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Pure Math, RPI, and You, a Guide.

A PDF version of this guide can be found here
Oh, before I even begin I can hear the comments. “What can you do with pure math? It’s useless.” “RPI is an applied school, you should just transfer if you want to do pure math.” “Well I’m an engineephysicist/computer scientist, so taking applied math would work out better for me.” “Pure math is so boring, you overly complicate yourself on the most basic things and abstract them past all understanding.”
These are common misunderstandings, so let me disabuse you of them before considering what opportunities exist for pure math at our lovely university. Primarily, while RPI does shy away from pure topics and heavily slants things towards applied, there are certainly some professors that can teach pure math topics and at times even use them in their own work.
Abstract Algebra? Used in quantum physics, in Noether’s Theorem, cryptography, robotics, graphics, imaging, even in chemistry (scroll down for english links). Number theory? One could argue that computers wouldn’t even exist without number theory, let alone cryptography. Topology? String theory, protein folding, topological insulators, chirality, robotics. Linear algebra has so many uses that it could be considered applied math just by itself. Analysis? Fourier series, wavelets, quantum physics uses Hilbert spaces - hell, functional analysis is basically just a quantum physics course in disguise, signal processing, imaging, weak solutions to differential equations, the basis for finite element analysis, measure theory is developed from this, giving us statistics, probability, stochastic modeling, and the list goes on and on.
Let me be abundantly clear. Pure math is not called that because it’s not applicable to real problems in the world. It’s called that because that’s not why we do pure math. I promise you, if you’re at RPI, with very few exceptions (such as STS), there is something in your field that you can better understand if you have some background in pure math.
Before we proceed, I’m just going to get some notation out of the way. If I mark something with italics, it’s not something I’ve had personal experience with and so am less confident of. If I mark something with bold, I feel strongly about it.


Undergraduate Classes

The first course on the catalog at RPI I’d really consider to be pure math is Abstract Algebra. This course has been, well, long neglected by the department, given who they had teaching it. He is no longer teaching it, and so everything I say needs to be taken with the caveat that I took a previous version of this class with a different instructor. That said, I actually really enjoyed the class modulo the instructor, coming out with a decent grade and a decent understanding of the material involved. The basic description I’d give of this course is you look at certain structures that do less than the reals that we normally work with, and show how these structures still have a lot of interesting properties, and that considering them gives us some insight into properties of underlying symmetries and of prime numbers. The workload was not that hard at all, though the book used is not one I would recommend (to my knowledge the book has not changed). It’s a really interesting course, though I do admit that it doesn’t have much immediate application, and some self study into the specific area you want to apply it to will be required. Think of it as background for your own self study into [algebra applied to x] if you take it looking for applications.
Number Theory, a review a friend of mine wrote up, “The course itself was a great deal of learning all about each topic, the theories, the processes, and then the application as well. We went over a number of uses for each topic and were tested on the application. The classes themselves consisted mostly on going through a proof of a theorem, and then applying the topic. - Arithmetical Functions and Dirichlet Multiplication, Congruences, Dirichlet Theorem on Primes in Arithmetic Progression, Quadratic Residues and the Quadratic Reciprocity Law, Primitive Roots, Analytic Proof of the Prime Number Theorem“.
Computability and Logic. Don’t take this course as a math course, take it as a humanities course. Why? You get HASS credit for taking a math/compsci course. I feel like both math and compsci have more options available to you for other courses to take than philosophy does. That said, the main difficulties with this course will be with the software you use. I’m in it currently and understand all the material fine, but the software is a pain in the ass. You cover topics like why there’s no way to prove computationally in certain systems of logic whether a proof is valid or invalid. Gödel’s incompleteness theorems do come up, as do various topics in number theory (Peano Axioms) and computer science (Turing Machines). I wouldn’t say this is a course you take if you want applications outside of computer science or mathematics.
Topology. This course no longer exists, no matter what the catalog says. It will be discussed further in a later section.
Foundations of Analysis. Basic introduction to set theory, logic, proofs, and LaTeX. Do not think of this as a math course, think of it as a course tax to get into other pure math courses. If you can do basic proofs and can use LaTeX you’ll be fine.
Linear Algebra. While I’ve taken this course I have not taken it at RPI, so my comments will be somewhat limited. The topic is incredibly interesting and useful. You mainly cover how vectors in arbitrarily dimensioned spaces behave and how you can move them from one such space to a similar vector in a similar space and how vectors behave under continuous transformations. Consider this either applied or pure, it’s up to you, but realize if you’re going into it for applied reasons and not doing any other pure courses it’ll probably the most pure course you’ll ever take, and if you’re taking pure courses and take it it’ll probably be the most applied course you’ll ever take.
Graph Theory. Could be considered pure. However, this course seems to not exist as well, a similar story to topology, will be discussed later.
Mathematical Analysis 1. Like calculus but better. You know how I said that Foundations of Analysis wasn’t actually a proof course but was just a course tax? This is the actual proof course. You’ll be concerned with justifying why we can do calculus at all, you’ll actually prove a lot of the things calculus takes for granted, and you’ll show when these things fail. If Abstract Algebra is the study of symmetry and Linear Algebra is the study of vector spaces, this is the study of functions and continuity. It was the best undergraduate math class at RPI when I took it. Since then a new professor has taken over and I’m told she grades harsher and the class is less fun and interesting. I’ve been told by certain professors that she might tone things down in the future, as they’re going to ask her to. But don’t take this as a promise, fair warning, it might be just as harshly graded this coming fall.
Mathematical Analysis 2 - I’ve reached out to some people and not heard anything back. Part of the issue is that the professor recently changed and people would specifically avoid the last one. Kovacic’s teaching it now and from what I’ve heard it’s much better. I’d recommend it just based on this info, but wouldn’t say it’s necessary to move onto grad courses, I didn’t, and did fine.

Grad Classes

Real Analysis. Haven’t taken it nor do I know someone who has so cannot comment at all. That said it looks like a ton of fun and I’m taking it next spring.
Introduction to Functional Analysis. The most fun math course I’ve taken at RPI. It’s taught by the most rigorous professor in the department. This subject area is in some sense infinite dimensional linear algebra. Everything you do here is relatable to quantum mechanics, and I can’t recommend this course enough if you have the prereqs or think you can do the work. That said, it’s a tough course, and it wasn’t uncommon to spend 5 hours on a single part of the homework. Due to who teaches it there’s going to be some emphasis on signal analysis and optimization at the very end, but aside from that you mainly cover operators and the properties they’ll have in infinite dimensional function spaces.
Nonlinear Functional Analysis. Like the above but you remove the linear component, it’s just more difficult. A lot of the class is about optimization, finding minimizing solutions to an equation in infinite dimensional space, and there are callbacks to topics in calculus involving optimization, such as Lagrange multipliers or implicit function theorems. If you enjoy functional analysis you’ll at least tolerate this.
Complex Analysis, this course no longer exists, even officially.

A Note on Independent Study

So I said I’d come back to topology and such, and here we are. The unofficial policy of the department is that topology (I know this specifically with topology, and I suspect other classes that haven’t been offered recently fall in this same category) isn’t worth a professor’s time, as very few people ever took it. Instead, they unofficially push those students who are interested in topology or other topics in pure math to talking to Professor Piper about doing reading courses. If you’re interested in a topic not available at RPI, such as topology, further reading in abstract algebra, gauge integration, measure theory, etc, then the way to take these courses is to work on a reading course with Professor Piper. Which brings us to our next section.


So not all professors will be relevant to you as someone engaged in pure math, at least not qua pure math student. So let’s go over those that are relevant.
Professor Piper is going to be your go to guy for reading courses, and as he’s the person who approves undergraduate degrees, being tight with him can’t possibly hurt. Taking a reading course would be better than taking an actual course with him, as I find him to be pure monotone. And his own specific interests in geometry did shine through in Fundamentals of Geometry when I took it, I think to the detriment of the course. He’s a great guy, just not the most engaging lecturer.
Professor Mitchell is someone I doubt you’ll ever take a course with, but he runs point on graduate studies things for the math department. If you want to take math grad courses, similarly to before, getting to know him at colloquia and such can only be a good thing.
Professor Holmes teaches Real Analysis, I believe. If you plan on taking real analysis, getting along with him is, of course, a good idea. I can’t comment much more than this, having not taken Real Analysis. But he seems nice enough.
Professor Kovacic currently teaches Analysis 2. I find he’s a very binary professor, you either love courses with him or you hate them. He’s not the most organized, and is prone to calling things trivial in office hours. Similar to Piper in that he’s a great guy but might not fit your style of learning as a lecturer.
Professor Kramer tends not to teach pure math courses. But he’s certainly no slouch in the area, I’m auditing his stochastic diffEQ class and there’s a decent amount of measure theory and abstract integration. He’s not directly relevant for pure math at this time, but he’s certainly a professor that you can talk to about pure math to a certain extent (more measure theory than any other area, of course), and he runs the MCM thing at this school, which even as a pure math student you should do.
[Continued in the comments]
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