Past

We discuss a class of high-order spectral-Galerkin surface integral algorithms with specific focus on simulating the scattering of electromagnetic waves by a collection of three dimensional deterministic and stochastic particles.

The talk concerns experiments that study strategic thinking by eliciting subjects’ initial responses to series of different but related games, while monitoring and analyzing the patterns of subjects’ searches for hidden but freely accessible payoff information along with their decisions.

This talk will largely be a survey and so will gloss over technicalities. After introducing the basics of the theory of the étale fundamental group I will state the theorems and conjectures related to Grothendieck's famous "anabelian" letter to Faltings. The idea is that the geometry and arithmetic of certain varieties is in some sense governed by their non-abelian (anabelian) fundamental group. Time permitting I will discuss current work in this area, particularly the work of Minhyong Kim relating spaces of (Hodge, étale) path torsors to finiteness theorems for rational points on curves leading to a conjectural proof of Faltings' theorem which has been much discussed in recent years.

tba

What do historians of mathematics do? What sort of questions do they ask? What kinds of sources do they use? This series of four informal lectures will demonstrate some of the research on history of mathematics currently being done in Oxford. The subjects range from the late Renaissance mathematician Thomas Harriot (who studied at Oriel in 1577) to the varied and rapidly developing mathematics of the seventeenth century (as seen through the eyes of Savilian Professor John Wallis, and others) to the emergence of a new kind of algebra in Paris around 1830 in the work of the twenty-year old Évariste Galois.

Each lecture will last about 40 minutes, leaving time for questions and discussion. No previous knowledge is required: the lectures are open to anyone from the department or elsewhere, from undergraduates upwards.

Consider the following simple question:

Is there a subcategory of Top that is dually equivalent to Lat?

where Top is the category of topological spaces and continuous maps and Lat is the category

of bounded lattices and bounded lattice homomorphisms.

Of course, the question has been answered positively by specializing Lat, and (less

well-known) by generalizing Top.

The earliest examples are of the former sort: Tarski showed that every complete atomic

Boolean lattice is represented by a powerset (discrete topological space); Birkhoff showed

that every finite distributive lattice is represented by the lower sets of a finite partial order

(finite T0 space); Stone generalized Tarski and then Birkhoff, for arbitrary Boolean and

arbitrary bounded distributive lattices respectively. All of these results specialize Lat,

obtaining a (not necessarily full) subcategory of Top.

As a conceptual bridge, Priestley showed that distributive lattices can also be dually

represented in a category of certain topological spaces augmented with a partial order.

This is an example of the latter sort of result, namely, a duality between a category of

lattices and a subcategory of a generalization of Top.

Urquhart, Hartung and Hartonas developed dualities for arbitrary bounded lattices in

the spirit of Priestley duality, in that the duals are certain topological spaces equipped with

additional structure.

We take a different path via purely topological considerations. At the end, we obtain

an affirmative answer to the original question, plus a bit more, with no riders: the dual

categories to Lat and SLat (semilattices) are certain easily described subcategories of Top

simpliciter. This leads directly to a very natural topological characterization of canonical

extensions for arbitrary bounded lattices.

Building on the topological foundation, we consider lattices expanded with quasioperators,

i.e., operations that suitably generalize normal modal operatos, residuals, orthocomplements

and the like. This hinges on both the duality for lattices and for semilattices

in a natural way.

This talk is based on joint work with Peter Jipsen.

Date: May 2010.

1

In this talk we will first survey results which guarantee the existence of

spanning subgraphs in dense graphs. This will lead us to the proof of the

bandwidth-conjecture by Bollobas and Komlos, which states that any graph

with minimum degree at least $(1-1/r+\epsilon)n$ contains every r-chromatic graph

with bounded maximum degree and sublinear bandwidth as a spanning subgraph.

We will then move on to discuss the analogous question for a host graph that

is obtained by starting from a sparse random graph G(n,p) and deleting a

certain portion of the edges incident at every vertex.

This is joint work with J. Boettcher, Y. Kohayakawa and M. Schacht.

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.