CS153: Current Topics in Theoretical Computer Science (Spring 2012)

Instructor: Chris Umans

Office: Annenberg 311

Times: Tu/Th 1:00-2:25 in Annenberg 314

Office hours: email me

Announcements:

• (6/8/2012) Please stop by my office if you'd like to pick up your graded problem sets. Have a great summer!

Handouts:

• Syllabus (pdf)

Lectures:

• Lecture 1: introduction; models of algebraic computation
• Lecture 2: bilinear circuits and tensor rank; Strassen's algorithm
• Lecture 3: border rank
• Lecture 4: Asymptotic Sum Inequality
• Lecture 5: groups, representations and the group-theoretic approach
• Lecture 6: examples satisfying the triple product property
• Lecture 7: beating the sum of the cubes
• Lecture 8: construction based on two families
• Lecture 9: constructions based on USPs and strong USPs
• Lecture 10: optimal USPs via hashing and pruning
• Lecture 11: the basic Coppersmith-Winograd algorithm
• Lecture 12: analyzing the square of the basic algorithm
• Lecture 13: problems equivalent to matrix multiplication
• Lecture 14: VP vs. VNP, lower bound for general arithmetic circuits via partial derivatives
• Lecture 15: depth reduction for arithmetic formulas and circuits
• Lecture 16: "chasm at depth 4"; universality of determinant
• Lecture 17: lower bounds for monotone circuits, and non-commutative formulas

Problem sets:

• Problem Set 1: (pdf) Solutions: (pdf)
• Problem Set 2: (pdf) Solutions: (pdf)

Papers:

• H. Cohn and C. Umans. A Group-theoretic Approach to Fast Matrix Multiplication. 2003.
• H. Cohn, R. Kleinberg, B. Szegedy, and C. Umans. Group-theoretic Algorithms for Matrix Multiplication. 2005.
• A. Stothers. On the complexity of matrix multiplication. 2010.
• V. V. Williams. Breaking the Coppersmith-Winograd barrier 2012.

Surveys and course notes:

• M. Blaser. Complexity of Bilinear Problems. 2009.
• A. Shpilka and A. Yehudayoff. Arithmetic circuits: a survey of recent results and open questions. 2010.

Possible papers for presentation:

• N. Alon, A. Shpilka and C. Umans. On Sunflowers and Matrix Multiplication. (claimed by Jing)
• F. Le Gall. Faster Algorithms for Rectangular Matrix Multiplication. (claimed by Zeyu)
• P. Koiran. Arithmetic Circuits: The Chasm at Depth Four Gets Wider. (claimed by Timothy)
• More to come...