Week 18.01.2009 – 24.01.2009

A brane tiling is a bicolored graph on an oriented real 2torus, which conjecturally describes both the derived category of coherent sheaves on a 2dimensional toric Fano stack and the derived category of the directed Fukaya category of the mirror. When the toric Fano stack is the projective plane, the corresponding brane tiling divides the torus into three hexagons. In the talk, based on a joint work in progress with Masahiro Futaki, I will describe the analogue of brane tiling for the projective space, which divides the real 3torus into four truncated octahedra, and explain how it helps to study a torus-equivariant version of homological mirror symmetry.

The eigenvalue statistics at the edge of the spectrum of large random matrices from the Gaussian unitary ensemble (GUE) are described by the Airy point process and the maximal eigenvalue is asymptotically Tracy-Widom distributed. In contrast, for the complex Ginibre ensemble (consisting of matrices with iid complex Gaussian entries), extreme eigenvalues behave like a Poisson process and the maximal modulus (or maximal real part) of the eigenvalues converges to a Gumbel-distributed random variable. In this talk, a family of ensembles interpolating between these models is considered, and we show how a non-trivial transition between Airy and Poisson statistics occurs for the eigenvalues near the edge of the spectrum.

In this talk we will consider dynamics with infinitely many time derivatives, such equations follow directly from string field theory and have many interesting properties. First we will review results for the case of Minkowski background and then consider coupling to nontrivial background, in particular, to Friedmann-Robertson-Walker metric. New methods for solving corresponding nonlocal Friedmann equations will be presented and resulting solutions in the view of cosmological applications will be discussed.

This is the first lecture of a short course on non-geometric backgrounds. For more information on the course and the schedule, please visit http://www.strings.ph.qmw.ac.uk/index.htm and follow the link to the Graduate Program in String/Field Theory.