Lectures related to Cantor's diagonal argument

Relations, Sequences and Summations

Covers strings, countable sets, cardinality, and the concept of countability, exploring the countability of various sets and Cantor diagonalization.

Mapping Functions and Surjections

Explores mapping functions, surjections, injective and surjective functions, and bijective functions.

Relations, Sequences, Summation: Cantor Diagonalization

Covers countable and uncountable sets, sequences, summation, and Cantor's Diagonalization proof.

Holomorphic Functions: Taylor Series Expansion

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

Additional Properties of Real Numbers

Explores the countability of subsets of real numbers and demonstrates that the set of real numbers is uncountable.

Analysis IV: Measurable Sets and Functions

Introduces measurable sets, functions, and the Cantor set properties, including ternary development of numbers.

Classification of Extensions

Explores the classification of extensions in group theory, emphasizing split extensions and semi-direct products.

Computing leading eigenvalues of the transfer operator

Explores computing the leading eigenvalue of a transfer operator beyond periodic points, focusing on mathematical settings, spectral radius estimation, and the Zaremba Conjecture.

Fractal Uncertainty Principle and Spectral Gaps

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

Distribution to Spaces: Interpolation and Continuity

Covers weak derivatives, interpolation spaces, and continuity in function spaces.

Different Infinities: Cantor's Theorem

Explains Cantor's theorem comparing cardinalities of different number sets.

Recurrence on Two Variables

Covers methods of proof, such as recurrence on two variables and the gradient.

Change of Variable Formula: Devil's Staircase

Explores a change of variable formula and the intriguing Devil's Staircase.

Diagonalization of Matrices: Theory and Examples

Covers the theory and examples of diagonalizing matrices, focusing on eigenvalues, eigenvectors, and linear independence.

Linear Applications: Definitions and Properties

Explores the definition and properties of linear applications, focusing on injectivity, surjectivity, kernel, and image, with a specific emphasis on matrices.

Two Definitions of Group Action

Explores two definitions of group action and their equivalence through bijection in group homomorphisms.

Properties of Real Numbers

Covers countability and bijections between sets, demonstrating the uncountability of real numbers.

Nonlinear Dynamics: Chaos and Complex Systems

Explores countable and uncountable sets, Cantor set, Mandelbrot set, and Box dimension in nonlinear dynamics and complex systems.

Hausdorff Dimension and Brownian Motion

Explores the Hausdorff dimension of Brownian motion zeros, demonstrating a dimension of 1/2 with certainty.

The Discriminant and Ideal Class Group in Mathematics

Explores the discriminant in matrices, ideal class groups, and optimal embeddings in mathematics.