Past

• (a) nonconvergence for the unregularized mathematical problem or
• (b) nonuniqueness of the limit if it exists, or
• (c) limiting solutions only in the very weak form of a space time dependent probability distribution.

We have seen that L-functions of elliptic curves of conductor N coincide exactly with L-functions of weight 2 newforms of level N from the Modularity Theorem. We will show how, using modular symbols, we can explicitly compute bases of newforms of a given level, and thus investigate L-functions of an elliptic curve of given conductor. In particular, such calculations allow us to numerically test the Birch-Swinnerton-Dyer conjecture.

This will be a review of recent work that obtains amplitudes at strong coupling from certain minimal surfaces in AdS.

9:50am Welcome \\

10:00am Malcolm McCulloch (Engineering, Oxford), "Dual usage of land: Solar power and cattle grazing"; \\

10:45am Jonathan Moghal (Materials, Oxford), “Anti-reflectance coatings: ascertaining microstructure from optical properties”; \\

11:15am (approx) Coffee \\

11:45am Agnese Abrusci (Physics, Oxford), "P3HT based dye-sensitized solar cells"; \\

12:15pm Peter Foreman (Destertec UK), "Concentrating Solar Power and Financial Issues" \\

1:00pm Lunch.

One may answer a question of Macintyre by showing that there are recursive existentially closed dimension groups.  One may build such groups having most of the currently known special properties of finitely generic dimension groups, though no finitely generic dimension group is arithmetic.

We report the numerical realization and demonstration of robustness of certain 2-component structures in Bose-Einstein Condensates in 2 and 3 spatial dimensions with non-trivial topological charge in one of the components. In particular, we identify a stable symbiotic state in which a higher-dimensional bright soliton exists even in a homogeneous setting with defocusing interactions, as a result of the effective potential created by a stable vortex in the other component. The resulting vortex-bright solitary waves, which naturally generalize the recently experimentally observed dark-bright solitons, are examined both in the homogeneous medium and in the presence of parabolic and periodic external confinement and are found to be very robust.

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.

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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.

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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