Past

In the first half of the talk we explain - in very broad terms - how the objects defined in the previous meetings are linked with each other. We will motivate this 'big picture' by briefly discussing class field theory and the Artin conjecture for L-functions. In the second part we focus on a particular aspect of the theory, namely the L-function preserving construction of elliptic curves from weight 2 newforms via Eichler-Shimura theory. Assuming the Modularity theorem we obtain a proof of the Hasse-Weil conjecture.

Convertible bonds are hybrid securities that embody the characteristics of both straight bonds and equities. The conflict of interests between bondholders and shareholders affects the security prices significantly. In this paper, we investigate how to use a non-zero-sum game framework to model the interaction between bondholders and shareholders and to evaluate the bond accordingly. Mathematically, this problem can be reduced to a system of variational inequalities. We explicitly derive a unique Nash equilibrium to the game.

Our model shows that credit risk and tax benefit have considerable impacts on the optimal strategies of both parties. The shareholder may issue a call when the debt is in-the-money or out-of-the-money. This is consistent with the empirical findings of “late and early calls"

(Ingersoll (1977), Mikkelson (1981), Cowan et al. (1993) and Ederington et al. (1997)). In addition, the optimal call policy under our model offers an explanation for certain stylized patterns related to the returns of company assets and stock on calls.

This paper uses data on house transactions in the state of Massachusetts over the last 20 years

to show that houses sold after foreclosure, or close in time to the death or bankruptcy of at least

one seller, are sold at lower prices than other houses. Foreclosure discounts are particularly large on

average at 27% of the value of a house. The pattern of death-related discounts suggests that they may

result from poor home maintenance by older sellers, while foreclosure discounts appear to be related

to the threat of vandalism in low-priced neighborhoods. After aggregating to the zipcode level and

controlling for regional price trends, the prices of forced sales are mean-reverting, while the prices

of unforced sales are close to a random walk. At the zipcode level, this suggests that unforced sales

take place at approximately ecient prices, while forced-sales prices re

ect time-varying illiquidity in

neighborhood housing markets. At a more local level, however, we nd that foreclosures that take

place within a quarter of a mile, and particularly within a tenth of a mile, of a house lower the price

at which it is sold. Our preferred estimate of this eect is that a foreclosure at a distance of 0.05 miles

lowers the price of a house by about 1%.

Abstract: We will review Kreimer's construction of a Hopf algebra for Feynman graphs, and explore several aspects of this structure including its relationship with renormalization and the (trivial) Hochschild cohomology of the algebra.  Although Kreimer's construction is heavily tied with the language of renormalization, we show that it leads naturally to recursion relations resembling the BCFW relations, which can be expressed using twistors in the case of N=4 super-Yang-Mills (where there are no ultra-violet divergences).  This could suggest that a similar Hopf algebra structure underlies the supersymmetric recursion relations...

Clustering phenomena occur in numerous areas of science.

This series of lectures will discuss:

(i) basic kinetic models for clustering- Smoluchowski's coagulation equation, random shock clustering, ballistic aggregation, domain-wall merging;

(ii) Criteria for approach to self-similarity- role of regular variation;

(iii) The scaling attractor and its measure representation.

A particular theme is the use of methods and insights from probability in tandem with dynamical systems theory. In particular there is a

close analogy of scaling dynamics with the stable laws of probability and infinite divisibility.

In the Said Business School

The covariance between nominal bonds and stocks has varied considerably over recent decades and has even switched sign. It has been predominantly positive in periods such as the late 1970s and early 1980s when the economy has experienced supply shocks and the central bank has lacked credibility. It has been predominantly negative in periods such as the 2000s when investors have feared weak aggregate demand and deflation. This lecture discusses the implications of changing bond risk for the shape of the yield curve, the risk premia on bonds, and the relative pricing of nominal and inflation-indexed bonds.

As everyone knows, one popular notion of a (Scott-Ersov) domain is defined as a bounded complete algebraic cpo. These are closely related to algebraic lattices: (i) A domain becomes an algebraic lattice with the adjunction of an (isolated) top element. (ii) Every non-empty Scott-closed subset of an algebraic lattice is a domain. Moreover, the isolated (= compact) elements of an algebraic lattice form a semilattice (under join). This semilattice has a zero element, and, provided the top element is isolated, it also has a unit element. The algebraic lattice itself may be regarded as the ideal completion of the semilattice of isolated elements. This is all well known. What is not so clear is that there is an easy-to-construct domain of countable semilattices giving isomorphic copies of all countably based domains. This approach seems to have advantages over both the so-called "information systems" or more abstract lattice formulations, and it makes definitions of solutions to domain equations very elementary to justify. The "domain of domains" also has a natural computable structure

An emerging set of methods enables an experimental dialogue with biological systems composed of many interacting cell types---in particular, with neural circuits in the brain. These methods are sometimes called “optogenetic” because they employ light-responsive proteins (“opto-“) encoded in DNA (“-genetic”). Optogenetic devices can be introduced into tissues or whole organisms by genetic manipulation and be expressed in anatomically or functionally defined groups of cells. Two kinds of devices perform complementary functions: light-driven actuators control electrochemical signals; light-emitting sensors report them. Actuators pose questions by delivering targeted perturbations; sensors (and other measurements) signal answers. These catechisms are beginning to yield previously unattainable insight into the organization of neural circuits, the regulation of their collective dynamics, and the causal relationships between cellular activity patterns and behavior.

In the eighteenth century Gaspard Monge considered the problem of finding the best way of moving a pile of material from one site to another. This optimal transport problem has many applications such as mesh generation, moving mesh methods, image registration, image morphing, optical design, cartograms, probability theory, etc. The solution to an optimal transport problem can be found by solving the Monge-Amp\`{e}re equation, a highly nonlinear second order elliptic partial differential equation. Leonid Kantorovich, however, showed that it is possible to analyse optimal transport problems in a framework that naturally leads to a linear programming formulation. In recent years several efficient methods have been proposed for solving the Monge-Amp\`{e}re equation. For the linear programming problem, standard methods do not exploit the special properties of the solution and require a number of operations that is quadratic or even cubic in the number of points in the discretisation. In this talk I will discuss techniques that can be used to obtain more efficient methods.

Joint work with Chris Budd (University of Bath).

In this talk, we describe new profile decompositions for bounded sequences in Banach spaces of functions defined on $\mathbb{R}^d$. In particular, for "critical spaces" of initial data for the Navier-Stokes equations, we show how these can give rise to new proofs of recent regularity theorems such as those found in the works of Escauriaza-Seregin-Sverak and Rusin-Sverak. We give an update on the state of the former and a new proof plus new results in the spirit of the latter. The new profile decompositions are constructed using wavelet theory following a method of Jaffard.

\paragraph{} Poisson quasi-Nijenhuis structures with background (PqNb structures) were recently defined and are one of the most general structures within Poisson geometry. On one hand they generalize the structures of Poisson-Nijenhuis type, which in particular contain the Poisson structures themselves. On the other hand they generalize the (twisted) generalized complex structures defined some years ago by Hitchin and Gualtieri. Moreover, PqNb manifolds were found to be appropriate target manifolds for sigma models if one wishes to incorporate certain physical features in the model. All these three reasons put the PqNb structures as a new and general object that deserves to be studied in its own right.

\paragraph{} I will start the talk by introducing all the concepts necessary for defining PqNb structures, making this talk completely self-contained. After a brief recall on Poisson structures, I will define Poisson-Nijenhuis and Poisson quasi-Nijenhuis manifolds and then move on to a brief presentation on the basics of generalized complex geometry. The PqNb structures then arise as the general structure which incorporates all the structures referred above. In the second part of the talk, I will define gauge transformations of PqNb structures and show how one can use this concept to construct examples of such structures. This material corresponds to part of the article arXiv:0912.0688v1 [math.DG].\\

\paragraph{} Also, if time permits, I will shortly discuss the appearing of PqNb manifolds as target manifolds of sigma models.

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The Wulff droplet arises by conditioning a spin system in a dominant

phase to have an excess of signs of opposite type. These gather

together to form a droplet, with a macroscopic Wulff profile, a

solution to an isoperimetric problem.

I will discuss recent work proving that the phase boundary that

delimits the signs of opposite type has a characteristic scale, both

at the level of exponents and their logarithmic corrections.

This behaviour is expected to be shared by a broad class of stochastic

interface models in the Kardar-Parisi-Zhang class. Universal

distributions such as Tracy-Widom arise in this class, for example, as

the maximum behaviour of repulsive particle systems. time permitting,

I will explain how probabilistic resampling ideas employed in spin

systems may help to develop a qualitative understanding of the random

mechanisms at work in the KPZ class.

We report on joint work with Arend Bayer on the space of stability conditions for the canonical bundle on the projective plane.

We will describe a connected component of this space, generalizing and completing a previous construction of Bridgeland.

In particular, we will see how this space is related to classical results of Drezet-Le Potier on stable vector bundles on the projective plane. Using this, we can determine the group of autoequivalences of the derived category. As a consequence, we can identify a $\Gamma_1(3)$-action on the space of stability conditions, which will give a global picture of mirror symmetry for this example.

In the second hour we will give some details on the proof of the main theorem.

Given a graph we want to draw it in the plane; well we *want* to draw it in the plane, but sometimes we just can't. So we resort to various compromises. Sometimes we add crossings and try to minimize the crossings. Sometimes we add handles and try to minimize the number of handles. Sometimes we add crosscaps and try to minimize the number of crosscaps.

Sometimes we mix these parameters: add a given number of handles (or crosscaps) and try to minimize the number of crossings on that surface. What if we are willing to trade: say adding a handle to reduce the number of crossings? What can be said about the relative value of such a trade? Can we then add a second handle to get an even greater reduction in crossings? If so, why didn't we trade the second handle in the first place? What about a third handle?

The crossing sequence cr_1, cr_2, ... , cr_i, ... has terms the minimum number of crossings over all drawings of G on a sphere with i handles attached. The non-orientable crossing sequence is defined similarly. In this talk we discuss these crossing sequences.

By Dan Archdeacon, Paul Bonnington, Jozef Siran, and citing works of others.

We report on joint work with Arend Bayer on the space of stability conditions for the canonical bundle on the projective plane.

We will describe a connected component of this space, generalizing and completing a previous construction of Bridgeland.

In particular, we will see how this space is related to classical results of Drezet-Le Potier on stable vector bundles on the projective plane. Using this, we can determine the group of autoequivalences of the derived category. As a consequence, we can identify a $\Gamma_1(3)$-action on the space of stability conditions, which will give a global picture of mirror symmetry for this example.

In the second hour we will give some details on the proof of the main theorem.

This talk is the second in a series of an elementary introduction to the ideas unifying elliptic curves, modular forms and Galois representations. I will discuss what it means for an elliptic curve to be modular and what type of representations one associates to such objects.

In this talk, we introduce the (general) homotopy groups of spheres as link invariants for Brunnian-type links through the investigations on the intersection subgroup of the normal closures of the meridians of strongly nonsplittable links. The homotopy groups measure the difference between the intersection subgroup and symmetric commutator subgroup of the normal closures of the meridians and give the invariants of the links obtained in this way. Moreover all homotopy groups of any dimensional spheres can be obtained from the geometric Massey products on certain links.

Since the work of Feigenbaum and Coullet-Tresser on universality in the period doubling bifurcation, it is been understood that crucial features of unimodal (one-dimensional) dynamics depend on the behavior of a renormalization (and infinite dimensional) dynamical system. While the initial analysis of renormalization was mostly focused on the proof of existence of hyperbolic fixed points, Sullivan was the first to address more global aspects, starting a program to prove that the renormalization operator has a uniformly hyperbolic (hence chaotic) attractor. Key to this program is the proof of exponential convergence of renormalization along suitable ``deformation classes'' of the complexified dynamical system. Subsequent works of McMullen and Lyubich have addressed many important cases, mostly by showing that some fine geometric characteristics of the complex dynamics imply exponential convergence.

We will describe recent work (joint with Lyubich) which moves the focus to the abstract analysis of holomorphic iteration in deformation spaces. It shows that exponential convergence does follow from rougher aspects of the complex dynamics (corresponding to precompactness features of the renormalization dynamics), which enables us to conclude exponential convergence in all cases.

In this talk, we report some of our recent work on hybrid switching diffusions in which continuous dynamics and discrete events coexist. Motivational examples in singular perturbed Markovian systems, manufacturing, and financial engineering will be mentioned. After presenting criteria for recurrence and ergodicity, we consider numerical methods for controlled switching diffusions and related game problems. Rates of convergence of Markov chain approximation methods will also be studied.

Andrew Stewart -

The role of the complete Coriolis force in ocean currents that cross the equator

Large scale motions in the atmosphere and ocean are dominated by the Coriolis force due to the Earth's rotation. This tends to prevent fluid crossing the equator from one hemisphere to the other. We investigate the flow of a deep ocean current, the Antarctic Bottom Water, across the equator using a shallow water model that includes the Earth's complete Coriolis force. By contrast, most theoretical models of the atmosphere and ocean use the so-called traditional approximation that neglects the component of the Coriolis force associated with the locally horizontal component of the Earth's rotation vector. Using a combination of analytical and numerical techniques, we show that the cross-equatorial transport of the Antarctic Bottom Water may be substantially influenced by the interaction of the complete Coriolis force with bottom topography.

In the first part of my presentation, I plan to review several applications modelled by delay differential equations (DDEs) starting from familiar examples such as traffic flow problems to physiology and industrial problems. Although delay differential equations have the reputation to be difficult mathematical problems, there is a renewed interest for both old and new problems modelled by DDEs. In the second part of my talk, I’ll emphasize the need of developing asymptotic tools for DDEs in order to guide our numerical simulations and help our physical understanding. I illustrate these ideas by considering the response of optical optoelectronic oscillators that have been studied both experimentally and numerically.

Jointly with Number Theory

Consider a family of abelian varieties whose base is an algebraic variety. The union of all torsion groups over all fibers of the family will be called the set of torsion points of the family. If the base variety is a point then the family is just an abelian variety.

In this case the Manin-Mumford Conjecture, a theorem of Raynaud, implies that a subvariety of the abelian variety contains a Zariski dense set of torsion points if and only if it is itself essentially an abelian subvariety. This talk is on possible extensions to certain families where the base is a curve. Conjectures of André and Pink suggest considering "special points": these are torsion points whose corresponding fibers satisfy an additional arithmetic property. One possible property is for the fiber to have complex multiplication; another is for the fiber to be isogenous to an abelian variety fixed in advance.

We discuss some new results on the distribution of such "special points"

on the subvarieties of certain families of abelian varieties. One important aspect of the proof is the interplay of two height functions.

I will give a brief introduction to the theory of heights in the talk.