Lectures related to Linear Projective Groups

Explores the torus, linear groups, stability properties of matrices, and homomorphisms.

Covers linear algebraic groups, including special sets and matrix examples.

Explores symplectic groups, linear transformations preserving non-degenerate alternating bilinear forms on vector spaces.

Self-similar groups II

Explores self-similar groups acting on rooted trees and their properties.

Matrix Operations: Scaling and Invertibility

Covers scaled matrices, ladder matrices, and the Gauss method for matrix operations.

Dynamics on Homogeneous Spaces and Interactions with Number Theory

Explores dynamics on homogeneous spaces and their interactions with number theory, focusing on modular lattices and the Iwasawa decomposition theorem.

Group Actions: Examples

Showcases examples of group actions on vector spaces and topological spaces.

Conductor of X: Waldsparger Formula and Subconvexity Bound

Covers the equidistribution of representations of integers by ternary forms and the conductor of X.

Linear Algebra: Elementary Row Operations

Covers elementary row operations, equivalent matrices, invertibility, and bases in linear algebra.

3D Rotations and Quaternions

Covers 3D rotations, quaternions, and matrices of direction cosines, providing an elegant solution to the limitations of rotation matrices.

Quaternion Algebra: Properties and Applications

Covers the properties and applications of quaternion algebra, including norm calculations and distribution in different spaces.

Quadratic Spaces and Quaternion Algebra

Explores quadratic spaces, norms, and quaternion algebra over fields.

The Regular Representation

Introduces the regular representation, a key tool for studying group actions on varieties.

Proof of Lagrange Theorem

Covers the proof of Lagrange Theorem and explores quaternions, prime numbers, and equations.

Connected Components

Covers the concept of connected components in linear algebraic groups and their relationship to singular groups.

Fourier Expansion and Quaternion Algebra

Discusses Fourier expansion, quaternion algebra, and Whittaker model in automorphic forms.

Determinants: Special Linear Groups and Matrix Properties

Explores the special linear group, matrix properties, and determinant theorems.

Diagonalizable Groups

Explores the concept of diagonalizable groups and their properties in linear algebraic groups.

Fermat's Theorem: Sums of Squares

Explores Fermat's Theorem, factorization of integers, properties of Z[i], and Hurwitz quaternions.

Linear Applications: Matrices and Transformations

Covers linear applications, matrices, transformations, and the principle of superposition.