therefore is to balance innovative ideas with practicability and tangibility.

Some of the problems for the workshop are:

1.) Finding the best fitting virtual living populations for the excavated cemeteries

2.) Sensitivity analyses of palaeodemographic variables

3.) General methodologies to evaluate the outcome of agent based models

4.) Present data in a way that is both statistically correct and up to date & clear for archaeologists like me

5.) Explore how to include analytical procedures in the model to present the archaeological community with a user-friendly and not necessarily overwhelming toolkit

We propose a model to reproduce qualitatively and quantitatively the experimental behavior obtained by the AFM techniques for the titin. Via an energetic based minimization approach we are able to deduce a simple analytical formulations for the description of the mechanical behavior of multidomain proteins, giving a physically base description of the unfolding mechanism. We also point out that our model can be inscribed in the led of the pseudo-elastic variational damage model with internal variable and fracture energy criteria of the continuum mechanics. The proposed model permits simple analytical calculations and

to reproduce hard-device experimental AFM procedures. The proposed model also permits the continuum limit approximation which maybe useful to the development of a three-dimensional multiscale constitutive model for biological tissues.

In this talk we show how Teichmüller curves can be used to compute

quantum invariants of certain Pseudo-Anasov mapping tori. This involves

computing monodromy of the Hitchin connection along closed geodesics of

the Teichmüller curve using iterated integrals. We will mainly focus on

the well known Teichmüller curve generated by a pair of regular

pentagons. This is joint work with J. E. Andersen.

To any complex smooth variety Y with an action of a finite group G, Etingof associates a global Cherednik algebra. The usual rational Cherednik algebra corresponds to the case of Y= C^n and a finite Coxeter group G

Domain decomposition methods were introduced as techniques for solving partial differential equations based on a decomposition of the spatial domain of the problem into several subdomains. The initial equation restricted to the subdomains defines a sequence of new local problems. The main goal is to solve the initial equation via the solution of the local problems. This procedure induces a dimension reduction which is the major responsible of the success of such a method. Indeed, one of the principal motivations is the formulation of solvers which can be easily parallelized.

In this presentation we shall develop a domain decomposition algorithm to the minimization of functionals with total variation constraints. In this case the interesting solutions may be discontinuous, e.g., along curves in 2D. These discontinuities may cross the interfaces of the domain decomposition patches. Hence, the crucial difficulty is the correct treatment of interfaces, with the preservation of crossing discontinuities and the correct matching where the solution is continuous instead. I will present our domain decomposition strategy, including convergence results for the algorithm and numerical examples for its application in image inpainting and magnetic resonance imaging.

A number of pricing models for electricity and carbon credit pricing involve nonlinear dependencies between two, or more, of the processes involved; for example, the models developed by Schwarz and Howison. The consequences of these nonlinearities are not well understood.

In this talk I will discuss some much simpler models, namely options whose values are defined self-referentially, which have been looked at in order to better understand the effects of these non-linear dependencies.

Let F be a free group, and N a normal subgroup of F with derived subgroup N'. The Magnus embedding gives a way of seeing F/N' as a subgroup of a wreath product of a free abelian group over over F/N. The aim is to show that the Magnus embedding is a quasi-isometric embedding (hence "Q.I." in the title). For this I will use an alternative geometric definition of the embedding (hence "picture"), which I will show is equivalent to the definition which uses Fox calculus. Please note that we will assume no prior knowledge of calculus.

In our 2004 paper, Fritz Grunewald and I constructed the first
pairs of finitely presented, residually finite groups $u: P\to G$
such that $P$ is not isomorphic to $G$ but the map that $u$ induces on
profinite completions is an isomorphism. We were unable to determine if
there might exist finitely presented, residually finite groups $G$ that
with infinitely many non-isomorphic finitely presented subgroups $u_n:
P_n\to G$ such that $u_n$ induces a profinite isomorphism. I shall
discuss how two recent advances in geometric group theory can be used in
combination with classical work on Nielsen equivalence to settle this
question.

SEMINAR CANCELLED

Given a finite list of vectors X in $\R^d$, one can define the box spline $B_X$. Box splines are piecewise polynomial functions that are used in approximation theory. They are also interesting from a combinatorial point of view and many of their properties solely depend on the structure of the matroid defined by the list X. The support of the box spline is a certain polytope called zonotope Z(X). We will show that if the list X is totally unimodular, any real-valued function defined on the set of lattice points in the interior of Z(X) can be extended to a function on Z(X) of the form $p(D)B_X$ in a unique way, where p(D) is a differential operator that is contained in the so-called internal P-space. This was conjectured by Olga Holtz and Amos Ron. The talk will focus on combinatorial aspects and all objects mentioned above will be defined. (arXiv:1211.1187)

We analyze the fluctuation of the loss from default around its large portfolio limit in a class of reduced-form models of correlated default timing. We prove a weak convergence result for the fluctuation process and use it for developing a conditionally Gaussian approximation to the loss distribution. Numerical results illustrate the accuracy of the approximation.

This is joint work with Kostas Spiliopoulos (Boston University) and Justin Sirignano (Stanford).

The question of estimating the length of short conjugators in between
elements in a group could be described as an effective version of the
conjugacy problem. Given a finitely generated group $G$ with word metric
$d$, one can ask whether there is a function $f$ such that two elements
$u,v$ in $G$ are conjugate if and only if there exists a conjugator $g$ such
that $d(1,g) \leq f(d(1,u)+d(1,v))$. We investigate this problem in free
solvable groups, showing that f may be cubic. To do this we use the Magnus
embedding, which allows us to see a free solvable group as a subgroup of a
particular wreath product. This makes it helpful to understand conjugacy
length in wreath products as well as metric properties of the Magnus
embedding.

I will speak about the of weak (and the existence and uniqueness of strong solutions) to the stochastic
Landau-Lifshitz equations for multi (one)-dimensional spatial domains. I will also describe the corresponding Large Deviations principle and it's applications to a ferromagnetic wire. The talk is based on a joint works with B. Goldys and T. Jegaraj.

Motived by the desire to study geometry over the 'field with one element', in the past decade several authors have constructed extensions of scheme theory to geometries locally modelled on algebraic objects more general than rings. Semi-ring schemes exist in all of these theories, and it has been suggested that schemes over the semi-ring T of tropical numbers should describe the polyhedral objects of tropical geometry. We show that this is indeed the case by lifting Payne's tropicalization functor for subvarieties of toric varieties to the category of T-schemes. There are many applications such as tropical Hilbert schemes, tropical sheaf theory, and group actions and quotients in tropical geometry. This project is joint work with N. Giansiracusa (Berkeley).

I will report on joint work in progress with David Aldous, concerning a curious random metric space on the plane which can be constructed with the help of an improper Poisson line process.

Abstract: Sharp asymptotic lower bounds of the expected quadratic

variation of discretization error in stochastic integration are given.

The theory relies on inequalities for the kurtosis and skewness of a

general random variable which are themselves seemingly new.

Asymptotically efficient schemes which attain the lower bounds are

constructed explicitly. The result is directly applicable to practical

hedging problem in mathematical finance; it gives an asymptotically

optimal way to choose rebalancing dates and portofolios with respect

to transaction costs. The asymptotically efficient strategies in fact

reflect the structure of transaction costs. In particular a specific

biased rebalancing scheme is shown to be superior to unbiased schemes

if transaction costs follow a convex model. The problem is discussed

also in terms of the exponential utility maximization.