University of Oxford

Words are building blocks of sentences, yet the meaning of a sentence goes well beyond meanings of its words. Formalizing the process of meaning assignment is proven a challenge for computational and mathematical linguistics; with the two most successful approaches each missing on a key aspect: the 'algebraic' one misses on the meanings of words, the vector space one on the grammar.

I will present a theoretical setting where we can have both! This is based on recent advances in ordered structures by Lambek, referred to as pregroups and the categorical/diagrammatic approach used to model vector spaces by Abramsky and Coecke. Surprisingly. both of these structures form a compact category! If time permits, I will also work through a concrete example, where for the first time in the field we are able to compute and compare meanings of sentences compositionally. This is collaborative work with E. Greffenstete, C. Clark, B. Coecke, S. Pulman.

Solving partial differential equations (PDEs) on curved surfaces is

important in many areas of science. The Closest Point Method is a new

technique for computing numerical solutions to PDEs on curves,

surfaces, and more general domains. For example, it can be used to

solve a pattern-formation PDE on the surface of a rabbit.

A benefit of the Closest Point Method is its simplicity: it is easy to

understand and straightforward to implement on a wide variety of PDEs

and surfaces. In this presentation, I will introduce the Closest

Point Method and highlight some of the research in this area. Example

computations (including the in-surface heat equation,

reaction-diffusion on surfaces, level set equations, high-order

interface motion, and Laplace--Beltrami eigenmodes) on a variety of

surfaces will demonstrate the effectiveness of the method.

The space BD of functions of bounded deformation consists of all L1-functions whose distributional symmetrized derivative (defined by duality with the symmetrized gradient ($\nabla u + \nabla u^T)/2$) is representable as a finite Radon measure. Such functions play an important role in a variety of variational models involving (linear) elasto-plasticity. In this talk, I will present the first general lower semicontinuity theorem for symmetric-quasiconvex integral functionals with linear growth on the whole space BD. In particular we allow for non-vanishing Cantor-parts in the symmetrized derivative, which correspond to fractal phenomena. The proof is accomplished via Jensen-type inequalities for generalized Young measures and a construction of good blow-ups, based on local rigidity arguments for some differential inclusions involving symmetrized gradients. This strategy allows us to establish the lower semicontinuity result even without a BD-analogue of Alberti's Rank-One Theorem in BV, which is not available at present. A similar strategy also makes it possible to give a proof of the classical lower semicontinuity theorem in BV without invoking Alberti's Theorem.

Many classes of polarised projective algebraic varieties can be constructed via explicit constructions of corresponding graded rings. In this talk we will discuss two methods, namely Basket data method and Key varieties method, which are often used in such constructions. In the first method we will construct graded rings corresponding to some topological data of the polarised varieties. The second method is based on the notion of weighted flag variety, which is the weighted projective analogue of a flag variety. We will describe this notion and show how one can use their graded rings to construct interesting classes of polarised varieties.

Many problems in computer science can be modelled as metric spaces, whereas for mathematicians they are more likely to appear as the opening question of a second year examination. However, recent interesting results on the geometry of finite metric spaces have led to a rethink of this position. I will describe some of the work done and some (hopefully) interesting and difficult open questions in the area.

The topology of the moduli space of stable bundles (of coprime rank and degree) on a smooth curve can be understood from different points of view. Atiyah and Bott calculated the Betti numbers by gauge-theoretic methods (using equivariant Morse theory for the Yang-Mills functional), arriving at the same inductive formula which had been obtained previously by Harder and Narasimhan using arithmetic techniques. An intermediate interpretation (algebro-geometric in nature but dealing with infinite-dimensional parameter spaces as in the gauge theory picture) comes from thinking about vector bundles in terms of matrix divisors, generalising the Abel-Jacobi map to higher rank bundles.

I'll sketch these different approaches, emphasising their parallels, and in the end I'll speculate about how (some of) these methods could be made to work when the underlying curve acquires nodal singularities.

I will talk about a recent paper of Huh, who, building on a wealth of pretty geometry and topology, has given a proof of a conjecture dating back to 1968 regarding the chromatic polynomial (the polynomial that determines how many ways there are of colouring the vertices of a graph with n colours in such a way that no vertices which are joined by an edge have the same colour). I will mainly talk about the way in which a problem that is explicitly a combinatorics problem came to be encoded in algebraic geometry, and give an overview of the geometry and topology that goes into the solution. The talk should be accessible to everyone: no stacks, I promise.

Consider the action of a complex reductive group on a complex projective variety X embedded in projective space. Geometric Invariant Theory allows us to construct a 'categorical' quotient of an open subset of X, called the semistable set. If in addition X is smooth then it is a symplectic manifold and in nice cases we can construct a moment map for the action and the Marsden-Weinstein reduction gives a symplectic quotient of the group action on an open subset of X. We will discuss both of these constructions and the relationship between the GIT quotient and the Marsden-Weinstein reduction. The quotients we have discussed provide a quotient for only an open subset of X and so we then go on to discuss how we can construct quotients of certain subvarieties contained in the complement of the semistable locus.

I will first introduce and motivate the notion of 'homological stability' for a sequence of spaces and maps. I will then describe a method of proving homological stability for configuration spaces of n unordered points in a manifold as n goes to infinity (and applications of this to sequences of braid groups). This method also generalises to the situation where the configuration has some additional local data: a continuous parameter attached to each point.

However, the method says nothing about the case of adding global data to the configurations, and indeed such configuration spaces rarely do have homological stability. I will sketch a proof -- using an entirely different method -- which shows that in some cases, spaces of configurations with additional global data do have homological stability. Specifically, this holds for the simplest possible global datum for a configuration: an ordering of the points up to even permutations. As a corollary, for example, this proves homological stability for the sequence of alternating groups.