Georgia Institute of Technology College of Sciences

The importance of analyzing big data and in particular very large networks has shown that the traditional notion of a fast algorithm, one that runs in polynomial time, is often insufficient. This is where property testing comes in, whose goal is to very quickly distinguish between objects that satisfy a certain property from those that are ε-far from satisfying that property. It turns out to be closely related to major developments in combinatorics, number theory, discrete geometry, and theoretical computer science. Some of the most general results in this area give "constant query complexity" algorithms, which means the amount of information it looks at is independent of the input size. These results are proved using regularity lemmas or graph limits. Unfortunately, typically the proofs come with no explicit bound for the query complexity, or enormous bounds, of tower-type or worse, as a function of 1/ε, making them impractical. We show by entirely new methods that for permutations, such general results still hold with query complexity only polynomial in 1/ε. We also prove stronger results for graphs through the study of new metrics. These are joint works with Jacob Fox.

Since Erdős–Rényi introduced random graphs in 1959, two closely related models for random graphs have been extensively studied. In the G(n,m) model, a graph is chosen uniformly at random from the collection of all graphs that have n vertices and m edges. In the G(n,p) model, a graph is constructed by connecting each pair of two vertices randomly. Each edge is included in the graph G(n,p) with probability p independently of all other edges. Researchers have studied when the random graph G(n,m) (or G(n,p), resp.) satisfies certain properties in terms of n and m (or n and p, resp.). If G(n,m) (or G(n,p), resp.) satisfies a property with probability close to 1, then one may say that a `typical graph’ with m edges (or expected edge density p, resp.) on n vertices has the property. Random graphs and their variants are also widely used to prove the existence of graphs with certain properties. In this talk, two problems for these categories will be discussed. First, a new approach will be introduced for the problem of the emergence of a giant component of G(n,p), which was first considered by Erdős–Rényi in 1960. Second, a variant of the graph process G(n,1), G(n,2), …, G(n,m), … will be considered to find a tight lower bound for Ramsey number R(3,t) up to a constant factor. (No prior knowledge of graph theory is needed in this talk.)

(1) A set D of natural numbers is called t-intersective if every positive upper density subset A of natural numbers contains a (t+1)-length arithmetic progression (AP) whose common differences is in D. Szemeredi's theorem states that the set of all natural numbers is t-intersective for every t. But there are other non-trivial examples like {p-1: p prime}, {1^k,2^k,3^k,\dots} for any k etc. which are t-intersective for every t. A natural question to study is at what density random subsets of natural numbers become t-intersective? (2) Let X_t be the number of t-APs in a random subset of Z/NZ where each element is selected with probability p independently. Can we prove precise estimates on the probability that X_t is much larger than its expectation? (3) Locally decodable codes (LDCs) are error correcting codes which allow ultra fast decoding of any message bit from a corrupted encoding of the message. What is the smallest encoding length of such codes? These seemingly unrelated problems can be addressed by studying the Gaussian width of images of low degree polynomial mappings, which seems to be a fundamental tool applicable to many such problems. Adapting ideas from existing LDC lower bounds, we can prove a general bound on Gaussian width of such sets which reproves the known LDC lower bounds and also implies new bounds for the above mentioned problems. Our bounds are still far from conjectured bounds which suggests that there is plenty of room for improvement. If time permits, we will discuss connections to type constants of injective tensor products of Banach spaces (or chernoff bounds for tensors in simpler terms). Joint work with Jop Briet.