Past

\ \ The cluster category of Dynkin type $A_\infty$ is a ubiquitous object with interesting properties, some of which will be explained in this talk.

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\ \ Let us denote the category by $\mathcal{D}$. Then $\mathcal{D}$ is a 2-Calabi-Yau triangulated category which can be defined in a standard way as an orbit category, but it is also the compact derived category $D^c(C^{∗}(S^2;k))$ of the singular cochain algebra $C^*(S^2;k)$ of the 2-sphere $S^{2}$. There is also a “universal” definition: $\mathcal{D}$ is the algebraic triangulated category generated by a 2-spherical object. It was proved by Keller, Yang, and Zhou that there is a unique such category.

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\ \ Just like cluster categories of finite quivers, $\mathcal{D}$ has many cluster tilting subcategories, with the crucial difference that in $\mathcal{D}$, the cluster tilting subcategories have infinitely many indecomposable objects, so do not correspond to cluster tilting objects.

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\ \ The talk will show how the cluster tilting subcategories have a rich combinatorial

structure: They can be parametrised by “triangulations of the $\infty$-gon”. These are certain maximal collections of non-crossing arcs between non-neighbouring integers.

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\ \ This will be used to show how to obtain a subcategory of $\mathcal{D}$ which has all the properties of a cluster tilting subcategory, except that it is not functorially finite. There will also be remarks on how $\mathcal{D}$ generalises the situation from Dynkin type $A_n$ , and how triangulations of the $\infty$-gon are new and interesting combinatorial objects.

A categorification of cycle class maps consists to define

realization functors from constructible motivic sheaves to other

categories of coefficients (e.g. constructible $l$-adic sheaves), which are compatible with the six operations. Given a field $k$, we

will describe a systematic construction, which associates,

to any cohomology theory $E$, represented in $DM(k)$, a

triangulated category of constructible $E$-modules $D(X,E)$, for $X$

of finite type over $k$, endowed with a realization functor from

the triangulated category of constructible motivic sheaves over $X$.

In the case $E$ is either algebraic de Rham cohomology (with $char(k)=0$), or $E$ is $l$-adic cohomology, one recovers in this way the triangulated categories of $D$-modules or of $l$-adic sheaves. In the case $E$ is rigid cohomology (with $char(k)=p>0$), this construction provides a nice system of $p$-adic coefficients which is closed under the six operations.

Suppose that $A \subset \mathbb F_2^n$ has density $\Omega(1)$. How

large a subspace is $A+A:=\{a+a’:a,a’ \in A\}$ guaranteed to contain? We

discuss this problem and how the the result changes as the density

approaches $1/2$.

Starting from Morel and Voevodsky's stable homotopy theory of schemes, one defines, for each noetherian scheme of finite dimension $X$, the triangulated category $DM(X)$ of motives over $X$ (with rational coefficients). These categories satisfy all the the expected functorialities (Grothendieck's six operations), from

which one deduces that $DM$ also satisfies cohomological proper

descent. Together with Gabber's weak local uniformisation theorem,

this allows to prove other expected properties (e.g. finiteness

theorems, duality theorems), at least for motivic sheaves over

excellent schemes.

Colorectal cancer (CRC) is one of the leading causes of cancer-related death worldwide, demanding a response from scientists and clinicians to understand its aetiology and develop effective treatment. CRC is thought to originate via genetic alterations that cause disruption to the cellular dynamics of the crypts of Lieberkűhn, test-tube shaped glands located in both the small and large intestine, which are lined with a monolayer of epithelial cells. It is believed that during colorectal carcinogenesis, dysplastic crypts accumulate mutations that destabilise cell-cell contacts, resulting in crypt buckling and fission. Once weakened, the corrupted structure allows mutated cells to migrate to neighbouring crypts, to break through to the underlying tissue and so aid the growth and malignancy of a tumour. To provide further insight into the tissue-level effects of these genetic mutations, a multi-scale model of the crypt with a realistic, deformable geometry is required. This talk concerns the progress and development of such a model, and its usefulness as a predictive tool to further the understanding of interactions across spatial scales within the context of colorectal cancer.

• (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.