Michael Penn Courses

Differential Forms - The Minkowski Metric and the Hodge Operator

Explore the Minkowski inner product's lifting to 2 and 3 forms and its impact on the Hodge operator in this advanced mathematics video lecture on differential geometry.

• YouTube
• 32 minutes
• Self-Paced
• Free Video

Differential Forms - The Hodge Operator via an Inner Product

Explore the generalized Hodge operator for m-forms using inner products, advancing understanding of differential geometry and mathematical physics concepts.

• YouTube
• 29 minutes
• Self-Paced
• Free Video

A Sum Two Ways

Explore two distinct methods for calculating the closed form of an intriguing infinite series, demonstrating advanced mathematical techniques and problem-solving approaches.

• YouTube
• 31 minutes
• Self-Paced
• Free Video

A Gentle Description of a Vertex Algebra

Explore the concept of vertex algebra through associative, Lie, and Poisson algebras. Learn its definition and gain intuition for understanding its components in this mathematical journey.

• YouTube
• 31 minutes
• Self-Paced
• Free Video

So Many Factorials

Explore advanced mathematical techniques to evaluate complex factorial series, utilizing central binomial coefficients, generating functions, and trigonometric integrals for problem-solving.

• YouTube
• 29 minutes
• Self-Paced
• Free Video

Real Analysis - Continuity

Explore the precise definition of continuity in real analysis, its equivalence to sequential continuity, and its connection to Calculus 1 limits. Gain a deeper understanding of this fundamental concept.

• YouTube
• 22 minutes
• Self-Paced
• Free Video

Problem Solving - The Pigeonhole Principle

Explore the pigeonhole principle through examples, including famous problems from Erdos and the 1994 Putnam Exam. Learn problem-solving techniques in mathematics.

• YouTube
• 29 minutes
• Self-Paced
• Free Video

Pi Is Irrational

Explore a simple proof demonstrating pi's irrationality, delving into mathematical concepts and reasoning behind this fundamental property of the famous constant.

• YouTube
• 23 minutes
• Self-Paced
• Free Video

Real Analysis - Subsequences

Explore subsequences in Real Analysis, including the Bolzano-Weirstrass Theorem. Learn key concepts and proofs to deepen your understanding of this fundamental mathematical topic.

• YouTube
• 22 minutes
• Self-Paced
• Free Video

Differential Forms - The Exterior Derivative and Vector Calculus

Explore the connection between classical vector calculus operators and differential forms, focusing on the exterior derivative and Hodge operator in this concise mathematical explanation.

• YouTube
• 20 minutes
• Self-Paced
• Free Video

Differential Forms - Integrating M-Forms

Explore the concept of integrating differential m-forms on m-dimensional hypersurfaces in R^n, including theory and practical examples to enhance understanding of this advanced mathematical topic.

• YouTube
• 20 minutes
• Self-Paced
• Free Video

An Infinite Tangent Product

Explore the evaluation of an infinite tangent product using identities and telescoping techniques, delving into advanced mathematical concepts and problem-solving strategies.

• YouTube
• 23 minutes
• Self-Paced
• Free Video

Real Analysis - Algebraic Properties of Limits

Explore algebraic properties of limits in real analysis, proving classic theorems and enhancing understanding of fundamental mathematical concepts.

• YouTube
• 21 minutes
• Self-Paced
• Free Video

Representations of Finite Groups - A Few More Common Examples

Explore representations of finite groups including cyclic, dihedral, quaternions, and symmetric groups. Gain insights into advanced abstract algebra concepts and their applications.

• YouTube
• 21 minutes
• Self-Paced
• Free Video

Do We Even Need the Real Numbers?

Explore the possibility of calculus using only rational numbers and discover the crucial role of the axiom of completeness in fundamental calculus theorems.

• YouTube
• 26 minutes
• Self-Paced
• Free Video