Lectures related to Riemann mapping theorem

Covers the topology of Riemann surfaces and the concept of triangulation using finitely many triangles.

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

Modular Curves: Genus and Mapping Theorems

Explores holomorphic maps, ramification points, and the genus of a modular curve.

Covers the concepts of local homeomorphisms and coverings in manifolds, emphasizing the conditions under which a map is considered a local homeomorphism or a covering.

Building surfaces from equilateral triangles

Explores the construction of Riemann surfaces from equilateral triangles and the dynamics of finite-type maps.

Conformal Applications: Theory and Examples

Covers the theory and examples of conformal applications, focusing on the concept of conformal mappings and Moebius transformations.

Introduction to Quantum Chaos

Covers the introduction to Quantum Chaos, classical chaos, sensitivity to initial conditions, ergodicity, and Lyapunov exponents.

Proper Actions and Quotients

Covers proper actions of groups on Riemann surfaces and introduces algebraic curves via square roots.

Algebraic Curves: Normalization

Covers the normalization process of plane algebraic curves, focusing on irreducible polynomials and affine curves.

Harmonic Forms: Main Theorem

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

Conformal Transformations

Explores conformal transformations, including holomorphic functions and Moebius transformations.

Homology of Riemann Surfaces

Explores the homology of Riemann surfaces, including singular homology and the standard n-simplex.

Mapping Theorems: Poisson Processes and Intensity Functions

Explores mapping theorems for Poisson processes and their intensity functions.

Fractal Uncertainty Principle and Spectral Gaps

Explores transfer operators, Patterson-Sullivan measure, FUP, spectral gaps, and resonances in Schottky groups.

FK-Percolation and the Six-Vertex Model: Critical Phenomena

Covers the transition from the six-vertex model to FK-percolation, focusing on critical phenomena and phase transitions in two-dimensional systems.

Fundamental Groups

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

Canonical Divisors and Modular Forms

Covers canonical divisors on Riemann surfaces and properties of modular forms.

Holomorphic Functions: Cauchy-Riemann Equations and Applications

Discusses holomorphic functions, focusing on the Cauchy-Riemann equations and their applications in complex analysis.

Riemann Sphere: Correlations and Conditions

Covers the Riemann sphere, focusing on its conditions and correlation functions in mathematical and physical contexts.

Modular Forms: Dimension Formula

Explores modular forms, discussing pullback maps, meromorphic differentials, and the Riemann-Roch theorem.