236602 Theory Lab Winter 2017/2018

This is a discovery-style class, based on work in groups during class time.

The students will work on weekly sheets:

- Week 1 (22 Oct 2017): Fourier analysis: introduction.
- Week 2 (29 Oct 2017): Fourier analysis: influence.
- Week 3 (5 Nov 2017): Fourier analysis: noise.
- Week 4 (12 Nov 2017): Fourier analysis: KKL.
- Week 5 (19 Nov 2017): Information theory: prefix codes.
- Week 6 (26 Nov 2017): Information theory: basic definitions.
- Week 7 (3 Dec 2017): Information theory: applications.
- Week 8 (10 Dec 2017): Information theory: coding.
- Week 9 (24 Dec 2017): Chernoff bound.
- Week 10 (31 Dec 2017): Coding theory: basic notions.
- Week 11 (7 Jan 2018): Coding theory: some codes.
- Week 12 (14 Jan 2018): Linear programming.
- Week 13 (21 Jan 2018): Bonus questions and open discussion.

Bonus sheets:

Possible topics for final assignment (other topics are welcome):

- Fourier analysis:
- Friedgut’s junta theorem and applications or related results.
- $p$-biased analysis and applications, for example the Friedgut–Kalai sharp threshold theorem.
- Gaussian space, the invariance principle, and Majority is Stablest.

- Information theory:
- Pinsker’s inequality and applications.
- Lower bounds in communication complexity.
- Proof of the parallel repetition theorem due to Ankit Garg and Mark Braverman.

- Concentration of measure and large deviation bounds:
- Martingales, Azuma’s inequality, and McDiarmid’s inequality.
- Martingales per se (for example, the optional stopping theorem).

- Coding theory:
- Efficient decoding algorithms.
- List decoding.
- Applications in theory of computation.

- Linear programming:
- Proof of strong duality via Farkas’ lemma.
- Simplex algorithm (and another proof of strong duality).
- Separation oracles.
- Applications to approximation algorithms.
- Semidefinite programming.

- Approximation theory:
- Proof of Markov’s inequality.
- Other applications.

- Algorithms in number theory:
- Primality testing.
- Fast integer multiplication.
- Fast matrix multiplication.
- Integer factorization.
- Polynomial factorization.

- Expander graphs:
- Basic definitions, (double-sided) Cheeger’s inequality, Alon–Boppana.
- Applications, for example Bourgain’s embedding theorem and its tightness.