# Seminars & Colloquia Calendar

## General Strong Polarization

#### Madhu Sudan, Harvard John A. Paulson School of Engineering and Applied Sciences

Location: ** CoRE 301**

Date & time: Wednesday, 04 April 2018 at 11:00AM - 12:00PM

Abstract: A martingale is a sequence of random variables that maintain their future expected value conditioned on the past. A \([0,1]\)-bounded martingale is said to polarize if it converges in the limit to either \(0\) or \(1\) with probability \(1\). A martingale is said to polarize strongly, if in \(t\) steps it is sub-exponentially close to its limit with all but exponentially small probability. In 2008, Arikan built a powerful class of error-correcting codes called Polar codes. The essence of his theory associates a martingale with every invertible square matrix over a field (and a channel) and showed that polarization of the martingale leads to a construction of codes that converge to Shannon capacity. In 2013, Guruswami and Xia, and independently Hassani et al. showed that strong polarization of the Arikan martingale leads to codes that converge to Shannon capacity at finite block lengths, specifically at lengths that are inverse polynomial in the gap to capacity, thereby resolving a major mathematical challenge associated with the attainment of Shannon capacity.

We show that a simple necessary condition for an invertible matrix to polarize over any non-trivial channel is also sufficient for strong polarization over all symmetric channels over all prime fields. Previously the only matrix which was known to polarize strongly was the \(2 \times 2\) Hadamard matrix. In addition to the generality of our result, it also leads to arguably simpler proofs. The essence of our proof is a ``local definition'' of polarization which only restricts the evolution of the martingale in a single step, and a general theorem showing the local polarization suffices for strong polarization.

In this talk I will introduce polarization and polar codes and, time permitting, present a full proof of our main theorem. No prior background on polar codes will be assumed.

Based on joint work with Jaroslaw Blasiok, Venkatesan Guruswami, Preetum Nakkiran and Atri Rudra.

R. Shapiro Organizer's Page

Chiara Damiolini, Ian Coley and Franco Rota -Charles Weibel Organizer's Page

Brooke Logan

Wujun Zhang Organizer's webpage

P. Gupta, X.Huang and J. Song Organizer's webpage

Swastik Kopparty, Sepehr Assadi Seminar webpage

Jeffry Kahn, Bhargav Narayanan, Jinyoung Park Organizer's webpage

Brooke Ogrodnik, Website

Robert Dougherty-Bliss and Doron Zeilberger --> homepage

Paul Feehan, Daniel Ketover, Natasa Sesum Organizer's webpage

Lev Borisov, Emanuel Diaconescu, Angela Gibney, Nicolas Tarasca, and Chris Woodward Organizer's webpage

Jason Saied Seminar webpage

Brian Pinsky, Rashmika Goswami website

Quentin Dubroff Organizer's webpage

James Holland; Organizer website

Edna Jones Organizer's webpage

Brooke Ogrodnik website

Yanyan Li, Zheng-Chao Han, Jian Song, Natasa Sesum Organizer's Webpage

Organizer: Luochen Zhao

Yanyan Li, Zheng-Chao Han, Natasa Sesum, Jian Song Organizer's Page

Lisa Carbone, Yi-Zhi Huang, James Lepowsky, Siddhartha Sahi Organizer's webpage

Simon Thomas website

Kasper Larsen, Daniel Ocone and Kim Weston Organizer's page

Joel Lebowitz, Michael Kiessling

Yanyan Li, Haim Brezis Organizer's Webpage

Stephen D. Miller, John C. Miller, Alex V. Kontorovich, Alex Walker seminar website

Stephen D. Miller

Brooke Ogrodnik, Website

Organizers: Yanyan Li, Z.C. Han, Jian Song, Natasa Sesum

Yael Davidov Seminar webpage

Kristen Hendricks, Xiaochun Rong, Hongbin Sun, Chenxi Wu Organizer's page

Fioralba Cakoni Seminar webpage

Ebru Toprak, Organizer

Organizer's webpage: Organizer's webpage

- Show events from all categories

## Special Note to All Travelers

Directions: map and driving directions. If you need information on public transportation, you may want to check the New Jersey Transit page.

*Unfortunately, cancellations do occur from time to time. Feel free to call our department: 848-445-6969 before embarking on your journey. Thank you.*