## Title: Locally decodable codes and arithmetic progressions in random settings

 dc.contributor.author Gopi, Sivakanth dc.contributor.corporatename Georgia Institute of Technology. School of Mathematics en_US dc.contributor.corporatename Microsoft Research en_US dc.date.accessioned 2018-11-26T18:59:22Z dc.date.available 2018-11-26T18:59:22Z dc.date.issued 2018-11-09 dc.description Presented on November 9, 2018 at 3:00 p.m. in Skiles 005. en_US dc.description Sivakanth Gopi is a postdocotoral researcher in the theory group at Microsoft Research Redmond. He recently graduated with a PhD from Princeton University where he was advised by Prof. Zeev Dvir. He is mainly interested in coding theory, pseudorandomness, complexity theory and additive combinatorics. en_US dc.description Runtime: 60:59 minutes en_US dc.description.abstract (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. en_US dc.format.extent 60:59 minutes dc.identifier.uri http://hdl.handle.net/1853/60557 dc.language.iso en_US en_US dc.relation.ispartofseries Combinatorics Seminar en_US dc.subject Arithmetic progressions en_US dc.subject Gaussian width en_US dc.subject Locally decodable codes en_US dc.title Locally decodable codes and arithmetic progressions in random settings en_US dc.type Moving Image dc.type.genre Lecture dspace.entity.type Publication local.contributor.corporatename College of Sciences local.contributor.corporatename School of Mathematics local.relation.ispartofseries School of Mathematics Colloquium relation.isOrgUnitOfPublication 85042be6-2d68-4e07-b384-e1f908fae48a relation.isOrgUnitOfPublication 84e5d930-8c17-4e24-96cc-63f5ab63da69 relation.isSeriesOfPublication 81718127-5196-4bb7-908f-64c744a91099
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