Oxford

In 1992 (or thereabouts) Bloch and Kriz gave the first explicit definition of the category of mixed Tate motives (MTM). Their definition relies heavily on the theory of algebraic cycles. Unfortunately, traditional methods of representing algebraic cycles (such as in terms of formal linear combinations of systems of polynomial equations) are notoriously difficult to work with, so progress in capitalizing on this description of the category to illuminate outstanding conjectures in the field has been slow. More recently, Gangl, Goncharov, and Levin suggested a simpler way to understand this category (and by extension, algebraic cycles more generally) by relating specific algebraic cycles to rooted, decorated, planar trees. In our talks, describing work in progress, we generalize this correspondence and attempt to systematize the connection between algebraic cycles and graphs. We will construct a Lie coalgebra L from a certain algebra of admissible graphs, discuss various properties that it satisfies (such as a well defined and simply described realization functor to the category of mixed Hodge structures), and relate the category of co-representations of L to the category MTM. One promising consequence of our investigations is the appearance of alternative bases of rational motives that have not previously appeared in the literature, suggesting a richer rational structure than had been previously suspected. In addition, our results give the first bounds on the complexity of computing admissibility of algebraic cycles, a previously unexplored topic.

Many interesting properties of groups are inherited by their subgroups examples of such are finiteness, residual finiteness and being free. People have asked whether hyperbolicity is inherited by subgroups, there are a few counterexamples in this area. I will be detailing the proof of some of these including a construction due to Rips of a finitely generated not finitely presented subgroup of a hyperbolic group and an example of a finitely presented subgroup which is not hyperbolic.

This is another opportunity to hear the 2013 LMS Presidential Address:

Abstract: The idea of space is central to the way we think.  It is the technology we have evolved for interpreting our experience of the world.  But space is presumably a human creation, and even inside mathematics it plays a variety of different roles, some modelling our intuition very closely and some seeming almost magical.  I shall point out how the homotopy category in particular breaks away from its own roots.  Then I shall describe how quantum theory leads us beyond the well-established notion of a topological space into the realm of noncommutative geometry.  One might think that noncommutative spaces are not very space-like, and yet it is noncommutativity that makes the world look as it does to us, as a collection of point particles.

I will explain how to compute the symplectic cohomology of a manifold $M$ conical at infinity, whose Reeb flow at infinity arises from a Hamiltonian circle-action on $M$. For example, this allows one to compute the symplectic cohomology of negative line bundles in terms of the quantum cohomology, and (in joint work with Ivan Smith) via the open-closed string map one can determine the wrapped Fukaya category of negative line bundles over projective space. In this talk, I will show that one can explicitly compute the quantum cohomology and symplectic cohomology of Fano toric negative line bundles, which are in fact different cohomology groups, and surprisingly it is actually the symplectic cohomology which recovers the Jacobian ring of the Landau-Ginzburg superpotential.

 We'll discuss the connection between irreducibilty, D- and 
aD-spaces.

Noncommutative projective geometry is the study of quantum versions of projective space and other projective varieties.  Starting with the celebrated work of Artin, Tate and Van den Bergh on noncommutative projective planes, a substantial theory of noncommutative curves and surfaces has been developed, but the classification of noncommutative versions of projective three-space remains unknown.  I will explain how a portion of this classification can be obtained, via deformation quantization, from a corresponding classification of holomorphic foliations due to Cerveau and Lins Neto.  In algebraic terms, the result is an explicit description of the deformations of the polynomial ring in four variables as a graded Calabi--Yau algebra.

We produce new families of steady and expanding Ricci solitons
that are not of Kahler type. In the steady case, the asymptotics are
a mixture of the Hamilton cigar and the Bryant soliton paraboloid
asymptotics. We obtain some examples of Ricci solitons on homeomorphic
but non-diffeomorphic spaces. We also find numerical evidence of solitons
with more complicated topology.

I will start by introducing Somos sequences, defined by innocent-looking quadratic recursions which, surprisingly, always return integer values. I will then explain how they can be viewed in a much larger context, that of the Laurent phenomenon in the theory of cluster algebras. Some further steps take us to the the quantum cluster positivity conjecture of Berenstein and Zelevinski. I will finally explain how, following Nagao and Efimov, cohomological Donaldson-Thomas theory leads to a proof of this conjecture in some, perhaps all, cases. This is joint work with Davison, Maulik, Schuermann.

Computational structures---from simple objects like bits and qubits,

to complex procedures like encryption and quantum teleportation---can

be defined using algebraic structures in a symmetric monoidal

2-category. I will show how this works, and demonstrate how the

representation theory of these structures allows us to recover the

ordinary computational concepts. The structures are topological in

nature, reflecting a close relationship between topology and

computation, and allowing a completely graphical proof style that

makes computations easy to understand. The formalism also gives

insight into contentious issues in the foundations of quantum

computing. No prior knowledge of computer science or category theory

will be required to understand this talk.