WHAT COMES AFTER CALCULUS? : A Look at My Higher Level Math Courses (I Took 22 of them).
TLDRThe speaker reflects on their extensive undergraduate mathematics education, detailing the 22 math courses they took, including subjects like linear algebra, differential forms, real and complex analysis, abstract algebra, number theory, topology, and fractal geometry. They also discuss their independent research in symbolic dynamics and their experience with computational complexity theory. The video serves as an insightful overview of advanced mathematics topics and the speaker's personal academic journey.
Takeaways
- 🎓 The speaker reflects on their math degree and the courses they took, totaling 22, with 5 not completed due to a term off for family reasons.
- 📚 Self-study was a significant part of their education, particularly in calculus, where they prepared over the summer and skipped the college course.
- 🔢 The first math course in undergrad was linear algebra, focusing on computation, matrix operations, and linear transformations.
- 📈 Introduction to mathematical structures involved proof methods, logic, and various mathematical concepts like sets, relations, and equivalence classes.
- 📜 Studying ancient Greek involved reading Euclid's Elements in the original language, providing a unique perspective on geometry.
- 🌀 Differential forms and vector calculus were explored, with a focus on the mathematical rigor behind integrals and differential forms.
- 📊 Real analysis introduced the rigor of calculus and set theory, covering sequences, series, convergence tests, and the Riemann integral.
- 🔍 Abstract algebra covered groups, rings, and fields, offering a puzzle-like approach to algebraic structures.
- 📊 Ordinary differential equations focused on harmonic oscillators, providing a slower-paced introduction to applied mathematics.
- 🔢 Field theory and Galois theory were深入, discussing field extensions, constructability theorems, and the fundamental theorem of Galois Theory.
- 🌐 Complex analysis delved into functions of a complex variable, integration techniques, and conformal mapping theory, including the Riemann mapping theorem.
Q & A
What was the speaker's primary focus during their undergraduate studies?
-The speaker's primary focus during their undergraduate studies was mathematics, as they took 22 math courses and even planned on taking 27.
Which course did the speaker find to be the most challenging?
-The speaker found the course on the theory of L-Functions and elliptic curves to be the most challenging.
What did the speaker think about the course on differential forms and vector calculus?
-Initially, the speaker thought the course on differential forms and vector calculus would be terrible but ended up finding it amazing because it focused on differential forms and the 'dX' at the end of the integral.
How did the speaker's experience with real analysis differ from their expectations?
-The speaker expected real analysis to be the most difficult class in their undergraduate career, but they found other classes to be harder than this one.
What did the speaker learn about in their course on abstract algebra?
-In the abstract algebra course, the speaker learned about groups, rings, and fields, as well as various mathematical objects related to these core algebraic structures.
Which course allowed the speaker to read Euclid's Elements in the original Greek?
-The course on Euclidean geometry allowed the speaker to read Euclid's Elements in the original Greek.
What was the main topic of the speaker's senior comprehensive project?
-The main topic of the speaker's senior comprehensive project was the topological entropy of two-dimensional symbolic spaces.
Which mathematical concept did the speaker find particularly fascinating?
-The speaker found the concept of topological entropy particularly fascinating, as it was related to their senior comprehensive project.
What was the focus of the speaker's independent research courses?
-The focus of the speaker's independent research courses was one-dimensional symbolic dynamics and generating special spaces in one-dimensional symbolic dynamics.
Which course did the speaker take that was an introduction to proof methods and logic?
-The course on mathematical structures was an introduction to proof methods and logic.
What was the speaker's experience with computational complexity theory?
-The speaker took a course on computational complexity theory, which was technically a theoretical computer science class but was approached with mathematical proofs, and they found it very cool.
Outlines
🎓 Reflecting on a Mathematics Degree
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.
🧮 Journey Through 22 Math Courses
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.
🔢 Advanced Math and Its Applications
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.
📚 Diving Deeper into Mathematical Complexity
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.
🎓 Final Stages and Reflections on a Math Degree
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.
Mindmap
Keywords
💡Calculus
💡Linear Algebra
💡Proof Methods and Logic
💡Differential Forms and Vector Calculus
💡Real Analysis
💡Abstract Algebra
💡Ordinary Differential Equations
💡Topology
💡Fractal Geometry
💡Fourier Series
Highlights
Reflections on what comes after calculus, a question many students wonder about.
Summary of 22 math courses taken in undergrad, including plans for 5 additional ones.
Self-study in calculus to skip college-level courses and dive into multivariable calculus.
First math course post-calculus: Linear Algebra, covering matrix operations, linear transformations, and eigenvalues.
Introduction to Mathematical Structures focusing on proof methods, logic, and foundational mathematical concepts.
Exploring differential forms and vector calculus, discussing differential forms, and proving Fubini's and generalized Stokes' theorems.
Real Analysis I, diving into the rigorous assessment of calculus and set theory.
Studying ancient texts in original Greek for a unique perspective on Euclidean geometry.
Abstract Algebra's exploration of groups, rings, and fields, challenging but rewarding.
Ordinary Differential Equations focusing on harmonic oscillators, critiqued for its slow pace.
Advanced topics in algebra, like field extensions and Galois Theory, and their complexity.
Complex Analysis, covering functions of a complex variable and various integration techniques.
Number Theory, revisiting basics like Fermat's Little Theorem and exploring new concepts like cyclotomic polynomials.
Real Analysis II, an introduction to measure theory, Lebesgue integration, and related important theorems.
Theory of Elliptic Curves, described as the hardest course, focusing on computational methods and cryptographic applications.
Probability and its combinatorial and continuous aspects, including coding in R.
Advanced Linear Algebra, delving into vector spaces, and culminating in Cayley-Hamilton theorem discussions.
Independent research on topological entropy and symbolic dynamics, leading to presentations at mathematical conferences.
Topology, learning about different topologies, fundamental groups, and knot theory.
An introduction to computability and complexity theory, approaching theoretical computer science with a mathematical lens.
Fractal Geometry, studying the mathematics behind fractals and iterated function systems.
Fourier Series and partial differential equations, emphasizing computational aspects of applied mathematics.
Courses not taken due to taking a term off for family reasons, including combinatorics and differential geometry.
Transcripts
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