Variational Problems: Convexity and Coercivity

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Covers the approximation of functions in Sobolev spaces using smooth functions.

Covers the formulation of optimization programs for minimizing piecewise linear cost functions.

Covers the KKT conditions and convex optimization, discussing constraint qualifications and tangent cones of convex sets.

Discusses proximal gradient descent and its applications in optimizing machine learning algorithms.

Covers the basics of constrained optimization, including tangent directions, trust-region subproblems, and necessary optimality conditions.

Covers distributions, derivatives, convergence, and continuity criteria in function spaces.

Covers weak derivatives, their properties, and applications in functional analysis.

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Covers Distributionally Robust Portfolio Optimization for maximizing expected utility with uncertain asset returns.

Covers Dijkstra's algorithm for shortest path problems and its application in ALL-TO-ONE and ALL-PAIRS algorithms.

Covers the concept of the integer hull in the general case, focusing on the proof of claim and the properties of cells.

Explores the equidistribution of CM points and the implications of ergodicity in measure-preserving systems.

Explores Mertens' theorems on prime estimates and the behavior of the Mobius function in relation to the prime number theorem.

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