 Introduction and Review
 Brief review of prerequisites
 Selected Topics in Convex Analysis and Optimization
 Introduction to convex sets and functions
 Convex set results: Separating hyperplane and Caratheodory's theorems
 Convex function results: Jensen's inequality and applications, LegendreFenchel transformation
 Lagrangian duality and KarushKuhnTucker (KKT) conditions
 Strong convexity and oracle complexity of gradient descent
 Gradient descent and its variants [optional]
 Foundations of Spectral Methods
 Positive semidefiniteness. Spectral and singular value decompositions
 CourantFischerWeyl minimax theorems
 PerronFrobenius theory [optional]
 Matrix norms and perturbation theory [optional]
 Concentration Inequalities and their Applications in CS
 Introduction to measure theory
 Markov, Chebyshev, and Chernoff Bounds
 Hoeffding and Bernstein inequalities
 Martingales, AzumaHoeffding bound, and McDiarmid's inequality
 Applications to learning theory (e.g., GlivenkoCantelli)
 Talagrand's Inequality [optional]
 Discrete Fourier Analysis
 Introduction to discrete Fourier Analysis and Abelian groups
 Characters, bias, Fourier transform
 Parseval's and Plancherel's identities
 BlumLubyRubinfeld (BLR) linearity testing
 Convolution and translation invariant systems
 Learning: The lowdegree algorithms
 Selected Topics
 Applied analysis for Learning Theory
 Coding Theory
 Additional Randomized Techniques
 Additional Topics in Discrete Fourier Analysis
