Minimal Discrepancy of Isolated Singularities and Reeb Orbits

Mark McLean
Stony Brook University
April 4, 2014
Let A be an affine variety inside a complex N dimensional vector space which either has an isolated singularity at the origin or is smooth at the origin. The intersection of A with a very small sphere turns out to be a contact manifold called the link of A. Any contact manifold contactomorphic to the link of A is said to be Milnor fillable by A. If the first Chern class of our link is 0 then we can assign an invariant of our singularity called the minimal discrepancy, which is an important invariant in birational geometry.

Do NP-Hard Problems Require Exponential Time?

Andrew Drucker
Institute for Advanced Study; Member, School of Mathematics
April 8, 2014
The P != NP conjecture doesn't tell us what runtime is needed to solve NP-hard problems like 3-SAT and Hamiltonian Path. While some clever algorithms are known, they all require exponential time, and some researchers suspect that this is unavoidable. This idea is expressed in the influential "Exponential Time Hypothesis" (ETH) of Impagliazzo, Paturi, and Zane. In this survey talk, I will describe the ETH and its consequences (some of which are rather subtle). We will see that this hypothesis holds considerable explanatory power.

Applications of additive combinatorics to Diophantine equations

Alexei Skorobogatov
Imperial College London
April 10, 2014
The work of Green, Tao and Ziegler can be used to prove existence and approximation properties for rational solutions of the Diophantine equations that describe representations of a product of norm forms by a product of linear polynomials. One can also prove that the Brauer-Manin obstruction precisely describes the closure of rational points in the adelic points for pencils of conics and quadrics over \(\mathbb Q\) when the degenerate fibres are all defined over \(\mathbb Q\).