The course will cover the basics as well as some of the state-of-the-art results in pseudoranodmness, explicit constructions in combinatorics, and their applications. The main goal is to introduce the many fascinating questions and ideas in pseudorandomness and to cover versatile tools that are useful in other areas such as discrete mathematics (e.g., expanders), algorithms (e.g., hashing, streaming algorithms), and cryptography (extractors, hardness vs randomness).

Hours & Location: MW 4-5:50, Bunche 2156. Office hours: Monday 2:30 - 3:30, 3732 BH.



Assignments:
Assignment 1.

The following is a tentative list of topics to be covered.
Pseudorandomness: why, what, and how? (3 lectures)
Randomized algorithms
Probabilistic method
Probability primer
What is pseudorandomness?
Derandomization of concrete algorithms (2 lecture)
Conditional expectations
Derandomizing by pairwise independence
Limited independence: the swiss-army knife (2 lectures)
Universal hashing, k-wise independence
Applications and constructions
Error correcting codes (2 lectures)
Reed-Solomon codes
Small-bias spaces
Expander graphs (3 lectures)
The many views of expansion
Applications of expanders: Expander samplers
Zig-Zag product construction of expanders
Interlacing polynomials (3 lectures)
Ramanujan Graphs from interlacing polynomials
Kadison-Singer problem
Extractors and Ramsey Graphs (3 lectures)
Expansion beating eigenvalue bounds
Extractors and their applications
Ramsey graphs and 2-source extractors


Prerequisites: Basic knowledge of probability will be the most helpful. The students are expected to be mathematically mature and be able to follow and write down formal proofs. Familiarity with discrete mathematics, combinatorics, and undergraduate-level algorithms will be helpful.

Course work: We will have three assignments (15% each), one mid-term (25%), and one final (30%). The grading will also be flexible: students can, if they choose to, exchange an homework for more scribing duties or an approved research project (that can also be counted towards the report) for the final exam.

Policy and required text: There will be no makeup exams for the course; the mid-term will be held in class. A reference for several lectures is the book Pseudorandomness by Salil Vadhan. Links to appropriate papers or other online material (typically other lecture notes) will be provided for each lecture.

Academic honesty: The students are expected to fully abide by UCLA's student conduct policies, including Section 102.01 on academic honesty. You will find a wealth of helpful materials here, including the Student Guide to Academic Integrity. Academic dishonesty will be promptly reported to the Dean of Students' Office for adjudication and disciplinary action. Remember, cheating will have significant and irrevocable consequences for your academic record and professional future. Please don't cheat.

While collaboration with other students on assignments is fine, each student must write their solutions independently and should clearly mention the collaborators. You should never share your written solutions with someone else or copy from someone else's written solutions. Under no circumstances may you use solution sets to problems from previous year courses or from similar courses elsewhere or other resoruces on the web.