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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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[OT]Ask Nate: Field Manual FM-02W SciFi Military Fiction

    Field Manual FM-02W

       SciFi Military Fiction

"What a gang of apes! Maybe if you'd all buy it this drop, they could start over and build the kind of outfit the Lieutenant expected you to be."      - Career Ship's Sergeant Jelal Starship Troopers 1959. Robert A. Heinlein


     Before we get started here, I want to send a huge shoutout to everyone who participated in FM-01W. Especially, WarriorPoet02 who has a in-depth knowledge of both modern and historical combat, as well as a validated expert (with actual experience) in modern Marine Combat. He was able to fill in some gaps in my knowledge and even I learned some stuff. On that point, while I may have a wide-range of general knowledge, I'm not an expert in things I'm not an expert in. Don't ever be afraid, no matter how much you think you know, to consult others. You can always learn more.
     Time for Hot Shots: Part Deux; SciFi. As for the quote, it comes from one of my all-time favorite SciFi MilFic books; which was turned into a campy movie that pretty much only shares the same title. The book itself was once on the Commandant of the Marine Corps reading list, because despite being an old SciFi action romp; there are a number of deep reflective themes in the book that transcended genera fiction. Themes that did not make it to the film. Neither did their awesome power armor.
     Enough about Starship Troopers. What am I going to cover here today? This guide and the Fantasy one are designed more to provoke inspiration and creativity, rather than to give you established precedent. These are just my recommendations, so feel free to pick and choose what you want. SciFi and Fantasy universes have their own rules, just stay consistent. I'll tell you how I (and a few others) have gone about using existing military concepts and apply them to straight up fiction. And remember, no matter how cool your tech is, the best stories are always about the people and their struggles.

Frame the Universe

     Before you take your first step out that airlock, you need to decide what your restrictions are. Unlike the other three guides which are locked to a single planet (Earth or your Fantasy world, don't make me go down a magic portal rabbit hole for other worlds there...) SciFi is usually out in space. But not always.
     Why is there conflict? You can write SciFi all day without political conflict (personal conflict, not so much). But military SciFi needs an excuse to break out diplomacy through other means. Just having a squad run around breaking sh*t and engineering chaos, while exciting, isn't very interesting. Who are your aggressors and why? Develop the reasons, they tend to make for a better read than just blatant xenophobia. Old Man's War has some interesting angles to this regard. I'm sure you can all cite a few more (Dune, also comes to mind).
     And as goes with all stories, your characters need to be relatable. So if you have no humans, there better be some threads you can attach to as a reader.

Force Building

It's like world building, but for the military.
Me, I like to ground things in a heritage of the existing. Why? Because it's what humans do, either because of heritage, relatability, or because we aren't very original. This is more relevant (I feel) in SciFi based off human futures (moreso than let's say High Fantasy), deviate accordingly.
Rank and Structure      Most of the best SciFi/Fantasy militaries are based on real world examples. Yes, you can start from scratch and create completely alien military structures. However, the more off-the-beaten-path you go, the harder it will be for the reader to relate to and follow. As much fun as it is to create a 27-tiered rank structure of a thirty-seven layered hierarchy, your readers aren't going to be able to keep track without constantly referring back to the 30-page appendix in the back of your novel.
So… Kreckel Jip Paccku of the 4th Gregglan Raggers… ok so a Kreckl outranks a Jiggag, but not a Opperg. And a Ragger unit is bigger than the J'hest, but subordinate to the Max Headroom?
     Yeah, confusing as hell without constant references. Titles like "Lord Imperator" or "Knight Commander" might not be modern ranks, but at least make enough sense for someone with an average IQ to follow. This is why you generally see authors stick to basic concepts of Private, Corporal, Sergeant, Lieutenant in their works. It's not just that future military concepts are rooted in history, it's that the reader has to follow it.
     Is it ok to make up new unit types and ranks? Sure, just be careful not to make it insanely confusing. I have a universe where I replaced line company Captain (O-3) with a rank called "Aegis". It's the same rank, I just did away with the confusion created by also having ship "Captains" (O-6). The other officer ranks mostly follow the traditional Marine/Navy structure. (Ignore the two "non-combat" columns on the right.) This problem actually exists in our own military, though it seldom causes any actual problems. Navy Captains out rank Army, Air Force, and Marine Captains by three levels.
Navy or AirForce?      Who took the lead? In SG-1, it was the Air Force, so they were very focused around their behavior and structure (though the other US DoD branches and Russians were also in the picture). In many SciFi tales they try to shape their fleets around Naval traditions. Feel free to blend/meld them… just try not to be too confusing.
Mega-Stupendo Heroes      Is your MC overpowered? Overtrained? Do it, don't do it, I don't care. But seriously, be careful with it if you do. Master Chief is a great video game character whose story became relatable because he struggles with his humanity. Having said be cautious of making super soldiers, I've done it myself. Yet, they too struggle with what it means to be human and are vulnerable at the loss of those around them.
     I actually started this point to ponder on an old scifi show Space: Above and Beyond which struggled to try and stay "realistic". Real pilots take years to train. Typical pilots hit the fleet as senior First Lieutenants who're about to be promoted to Captain. You don't risk all that training sending pilots on infantry missions. The show did that a number of times. There were other weak plots I caught watching it as an adult who had served, that never dawned on me as a kid. (I still enjoy the hell out of S:AAB.) Yes, your MC might be an elite, super warrior… but there better be logical explanations for why before you end up with a magic Mary/Marty Sue.
Warrior Generals      I suppose this largely depends on the universe you build out, but even some of the most hardcore modern Generals don't see frontline combat. In the real world, you'd be lucky to see anyone above a Major on the ground/in the fight or a LtCol/Commander in the sky trading shots with the enemy. You tend to trade rank for boring desk jobs and rear-echelon leadership roles. The "warrior general" is really something of the bygone era and has more of a place in Fantasy, than SciFi. That's not to say the rules of your force can't dictate that, just be mindful. In Starship Troopers, everyone dropped and everyone fought. Pretty sure one of the big Generals bought it in a bug fight too.


      War, in the most extremely basic mathematical approach, is all about rendering more of the enemy combat ineffective than they can inflict similarly on to you. Conflict is all about the 5 D's: Defend, Delay, Disrupt, Destroy, or Divert. Technology has evolved over time to do those five things. I'll cover a handful of popular concepts, but this isn't even close to the full list.
Space Battles      Massive fleets of capital ships and squadrons of fighters are often woven into the fabric of an epic space opera. So how does one describe this dance of behemoths?
     Scarecrowsid, also pointed out: Using Navy structure as a base, the value of studying the ways in which a CIC and Bridge operate can have a significant effect on how battle sequences play out in Ship to Ship combat. There are a number of options here, but to name a few:
Drone warfare      On the note of space battles and the argument for human pilots vs drones: Light has a current finite speed. 299,792 kilometers per second. We'll assume no one is jamming your comms. That's still millisecond lag in close engagements. We put current drone operators as near as we can to avoid lag in terrestrial situations. Even then you get some lag. Not just data travel, it's processing too. I can't go into actual lag times or the differences in responsiveness for drone operators in CONUS vice in Theater (not only because it's likely classified, but because I don't know the details other than it exists). These are drones not engaged in 1-on-1 sorties. So imagine in your space combat drone vs piloted and how the signal-decision-command-execute delay is for a live pilot vs a drone that is getting further and further from the base station. If you have pushed the magic insta-communications "I believe" button, then go ahead…
Communications      Back to the speed of light… physics is a bitch. Ok, we can assume even in an FTL world, light still takes time to travel. Some EU gets around this a number of ways.
Power ArmoMECHs      I enjoy a nice suit (the HIMYM/Avengers gag was brought to you by MajorParadox). The US military is currently working on a few prototypes, but the biggest hangup is still power. It will likely continue to be a problem for the near future before we see Space Marines dropping on us. But this is your Nuka-Cola™ powered future. Bring on the Jeagers, Power Armor, and Battlemechs. Things you might want to toy with are scale issues (stepping on friendly forces), power (might still run out of those Fusion Cores), Ammo (it's still gonna run out), crossing large swaths of terrain (are you limited to human running speed, there still is a human in that suit), and the shortcomings of human anatomy (there still are restrictions on how much punishment a body can take even in the nicest padded cell and a body has to fit into the armor somehow).
Superweapons      I'll show you my Galaxy Gun if you let me see your Death Stars. (I'm already ashamed at that joke.) Yes, superweapons are a trope. As are the: Lost Superweapon, Forgotten Superweapon, Superweapon Surprise, and the Ancestral Weapon. I'd say not to, but the Mouse now owns an entertainment Empire born on the back of them. My advice…make sure your thermal exhaust port isn’t showing. (cringes) Ok, bad jokes aside; if you want to go the superweapon route, don't build me a third Death Star (cough, cough JJ?). What do I mean? Get to it, fight it, but don't dwell on it forever. (Seriously, don't spend pages on building it for me.)
Robotics      I actually was going to skip this for the sake of running long, but two of the reviewers pointed out it would be a good topic to cover. Macro and micro (regular and nano?) robots are likely to be a large part of space exploration since humans are fragile (we squish good) and take a lot of logistics to support. Human-sized or larger combat troops might just be the answer. On the other hand, smaller nanobots might be used for repairs/construction or in a swarm/cloud attack approach.
     (Will insert quote here if permission is given, don't like using people's words without their consent.) The gist of the argument made, however, is that continued human combat would be unlikely given a robust AI robo-troop force.
     It's a valid point. I could see it go a few ways as a story teller:
  1. You could write from a sentient AI robot perspective and keep to the above ConOps
  2. You could be one of the Robot Handlers. A commander of robotroops.
  3. Your society banned AI after a Terminator-like Rise of the Machines, so only basic (or no) robots are assisting humans in space.
     Also, from my buddy /Merklynn:
I'd throw Phillip K. Dick's short story "Second Variety" into the discussion. It's focus on a post-apocalyptic Earth, where a small rabble of surviving military from both sides try to rendezvous for a truce while avoiding the lethal "claws", disguised AGI killing machines, is about as close as I've seen PKD come to military sci-fi. The story isn't pure military sci-fi, but it is one of the earliest examples I can think of in which artificial intelligence has gone out of control, infiltrated humanity and constantly upgrading itself, making a terrifying enemy. The 1995 film adaptation Screamers is hit and miss, but the surviving enemy units being forced together and dealing with a total loss of communication with their superiors on Earth is one of the most compelling things about the story for me.
     Stuff I didn't cover: killer robots, nanobots, superviri, cyborgs, and more… I'd be here all week.


     Alright, back to your combat. So now you have your universe, how are you going to play in your sandbox? A lot of the stuff covered in the previous guide still applies (make sure to skim the comments of that guide for some great commentary on combat). Some additional things to consider:

BONUS: Mercs / Parma-Military Contractors

     penguinzeppelin asked me about Mercs. Well IRL, most Military Contractors (even the ones without guns) heavily draw from the retired/veteran pool. It's a steryotype for a reason. I would likely approach my characters as such, or conversely how difficult they had blending into a group of all vets -OR- as a company/grioup the hard time they had getting work without that on their CV.
     While entertainment media likes to portray corporate military orgs like crazy wildcards (yes, Blackwater was bad m-kay) the ones that don't adhere to strict business practices tend to flash and fade as they die the death of a million lawsuits. Blackwater did end up in very hot water.
     Mercs that don't have business acumen tend to become pirates, privateers, or freelancers at least in decent SciFi. Morals tend to get in the way (or become great story points). Is it a big team of hundreds and your MCs are just a small cog in the wheel (maybe they break off and go it alone)? Are they a small (12 or less) team that does independent contractor work? Do they look for a specific type of work that suits their personalities or are they so desperate for work that anything goes?
Big Orgs
Small Orgs
Again, this "article" is more designed to generate ideas and concepts to be applied to the normal rules of good story telling.
Questions, Comments, Complaints for your Congressman?
Ask your questions and I will get to them as soon as my day allows. Everyone is encouraged to participate and share your own thoughts. This is an open discussion. If people bring up good points, I will edit this accordingly. Also feel free to list your favorite SciFi military books, shows, etc in the comments.
the Military Fiction (MilFic) Field Manuals FM-01W - Modern Military Fiction FM-02W - SciFi Military Fiction - (this guide) FM-03W - Fantasy Military Fiction (High and Low) - TBD FM-04W - Historical Military Fiction - TBD – Will cover ancient armies (Roman/Egyptian) up to early-Industrial/pre-WWI
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