## Orientability and Open Gromov-Witten Invariants

Penka Georgieva
Princeton University
November 11, 2011

I will first discuss the orientability of the moduli spaces of J-holomorphic maps with Lagrangian boundary conditions. It is known that these spaces are not always orientable and I will explain what the obstruction depends on. Then, in the presence of an anti-symplectic involution on the target, I will give a construction of open Gromov-Witten disk invariants. This is a generalization to higher dimensions of the works of Cho and Solomon, and is related to the invariants defined by Welschinger

## An Isoperimetric Inequality for the Hamming Cube and Integrality Gaps in Bounded-Degree Graphs

Siavosh Benabbas
Institute for Advanced Study
November 21, 2011
In 1970s Paul Erdos asked the following question: Consider all the boolean strings of length n. Assume that one has chosen a subset S of the strings such that no two chosen strings are different in precisely n/4 (or its closest even integer) coordinates. How big can the set of chosen strings be? Erdos conjectured that the answer is small (in a precise sense) and put a \$250 prize for a solution. In 1987 Frankl and Rodl proved a strong form of this conjecture showing that such a set has to be exponentially smaller than the set of all strings of length n.

## Entropy-Based Bounds on Dimension Reduction in L_1

Oded Regev
CNRS-ENS-Paris and Tel Aviv University
November 28, 2011