Past

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.

Mikhail Borovoi's theorem states that any simply connected compact semisimple Lie group can be understood (as a group) as an amalgam of its rank 1 and rank 2 subgroups. Here we present a recent extension of this, which allows us to understand the same objects as a colimit of their rank 1 and rank 2 subgroups under a final group topology in the category of Lie groups. Loosely speaking, we obtain not only the group structure uniquely by understanding all rank 1 and rank 2 subgroups, but also the topology.

The talk will race through the elements of Lie theory, buildings and category theory needed for this proof, to leave the audience with the underlying structure of the proof. Little prior knowledge will be assumed, but many details will be left out.

We'll discuss 2 ways to decompose a 3-manifold, namely the Heegaard

splitting and the celebrated geometric decomposition. We'll then see

that being hyperbolic, and more in general having (relatively)

hyperbolic fundamental group, is a very common feature for a 3-manifold.

We highlight a technique for studying edge colourings of multigraphs, due to Tashkinov. This method is a sophisticated generalisation of the method of alternating paths, and builds upon earlier work by Kierstead and Goldberg. In particular we show how to apply it to a number of edge colouring problems, including the question of whether the class of multigraphs that attain equality in Vizing's classical bound can be characterised.

This talk represents joint work with Jessica McDonald.