Course Prerequisites: The contents of the course will be well-suited to a grad or highly motivated undergrad with mathematical maturity. Exposure theory of computation is a definite plus.
Course Content: The techniques employed towards complexity lower bounds and related topics in the last 25 years span a broad range of mathematics areas, from combinatorics and logic to geometry, algebra, analysis (see for example ), these topics )
Rough Syllabus: We will draw up a list of chapters from Arora and Barak as well as carefully selected papers chosen from 5 broad topics that have many interconnections including powerful common lemmas.
Week 1 , we will sketch the 5 broad topics (we have to drastically cut and choose which to concentrate on - i.e., spend at least a month on.. NOTE: If people want to delve broader or deeper, one option is to **informally** spillover into the Friday Algo-theory seminar -- NOT part of the class):
--Approximation: Between P and NP
--Derandomization: P vs. BPP
--Circuit and Communication complexity lowerbounds: P vs. NP
--Permanent vs. Determinant: P vs. #P
--Proof length complexity: NP vs. co-NP
Very rough approximate order of lectures starting Week 2.
----Chapter 3:
-- Baker Gill Solovay Diagonalization barrier
to proving complexity lower bounds
----Chapter 3:
-- Ladner's result on intermediate problems between P and NP-complete
(Graph isomorphism: we hope to visit Babai's result in the Algo-Theory seminar)
--Approximation complexity hierarchy,
by Bordewich (in the spirit of Ladner),
--Approximation resistance: by Austrin/Hastad
--
An early survey by Trevisan
--
A somewhat later survey by Escoffier and Paschos
------Chapters 8,18,19, (IP, PCP theorem, Inapproximabilty)
VERY FEW carefully chosen papers related to some subset of:
--Metric Space Embedding,
--Unique games conjecture,
--Semidefinite programming (rank bounds and matrix factorization)
--Lassere hierarchy,
--Sum of squares, positivenstellensatz, and
--Integrality gaps
-- We hope to visit a relevant powerful spectral graph theory technique like
real stable interlacing polynomials
used to solve the Kadison Singer - Weaver paving conjecture
in the Friday Algo-Theory seminar
----
Month 2 (TBD):
--Chapters: 7A/B, 10, 16,17 + papers on derandomization
--Additive combinatorics and pseudorandomness (Szemeredi's regularity,
Gower's uniformity etc.: Samorodnitsky + Trevisan)
----
Month 3 (TBD):
--
Chapters 12, 13, 14 + papers on polynomial method towards circuit
lowerbounds, sign rank lower bounds viewed as matrix
factorization, Dvoretzky's theorem
--Chapter 22: Naturalization barrier to proving complexity bound
(diagonalization avoids naturalization!)
A blog discussion
--Algebrization barrier (Aaronson and Wigderson)
----Month 4 (TBD):
--
Invariants and the Mulmuley-Sohoni program towards hardness of the permananent
(Mulmuley surveys)
--GCT Chasm, Derandomization of Noether's normalization and Determinantal identity testing (algebraic
circuits) (Forbes, Shpilka, Kayal)
--Connection to Circuit and Proof complexity??
(Grochow/Pitassi ??).
Format/Grading: As is customary in theoretical computer science courses taught across the country (and which I have adopted whenever I teach a topics course), each student is responsible for scribing lecture notes for 1 week approximately. I will help them edit it (lots of back and forth) AFTER the lectures, then it becomes a presentable set of lecture notes for the class. They will probably also read and present 1-2 lectures each.
Course Time change (hopefully): It is currently listed as T/R 2-3, but since I already know that some interested students have a clash with that time, I will try to move it (after consulting with students in the first week of class).