Lara Anderson.  Heterotic and Ftheory Compactifications and Geometry
 We systematically analyze a broad class of dual heterotic and
Ftheory models that give rise to six and fourdimensional supergravity
theories, and compare the geometric constraints on the two sides of the
duality. In this talk we will demonstrate that both theories together
give new insight into the space of possible effective theories arising
from string theory. We will describe recent explicit constructions of
all dual Ftheory/heterotic pairs in a broad class and discuss aspects
of Gflux in Ftheory.

Katrin Becker.  Mtheory and G_2 manifolds  This is an
introduction to Mtheory which is a theory of quantum gravity. It unifies
general relativity with quantum field theory. Mtheory includes besides
all string theories elevendimensional supergravity. In a Kaluza and Klein
approach to unification, a fourdimensional theory can be obtained by
assuming the additional seven dimensions to be small and compact. Unbroken
supersymmetry requires that the seven dimensional manifold admits a G2
holonomy metric. We will discuss how these classical solutions become
solutions of the quantum theory.

Jonathan Heckman.  Higher Derivative Holography, Estrings, and a 6D
Conformal Anomaly  In this talk we show how to use a recently
proposed formula for the conformal anomaly a_{6D} of 6D superconformal
field theories to probe the structure of higher derivative corrections
to 11D supergravity in the presence of nontrivial fourform fluxes and
a HoravaWitten 9brane.

Sheldon Katz.  The partition function of elliptically fibered
CalabiYau threefolds and Jacobi forms  I give evidence that the
all genus amplitudes of topological string theory on compact elliptically
fibered CalabiYau manifolds can be written in terms of meromorphic Jacobi
forms whose weight grows linearly and whose index grows quadratically
with the base degree. This talk is based on joint work with Albrecht
Klemm and Minxin Huang.

SiuCheong Lau.  Modular properties of SYZ mirrors  In this
talk we will construct the SYZ mirrors of certain infinitetype toric
CalabiYau manifolds and their quotients, and investigate their properties
under modular transformations of the global moduli. We will also study
the generalized SYZ mirrors of elliptic curve quotients. While an SYZ
mirror is a priori defined over a formal neighborhood around the large
volume limit, the study in all these cases suggests that it is indeed
the tip of the iceberg over the global moduli space.

Bong Lian.  Differential zeros of generalized hypergeometric
functions  I will discuss a new way to compute zero locus
of certain differential polynomials of period integrals of families
of algebraic varieties, including CalabiYau and general type
hypersurfaces in a Gspace. These zero locus generalizes zeros of
classical hypergeometric functions in one variable. The method is based
on a new algebraic description of PicardFuchs systems for the period
integrals, and generalized hypergeometric functions in particular. The
talk is based on joint work with J. Chen, A. Huang, S.T. Yau, and X. Zhu.

Laura Schaposnik.  Higgs bundles, branes, and applications.
 Higgs bundles (introduced by N. Hitchin in 1987) are
pairs of holomorphic vector bundles and holomorphic 1forms taking
values in the endomorphisms of the bundle. The moduli space of Higgs
bundles carries a natural Hyperkahler structure, through which we
can study Abranes (Lagrangian subspaces) or Bbranes (holomorphic
subspaces) with respect to each structure. We shall begin the talk by
first introducing Higgs bundles for complex Lie groups and the associated
Hitchin fibration, and recalling how to realize Langlands duality through
spectral data. We will then look at a natural construction of families
of subspaces which give different types of branes, and explain how the
topology of some of these branes can be completely determined via the
monodromy action of the Hitchin system. Finally, we shall give some
applications of the above approaches in relation to Langlands duality
and the study of character varieties. Some of the work presented during
the talk is in collaboration with David Baraglia (Adelaide).

Paul Seidel.  Fukaya categories and the enumerative geometry of
Lefschetz pencils  I will describe the conjectural interplay
between Fukaya categories of Lefschez fibrations and mirror maps for
CalabiYau hypersurfaces.

Nick Sheridan.  Counting curves using the Fukaya category 
In 1991, string theorists Candelas, de la Ossa, Green and Parkes made a
startling prediction for the number of curves in each degree on a generic
quintic threefold, in terms of periods of a holomorphic volume form on a
`mirror manifold'. Givental and Lian, Liu and Yau gave a mathematical
proof of this version of mirror symmetry for the quintic threefold
(and many more examples) in 1996. In the meantime (1994), Kontsevich
had introduced his `homological mirror symmetry' conjecture and stated
that it would `unveil the mystery of mirror symmetry'. I will explain
how to prove that the number of curves on the quintic threefold matches
up with the periods of the mirror via homological mirror symmetry. I
will also attempt to explain in what sense this is `less mysterious'
than the previous proof.

Cumrun Vafa. 
BPS degeneracies and Superconformal Index.  Abstract: I will discuss the
relation between BPS degeneracies and specialization of superconformal index in 2 and
4 dimensions and its relation with wallcrossing phenomena.
