Georgia Institute of Technology College of Sciences

Database privacy has garnered a recent surge in interest from the theoretical science community following the seminal work of Dwork 2006, which proposed the strong notion of differential privacy. In this setting, each row of a database corresponds to the data owned by some (distinct) individual. An analyst submits a database query to a differentially private mechanism, which replies with a noisy answer guaranteeing privacy for the data owners and accuracy for the analyst. The mechanism's privacy parameter \epsilon is correlated negatively with privacy and positively with accuracy.This work builds a framework for creating and analyzing a market that 1) solves for some socially efficient value of \epsilon using the privacy and accuracy preferences of a heterogeneous body of data owners and a single analyst, 2) computes a noisy statistic on the database, and 3) collects and distributes payments for privacy that elicit truthful reporting of data owners' preferences. We present a market for database privacy in this new framework expanding on the public goods market of Groves and Ledyard, 1977.

A sandpile on a graph is an integer-valued function on the vertices. It evolves according to local moves called topplings. Some sandpiles stabilize after a finite number of topplings, while others topple forever. For any sandpile s_0 if we repeatedly add a grain of sand at an independent random vertex, we eventually reach a sandpile s_\tau that topples forever. Statistical physicists Poghosyan, Poghosyan, Priezzhev and Ruelle conjectured a precise value for the expected amount of sand in this "threshold state" s_\tau in the limit as s_0 goes to negative infinity. I will outline the proof of this conjecture in http://arxiv.org/abs/1402.3283 and explain the big-picture motivation, which is to give more predictive power to the theory of "self-organized criticality".

Generalized triangle T_r is an r-graph with edges {1,2,…,r}, {1,2,…,r-1, r+1} and {r,r+1, r+2, …,2r-2}. The family \Sigma_r consists of all r-graphs with three edges D_1, D_2, D_3 such that |D_1\cap D_2|=r-1 and D_1\triangle D_2\subset D_3. In 1989 it was conjectured by Frankl and Furedi that ex(n,T_r) = ex(n,\Sigma_r) for large enough n, where ex(n,F) is the Tur\'{a}n function. The conjecture was proven to be true for r=3, 4 by Frankl, Furedi and Pikhurko respectively. We settle the conjecture for r=5,6 and show that extremal graphs are blow-ups of the unique (11, 5, 4) and (12, 6, 5) Steiner systems. The proof is based on a technique for deriving exact results for the Tur\'{a}n function from “local stability" results, which has other applications. This is joint work with Sergey Norin.

We discuss bi-parameter Calderon-Zygmund singular integrals from the point of view of modern probabilistic and dyadic techniques. In particular, we discuss their structure and boundedness via dyadic model operators. In connection to this we demonstrate, via new examples, the delicacy of the problem of finding a completely satisfactory product T1 theorem. Time permitting related non-homogeneous bi-parameter results may be mentioned.