Subject 
A main objective of theoretical computer science is to understand the amount of resources needed to solve computational problems. While the design and analysis of algorithms puts upper bounds on such amounts, computational complexity theory is mostly concerned with lower bounds; In this class, we will be largely interested in infeasible problems, that is computational problems that require impossibly large resources to be solved, even on instances of moderate size. It is very hard to show that a particular problem is infeasible, and in fact for a lot of interesting problems the question of their feasibility is still open. Another direction this class studies is the relations between different computational problems and between different “modes” of computation. For example what is the relative power of algorithms using randomness and deterministic algorithms, what is the relation between worstcase and averagecase complexity, how easier can we make an optimization problem if we only look for approximate solutions, and so on. A tentative syllabus can be found here. 
Prerequisites 
Mathematical maturity; exposure to advanced undergraduate material in Algorithms, and in Discrete Probability and Combinatorics. More specifically, CS 374, and MATH 461 or STAT 400 or equivalent are required. If you have not taken those classes but believe that your background is close to being sufficient, please make sure you have filled up any potential gaps by the end of the second week of classes. You can refer to the standard algorithms and probability textbooks for the classes above as supplementary material. If you are not sure whether your background suffices, please see the instructor. The course is designed for graduate students but may be suitable for advanced undergraduates. Undergraduate students who are interested in taking the course are advised to consult with the instructor before registering. 
InstructorTeaching Assistant 
Alexandra Kolla (akolla [at] illinois [dot] edu) 3222 SC [AK] Spencer Gordon (slgordo2 [at] illinois [dot] edu) 3111 [SG]

Times 
Wednesday 11:00AM  12:15PM and Friday 11:00AM  12:15PM  1109 SC 
Office Hours 
Alexandra Kolla  Wednesdays 4:00PM @ 3222 SC Spencer Gordon  Tuesdays 2:00PM @ Siebel 3rd Floor Lounge 
#  Date  Topic  Lecture Slides  Reading Material 

1  W January 18  Introduction to Complexity Theory  Slides  
2  F January 20  P vs NP, time hierarchy theorems  Slides  See these lecture notes and Chapters 2,3 from AroraBarak 
3  W January 25  Time Hierarchy Thoerems cont., Space Complexity  Slides  See Chapter 4 from AroraBarak 
4  F January 27  NL=coNL, Polynomial Hierarchy  Slides  See these lecture notes and Chapters 4,5 from AroraBarak 
5  W February 1  Polynomial Hierarchy  See previous lecture  See previous lecture 
6  F February 3  Boolean Circuits  Slides  See these lecture notes and Chapter 6 from AroraBarak 
7  W February 8  Randomized Computation  Slides  See these lecture notes and Chapter 7 from AroraBarak 
7  F February 10  ValiantVazirani  Slides  See these lecture notes 
8  W February 15  Counting Problems  Slides  See these lecture notes and Chapter 17 from AroraBarak 
9  F February 17  Interactive Proofs  Slides  See Chapter 8 from AroraBarak 
10  W February 22  IP=PSPACE Warmup  Slides 1 and Slides 2  See Luca's notes and Chapter 14 from AroraBarak 
11  F February 24  GoldwasserSipser Set Lower Bound, MIP  N/A  See Chapter 14 from AroraBarak 
12  W March 1  Undirected Connectivity, Introduction to Eigenvalues  Slides  See these lecture notes 
13  F March 3  More on Eigenvalues  N/A  See these lecture notes 
14  W March 8  No class.  
15  F March 10  Eigenvalues and Random Walks, UCONN in RL  Slides  See these lecture notes 
16  W March 15  QuasiRandom Properties of Expanders, PRGs  Slides  See these lecture notes 
17  F March 17  Linear Algebra Review  N/A  N/A 
18  W March 22  ZigZag Product and Expanders  N/A  See these lecture notes 
19  F March 24  ZigZag Product and Expanders, cont.  N/A  See these lecture notes 
20  W March 29  More on the ZigZag Product  N/A  See these lecture notes 
21  F March 31  More on the ZigZag Product, cont.  N/A  See these lecture notes 
22  W April 5  L=SL  N/A  See these lecture notes and this paper 
23  F April 7  Intro to Quantum Computing  N/A  See these lecture notes. 
Homework #  Due  Homework Solutions 

HW1  February 3 before the end of class  Solutions 
HW2  February 17 before the end of class   
HW3  March 3 before the end of class   
HW4  March 29 before the end of class   
HW5  April 14 before the end of class   
HW6  May 3 before the end of class   
Homework 

There will be a total of 56 homeworks. 
They can be turned in in groups of up to 3 students. All students in each group get the same grade. 
Please do not forget to cite your sources (you will get a zero if you use material from elsewhere and do not cite the source!) 
Project 

There will be a final project. More details to come. 
Here is a list of potential project ideas: Project Ideas. 
We can provide references to help you get started looking into any of these projects. You may also come up with a project idea of your own. 
Grading 

70% homeworks and 30% exam/final project. Extra points may be given for class participation at various occasions. 

Part 1: "Classical Complexity Theory" (about 5 weeks) 
 We will start with "basic" and "classical" material about time, space, P versus NP, polynomial hierarchy, circuit complexity and so on, including moderately modern and advanced material, such as the power of randomized algorithms, the complexity of counting problems, the averagecase complexity of problems, and interactive proofs. 
Part 2: "Spectral Techniques for Complexity Theory " (about 4 weeks)

 We will learn techniques from spectral graph theory and apply them to several key complexity theory results such as expander graphs, the Unique Games Conjecture, pseudorandom generators, and Reingold's L=SL result.

Part 3: "Quantum Computing" (about 2 weeks)

 We will focus on quantum complexity theory. 
Part 4 (tentative): "Final Presentations" (remaining time) 
Depending on attendance. 