Stochastic Processes: This is the biggest omission on this list, particularly given the emphasis on probability and AI. This is probability over time, or some time like construct like rounds in a game. The basics include random walks, Markov chains, Hidden Markov Models, and conditional probability. I studied the textbook Probability Models by Ross, which is great and is available for free download, although I’m sure other textbooks are good as well. Most applied probability uses stochastic processes.
Measure Theoretic Probability: If you are serious about probability and applications, this is important. This is a very mathematically rigorous treatment of probability and will give you a much deeper understanding of the subject. This is needed for a lot of probability applications. This post mentions Mathematical Finance, which uses measure theoretic probability heavily.
Prereqs: This post recommends skipping prereqs, but recommends taking linear algebra, probability, and a standard calculus sequence, which are arguably important prereqs. The idea of taking years of prereqs sounds awful. If you take a probability class or a linear algebra class and view it as just some prereq before the real classes start, that isn’t exciting. But often, particularly in math, when advanced concepts build on earlier concepts, you want to learn everything in the right order. Lots of schools, have non-technical prereqs, like you have to take a freshman seminar class before you are allowed to enroll in upper division stuff, and that’s intended to help orient 18-19 year olds to campus life, and is kind of off topic for this thread.
Modern Math Fundamentals: The three big areas are analysis, abstract algebra, and topology. Even if your interest is strictly practical + applied, learning the fundamentals of modern math is important. The OP mentions calculus; real analysis is the more advanced + theoretical version of calculus. And it’s necessary if you plan to do any type of higher math, including practical applied stuff. You might view these as important prereqs.
Cryptography: Any applied math list should mention this. This list mentions “modular arithmetic”, that’s usually called Integer Number Theory. Also, some abstract algebra is important. A good math department should offer a good semester on mathematical cryptography that covers RSA, DSA, key exchange algorithms, elliptic curve variants, etc.
Logic: This post mentions a book on Godel; taking a serious course on Godel’s logic contributions, most notably Godel’s incompleteness theorems is worthwhile. My undergrad school offered two core logic semesters: the first semester was basic deductive logic including formal proof systems, truth trees, etc. The second semester “mathematical logic” or “meta-logic” was much more rigorous, covers a ton of content, including theoretical computability, Turing computability, and Godel’s incompleteness theorems.
I personally covered the relevant parts of measure theory and a lot of stochastic processes in math finance, which I think is a good way to do it. I did take an OR class which spent about half the time on Markov chains, but I consider that stuff pretty straightforward if you have a good grounding in linear algebra.
Analysis/abstract/topology are exactly the sort of prereqs I recommend skipping. The intro classes usually spend a bunch of time on fairly boring stuff; intermediate-level classes will usually review the actually-useful parts as-needed.
The crypto recommendation makes sense. For logic, I don’t think there’s much value in diving into the full rigor; it’s mostly the concepts that matter, and proving it all carefully is extremely tedious. Definitely important to get the core concepts, though.
You recommend the basic math courses: linear algebra, probability, a standard calculus sequence. You just don’t recommend the more pure math type courses. In your view, pure math courses spend too much time digging into boring tedious details, and you advise more applied courses instead. That’s an entirely valid perspective. And it may be the most productive tactic.
Real analysis, abstract algebra, and topology are often the hardest and most advanced courses in the undergraduate math catalog. Those are considered the capstone courses of an undergraduate degree in pure mathematics. You reference them as introductory classes or prereqs which seems not correct. At almost any university, Real Analysis is the more advanced, theoretical, and difficult version of calculus.
Did you study martingales or stopped brownian motion? Are those useful or recommended? Those seem relevant to finance and applied probability?
I really enjoyed this post, and thank you for the awesome reply.
Real analysis, abstract algebra, and topology are often the hardest and most advanced courses in the undergraduate math catalog. Those are considered the capstone courses of an undergraduate degree in pure mathematics. You reference them as introductory classes or prereqs which seems not correct.
Yeah, fair. Harvey Mudd is probably unusual in this regard—it’s a very-top-tier exclusively-STEM school, so analysis and abstract algebra were typically late-sophomore-year/early-junior-year courses for the math majors (IIRC). I guess my corresponding advice for someone at a typical not-exclusively-undergrad university would be to jump straight into grad-level math courses.
(As with the post, this advice is obviously not for everyone.)
Did you study martingales or stopped brownian motion? Are those useful or recommended? Those seem relevant to finance and applied probability?
Yup, that comes up in math finance. I haven’t seen them come up much outside of finance, they’re kind of niche in the broader picture.
Stochastic Processes: This is the biggest omission on this list, particularly given the emphasis on probability and AI. This is probability over time, or some time like construct like rounds in a game. The basics include random walks, Markov chains, Hidden Markov Models, and conditional probability. I studied the textbook Probability Models by Ross, which is great and is available for free download, although I’m sure other textbooks are good as well. Most applied probability uses stochastic processes.
Measure Theoretic Probability: If you are serious about probability and applications, this is important. This is a very mathematically rigorous treatment of probability and will give you a much deeper understanding of the subject. This is needed for a lot of probability applications. This post mentions Mathematical Finance, which uses measure theoretic probability heavily.
Prereqs: This post recommends skipping prereqs, but recommends taking linear algebra, probability, and a standard calculus sequence, which are arguably important prereqs. The idea of taking years of prereqs sounds awful. If you take a probability class or a linear algebra class and view it as just some prereq before the real classes start, that isn’t exciting. But often, particularly in math, when advanced concepts build on earlier concepts, you want to learn everything in the right order. Lots of schools, have non-technical prereqs, like you have to take a freshman seminar class before you are allowed to enroll in upper division stuff, and that’s intended to help orient 18-19 year olds to campus life, and is kind of off topic for this thread.
Modern Math Fundamentals: The three big areas are analysis, abstract algebra, and topology. Even if your interest is strictly practical + applied, learning the fundamentals of modern math is important. The OP mentions calculus; real analysis is the more advanced + theoretical version of calculus. And it’s necessary if you plan to do any type of higher math, including practical applied stuff. You might view these as important prereqs.
Cryptography: Any applied math list should mention this. This list mentions “modular arithmetic”, that’s usually called Integer Number Theory. Also, some abstract algebra is important. A good math department should offer a good semester on mathematical cryptography that covers RSA, DSA, key exchange algorithms, elliptic curve variants, etc.
Logic: This post mentions a book on Godel; taking a serious course on Godel’s logic contributions, most notably Godel’s incompleteness theorems is worthwhile. My undergrad school offered two core logic semesters: the first semester was basic deductive logic including formal proof systems, truth trees, etc. The second semester “mathematical logic” or “meta-logic” was much more rigorous, covers a ton of content, including theoretical computability, Turing computability, and Godel’s incompleteness theorems.
I personally covered the relevant parts of measure theory and a lot of stochastic processes in math finance, which I think is a good way to do it. I did take an OR class which spent about half the time on Markov chains, but I consider that stuff pretty straightforward if you have a good grounding in linear algebra.
Analysis/abstract/topology are exactly the sort of prereqs I recommend skipping. The intro classes usually spend a bunch of time on fairly boring stuff; intermediate-level classes will usually review the actually-useful parts as-needed.
The crypto recommendation makes sense. For logic, I don’t think there’s much value in diving into the full rigor; it’s mostly the concepts that matter, and proving it all carefully is extremely tedious. Definitely important to get the core concepts, though.
You recommend the basic math courses: linear algebra, probability, a standard calculus sequence. You just don’t recommend the more pure math type courses. In your view, pure math courses spend too much time digging into boring tedious details, and you advise more applied courses instead. That’s an entirely valid perspective. And it may be the most productive tactic.
Real analysis, abstract algebra, and topology are often the hardest and most advanced courses in the undergraduate math catalog. Those are considered the capstone courses of an undergraduate degree in pure mathematics. You reference them as introductory classes or prereqs which seems not correct. At almost any university, Real Analysis is the more advanced, theoretical, and difficult version of calculus.
Did you study martingales or stopped brownian motion? Are those useful or recommended? Those seem relevant to finance and applied probability?
I really enjoyed this post, and thank you for the awesome reply.
Yeah, fair. Harvey Mudd is probably unusual in this regard—it’s a very-top-tier exclusively-STEM school, so analysis and abstract algebra were typically late-sophomore-year/early-junior-year courses for the math majors (IIRC). I guess my corresponding advice for someone at a typical not-exclusively-undergrad university would be to jump straight into grad-level math courses.
(As with the post, this advice is obviously not for everyone.)
Yup, that comes up in math finance. I haven’t seen them come up much outside of finance, they’re kind of niche in the broader picture.