L3

String field theory is a candidate for a full non-perturbative definition

of string theory. We aim to define string field theory on a space-time

lattice to investigate its behaviour at the quantum level. Specifically, we

look at string field theory in a one dimensional linear dilaton background,

using level truncation to restrict the theory to a finite number of fields.

I will report on our preliminary results at level-0 and level-1.

Much effort has been devoted to the study of weak scale particles, e.g. supersymmetric neutralinos, which have a relic abundance from thermal equilibrium in the early universe of order what is inferred for dark matter. This does not however provide any connection to the comparable abundance of baryonic matter, which must have a non-thermal origin. However "dark baryons" of mass ~5 GeV from a new strongly interacting sector would naturally provide dark matter and are consistent with recent putative signals in experiments such as CoGeNT and DAMA. Such particles would accrete in the Sun and affect heat transport in the interior so as to affect low energy neutrino fluxes and can possibly resolve the current conflict between helioseismological data and the Standard Solar Model.

Gravitational instantons are complete hyperkaehler 4-manifolds whose Riemann curvature tensor is square integrable. They can be viewed as Einstein geometry analogs of finite energy Yang-Mills instantons on Euclidean space. Classical examples include Kronheimer's ALE metrics on crepant resolutions of rational surface singularities and the ALF Riemannian Taub-NUT metric, but a classification has remained largely elusive. I will present a large, new connected family of gravitational instantons, based on removing fibers from rational elliptic surfaces, which contains ALG and ALH spaces as well as some unexpected geometries.

I will describe recent work on motivic DT invariants for 3-manifolds, which are expected to be a refinement of Chern-Simons theory. The conclusion will be that these should be possible to define and work with, but there will be some interesting problems along the way. There will be a discussion of the problem of upgrading the description of the moduli space of flat connections as a critical locus to the problem of describing the fundamental group algebra of a 3-fold as a "noncommutative critical locus," including a recent topological result on obstructions for this problem. I will also address the question of how a motivic DT invariant may be expected to pick up a finer invariant of 3-manifolds than just the fundamental group.

Sutured manifolds are compact oriented 3-manifolds with boundary, together with a set of dividing curves on the boundary. Sutured Floer homology is an invariant of balanced sutured manifolds that is a common generalization of the hat version of Heegaard Floer homology and knot Floer homology. I will define cobordisms between sutured manifolds, and show that they induce maps on sutured Floer homology groups, providing a type of TQFT. As a consequence, one gets maps on knot Floer homology groups induced by decorated knot cobordisms.

Consider the valued field $\mathbb{R}((\Gamma))$ of generalised series, with real coefficients and

monomials in a totally ordered multiplicative group $\Gamma$ . In a series of papers,

we investigated how to endow this formal algebraic object with the analogous

of classical analytic structures, such as exponential and logarithmic maps,

derivation, integration and difference operators. In this talk, we shall discuss

series derivations and series logarithms on $\mathbb{R}((\Gamma))$ (that is, derivations that

commute with infinite sums and satisfy an infinite version of Leibniz rule, and

logarithms that commute with infinite products of monomials), and investigate

compatibility conditions between the logarithm and the derivation, i.e. when

the logarithmic derivative is the derivative of the logarithm.

Vopenka's Principle is a very strong large cardinal axiom which can be used to extend ZFC set theory. It was used quite recently to resolve an important open question in algebraic topology: assuming Vopenka's Principle, localisation functors exist for all generalised cohomology theories. After describing the axiom and sketching this application, I will talk about some recent results showing that Vopenka's Principle is relatively consistent with a wide range of other statements known to be independent of ZFC. The proof is by showing that forcing over a universe satisfying Vopenka's Principle will frequently give an extension universe also satisfying Vopenka's Principle.