Lectures related to System Composition

Covers block diagrams and causal LTI systems with rational transfer functions.

Explores Wedderburn's theorem, group algebras, and Maschke's theorem in the context of finite dimensional simple algebras and their endomorphisms.

Schur's Lemma and Representations

Explores Schur's lemma and its applications in representations of an associative algebra over an algebraically closed field.

LTI Systems: Analysis and Properties

Covers the analysis and properties of Linear Time-Invariant (LTI) systems.

Lie Algebras and Representations

Explores Lie algebras, representations, tensor products, and commutation relations in mathematics.

Representation Theory: Algebras and Homomorphisms

Covers the goals and motivations of representation theory, focusing on associative algebras and homomorphisms.

Laplace Transform: Properties and Examples

Covers the Laplace Transform, including its properties and examples.

Subrepresentations of the regular representation

Explores subrepresentations of the regular representation in group theory, emphasizing properties and isomorphism between subrepresentations.

Connected groups of semisimple elements

Explores weight space decomposition and proves that a connected linear algebraic group with all semi-simple elements must be a torus.

Group Theory: Reducible and Irreducible Representations

Explores group theory in quantum physics, emphasizing reducible and irreducible representations, conservation laws, and group properties.

Density Theorem: Endomorphisms of Simple Modules

Covers the density theorem in representation theory, focusing on endomorphisms of simple finite-dimensional modules.

Symmetry and Group Theory: Exercise Set 4

Explores symmetry operations, isomorphic representations, and three-dimensional group representations.

Group Representations: Matrices and Symmetry

Explores group representations using matrices, discussing reducible vs. irreducible representations and the Great Orthogonality Theorem.

The Regular Representation

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

Signals & Systems Review

Provides a comprehensive review of signals and systems, covering topics such as time-domain analysis, frequency-domain analysis, and Fourier transform.

Complete Reducibility of Complex Representations

Covers the complete reducibility of complex representations and the relation between Lie algebras and Lie groups.

Kirillov Paradigm for Heisenberg Group

Explores the Kirillov paradigm for the Heisenberg group and unitary representations.

LTI System Properties

Explores the properties of Linear Time-Invariant systems through impulse response and convolution sum.

Machine Learning at the Atomic Scale

Explores simple models, electronic structure evaluation, and machine learning at the atomic scale.

Ideals and Representations

Covers ideals, representations, modules, and maximal ideals in associative algebras.