Georgia Institute of Technology College of Sciences

Pricing and allocating goods to buyers with complex preferences in order to maximize some desired objective (e.g., social welfare or profit) is a central problem in Algorithmic Mechanism Design. In this talk I will discuss some particularly simple algorithms that are able to achieve surprisingly strong guarantees for a range of problems of this type. As one example, for the problem of pricing resources, modeled as goods having an increasing marginal extraction cost to the seller, a simple approach of pricing the i-th unit of each good at a value equal to the anticipated extraction cost of the 2i-th unit gives a constant-factor approximation to social welfare for a wide range of cost curves and for arbitrary buyer valuation functions. I will also discuss simple algorithms with good approximation guarantees for revenue, as well as settings having an opposite character to resources, namely having economies of scale or decreasing marginal costs to the seller.

I will survey the major results in graph and hypergraph Ramsey theory and present some recent results on hypergraph Ramsey numbers. This includes a hypergraph generalization of the graph Ramsey number R(3,t) proved recently with Kostochka and Verstraete. If time permits some proofs will be presented.

Information-based complexity is an alternative to Turing complexity that is well-suited for understanding a broad class of convex optimization algorithms. The groundbreaking work of Nemirovski and Yudin provided the basis of the theory, establishing tight lower bounds on the running time of first-order methods in a number of settings. There has been a recent interest on these classical techniques, because of exciting new applications on Machine Learning, Signal Processing, Stochastic Programming, among others. In this talk, we will introduce the rudiments of the theory, some examples, and open problems. Based on joint work with Gábor Braun and Sebastian Pokutta.

In this series of talks I will begin by discussing the idea of studying smooth manifolds and their submanifolds using the symplectic (and contact) geometry of their cotangent bundles. I will then discuss Legendrian contact homology, a powerful invariant of Legendrian submanifolds of contact manifolds. After discussing the theory of contact homology, examples and useful computational techniques, I will combine this with the conormal discussion to define Knot Contact Homology and discuss its many wonders properties and conjectures concerning its connection to other invariants of knots in S^3.