**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]