Lectures related to Cauchy Equations and Integral Decomposition

Harmonic Forms: Main Theorem

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

Meromorphic Functions & Differentials

Explores meromorphic functions, poles, residues, orders, divisors, and the Riemann-Roch theorem.

Complex Integration and Cauchy's Theorem

Discusses complex integration and Cauchy's theorem, focusing on integrals along curves in the complex plane.

Harmonic Forms and Riemann Surfaces

Explores harmonic forms on Riemann surfaces, covering uniqueness of solutions and the Riemann bilinear identity.

Curve Integrals: Gauss/Green Theorem

Explores the application of the Gauss/Green theorem to calculate curve integrals along simple closed curves.

Residue Theorem: Applications in Complex Analysis

Discusses the residue theorem and its applications in complex analysis, including integral calculations and Laurent series.

Residue Theorem: Calculating Integrals on Closed Curves

Covers the application of the residue theorem in calculating integrals on closed curves in complex analysis.

Applications of Residue Theorem in Complex Analysis

Covers the applications of the Residue theorem in evaluating complex integrals related to real analysis.

Complex Analysis: Laurent Series and Residue Theorem

Discusses Laurent series and the residue theorem in complex analysis, focusing on singularities and their applications in evaluating complex integrals.

Residue Theorem: Cauchy's Integral Formula and Applications

Covers the residue theorem, Cauchy's integral formula, and their applications in complex analysis.

Complex Analysis: Derivatives and Integrals

Provides an overview of complex analysis, focusing on derivatives, integrals, and the Cauchy theorem.

Holomorphic Functions: Taylor Series Expansion

Covers the basic properties of holomorphic maps and Taylor series expansions in complex analysis.

Residue Theorem: Applications in Complex Analysis

Discusses the residue theorem and its applications in calculating complex integrals.

Equidistribution of CM Points

Covers the joint equidistribution of CM points in algebraic structures and quadratic forms.

Complex Analysis: Cauchy Theorem

Explores the Cauchy Theorem and its applications in complex analysis.

Differential Forms Integration

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

Differentiability of Functions of Several Variables

Covers the differentiability of functions of multiple variables and the significance of directional derivatives and gradients.

Open Mapping Theorem

Explains the Open Mapping Theorem for holomorphic maps between Riemann surfaces.

Orthogonality and Projection

Covers orthogonality, scalar products, orthogonal bases, and vector projection in detail.

Green's Theorem in 2D: Applications

Explores the applications of Green's Theorem in 2D, emphasizing the importance of regular domains for successful integration.