Concept

Discrete space

Related lectures (31)

Riemannian distance, geodesically convex sets

Covers the structure of Riemannian manifolds, geodesic convexity, and the Riemannian distance function.

Optimal Transport: Disintegration Theorem

Covers the Disintegration Theorem in the context of Optimal Transport.

Manifolds: Charts and Compatibility

Covers manifolds, charts, compatibility, and submanifolds with smooth analytic equations.

Open Mapping Theorem

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

Sub-Varieties and Topological Concepts

Covers sub-varieties in different dimensions and topological concepts with examples and applications of theorems.

Dimension Theory: Topological Space and Closed Subsets

Explores dimension theory in topological spaces and closed subsets, including height interpretation and additivity.

Base B for X: Connected Sets and Open Bases

Explores the concept of a base B for a topological space X and the properties of connected sets and open bases.

Lifting Criterion

Explores the lifting criterion for maps between path-connected and locally path-connected spaces.

Natural Transformations: Categories, Functors, and Equivalence

Covers natural transformations between functors, identity transformations, and equivalence of categories.

Modular curves: Riemann surfaces and transition maps

Covers modular curves as compact Riemann surfaces, explaining their topology, construction of holomorphic charts, and properties.

Homology of Riemann Surfaces

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

Markov Chains: Basics and Applications

Introduces Markov chains, covering basics, generation algorithms, and applications in random walks and Poisson processes.

Homotopy Lifting

Explains homotopy lifting, paths, proofs, constant paths, discrete spaces, and monomorphisms.

Advanced Numerical Analysis: Space Discretization

Explores advanced space discretization techniques in numerical analysis for solving differential systems efficiently and accurately.

Sheaves: Hartshorne I.1

Covers the concept of sheaves, emphasizing the unique determination of functions by local data and the importance of direct limits.

Image manipulations: Affine transforms and Interpolation

Covers affine transforms, rotation, shear, and bilinear interpolation for image manipulations.

Fractal Uncertainty Principle and Spectral Gaps

Delves into the Fractal Uncertainty Principle for the Fourier transform and spectral gaps, including specialized cases for Cantor sets.

Interior and Closure in Topology

Covers the concepts of interior and closure of a set in a topological space, as well as isolated points and accumulation points.

Metric Spaces: Topology and Continuity

Introduces metric spaces, topology, and continuity, emphasizing the importance of open sets and the Hausdorff property.

Dimension theory of rings

Covers the dimension theory of rings, including additivity of dimension and height, Krull's Hauptidealsatz, and the height of general complete intersections.