Tue, January 31, 2017
Public Access

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January 2017
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11:00am [11:00am] Math Colloquium
TITLE: Progress in Error-Correction: A Survey Speaker: Venkatesan Guruswami, Carnegie Mellon Univ. ABSTRACT: Error-correcting codes play a crucial role in safeguarding data against the adverse effects of noise during communication and storage. They are also powerful tools underlying several recent advances in theoretical computer science and combinatorics. The central challenge in coding theory is to construct codes with minimum possible redundancy for different error models and requirements on the decoder, along with efficient algorithms for error-correction using those codes. Much progress has been made toward this quest in the nearly seven decades since the birth of coding theory. Several fundamental problems, however, continue to challenge us, and exciting new directions routinely emerge to address current technological demands as well as applications in computational complexity and cryptography. This talk will survey some of our recent works on error-correction in various models, such as: - worst-case errors, where we construct list decodable codes with redundancy as small as the target error fraction; - i.i.d. errors, where we show polar codes enable efficient error-correction even as the redundancy approaches Shannon capacity; - bit deletions, where we give codes that can correct the largest known fraction of deletions; - single symbol erasure, a model of substantial current interest for tackling node failures in distributed storage, where we give novel repair algorithms for Reed-Solomon codes as well as simple new codes with low-bandwidth repair mechanisms.

3:00pm [3:30pm] S.G. Dani
Title: Values of quadratic forms at integer points II Abstract: This will be a continuation of the overview from the last week. Some details will be briefly recalled from the last time, for continuity and the benefit of new audience if any.

5:00pm [5:00pm] Saurav Bhaumik
Title: Some consequences of the Riemann hypothesis for varieties over finite field Abstract: We will talk about a result of M. Katz and W. Messing, which says the following. From the Riemann hypothesis and the hard Lefschetz theorem in l-adic cohomology, the corresponding facts for any Weil cohomology follow.