60% of the grade is based on a project. The project (which can be done individually or in groups of two) can be a literature review, but I would recommended attempting to do original research, either by trying to make partial progress on (or completely solve!) one of the open problems below, or by pursuing another research direction. There are forty-two open problems (now 41, as in meanwhile Open Problem 1.3 has been solved [MS15]!).

Open Problems

0.1: Komlos Conjecture (PDF)

0.2: Matrix AM-GM Inequality (PDF)

1.1: Mallat and Zeitouni's Problem (PDF)

1.2: Monotonicity of Eigenvalues (PDF)

1.3: Cut SDP Spike Model Conjecture (PDF)Solved here

2.1: Ramsey Numbers (PDF)

2.2: Erdos-Hajnal Conjecture (PDF)

2.3: Planted Clique Problems (PDF)

3.1: Optimality of Cheeger's Inequality (PDF)

3.2: Certifying Positive-semidefiniteness (PDF)

3.3: Multy-way Cheeger's Inequality (PDF)

4.1: Non-commutative Khintchine Improvement (PDF)

4.2: Latala-Riemer-Schutt Problem (PDF)

4.3: Matrix Six Deviations Suffice (PDF)

4.4: OSNAP Problem (PDF)

4.5: Random k-lifts of Graphs (PDF)

4.6: Feige's Conjecture (PDF)

5.1: Deterministic Restricted Isometry Property Matrices (PDF)

5.2: Certifying the Restricted Isometry Property (PDF)

6.1: Random Partial Discrete Fourier Transform (PDF)

6.2: Mutually Unbiased Bases (PDF)

6.3: Zauner's Conjecture (SIC-POVM) (PDF)

6.4: The Paley ETF Conjecture (PDF)

6.5: Constructive Kadison-Singer (PDF)

7.1: Gilbert-Varshamov Bound (PDF)

7.2: Boolean Classification and Annulus Conjecture (PDF)

7.3: Shannon Capacity of 7 Cycle (PDF)

7.4: The Deletion Channel (PDF)

8.1: The Unique Games Conjecture (PDF)

8.2: Sum of Squares Approximation Ratio for Max-Cut (PDF)

8.3: The Grothendieck Constant (PDF)

8.4: The Paley Clique Problem (PDF)

8.5: Maximum and Minimum Bisections on Random Regular Graphs (PDF)

9.1: Detection Threshold for SBM for Three of More Communities (PDF)

9.2: Recovery Threshold for SBM for Logarithmic Many Communities (PDF)

9.3: Tightness of k-median LP (PDF)

9.4: Stability Conditions for Tightness of k-median LP and k-means SDP (PDF)

9.5: Positive PCA Tightness (PDF)

10.1: Angular Synchronization via Projected Power Method (PDF)

10.2: Sharp Tightness of the Angular Synchronization SDP (PDF)

10.3: Tightness of the Multireference Alignment SDP (PDF)

10.4: Consistency and Sample Complexity of Multireference Alignment (PDF)

Course Info

Learning Resource Types

assignment Problem Sets
notes Lecture Notes
co_present Instructor Insights