# Projects -- CS 254

The final project involves: studying a topic not covered in the course; writing a short report explaining the context of the problem, the main ideas of the proof and, in more details, a core technical statement in the proof; giving a 25-minute blackboard presentation. A group of two people can collaborate on a project. In such a case they will take on a more challenging topic (the list below includes subjects of varying level of difficulty), go into more details in the report, and split the time in a 40-minute presentation. Here are some ideas for projects:
• Toda's Theorem. The proof is in Section 17.4 of Arora-Barak
• Proof Complexity. The notion of proof by 'resolution' and exponential lower bounds on resolution proof length for random 3SAT. A very powerful (and relatively simple) approach is via the notion of `width' of a proof. See this paper by Ben-Sasson and Wigderson.
• Quantum Computing. We didn't have time to talk about Shor's algorithm for factoring, whose existence is probably the main reason for the interest in quantum algorithms. Section 10.6 of Arora-Barak
• Monotone Circuit Lower Bounds. Razborov's proof that clique requires superpolynomial monotone circuits is one of the landmarks of circuit complexity. Section 14.3 of Arora-Barak
• Explicit Construction of Expanders. There are two constructions that are relatively simple to describe and analyse.
• The one by Gabor and Galil is simpler to describe but requires a bit of harmonic analysis to be proved correct. It is described in Lecture 7 of these notes, although you should also look at the previous lecture for the basics of spectral graph theory. (It's simpler than it sounds.)
• The one of Reingold, Vadhan and Wigderson via the zig-zag graph product is more complicated to describe but has a completely "elementary" analysis that needs only basic linear algebra. It is described in lecture 8 of the same set of notes.
• The Hastad Switching Lemma is a very powerful tool to prove circuit lower bounds, and has several other applications, including to learning theory. Paul Beame's exposition "A switching lemma primer" is a good source to learn the proof.