Past

he Iwasawa theory of elliptic curves over the rationals, and more
generally of modular forms, has mostly been studied with the
assumption that the form is "ordinary" at p -- i.e. that the Hecke
eigenvalue is a p-adic unit. When this is the case, the dual of the
p-Selmer group over the cyclotomic tower is a torsion module over the
Iwasawa algebra, and it is known in most cases (by work of Kato and
Skinner-Urban) that the characteristic ideal of this module is
generated by the p-adic L-function of the modular form.

I'll talk about the supersingular (good non-ordinary) case, where
things are slightly more complicated: the dual Selmer group has
positive rank, so its characteristic ideal is zero; and the p-adic
L-function is unbounded and hence doesn't lie in the Iwasawa algebra.
Under the rather restrictive hypothesis that the Hecke eigenvalue is
actually zero, Kobayashi, Pollack and Lei have shown how to decompose
the L-function as a linear combination of Iwasawa functions and
explicit "logarithm-like" series, and to modify the definition of the
Selmer group correspondingly, in order to formulate a main conjecture
(and prove one inclusion). I will describe joint work with Antonio Lei
and Sarah Zerbes where we extend this to general supersingular modular
forms, using methods from p-adic Hodge theory. Our work also gives
rise to new phenomena in the ordinary case: a somewhat mysterious
second Selmer group and L-function, which is related to the
"critical-slope L-function" studied by Pollack-Stevens and Bellaiche.


This talk is about the Induced Dimension Reduction (IDR) methods developed by Peter Sonneveld and, more recently, Martin van Gijzen. We sketch the history, outline the underlying principle, and give a few details about different points of view on this class of Krylov subspace methods. If time permits, we briefly outline some recent developments in this field and the benefits and drawbacks of these and IDR methods in general.

In this talk I want to ask how to create a coherent mathematical framework for pricing and hedging which starts with the information available in the market and does not assume a given probabilistic setup. This calls for re-definition of notions of arbitrage and trading and, subsequently, for a ``probability-free first fundamental theorem of asset pricing". The new setup should also link with a classical approach if our uncertainty about the model vanishes and we are convinced a particular probabilistic structure holds. I explore some recent results but, predominantly, I present the resulting open questions and problems. It is an ``internal talk" which does not necessarily present one paper but rather wants to engage into a discussion. Ideas for the talk come in particular from joint works with Alex Cox and Mark Davis.

Solutions which are time-bounded in L^3 up to time T can be continued

past this time, by a landmark result of Escauriaza-Seregin-Sverak,

extending Serrin's criterion. On the other hand, the local Cauchy

theory holds up to solutions in BMO^-1; we aim at describing how one

can obtain intermediate regularity results, assuming a priori bounds

in negative regularity Besov spaces.

This is joint work with J.-Y. Chemin, Isabelle Gallagher and Gabriel

Koch.

First of all, we are going to recall some basic facts and definitions about homogeneous Riemannian manifolds. Then we are going to talk about existence and non-existence of invariant Einstein metrics on compact homogeneous manifolds. In this context, we have that it is possible to associate to every homogeneous space a graph. Then, the graph theorem of Bohm, Wang and Ziller gives an existence result of invariant Einstein metrics on a compact homogeneous space, based on properties of its graph. We are going to discuss this theorem and sketch its proof.

Given a deterministically time-changed Brownian motion Z starting from 1, whose time-change V (t) satisfies V (t) > t for all t > 0, we perform an explicit construction of a process X which is Brownian motion in its own filtration and that hits zero for the first time at V (S), where S := inf {t > 0 : Z_t = 0}. We also provide the semimartingale decomposition of X under the filtration jointly generated by X and Z. Our construction relies on a combination of enlargement of filtration and filtering techniques. The resulting process X may be viewed as the analogue of a 3-dimensional Bessel bridge starting from 1 at time 0 and ending at 0 at the random time V (S). We call this a dynamic Bessel bridge since S is not known at time 0 but is slowly revealed in time by observing Z. Our study is motivated by insider trading models with default risk. (this is a joint work with Luciano Campi and Albina Danilova)

In 1968, Dixmier posed six problems for the algebra of polynomial

  differential operators, i.e. the Weyl algebra. In 1975, Joseph

solved the third and sixth problems and, in 2005, I solved the

  fifth problem and gave a positive solution to the fourth problem

  but only for homogeneous differential operators. The remaining three problems are still open. The first problem/conjecture of Dixmier (which is equivalent to the Jacobian Conjecture as was shown in 2005-07 by Tsuchimito, Belov and Kontsevich) claims that the Weyl algebra `behaves'

like a finite field. The first problem/conjecture of

  Dixmier:   is it true that an algebra endomorphism of the Weyl

  algebra an automorphism? In 2010, I proved that this question has

  an affirmative answer for the algebra of polynomial

  integro-differential operators. In my talk, I will explain the main

  ideas, the structure of the proof and recent progress on the first problem/conjecture of Dixmier.

A random planar graph $P_n$ is a graph drawn uniformly at random from the class of all (labelled) planar graphs on $n$ vertices. In this talk we show that with probability $1-o(1)$ the number of vertices of degree $k$ in $P_n$ is very close to a quantity $d_k n$ that we determine explicitly. Here $k=k(n) \le c \log n$. In the talk our main emphasis will be on the techniques for proving such results. (Joint work with Kosta Panagiotou.)

In this talk, we will present some recent mathematical features around two-fluid models. Such systems may be encountoured for instance to model internal waves, violent aerated flows, oil-and-gas mixtures. Depending on the context, the models used for simulation may greatly differ. However averaged models share the same structure. Here, we address the question whether available mathematical results in the case of a single fluid governed by the compressible barotropic equations for single flow may be extended to two phase model and discuss derivations of well-known multi-fluid models from single fluid systems by homogeneization (assuming for instance highly oscillating density). We focus on existence of local existence of strong solutions, loss of hyperbolicity, global existence of weak solutions, invariant regions, Young measure characterization.

Leveraged ETFs are funds that target a multiple of the daily return of a reference asset; eg UYG (Proshares) targets twice the daily return of XLF (Financial SPDR) and SKF targets minus twice the daily return of XLF.

 It is well known that these leveraged funds have exposure to realized volatility. In particular, the relation between the leveraged and the unleveraged funds over a given time-horizon (larger than 1 day) is uncertain and will depend on the realized volatility. This talk examines this phenomenon theoretically and empirically first, and then uses this to price options on leveraged ETFs in terms of the prices of options on the underlying ETF. The resulting model allows to model the volatility skews of the leveraged and unleveraged funds in relation to each other and therefore suggest an arbitrage relation that could prove useful for traders and risk-managers.

The goal of this talk is to construct new examples of hyperbolic

aspherical complexes. More precisely, given an aspherical simplicial

complex P and a subcomplex Q of P, we are looking for conditions under

which the complex obtained by attaching a cone of base Q on P remains

aspherical. If Q is a set of loops of a 2-dimensional complex, J.H.C.

Whitehead proved that this new complex is aspherical if and only if the

elements of the fundamental group of P represented by Q do not satisfy

any identity. To deal with higher dimensional subcomplexes we use small

cancellation theory and extend the geometric point of view developed by

T. Delzant and M. Gromov to rotation families of groups. In particular

we obtain hyperbolic aspherical complexes obtained by attaching a cone

over the "real part" of a hyperbolic complex manifold.

Recently, Liang, Lyons and Qian developed a new methodology for the study of backward stochastic differential equations (BSDEs) on general filtered probability spaces. Their approach is based on the analysis of a particular class of functional differential equations, where the driver of the equation does not depend only on the present, but also on the terminal value of the solution.

The purpose of this work is to study fully coupled systems of forward functional differential equations, which are related to a broad class of fully coupled forward-backward stochastic dynamics with respect to general filtrations. In particular, these systems of functional differential equations have a more homogeneous structure with respect to the underlying forward-backward problems, allowing to partly avoid the conflicting nature between the forward and backward components.

Another advantage of the approach is that its generality allows to consider many other types of forward-backward equations not treated in the classical literature: this is shown with the help of several examples, which have interesting applications to mathematical finance and are related to parabolic integro-partial differential equations. In the second part of the talk, we introduce a numerical scheme for the approximation of decoupled systems, based on a time discretization combined with a local iteration approach.

I will introduce the notion of a nonlinear Levy process, discuss basic well-posednes, SDE links and the connection with interacting particles. The talk is aimed to be an introduction to the topic of my recent CUP monograph 'Nonllinear Markov processes and kinetic equations'.

We argue that a natural extension of the well known structural credit risk framework of Black and Cox is to model both the firm's assets and liabilities as correlated geometric Brownian motions. This financially reasonable assumption leads to a unification of equity derivatives (written on the stock price), and credit securities like bonds and credit default swaps (CDS), nesting the Black-Cox credit model with a particular stochastic volatility model for the stock. As we will see, it yields reasonable pricing performance with acceptable computational efficiency. However, it has been well understood how to extend a credit framework like this quite dramatically by the trick of time- changing the Brownian motions. We will find that the resulting two factor time-changed Brownian motion framework can encompass well known equity models such as the variance gamma model, and at the same time reproduce the stylized facts about default stemming from structural models of credit. We will end with some encouraging calibration results for a dataset of equity and credit derivative prices written on Ford Motor Company.

In this talk I shall present some ongoing work on principal G-Higgs bundles, for G a simple Lie group. In particular, we will consider two non-compact real forms of GL(p+q,C) and SL(p+q,C), namely U(p,q) and SU(p,q). By means of the spectral data that principal Higgs bundles carry for these non-compact real forms, we shall give a new description of the moduli space of principal U(p,q) and SU(p,q)-Higgs bundles. As an application of our method, we will count the connected components of these moduli spaces.

The notion of critical mass in research is one that has been around for a long time without proper definition. It has been described as some kind of threshold group size above which research standards significantly improve. However no evidence for such a threshold has been found and critical mass has never been measured -- until now.

We present a new, simple, sociophysical model which explains how research quality depends on research-group structure and in particular on size. Our model predicts that there are, in fact, two critical masses in research, the values of which are discipline dependent. Research quality tends to be linearly dependent on group size, but only up to a limit termed the 'upper critical mass'. The upper critical mass is interpreted as the average maximum number of colleagues with whom a given individual in a research group can meaningfully interact. Once the group exceeds this size, it tends to fragment into sub-groups and research quality no longer improves significantly with increasing size. There is also a

lower critical mass, which small research groups should strive to achieve for stability.

Our theory is tested using empirical data from RAE 2008 on the quantity and quality of research groups, for which critical masses are determined. For pure and applied mathematics, the lower critical mass is about 2 and 6, respectively, while for statistics and physics it is 9 and 13. The upper critical mass, beyond which research quality does not significantly improve with increasing group size, is about twice the lower value.

Many continuum models have been derived in recent years which describe the self-assembly of industrially utilisable crystalline films to a level of detail that allows qualitative comparisons with experiments. For thin-film problems, where the characteristic length scales in vertical and horizontal directions differ significantly, the governing surface diffusion equations can be reduced to simpler PDEs by making use of asymptotic expansions. Many mathematical problems and solutions emerge from such new evolution equations and many of them remind of Cahn-Hilliard type equations. The surface diffusion models are of high, of fourth or even sixth, order.

We present the modeling, model reduction and simulation results for heteroepitaxial growth as for Ge/Si quantum dot self-assembly. The numerical methods we are using are based on trigonometric interpolation. These kind of pseudospectral methods seem very well suited for simulating the coarsening of large quantum dot arrays. When the anisotropy of the growing crystalline film is strong, it might become necessary to add a corner regularisation to the model. Then the transition region between neighboring facets is still smooth, but its scale is rather small. In this case it might be useful to think about an adaptive extension of the existing method.

Figure 1: Ostwald ripening process of quantum dots depicted at consecutive time points. One fourth of the whole, periodic, simulated domain is shown.

Joint work with Peter Evans and Barbara Wagner

We prove the existence of weak solutions to steady Navier Stokes equations

$$\text{div}\, \sigma+f=\nabla\pi+(\nabla u)u.$$

Here $u:\mathbb{R}^2\supset \Omega\rightarrow \mathbb{R}^2$ denotes

the velocity field satisfying $\text{div}\, u=0$,

$f:\Omega\rightarrow\mathbb{R}^2$ and

$\pi:\Omega\rightarrow\mathbb{R}$ are external volume force and

pressure, respectively. In order to model the behavior of

Prandtl-Eyring fluids we assume

$$\sigma= DW(\varepsilon (u)),\quad W(\varepsilon)=|\varepsilon|\log

(1+|\varepsilon|).$$

A crucial tool in our approach is a modified Lipschitz truncation

preserving the divergence of a given function.

We begin by showing the underlying ideas Bourgain used to prove that the Cayley graph of the free group of finite rank can be embedded into a Hilbert space with logarithmic distortion. Equipped with these ideas we then tackle the same problem for other metric spaces. Time permitting these will be: amalgamated products and HNN extensions over finite groups, uniformly discrete hyperbolic spaces with bounded geometry and Cayley graphs of cyclic extensions of small cancellation groups.

We discuss a backward stochastic differential equation, (BSDE), approach to a risk-based, optimal investment problem of an insurer. A simplified continuous-time economy with two investment vehicles, namely, a fixed interest security and a share, is considered.

The insurer's risk process is modeled by a diffusion approximation to a compound Poisson risk process. The goal of the insurer is to select an optimal portfolio so as to minimize the risk described by a convex risk measure of his/her terminal wealth. The optimal investment problem is then formulated as a zero-sum stochastic differential game between the insurer and the market. The BSDE approach is used to solve the game problem. This leads to a simple and natural approach for the existence and uniqueness of an optimal strategy of the game problem without Markov assumptions. Closed-form solutions to the optimal strategies of the insurer and the market are obtained in some particular cases.

In 1939, Wilhelm Magnus gave a characterization of free groups in terms of their rank and nilpotent quotients. Our goal in this talk is to present results giving both positive and negative answers to the following question: does a similar characterization hold within the class of finite-extensions of finitely generated free groups? This talk covers joint work with Brandon Seward.

I will discuss a conjectural Bogomolov-Gieseker type inequality for "tilt-stable" objects in the derived category of coherent sheaves on smooth projective threefolds. The conjecture implies the existence of Bridgeland stability conditions on threefolds, and also has implications to birational geometry: it implies a slightly weaker version of Fujita's conjecture on very ampleness of adjoint line bundles.

I will present some new results on classifying 123 TQFTs,
using a 2-categorical approach. The invariants defined by a TQFT are
described using a new graphical calculus, which makes them easier to
define and to work with. Some new and interesting physical phenomena
are brought out by this perspective, which we investigate. I will
finish by banishing some TQFT myths! This talk is based on joint work
with Bruce Bartlett, Chris Schommer-Pries and Chris Douglas.

Given a film of viscous heavy liquid with upper free boundary over an inclined plane, a steady laminar motion develops parallel to the flat bottom ofthe layer. We name this motion\emph{ Poiseuille Free Boundary} PFBflow because of its (half) parabolic velocity profile. In flowsover an inclined plane the free surface introduces additionalinteresting effects of surface tension and gravity. These effectschange the character of the instability in a parallel flow, see{Smith} [1]. \par\noindentBenjamin [2], and Yih [3], have solved the linear stabilityproblem of a uniform film on a inclined plane. Instability takesplace in the form of an infinitely long wave, however\emph{surface waves of finite wavelengths are observed}, see e.g.Yih [3]. Up to date direct nonlinear methods for the study ofstability seem to be still lacking.
Aim of this talk is the investigation of nonlinear stability ofPFB providing \emph{ a rigorous formulation of the problem by theclassical direct Lyapunov method assuming periodicity in theplane}, when above the liquid there is a uniform pressure due tothe air at rest, and the liquid is moving with respect to the air.Sufficient conditions on the non dimensional Reynolds, Webernumbers, on the periodicity along the line of maximum slope, onthe depth of the layer and on the inclination angle are computedensuring Kelvin-Helmholtz \emph{nonlinear stability}. We use\emph{a modified energy method, cf. [4],[5], which providesphysically meaningful sufficient conditions ensuring nonlinearexponential stability}. The result is achieved in the class ofregular solutions occurring in simply connected domains havingcone property.\par\noindentNotice that the linear equations, obtained by linearization of ourscheme around the basic Poiseuille flow, do coincide with theusual linear equations, cf. {Yih} [3]. \\ {\bf References}\\ [1]  M.K. Smith, \textit{The mechanism for the long-waveinstability in thin liquid films} J. Fluid Mech., \textbf{217},1990, pp.469-485.
\\ [2]  Benjamin T.B., \textit{Wave formation in laminar flow down aninclined plane}, J. Fluid Mech. \textbf{2}, 1957, 554-574.
\\ [3]  Yih Chia-Shun, \textit{Stability of liquid flow down aninclined plane}, Phys. Fluids, \textbf{6}, 1963, pp.321-334.
\\ [4] Padula M., {\it On nonlinear stability of MHD equilibriumfigures}, Advances in Math. Fluid Mech., 2009, 301-331.
\\ [5] Padula M., \textit{On nonlinear stability of linear pinch},Appl. Anal.  90 (1), 2011, pp. 159-192.

We give an introduction to the Kakimizu complex of a link,

covering a number of recent results. In particular we will see that the

Kakimizu complex of a knot may be locally infinite, that the Alexander

polynomial of an alternating link carries information about its Seifert

surfaces, and that the Kakimizu complex of a special alternating link is

understood.

The paper analyses structural models for the evaluation of risky debt following H.E. LELAND [2], with an approach of optimal stopping problem (for instance cf. N. EL KAROUI [1]) and within a more general context: a dividend is paid to equity holders, moreover a different tax schedule is introduced, depending on the firm current value. Actually, an endogenous default boundary is introduced and a nonlinear convex tax schedule allowing for a possible switching in tax benefits. The aim is to find optimal capital structure such that the failure is delayed, meaning how to decrease the failure level VB, anyway preserving D debtholders and E equity holders’interests: for the firm VB is needed as low as possible, for the equity holder, an optimal equity is requested, finally an optimal coupon C is asked  for the total value.

Keywords: corporate debt, optimal capital structure, default,

• Wan Chen - “From Brownian Dynamics to Transition Rate Theory: An Ion Channel Example”
• Thomas Lessinnes - "Neuronal growth: a mechanical perspective"
• Savina Joseph - “Current generation in solar cells”
• Shengxin Zhu - “The Numerical Linear Algebra of Approximation involving Radial Basis Functions”

In the Examination Schools.

Over the last decades, advanced probabilistic methods have played an increasing role in Finance, both in Academia and in the financial industry. In view of the recent financial crisis it has been asked to which extent "misplaced reliance on sophisticated maths" has been part of the problem. We will focus on the foundational issue of model uncertainty, also called "Knightian uncertainty". This will be illustrated

by the problem of quantifying financial risk. We discuss recent advances

in the theory of convex risk measures and a corresponding robustification of classical problems of optimal portfolio choice, where model uncertainty is taken into account explicitly.

Biography: Hans Follmer is Professor Emeritus of Mathematics at Humboldt-Universitat zu Berlin, Andrew D. White Professor-at-Large at Cornell University, and Visiting Professor at the National University of Singapore. Before joining Humboldt University in 1994, he has been professor at the universities of Frankfurt and Bonn and at ETH Zurich.

Hans Follmer is widely known for his contributions to probability theory and mathematical finance. He received numerous awards, including the Prix Gay-Lussac/Humboldt of the French Government, the Georg-Cantor medal of the German Mathematical Society, and a honorary degree of the University Paris-Dauphine. He is a member of the Berlin-Brandenburgische Akademie der Wissenschaften, the German National Academy of Sciences Leopoldina, and the European Academy of Sciences Academia Europaea.

We develop a theory of impact of viscoelastic spheres with adhesive

interactions. We assume that the collision velocities are not large to

avoid the fracture and plastic deformation of particles material and

microscopic relaxation time is much smaller than the collision duration.

The adhesive interactions are described with the use of Johnson, Kendall

and Roberts (JKR) theory, while dissipation is attributed to the

viscoelastic behavior of the material. For small impact velocities we

apply the condition of a quasi-static collision and obtain the

inter-particle force. We show that this force is a sum of four

components, having in addition to common elastic, viscous and adhesive

force, the visco-adhesive cross term. Using the derived force we compute

the coefficient of normal restitution and consider the application of our

theory to the collisions of macro and nano-particles.

The points in question can be found on  any semi-abelian surface over an elliptic curve with complex multiplication. We will show that they provide counter-examples to natural expectations in a variety of fields :  Galois representations (following K. Ribet's initial study from the 80's), Lehmer's problem on heights, and more recently, the relative  analogue of the Manin-Mumford conjecture. However, they do support Pink's general conjecture on special subvarieties of mixed Shimura varieties.

The points in question can be found on any semi-abelian surface over an

elliptic curve with complex multiplication. We will show that they provide

counter-examples to natural expectations in a variety of fields : Galois

representations (following K. Ribet's initial study from the 80's),

Lehmer's problem on heights, and more recently, the relative analogue of

the Manin-Mumford conjecture. However, they do support Pink's general

conjecture on special subvarieties of mixed Shimura varieties.

This seminar will be held at the Rutherford Appleton Laboratory near Didcot.

Abstract:

Numerical calculations of laminar flow in a two-dimensional channel with a sudden expansion exhibit a symmetry-breaking bifurcation at Reynolds number 40.45 when the expansion ratio is 3:1. In the experiments reported by Fearn, Mullin and Cliffe [1] there is a large perturbation to this bifurcation and the agreement with the numerical calculations is surprisingly poor. Possible reasons for this discrepancy are explored using modern techniques for uncertainty quantification.

When experimental equipment is constructed there are, inevitably, small manufacturing imperfections that can break the symmetry in the apparatus. In this work we considered a simple model for these imperfections. It was assumed that the inlet section of the channel was displaced by a small amount and that the centre line of the inlet section was not parallel to the centre line of the outlet section. Both imperfections were modelled as normal random variables with variance equal to the manufacturing tolerance. Thus the problem to be solved is the Navier-Stokes equations in a geometry with small random perturbations. A co-ordinate transformation technique was used to transform the problem to a fixed deterministic domain but with random coefficient appearing in the transformed Navier-Stokes equations. The resulting equations were solved using a stochastic collocation technique that took into account the fact that the problem has a discontinuity in parameter space arising from the bifurcation structure in the problem.

The numerical results are in the form of an approximation to a probability measure on the set of bifurcation diagrams. The experimental data of Fearn, Mullin and Cliffe are consistent with the computed solutions, so it appears that a satisfactory explanation for the large perturbation can be provided by manufacturing imperfections in the experimental apparatus.

The work demonstrates that modern methods for uncertainty quantification can be applied successfully to a bifurcation problem arising in fluid mechanics. It should be possible to apply similar techniques to a wide range of bifurcation problems in fluid mechanics in the future.

References:

[1] R M Fearn, T Mullin and K A Cliffe Nonlinear flow phenomena in a symmetric sudden expansion, J. Fluid Mech. 211, 595-608, 1990.