Conference

We investigate the feasibility of explosive particle production via parametric resonance or tachyonic preheating in multi-field inflationary models by means of lattice simulations. We observe a strong suppression of resonances in the presence of four-leg interactions between the inflaton fields and a scalar matter field, leading to insufficient preheating when more than two inflatons couple to the same matter field. This suppression is caused by a dephasing of the inflatons that increases the effective mass of the matter field. Including three-leg interactions leads to tachyonic preheating, which is not suppressed by an increase in the number of fields. If four-leg interactions are sub-dominant, we observe a slight enhancement of tachyonic preheating. Thus, in order for preheating after multi-field inflation to be efficient, one needs to ensure that three-leg interactions are present. If no tachyonic contributions exist, we expect the old theory of reheating to be applicable.

Modifications of the initial-state of the inflaton field can induce a departure from Gaussianity and leave a testable imprint on the higher order correlations of the CMB and large scale structures in the Universe. I will discuss general vacuum state modifications in the case of a canonical single-field action, after adding a dimension 8 higher order derivative term, and DBI models of inflation. Observed bounds on local and equilateral non-Gaussianities, even though they correspond to template shapes that are far from optimal, can lead to constraints that are already competing to those derived from the power spectrum alone, due to enhancement effects. We emphasize that the construction and application of especially adapted templates could lead to significant improvements in the CMB bispectrum constraints on modified initial states.

"A Hamiltonian action of a Lie group on a symplectic manifold $(M,\omega)$ gives rise to a gauge theoretic deformation of the Cauchy-Riemann equations, called the symplectic vortex equations. Counting solutions of these equations over the complex plane leads to a quantum version of the Kirwan map. In joint work with Christopher Woodward, we interpret this map as a weak morphism of cohomological field theories."

This talk will discuss, illustrated by a toy example, how to construct "higher-algebraic" quantum field theories using groupoids. In particular, the groupoids describe configuration spaces of connections, together with their gauge symmetries, on spacetime, space, and boundaries of regions in space. The talk will describe a higher-algebraic "sum over histories", and how this construction is related to usual QFT's, and particularly the relation to the case of the Chern-Simons theory.

The space of regular noncommutative algebras includes regular graded Clifford algebras, which correspond to base point free linear systems of quadrics in dimension n in P^n. The schemes of linear modules for these algebras can be described in terms of this linear system. We show that the space of line modules on a 4 dimensional algebra is an Enriques surface called the Reye congruence, and we extend this result to higher dimensions.

The classical "split" rational R-matrix Poisson bracket structure on the space of rational connections over the Riemann sphere provides a natural setting for studying deformations. It can be shown that a natural set of Poisson commuting spectral invariant Hamiltonians, which are dual to the Casimir invariants of the Poisson structure, generate all deformations which, when viewed as nonautonomous Hamiltonian systems, preserve the generalized monodromy of the connections, in the sense of Birkhoff (i.e., the monodromy representation, the Stokes parameters and connection matrices). These spectral invariants may be expressed as residues of the trace invariants of the connection over the spectral curve. Applications include the deformation equations for orthogonal polynomials having "semi-classical" measures. The $\tau$ function for such isomonodromic deformations coincides with the Hankel determinant formed from the moments, and is interpretable as a generalized matrix model integral. They are also related to Seiberg-Witten invariants. (This talk is based in part on joint work with: Marco Bertola, Gabor Pusztai and Jacques Hurtubise)

I will discuss the metric behavior of the Kahler-Ricci flow on Hirzebruch surfaces assuming that the initial metric is invariant under a maximal compact subgroup of the automorphism group. I will describe how, in the sense of Gromov-Hausdorff, the flow either shrinks to a point, collapses to P^1 or contracts an exceptional divisor. This confirms a conjecture of Feldman-Ilmanen-Knopf. This is a joint work with Jian Song.

"In this joint work with Jingyi Chen and Weiyong He, we prove existence of longtime smooth solutions to mean curvature flow of entire Lipschitz Lagrangian graphs. A Bernstein type result for translating solitons is also obtained."

I will describe some combinatorial problems which arise when computing various types of partition functions for the Donaldson-Thomas theory of a space with a torus action. The problems are all variants of the following: give a generating function which enumerates the number of ways to pile n cubical boxes in the corner of a room. Often the resulting generating functions are nice product formulae, as predicted by the recent wall-crossing formulae of Kontsevich-Soibelman. There are now a variety of techniques, both geometric and combinatorial, to compute these formula. My work uses the entirely combinatorial techniques, namely vertex operators and the planar dimer model; these techniques can be applied essentially "bare-handed" and rely very little upon the underlying algebraic geometry.