Michael Penn Courses

Another Riemann-Zeta Function Identity

Explore an intriguing identity involving even values of the Riemann-Zeta function, delving into limits, sums, and transformations to uncover mathematical insights.

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

Ramanujan's 723rd Problem

Explore Ramanujan's 4th notebook identity involving floor function and square root. Proof and mathematical insights provided in this engaging mathematical exploration.

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

Differential Forms - What Is an M-Form?

Explore the formal definition, examples, and general structure of m-forms in differential geometry, enhancing understanding of this advanced mathematical concept.

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

Real Analysis - The Countability of the Rational Numbers

Explore a rigorous proof demonstrating the countability of rational numbers using a unique approach of representing them as a countable union of disjoint finite sets in real analysis.

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

Differential Forms - 2-Forms

Introduction to 2-forms in differential geometry, defining the concept and providing examples to illustrate their properties and applications in mathematical analysis.

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

Differential Forms - The Geometry of Multiplying 1-Forms

Explore the geometric interpretation of multiplying 1-forms in differential geometry, enhancing understanding of this advanced mathematical concept through examples and visual explanations.

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

Lobachevsky's Integral Formula and a Nice Application

Explore Lobachevsky's integral formula and its application to calculate specific integrals using Fourier expansion, with a focus on sin(x)/x and abs(sin x)sin(x)/x.

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

Abstract Algebra - Ideals of Quotients of PIDs

Explore the proof that ideals in quotients of principal ideal domains are principal, delving into the intricacies of abstract algebra and ring theory.

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

Rogers-Ramanujan Identities - The Empirical Hypothesis

Proof of the empirical hypothesis for q-series constructed from Rogers-Ramanujan identities, exploring structural properties and advancing understanding of these mathematical relationships.

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

Abstract Algebra - A PID That Is Not a Euclidean Domain

Explore a principal ideal domain that's not Euclidean, using Dedekind-Hasse Norm and universal side divisor concepts to demonstrate this intriguing algebraic property.

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

A Trigonometric Integral

Explore strategies for solving complex trigonometric integrals using recursively defined sequences, enhancing your mathematical problem-solving skills.

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

Abstract Algebra - If D is a UFD then D[x] is a UFD

Explore the proof that polynomial rings over unique factorization domains are also UFDs, covering key concepts like Gauss' lemma and polynomial factorization in integral domains.

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

The Millin Series - A Nice Fibonacci Sum

Explore the Millin series, involving reciprocals of 2^nth Fibonacci numbers, using Catalan's identity and the golden ratio to derive its closed form.

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

Abstract Algebra - Every PID is a UFD

Explore the connection between principal ideal domains and unique factorization domains, introducing key concepts like ascending chain condition and Noetherian rings.

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

A Classic Trig Identity - Featuring Chebyshev Polynomials

Explore a classic trigonometric identity derivation using Chebyshev polynomials, enhancing your understanding of advanced mathematical concepts and their applications.

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