WHAT COMES AFTER CALCULUS? : A Look at My Higher Level Math Courses (I Took 22 of them).

The speaker shares their journey of reflecting on their math degree, specifically focusing on what lies beyond calculus, a question that piqued their curiosity before and during their undergraduate studies. Despite exploring calculus and differential equations independently and through school up to AP BC calculus, the true expanse of mathematical studies remained largely unexplored until college. They reveal having taken 22 undergraduate math courses, a feat driven by passion and a brief hiatus to support their family. The video promises a detailed exploration of these courses, starting with a decision to self-study calculus to skip directly to multivariable calculus in college, effectively bypassing conventional routes. The speaker encourages viewers to subscribe for more math content, setting the stage for an in-depth discussion on the breadth of mathematical studies post-calculus.

This segment delves into the initial set of math courses taken by the speaker, beginning with linear algebra, which laid the foundation with computation-based learning and moved onto concepts like linear transformations and eigenvalues. Introduction to Mathematical Structures followed, focusing on proof methods and logic, exploring sets, functions, and cardinality, and touching on topics like Russell's paradox and the four-color theorem. The speaker then transitioned to differential forms and vector calculus, highlighting the course's exploration of differential forms, generalized divergence, and Fubini's theorem. Real Analysis I was noted for its rigorous approach to calculus and set theory. A unique turn was taken with a course on reading Euclid's Elements in Greek, blending the speaker’s love for languages with math, leading to insights into Euclidean geometry and ancient proof methods.

The narrative progresses into more advanced and specialized courses. Abstract algebra introduced the study of groups, rings, and fields, emphasizing the speaker’s shift from analysis to algebra. This was followed by Ordinary Differential Equations, which, despite its slow pace, covered fundamental topics like harmonic oscillators. A course without assigned readings delved into field extensions and Galois Theory, offering a deep dive into algebraic structures and constructability theorems. Complex analysis expanded on functions of a complex variable and integration techniques, including keyhole integration and conformal mapping theory. The segment underscores the speaker’s transition into higher complexity topics, reflecting a growing expertise and the challenges of navigating such diverse mathematical landscapes.

The speaker continues with an exploration of topics that push the boundaries of their mathematical understanding, starting with topology, which introduced a new perspective through the study of topologies and fundamental groups. The journey through math became increasingly specialized with courses like the Theory of Elliptic Curves, focusing on highly specific non-singular curves and their computational challenges. Advanced linear algebra revisited vector spaces with a more proof-oriented approach. The speaker also touched on their comprehensive project and independent research in symbolic dynamics, highlighting their deep dive into combinatorial aspects of mathematics and the creative freedom of their academic endeavors. This segment reflects the speaker’s commitment to exploring the depths of mathematics, engaging with complex concepts, and contributing to research.

In the concluding segment, the speaker reflects on the final courses of their undergraduate journey, touching on subjects like computability and complexity theory, which bridged mathematics with theoretical computer science through rigorous proof-based approaches. Statistics provided a lighter coursework focus, contrasting with the intensity of previous studies. Fractal geometry offered a fascinating look into the mathematical underpinnings of fractals, exploring iterated function systems and dimension theories. The journey culminated with Fourier series, challenging the speaker with complex applied mathematics and computational emphases. The speaker regretfully notes missed opportunities in courses like combinatorics and differential geometry due to a term off but concludes with a reflection on the vast array of mathematical knowledge and skills acquired, inviting viewers to engage further with their channel for more math content.