Concept

Pseudo-Euclidean space

Related lectures (30)

Covers the definition of scalar product, properties, examples, and applications in Euclidean spaces, including the Cauchy-Schwartz inequality.

Optimization on Manifolds

Covers optimization on manifolds, focusing on smooth manifolds and functions, and the process of gradient descent.

Orthogonality and Gram-Schmidt Process

Explores orthogonality, Gram-Schmidt process, dot products, and solution minimization in systems.

Grassmann manifold and Retractions

Covers the Grassmann manifold and retractions on submanifolds.

Orthogonal Projection: Euclidean Space

Explores orthogonal projection in Euclidean space, emphasizing uniqueness and calculation methods.

Introduction to Optimization

Introduces linear algebra, calculus, and optimization basics in Euclidean spaces, emphasizing the power of optimization as a modeling tool.

Weyl Characters and Borcherds Products

Explores Weyl characters, Macdonald's identities, and Borcherds products in Euclidean space.

Gaussian Measure and Perimetric Inequalities

Covers isoperimetric inequalities and Gaussian measure on Euclidean space.

Embedding Space: AdS and Euclidean Geometry

Covers the concept of embedding space for AdS and Euclidean geometry, discussing AdS isometries and Euclidean Ads.

Vector Spaces: Properties and Examples

Covers the definition and properties of vector spaces, along with examples like Euclidean spaces and matrix spaces.

Orthogonal Matrices, Equivalences

Explores the equivalence conditions for orthogonal matrices and includes examples of rotations.

Matrices and Orthogonal Transformations

Explores orthogonal matrices and transformations, emphasizing preservation of norms and angles.

Introduction to Euclidean Elements

Introduces the 5th postulate in Euclidean geometry and explores non-Euclidean geometries and their implications.

Conformal Symmetries in Euclidean and AdS Spaces

Explores conformal symmetries in Euclidean and AdS spaces, isometries, induced metric, Poincaré coordinates, and boundary structure.

Constructing Correlators with Path Integrals

Explores constructing correlators using path integrals in quantum mechanics, focusing on the Euclidean and Minkowski spaces and the significance of imaginary time evolution.

Axioms of Connection

Introduces the axioms of connection in Euclidean geometry, emphasizing unique lines and non-collinear points.

Regular Polyhedra in Euclidean Geometry

Explores the historical background and properties of regular polyhedra in Euclidean geometry, including the construction of perfect even numbers and the proportionality of arcs and angles.

Euclidean Geometry Fundamentals

Introduces the fundamentals of Euclidean geometry, covering equilateral triangles, symmetries, radical axes, and ancient architectural figures.

Manifolds and Tangent Space

Introduces manifolds, charts, atlases, tangent space, tensors, and the metric in curved spaces.

E8 Lattice Optimality

Delves into the construction and optimality of the E8 lattice as the densest sphere packing in dimension eight.