Lectures related to Regularity of Weak Solutions

Construction of Solutions by Dirichlet-Laplace

Explores the construction of solutions by Dirichlet-Laplace in general domains.

Geodesic Curves: Minimizing Distance and Constant Speed

Covers geodesic curves parameterized at constant speed to minimize distances between points.

Covers fundamental topics in calculus of variations, including minimizers and the Euler-Lagrange equation.

Explores fundamental solutions in partial differential equations, highlighting their significance in mathematical applications.

General Existence Results for PDEs

Covers the general existence result for the Laplace-Dirichlet problem and the properties of the Newtonian potential.

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Sparsest Cut: ARV Theorem

Covers the proof of the Bourgain's ARV Theorem, focusing on the finite set of points in a semi-metric space and the application of the ARV algorithm to find the sparsest cut in a graph.

Analysis IV: Convergence Theorems and Integrable Functions

Covers convergence theorems and integrable functions, including the Lebesgue integral and Borel-Cantelli sets.

Elliptic Partial Differential Equations

Covers the model problem of elliptic PDEs with weak formulation and classical solutions.

Fundamental Groups

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

Harmonic Forms: Main Theorem

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

Differential Forms Integration

Covers the integration of differential forms on smooth manifolds, including the concepts of closed and exact forms.

Proofs: Logic, Mathematics & Algorithms

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

Impossibility Result: Atomic SRSW Register

Explores the impossibility of implementing an atomic SRSW register using bounded regular registers.

Sparsest Cut: Leighton-Rao Algorithm

Covers the Leighton-Rao algorithm for finding the sparsest cut in a graph, focusing on its steps and theoretical foundations.

Modular Arithmetic: Inverses and Equations

Explores modular arithmetic, emphasizing inverses and equations in Z/mZ, with practical examples and exercises.

Sparsest Cut: Bourgain's Theorem

Explores Bourgain's theorem on sparsest cut in graphs, emphasizing semimetrics and cut optimization.

Weak Solutions of Differential Equations

Explores weak solutions of differential equations and their properties.

Zig Zag Lemma

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