Lecture

Caratheodory Bounds: Integer Programming

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Covers approximation algorithms for optimization problems, LP relaxation, and randomized rounding techniques.

Introduces Cartesian product and induction for proofs using integers and sets.

Covers the Hedge algorithm for minimizing loss in linear programming problems.

Explores proof concepts, techniques, and applications in logic, mathematics, and algorithms.

Covers semi-definite programming and optimization over positive semidefinite cones.

Explores optimal transport in heat equations and metric spaces.

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

Explores linear programming duality, covering constraints, variables, solutions, and the relationship between primal and dual LP.

Covers the concept of outer measure and properties of measurable sets.

Explores harmonic forms on Riemann surfaces and the uniqueness of solutions to harmonic equations.

Explores fundamental groups, homotopy classes, and coverings in connected manifolds.

Explores rounding the relaxation and Max Sum Diversification, discussing effects, integral solutions, and loss bounds.

Covers the Zig Zag Lemma and the long exact sequence of relative homology.

Covers Dognion's Theorem, explaining the proof of infeasible Integer Programming problems.

Explores convergence in law for random variables, including Kolmogorov's theorem and proofs based on probability lemmas.

Covers the Laplace equation, decomposition of linear problems, and solutions through separation of variables.

Explores the proof of the Prime Number Theorem and its implications in number theory.

Covers the concept of compression and distance boards.

Explores the cutset formulation for the MST Problem and Gomory Cutting Planes method.

Covers the Fourier inversion formula, exploring its mathematical concepts and applications, emphasizing the importance of understanding the sign.