Georgia Institute of Technology College of Sciences

In a paper published in 2012, Nathan Dunfield and StavrosGaroufalidis gave simple, sufficient conditions for a spunnormal surface tobe essential in a compact, orientable 3-manifold with torus boundary. Thistalk will discuss a generalization of this result which utilizes a theoremfrom tropical geometry.

Given f: \C \times T^1 to itself, an analytic perturbation of a fibered rotation map , we will present two proofs of existence of an analytic conjugation of f to the fibered rotation , on a neighborhood of {0} \times T^1. This talk will be self- contained except for some usual "tricks" from KAM theory and which will be explained better in another talk. In the talk we will discuss carefully the number theoretic conditions on the fibered rotation needed to obtain the theorem.

Erdos and Szemeredi conjectured that if one has a set of n numbers, one must have either the sumset or product set be of nearly maximal size, cn^2/log(n). In this talk, he will introduce the sum-product problem in the reals, show previous, beautiful geometric proofs by Solymosi and Elekes, and discuss some recent progress by Amirkhanyan, Croot, Pryby and Bush.

We will discuss Milnor's classic proof of the existence of exotic smooth structures on the 7-sphere.

The problem of finding large average submatrices of a real-valued matrix arises in the exploratory analysis of data from disciplines as diverse as genomics and social sciences. Motivated in part by previous work on this applied problem, this talk will present several new theoretical results concerning large average submatrices of an n x n Gaussian random matrix. We will begin by considering the average and joint distribution of the k x k submatrix having largest average value (the global maximum). We then turn our attention to submatrices with dominant row and column sums, which arise as the local maxima of a practical iterative search procedure for large average submatrices I will present a result characterizing the value and joint distribution of a local maximum, and show that a typical local maxima has an average value within a constant factor of the global maximum. In the last part of the talk I will describe several results concerning the *number* L_n(k) of k x k local maxima, including the asymptotic behavior of its mean and variance for fixed k and increasing n, and a central limit theorem for L_n(k) that is based on Stein's method for normal approximation. Joint work with Shankar Bhamidi (UNC) and Partha S. Dey (UIUC)

Although production is an integral part of the Arrow-Debreu market model, most of the work in theoretical computer science has so far concentrated on markets without production, i.e., the exchange economy. In this work, we take a significant step towards understanding computational aspects of markets with production. We first define the notion of separable, piecewise-linear concave (SPLC) production by analogy with SPLC utility functions. We then obtain a linear complementarity problem (LCP) formulation that captures exactly the set of equilibria for Arrow-Debreu markets with SPLC utilities and SPLC production, and we give a complementary pivot algorithm for finding an equilibrium. This settles a question asked by Eaves in 1975. Since this is a path-following algorithm, we obtain a proof of membership of this problem in PPAD, using Todd, 1976. We also obtain an elementary proof of existence of equilibrium (i.e., without using a fixed point theorem), rationality, and oddness of the number of equilibria. Experiments show that our algorithm runs fast on randomly chosen examples, and unlike previous approaches, it does not suffer from issues of numerical instability. Additionally, it is strongly polynomial when the number of goods or the number of agents and firms is constant. This extends the result of Devanur and Kannan (2008) to markets with production. Based on a joint work with Vijay V. Vazirani.

In this talk we will extend the sutured product disk decompositions from the last talk to construct foliations on some knot complements and see how this can help understand the minimal genus of Seifert surfaces for knots and links.