Past

Homology counts components and cycles, K-theory counts vector bundles and bundles of Clifford algebra modules.  What about geometric models for other generalized cohomology theories?  There is a vision, introduced by Segal, Stolz, and Teichner, that certain cohomology theories should be expressible in terms of topological field theories.

I will describe how the 0-th K-theory group can be formulated in terms of equivalence classes of 1-dimensional topological field theories.  Then I will discuss what it means to twist a topological field theory, and explain that the n-th K-theory group comes from twisted 1-dimensional topological field theories.

The expectation is that 2-dimensional topological field theories should be analogously related to elliptic cohomology.  I will take an extended digression to explain what elliptic cohomology is and why it is interesting.  Then I will discuss 2-dimensional twisted field theory and explain how it leads us toward a notion of higher

("2-dimensional") algebra.  

We will introduce both the classical Hanna Neumann Conjecture and its strengthened version, discuss Stallings' reformulation in terms of immersions of graphs, and look at some partial results. If time allows we shall also look at the new approach of Joel Friedmann.

We shall explain what is the weighted fundamental lemma and how it is related to the truncated Hitchin fibration.

Ordinary homology is a geometrically defined invariant of spaces: the 0-th homology group counts the number of components; the n-th homology group counts n-cycles, which correspond to an intuitive notion of 'n-dimensional holes' in a space.  K-theory, or more specifically the 0-th K-theory group, is defined in terms of vector bundles, and so also has an immediate relationship to geometry.  By contrast, the n-th K-theory group is typically defined homotopy-theoretically using the black box of Bott periodicity.

I will describe a more geometric perspective on K-theory, using Z/2-graded vector bundles and bundles of modules for Clifford algebras.  Along the way I will explain Clifford algebras, 2-categories, and Morita equivalence, explicitly check the purely algebraic 8-fold periodicity of the Clifford algebras, and discuss how and why this periodicity implies Bott periodicity.

The talk will not presume any prior knowledge of K-theory, Clifford algebras, Bott periodicity, or the like.

Starting from a definition of the cohomology of a group, we will define the bounded cohomology of a group. We will then show how quasi-homomorphisms lead to cocycles in the second bounded cohomology group, and use this to look at the second bounded cohomology of some of our favourite groups. If time permits we will end with some applications.

Consider a configuration of points in $d$-dimensional Euclidean space

together with a set of constraints

which fix the direction or the distance between some pairs of points.

Basic questions are whether the constraints imply that the configuration

is unique or locally unique up to congruence, and whether it is bounded. I

will describe some solutions

and partial solutions to these questions.

Consider the space M of parabolas y=ax^2+bx+c, with (a, b, c) as coordinates on M. Two parabolas generically intersect at two (possibly complex) points, and we can define a conformal structure on M by declaring two points to be null separated iff the corresponding parabolas are tangent. A simple calculation of discriminant shows that this conformal structure is flat.

In this talk (based on joint works with Godlinski and Sokolov) I shall show how replacing parabolas by rational plane curves of higher degree allows constructing curved conformal structures in any odd dimension. In dimension seven one can use this "twistor" construction to find G_2 structures in a conformal class.

We shall explain what is the weighted fundamental lemma and how it is related to the truncated Hitchin fibration.

In this talk we describe some recent work on shock

reflection problems for the potential flow equation. We will

start with discussion of shock reflection phenomena. Then we

will describe the results on existence, structure and

regularity of global solutions to regular shock reflection. The

approach is to reduce the shock reflection problem to a free

boundary problem for a nonlinear elliptic equation, with

ellipticity degenerate near a part of the boundary (the sonic

arc). We will discuss techniques to handle such free boundary

problems and degenerate elliptic equations. This talk is based

on joint works with Gui-Qiang Chen, and with Myoungjean Ba

Whether as the sudoku puzzles of popular culture or as

restricted coloring problems on graphs or hypergraphs, completing partial

Latin squares and cubes present a framework for a variety of intriguing

problems. In this talk we will present several recent results on

completing partial Latin squares and cubes.

In addition to existence, the excess-demand approach allows us to establish uniqueness and provide efficient computational algorithms for various complete- and incomplete-market stochastic financial equilibria.

A particular attention will be paid to the case when the agents exhibit constant absolute risk aversion. An overview of recent results (including those jointly obtained with M. Anthropelos and with Y. Zhao) will be given.

This will discuss the paper of Ricci, Tseytlin & Wolf from 2007.

'Compressive sampling' is a topic of current interest. It relies on data being sparse in some domain, which allows what is apparently 'sub Nyquist' sampling so that the quantities of data which must be handled become more closely related to the information rate. This principal would appear to have (at least) three applications for radar and electronic warfare: \\

The most modest application is to reduce the amount of data which we must handle: radar and electronic warfare receivers generate vast amounts of data (up to 1Gbit/second or even 10Gbit.sec). It is desirable to be able to store this data for future analysis and it is also becoming increasingly important to be able to share it between different sensors, which, prima facie, requires vast communication bandwidths and it would be valuable to be able to find ways to handle this more efficiently. \\

The second advantage is that if suitable data domains can be identified, it may also be possible to pre-process the data before the analogue to digital converters in the receivers, to reduce the demands on these critical components. \\

The most ambitious use of compressive sensing would be to find ways of modifying the radar waveforms, and the electronic warfare receiver sampling strategies, to change the domain in which the information is represented to reduce the data rates at the receiver 'front ends', i.e. make the data at the front end better match the information we really want to acquire.\\

The aim of the presentation will be to describe the issues with which we are faced, and to discuss how compressive sampling might be able to help. A particular issue which will be raised is how we might find domains in which the data is sparse.

Standard quantum logic, as intitiated by Birkhoff and von Neumann, suffers from severe problems which relate quite directly to interpretational issues in the foundations of quantum theory. In this talk, I will present some aspects of the so-called topos approach to quantum theory, as initiated by Isham and Butterfield, which aims at a mathematical reformulation of quantum theory and provides a new, well-behaved form of quantum logic that is based upon the internal logic of a certain (pre)sheaf topos.

Bloch Floquet waves are considered in structured media. Such waves are dispersive and the dispersion diagrams contain stop bands. For an example of a harmonic lattice, we discuss dynamic band gap Green’s functions characterised by exponential localisation. This is followed by simple models of exponentially localised defect modes. Asymptotic models involving uniform asymptotic approximations of physical fields in structured media are compared with homogenisation approximations.

I will discuss what is known about the cohomology of several moduli spaces coming from algebraic and differential geometry. These are: moduli spaces of non-singular curves (= Riemann surfaces) $M_g$, moduli spaces of nodal curves $\overline{M}_g$, moduli spaces of holomorphic line bundles on curves $Hol_g^k \to M_g$, and the universal Picard varieties $Pic^k_g \to M_g$. I will construct characteristic classes on these spaces, talk about their homological stability, and try to explain why the constructed classes are the only stable ones. If there is time I will also talk about the Picard groups of these moduli spaces.

Much of this work is due to other people, but some is joint with J. Ebert.

\ \ 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.