Instructor: Emanuele Viola
Meetings: Tuesday 10:30 - 12:10 in Room 166 WVH map and Friday 1:35 PM - 3:15 PM Room 429 Ryder Hall (RY), 11 Leon St., campus map, map.
Office Hours: by appointment.
This course covers some of the most exciting and recent
progress in theoretical computer science. It presents state-of-the-art results on active research areas, and teaches related proof techniques. A tentative list of topics includes:
Lower bounds for constant-depth circuits.
The Nisan-Wigderson pseudorandom generator.
Cryptography in constant parallel time.
The complexity of Nash equilibria.
Undirected connectivity in logarithmic space (SL = L).
Communication complexity.
Primes is in P.
Fast matrix multiplication.
No background is required for this class. However, this is a theoretical course with emphasis on theorems and proofs, so ``mathematical maturity'' is expected.
Each student is required to scribe (#lectures/#students) lectures.
| Lecture | Content | Lecture notes |
Extra |
| Jan 6 | Overview of the course. | Template for scribes. | Sanjeev Arora and Boaz Barak's book on complexity. Oded Goldreich's material on complexity, including his book. |
| Jan 9 | Preliminaries: Probability, correlation, circuits, and pseudorandom generators. | Lecture 1. | Problem 1. (*) |
| Jan 13 |
The Nisan-Wigderson pseudorandom generator: I Yao's next-bit predictor. |
Lecture 2. | |
| Jan 16 |
The Nisan-Wigderson pseudorandom generator: II |
Lecture 3. | Problems 2,3. (*) |
| Jan 20 |
Parity requires large constant-depth circuits:
Aspnes, Beigel, Furst, and Rudich's proof. I |
||
| Jan 23 |
Parity requires large constant-depth circuits:
Aspnes, Beigel, Furst, and Rudich's proof. II |
Lectures 4 and 5. | |
| Jan 27 |
Parity requires large constant-depth circuits:
Aspnes, Beigel, Furst, and Rudich's proof. III |
||
| Jan 30 |
Parity has exponentially small correlation with small
constant-depth circuits: Klivans and Vadhan's proof. | Lectures 6 and 7. | Problem 4. (*) |
| Feb 3 |
Arithmetic in log-depth circuits: addition, iterated addition,
multiplication. Beame, Cook, and Hoover's log-depth circuits for division. I |
Lecture 8. | |
| Feb 6 |
Beame, Cook, and Hoover's log-depth circuits for division. II Proof of the weak prime number theorem. |
Lecture 9. | |
| Feb 10 |
Valiant's result that log-depth linear-size is in depth-3
subexponential-size. |
Lecture 10. | |
| Feb 13 |
Barrington's theorem. |
Lecture 11. | |
| Feb 17 |
Applebaum, Ishai, and Kushilevitz's cryptograhy in constant depth: I |
Lecture 12. | |
| Feb 20 | Applebaum, Ishai, and Kushilevitz's cryptograhy in constant depth: II | ||
| Feb 24 |
Applebaum, Ishai, and Kushilevitz's cryptograhy in constant depth: III | Lectures 13 and 14. | |
| Feb 27 | Spectral graph theory. Undirected reachability in randomized log space. | ||
| Spring break. | Problems 5,6. (*) | ||
| Mar 10 | Undirected reachability in log space, Rozenman and Vadhan's proof: I | Course on expanders by Linial and Wigderson. | |
| Mar 13 | Undirected reachability in log space, Rozenman and Vadhan's proof: II | ||
| Mar 17 | Undirected reachability in log space, Rozenman and Vadhan's proof: III Primes in P: I | Lectures 15-18 on undirected reachability in log space. | |
| Mar 20 | Primes in P: II | Andrew Granville's survey, Victor Shoup's book . | |
| Mar 24 | Primes in P: III | Lectures 18-20 on Primes in P. | |
| Mar 27 | Group-theoretic algorithms for fast matrix multiplication: I | ||
| Mar 31 | Group-theoretic algorithms for fast matrix multiplication: II | ||
| Apr 3 | Group-theoretic algorithms for fast matrix multiplication: III Succinct data structures: I | Lectures 21-23 on matrix multiplication. | |
| Apr 7 | Succinct data structures: II | Lectures 23-24 on succinct data structures. | |
| Apr 10 | Communication complexity | ||
| Apr 14 | Multiparty communication complexity: I | ||
| Apr 17 | Multiparty communication complexity: II | Lectures 25-27 on communication complexity. | |
| Apr 21 | Natural Proofs | Lecture 28 on natural proofs. |