Mathematical Methods in Theoretical Computer Science (CS286a)

Instructor: Chris Umans

Office: Jorgensen 286

Times: Tu/Th 1:00-2:25 in Jorgensen 262

Office hours: email me

Announcements:

• Presentations will be 40 minutes apiece; they will be on Dec 2 (Tues), 4 (Thurs), 8 (Mon), and 11 (Thurs) starting at 1pm. I'd like everyone who is in town to attend. There WILL be food.
• Talk schedule:
  • Tuesday Dec. 2 at 1pm: 1. Chris 2. Bill
  • Thursday Dec. 4 at 1pm: 1. Raga 2. Brian
  • Monday Dec. 8 at 1pm: 1. Dave 2. Euiwoong
  • Thursday Dec. 11 at 1pm: 1. Shankar 2. Hwan-seung 3. Arda

Handouts:

• Syllabus (pdf)

Lectures:

• Lecture 1: introduction and algebra review
• Lecture 2: nearly-linear time algorithms for polynomials (FFT and multiplication)
• Lecture 3: nearly-linear time algorithms for polynomials (multiplication continued, division with remainder, multipoint evaluation, interpolation, GCD)
• Lecture 4: integer multiplication (De et al. 2008).
• Lecture 5: integer multiplication continued; matrix multiplication (Cohn et al. 2005)
• Lecture 6: matrix multiplication continued
• Lecture 7: matrix multiplication continued; quantum factoring algorithm (Shor)
• Lecture 8: quantum factoring algorithm continued
• Lecture 9: cancelled due to FOCS
• Lecture 10: Fourier analysis on the Boolean cube and constant-depth circuits (Linial et al.)
• Lecture 11: LMN continued; Bazzi's Theorem (Razborov 2008)
• Lecture 12: Bazzi's Theorem continued; PRGs against constant degree polynomials (Viola 2008)
• Lecture 13: PRGs against constant degree polynomials, continued
• Lecture 14: polynomial factorization and modular composition (Kedlaya/Umans 2008)
• Lecture 15: polynomial factorization and modular composition, continued
• Lecture 16: finishing up modular composition; PV codes (please read Parvaresh/Vardy 2005)
• Lecture 17: PV codes continued; Folded Reed-Solomon codes (please read Guruswami/Rudra 2006)

Paper list:

• A. De, P. Kurur, C. Saha, R. Saptharishi. Fast Integer Multiplication using Modular Arithmetic. 2008.
• See also: M. Furer. Faster Integer Multiplication. 2007.
• H. Cohn, R. Kleinberg, B. Szegedy, C. Umans. Group-theoretic algorithms for matrix multiplication. 2005.
• See also: H. Cohn, C. Umans. A group-theoretic approach to fast matrix multiplication. 2005.
• P. Shor. Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer. 2004.
• See also: R. Raz. Quantum Computation. (Lectures 1-3).
• N. Linial, Y. Mansour, N. Nisan. Constant depth circuits, Fourier transform, and learnability. 1993.
• A. Razborov. A simple proof of Bazzi's theorem. 2008.
• See also: L. Bazzi. Polylogarithmic independence can fool DNF formulas. 2007.
• D. Moshkovitz, R. Raz. Sub-constant error low-degree test of almost-linear size (Section 5). 2006.
• J. Hastad. Some optimal inapproximability results. 1997.
• See also: U. Vazirani. Fourier Transforms & Theoretical Computer Science. (Lectures 6-7).
• E. Viola. The sum of d small-bias generators fools polynomials of degree d. 2008.
• See also: A. Bogdanov, E. Viola. Pseudorandom bits for polynomials. 2007.
• See also: S. Lovett. Unconditional Pseudorandom Generators for Low Degree Polynomials. 2008.
• K. Kedlaya, C. Umans. Fast polynomial factorization and modular composition. 2008.
• See also: E. Kaltofen, V. Shoup. Subquadratic-time factoring of polynomials over finite fields. 1995.
• F. Parvaresh, A. Vardy. Correcting errors beyond the Guruswami-Sudan radius in polynomial time. 2005.
• V. Guruswami, A. Rudra. Explicit codes achieving list decoding capacity: Error-correction with optimal redundancy. 2006.
• V. Guruswami, C. Umans, S. Vadhan. Unbalanced Expanders and Randomness Extractors from Parvaresh-Vardy Codes. 2007.
• C. Umans. Pseudo-random Generators for All Hardnesses. 2002.
• See also: R. Shaltiel, C. Umans. Simple Extractors for All Min-Entropies and a New Pseudo-Random Generator. 2001.
• E. Ben-Sasson, S. Kopparty, J. Radhakrishnan. Subspace Polynomials and List Decoding of Reed-Solomon Codes. 2006.
• A. Ta-Shma, C. Umans. Better Lossless Condensers Through Derandomized Curve Samplers. 2006.

Useful online resources:

• Madhu Sudan: Algebra and Computation (2005 course)
• Umesh Vazirani: Fourier Transforms and Theoretical Computer Science (1999 course)
• Victor Shoup: A computational introduction to number theory and algebra (book)

Possible papers for presentation:

• either of two topics we skipped in the first part of the course [Moshkovitz/Raz claimed by Raga]
• any of the last three topics that we skipped in the second part of the course [Ben-Sasson/Kopparty/Radhakrishnan claimed by Chris]
• A. Klivans and R. Servedio. Learning DNF in time 2^{O(n^{1/3})} [claimed by Euiwoong]
• Z. Dvir.On the size of Kakeya sets in finite fields + followup papers by Dvir/Wigderson, and Saraf/Sudan [claimed by Dave]
• S. Yekhanin Towards 3-query locally decodable codes of subexponential length + followup papers by Kedlaya/Yekhanin and Efremenko [claimed by Hwan-seung]
• B. Barak, R. Impagliazzo, and A. Wigderson. Extracting Randomness Using Few Independent Sources and Bourgain's 2-source extractor (see A. Rao. An Exposition of Bourgain's 2-Source Extractor )
• Dimension expanders (Dvir, Shpilka)
• Extractors for algebraic sources (Dvir, Gabizon, Wigderson)
• M. Agrawal, N. Kayal, and N. Saxena PRIMES is in P [claimed by Brian]
• V. Guruswami and P. Raghavendra A 3-query PCP over the integers [claimed by Shankar]
• quantum factoring via phase estimation [claimed by Bill]
• Dixon's integer factorization algorithm [claimed by Arda]