# What should be offered in undergraduate mathematics that's currently not (or isn't usually)? [closed]

What's one class that mathematics that should be offered to undergraduates that isn't usually? One answer per post.

Ex: Just to throw some ideas out there Mathematical Physics (for math students, not for physics students) Complexity Theory

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## closed as no longer relevant by Felipe Voloch, Bill Johnson, Andrés Caicedo, Mark Meckes, Yemon ChoiJan 8 '12 at 20:11

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Programming. I think it varies a lot from department to department but some places seem to do a bad job of teaching programming and it can be a really important skill.

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I think programming is not only useful to be able to compute something with the aid of a computer, but programming implies a special way of thinking about problems. Debugging experience and software engineering has helped me much in mathematics. (So, my +1) – Konrad Voelkel Dec 7 '09 at 16:08
I would also add some bit of "Experimental mathematics" to programming---excellent combination. – Suvrit Aug 25 '11 at 23:55
And especially, programming well. There are a lot of techniques and tricks that make programming much less painful: source version control, how to write tests, good comments vs bad comments, how to refactor code so that it's readable and "makes sense", use the right language for the right problem and don't worry about premature optimizations... – Federico Poloni Aug 26 '11 at 7:43

I would have loved a class on how to write mathematical papers and what goes into mathematics research. Everything from neat Latex tricks to how to organize and structure ideas, theorems, etc, going over bad vs good papers, even perhaps discussing what makes good math books. As well, an overview of what goes into a PhD thesis would have been extremely useful. It's a shame that one usually has to pick up these various bits of info on their own. The class could coincide with a current undergraduate senior project for example and act as a supplement. I took a similar class like this in the physics department and it helped me immensely with my work.

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There's not a whole lot of magic when it comes to writing good papers or books. Usually the magic is an immense amount of hidden effort. Allen Hatcher's "Algebraic Topology" is considered a pretty good book. I have dot-matrix printouts of some of the first versions, dating back to about when Hatcher arrived at Cornell. So the book represents about 30 years of teaching algebraic topology, and many iterates of revising the lecture notes. – Ryan Budney Nov 6 '09 at 22:43
I now wish I had such a class as an undergrad, incidentally, although I think some of the undergrads actually in the class view it (incorrectly!) as "fluff." (Many students take 18.100B, which gives the analysis without the writing.) I disagree with Ryan Budney: just like giving presentations, or teaching, or even conducting research, there are lots of specific, teachable skills that can improve one's writing. Before helping to teach this class, I could diagnose a paragraph as unclear, and with enough work I could fix it. ... – JBL Mar 18 '10 at 15:50
The content of this class includes both a language to describe certain types of bad writing, and tools for quickly identifying and correcting what about it loses the reader. In other words, there are techniques to make good writing easier, and people who are good writers typically either know these techniques or have a strong intuitive feel for them. Just as is the case with any other endeavor, teaching the techniques makes it easier to do it well, and having really good heuristics lets one write better without having to work any harder at it. – JBL Mar 18 '10 at 15:58

Approximation/asymptotics. It amazes me how many otherwise good students don't have a sense for which parts of an expression are large and which are small.

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My EE education in Romania was 10% about how to approximate. Of course, that's not an exact figure. – rgrig Mar 15 '10 at 10:45

Inequalities!

I don't think I've ever seen a course on inequalities, and there's certainly enough elementary material to cover in a one-semester course. Very few undergrads know much about inequalities.

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I agree. And the book "Cauchy-Schwarz Master Class" by Michael Steele would be a great textbook for the class. – John D. Cook Nov 6 '09 at 22:41

Motivation.

I have seen many students dropping out of math because they didn't get an answer to the question "why should I learn this?". Of course, one could say, a good student should have intrinsic motivation and/or figure out the motivations by himself, but this seems to me like wasting potential.

I don't want to say that mathematics courses don't provide any motivation, but in undergraduate courses (and even textbooks), especially in linear algebra and calculus (when there isn't so much time), I haven't seen enough motivation.

This motivation should go beyond "we want to model the physical world" and/or "with this theorem you can calculate the Eigenvalues". Students need the story between these two extremal answers, they need to know how calculation of Eigenvalues is really applied in modelling the physical world. (This is just an example, I would appreciate to see more motivation for abstract, non-applied mathematics, too)

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When teaching eigenvalues the textbook I used gave a simple predator prey example and then looked at the diagrams of the phase space . (I think it was woodrats and spotted owls (oops that is a prey-predator problem!)) The oscillations in population levels could be seen to be interesting as a simple explanation of things students have heard about. That lead o to how on earth can you calculate these eigenvalues and eigen vectors and to some numerical methods. The point was that the example chosen was simple enough to comprehend without knowing a whole lot of some other subject area. – Tim Porter Mar 14 '10 at 19:38

Category theory!

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IMO category theory is best kept until the student has a context for it. So picking up bits and pieces as they learn algebraic topology is a natural thing to do, making it graduate level material for most students. – Ryan Budney Nov 6 '09 at 22:40
I think there is a big danger in making undergraduate classes (especially classes with a strong algebraic flavor) consist entirely of definitions and trivial lemmas. A good class needs some deep theorems! Dwelling on things like category theory often ends up making this problem worse. – Andy Putman Nov 9 '09 at 8:04
Okay, so make it "category theory done right"! But I wouldn't teach a class that was called "Category Theory". That would be like offering a class in, oh I don't know, "Rings and modules". What's the point of that? But category theory could easily be introduced into a more general course well before they reached the "Algebraic Topology" stage where we can finally use the word "functor" without shame. And I disagree completely with your statement "A good class needs some deep theorems". No! A good class needs a story. – Loop Space Nov 9 '09 at 8:54
Using categories achieves this unification at only a superficial level. A better strategy is to have "capstone" courses on subjects like, say, semisimple Lie algebras that are pretty accessible but use real ideas and results from several different undergraduate areas. – Andy Putman Nov 11 '09 at 6:51
The point is that category theory does enable you too see how much of what you are doing is trivial. It is a huge conceptual organizer: Free groups, tensor products, sheafification, ... These things used to be hard to understand (and compute with!) before people routinely used universal mapping properties. You don't have to teach a course on category theory, but emphasizing universal properties first, before giving the crazy constructions particular to each subject area, is a great unifying tool that helps to organize any subject. – Steven Gubkin Nov 12 '09 at 19:44

I think undergraduates should take problem-solving classes. I don't think such classes are widely available, but bright students who didn't do a lot of problem-solving in high school would definitely get a lot out of them.

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I agree that problem solving is good. But the current phenomenon of having "problem solving" classes implies that the rest of mathematics is not about problem solving. Problem solving should be incorporated into the standard curriculum. – Michael Lugo Nov 3 '09 at 18:27
Fair point. What would be ideal is if classes more strongly emphasized problems and less strongly emphasized the stating and proving of theorems without motivation. – Qiaochu Yuan Nov 3 '09 at 18:35
To be contrarian, I've found that problem solving classes tend to be prep classes for exams like the IMO and the Putnam. Although you can teach students to do well on exams, I think there's better things you can do with their time. – Ryan Budney Nov 6 '09 at 22:24
I was thinking more along the lines of the third excerpt from Halmos here: qchu.wordpress.com/2009/08/05/halmos-on-writing-and-education – Qiaochu Yuan Nov 6 '09 at 22:47

I'm going to change the question slightly. What topics do we all think are taught in undergraduate mathematics, but often leak out of the curriculum so that students see too little of them? I have in mind the standard situation at large research universities, where there is a mix of good and not-so-good students.

My pet peeves:

1. Complex numbers as they should appear in standard calculus and linear algebra. They tend to be postponed to upper division courses. Complex numbers greatly simplify both trig identities and partial fractions, but calculus students aren't told.
2. Complex analysis. It tends to float to the top of upper division and disappear.
3. Full multivariate calculus: The Jacobian of a general change of coordinates, the derivative of a multivariate inverse function, maybe also the multivariate Newton's method. The calculus sequence often chickens out and just does special cases of the first of these.
4. Higher-dimensional Euclidean geometry. Like, the definition of an n-cube and the fact that it has 2n vertices.
5. Multivariate probability, especially with both discrete and continuous features.
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Linear algebra. At least, it's not taught to math majors. They tend to see little bits of it in the calculus sequence (in order to solve systems of ODEs, for example) and then again in an abstract algebra class (matrices being a nice source of examples of groups) but never see a coherent treatment of it. At least this is true at the universities I'm familiar with. In both cases there is a linear algebra course mostly taken by nonmajors, but it's not possible for majors to get credit for both linear algebra and first-semester abstract algebra (basically the group theory course). – Michael Lugo Nov 4 '09 at 13:31
Adding emphasis to your comment (3), I find especially in the States there's a rather artificial distinction between "analysis" and "calculus". To the point that calculus is seen as mindless symbol manipulation, but in analysis thought is allowed. John Hubbard's textbook does a great job of blurring those boundaries, especially when it comes to things like the inverse function theorem and Newton's method. – Ryan Budney Nov 6 '09 at 22:32
My son took a really good AP calculus course at the local high school, using the most standard textbook and in some sense the most standard syllabus. So I've seen it from the other side. I don't think it's fair to call it "mindless symbol manipulation". But it is true that American universities sell calculus short in various ways. One reflection of that is that a good AP calculus course can be harder than the university product. – Greg Kuperberg Nov 6 '09 at 22:59
In Germany, linear algebra is emphasized very much. Every university I know has a compulsory one-year linear algebra course along with the one-year analysis course (whose second semester is devoted to multivariate calculus) in the first year. Complex numbers are usually one of the first things which are introduced. Sometimes, they are even introduced in the first week of both courses since the professors talk less to each other than they should ;). But then, in most courses in analysis one is not really told what to do to solve an integral. – Lennart Meier Jun 16 '10 at 8:35

Following Greg's lead, I wish that undergrads who want to become math majors didn't "skip out" of differential equations classes (in their eagerness to get to the good stuff like Drinfeld chtoukas, or whatever). I can't think of a more important foundational subject that tends to be systematically avoided by the "best" undergraduate mathematics students.

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I completely agree!!! Lots of places have "rigorous linear algebra" type classes (I'm teaching one right now) -- I wonder why more places don't have a differential equations class geared for math majors. There are even nice books available (like Hurewitz's beautiful little book, or Arnold's). – Andy Putman Nov 12 '09 at 19:35
I think part of the problem here is the types of course offerings universities have. For example, when I was an undergrad at U.Alberta, there was an honours intro ODEs course, but it was taught like a service course and was largely non-rigorous. It's difficult to attract students to material when the curricula is presented in such an unflattering light. – Ryan Budney Aug 31 '11 at 15:21

Computer Science. I know programming has been said already, but computer science isn't programming. (There's the famous Dijkstra quote: “Computer science is no more about computers than astronomy is about telescopes.”)

There is a vast and beautiful field of computer science out there that draws on algebra, category theory, topology, order theory, logic and other areas and that doesn't get much of a mention in mathematics courses (AFAIK). Example subjects are areas like F-(co)algebras for (co)recursive data structures, the Curry-Howard isomorphism and intuitionistic logic and computable analysis.

When I did programming as part of my mathematics course I gave it up. It was merely error analysis for a bunch of numerical methods. I had no idea that concepts I learnt in algebraic topology could help me reason about lists and trees (eg. functors), or that transfinite ordinals aren't just playthings for set theorists and can be immediately applied to termination proofs for programs, or that if my proof didn't use the law of the excluded middle then maybe I could automatically extract a program from it, or that there's a deep connection between computability and continuity, and so on.

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+1 for "Computer science isn't programming" and the great Dijkstra quote. – Gabe Cunningham May 20 '10 at 1:17
Numerical methods should be considered a very advanced topic, even though it was historically 'early'. The beauty of computer science, as far as I am concerned, generally lies in those parts where everything is exact. The connections between a multitude of areas of mathematics and computer science seem to be exploding in the last few years - and generally with no numbers in sight! – Jacques Carette May 20 '10 at 2:23
I'm one of a very large and growing number of graduate students whose training didn't require any serious computer science-and who deeply regrets it now. In many ways,computer science is one vast interrelated set of applications of mathematics to engineering.It DEFINITELY should be required of mathematics students and as early as possible in thier training.There's very good reason to unite the CS and mathematics departments as many universities do. – The Mathemagician Jun 12 '10 at 1:57
I agree wholeheartedly. I feel that Pure Maths is more relevant to, and has more in common with, CS than Stats, and yet my department is in the middle of combining with the Stats school. – ADL Jun 14 '10 at 13:36
@Alan Stats will make the joint department a lot more money then theoretical computer science will. It's all about the bottom line. – The Mathemagician Jun 14 '10 at 18:11

Computational Algebraic Geometry. Something like the book "Ideals, Varieties and Algorithms" by Cox, Little and O'Shea serves as a good bridge from high school algebra with lots of computations and polynomials to modern algebra with rings and groups, without assuming knowledge of the latter.

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At Bangor years ago we had an undergrad course on Grobner bases etc. Ronnie Brown taught it and it seemed to go down well with the students. He says: It is claimed that mathematics is so advanced that it is impossible for students to read research papers in the subject. However, in a Bangor course on Groebner Bases, one quarter of the continuous assessment was the following: use a bibiliographic database to search for papers on Groebner bases, choose on, and write an account of its use of Groebner bases to the best of your ability and knowledge in the time available. – Tim Porter Mar 14 '10 at 18:12

How to write on a blackboard! At the very least, how to write so that the chalk doesn't squeak.

(Declaration of interests: this was inspired by Kim Greene's answer to Tyler Lawson's question about getting fonts right on a blackboard.)

Slightly more seriously, we should teach our students how to present their ideas well.

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It won't be much longer ... classrooms won't have chalkboards at all. – Gerald Edgar Mar 14 '10 at 18:30
WHY NOT?!? Low tech is good sometimes. – The Mathemagician May 7 '10 at 5:01
Blackboards are threatened in many places (I'm aware of Oxford, and some places in Australia). The mathematicians I know in those places are very unhappy with this situation. – André Henriques May 11 '11 at 18:05

I was never offered a geometry course as an undergraduate, and there's so much lovely geometry, from Euclidean and non-Euclidean geometries, to algebraic and differential geometry, and the rest. So...geometry.

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Basic logic. Coming into university we all start from different backgrounds and some of us have been taught poorly in the past and haven't had the opportunity to learn the fundamentals of conditional statements or if and only ifs, etc. For example, knowing that proving the contrapositive is the same as proving the original statement is worthy knowledge indeed! Try proving that if x^2 is even then x is even without knowing this trick.

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The contrapositive is every mathematician's best friend. – Qiaochu Yuan Nov 4 '09 at 1:14
That is, anyone whose best friend isn't the contrapositive isn't a mathematician. – Tom Church Nov 4 '09 at 4:21

Courses in physics and complexity theory were certainly available in my undergrad days, and were mandatory for some undergrads. I guess it depends on the country, and possibly the college...?

One glaring omission that was prevalent in my time (and place) was number theory. It was typical for math graduates to never have seen even the statement of quadratic reciprocity, which I find crazy.

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A course that just attempts to define the current research areas of maths. If the landscape is so complex, why can't undergraduates be provided with a map, so to speak, in order to begin to decipher this subject?

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Bayesian Statistics. I think it's more useful in many practical situations than traditional statistics.

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Bayesian statistics is more coherent too. You can cover the basics in a week (and spend a career looking at the implications of those basics!) – John D. Cook Nov 6 '09 at 22:46

Basics of numerical methods: What computers can and can't do and how they operate in general.

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I think a great class for undergrads (in particular, for seniors planning on grad school) would be a capstone "Comparative Mathematics" course. In my imagining, this would be a mix of math history, the "greatest hits", contrasting the fundamental objects of study and proof techniques, and an introduction to the map of modern mathematics. Think the Princeton Companion to Mathematics distilled into a semester.

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Maybe this is an overbroad answer, but I'd like to see more specialized subjects that are just really fun. Computational geometry (in the classical/Euclidean sense, not the computational algebraic geometry sense) is the example that leaps to mind -- I'm not aware of anywhere that offers it as an explicitly undergrad-level course, despite the fact that it's amazingly fun, quite simple (I suspect that bright undergrads could get to Arora's PTAS for Euclidean TSP within a semester, and certainly Christofides' algorithm is within the reach of anyone who's taken basic algorithms), and practically useful, although I guess this is more (T)CS than straight math...

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I am actually thinking about functional analysis and modern Fourier analysis.

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I was a graduate student at Yale, and all but a few courses had single digit enrollment. I distinctly remember a course where I was the other student, and a course where our exchange student from Germany got stuck, being the only one attending and not wishing to offend the professor by dropping. – Victor Protsak May 20 '10 at 7:39

Quantum mechanics, as in understanding the mathematics behind its foundational issues, and not just as in computing the spectrum of the Hydrogen atom (though that's good too).

It's hard to think of a topic that shakes one's image of the physical world harder than quantum mechanics. General relativity is easy to digest once you are not scared of things like manifolds. Quantum mechanics remains a challenge to one's worldview no matter how hard one tries to get used to it. You cannot count yourself scientifically literate if you were not exposed to the foundational issues of quantum mechanics.

And it's a math course at least as much as a physics course. The pre-requisites are basic probability and logic and complex numbers and basic Hilbert space theory, and the content is philosophy and non-commutative probability theory and (may as well, at the end) some spectra of some differential operators.

Mermin article "Is the moon there when nobody looks? Reality and the quantum theory" was an eye opener for me, the year after I finished my undergraduate studies.

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Traditional Statistics. Many biology majors end up knowing more statistics than many mathematics majors which I think is a weird state of affairs.

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It's easy to teach statistics if you're sloppy, but if you want to be careful you quickly end up in deep water -- philosophical controversies, not to mention measure theory etc. So I think one reason stat is not often taught to math undergrads is that it's hard to do well. – John D. Cook Nov 6 '09 at 22:44

A calculus class that goes very slowly.

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If you don't teach calculus well, you can't have a decent math major. – Charlie Frohman Mar 15 '10 at 12:44

Igor Rivin has a whole diatribe on this topic!

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"What's one class that mathematics that should be offered to undergraduates that isn't usually?"

OK, I'll rephrase my earlier answer. A class that should be offered to undergraduates that usually isn't is a "what is mathematics" course for those liberal-arts majors who will take only one math course in their post-secondary schooling. It would be a truthful course that would avoid telling them that mathematics consists memorizing algorithms whose utility can be seen only by taking later courses that they won't take. It would acquaint them with the fact that mathematics, like physics, is a subject in which new discoveries are constantly being made. It would tell them that one doesn't generally do math by taking a problem and feeding it into an algorithm that was given to one by a prophet who came down from Mount Sinai. It would tell them that mathematics is a subject that, like music, relies heavily on technical skills but does not consist of those alone. Among the goals would be that a student who takes only that course and becomes a professor of some liberal arts subject would not be among the many such professors who don't suspect the existence of such a field as mathematics.

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Having been offered a not-all-that typical undergraduate curriculum, and having then proceeded to miss a lot of it through over-sleeping, I'm not sure what is or isn't usually offered up. Does Ramsey theory (or even just Ramsey's theorem) get a mention in undergrad-level combinatorics? If not, that'd be my suggestion: about the only mathematics I've succeeded in explaining to non-scientists in the pub, from R(3,3) to the idea of lower bounds via random colourings.

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Courses aimed towards applied, rather than pure, mathematics. Like a modeling course related to environmental sciences, perhaps. Most math majors prepare the students for graduate school in pure mathematics, but offer less support for applied tracks, and there be some good careers there.

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I did a course with Atomic and Nuclear Physics making up a third of the time. Apart from beautiful functional analysis and topology courses, one of my favorite courses was in Quantum Mechanics taught by an excellent Physics prof. I there understood about orthogonal functions, etc. He talked sense! The earlier courses on these subjects went straight past me. I could do the problems but did not grok them. – Tim Porter Mar 14 '10 at 18:27

DESIDERATA:

• asymptotical analysis and its applications
• analytic combinatorics
• analytic number theory
• complex analysis with focus on transforms (i.e laplace, inverse laplace, saddle-point method, stationary phase...)

WE NEED MORE OF THOSE BEFORE THE UGRADS THINK THAT CATEGORY THEORY IS THE PINNACLE OF MATH!!! (the last sentence is a joke, please do not get offended anybody out there).

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Combinatorics seems like one of those subjects that you really have to have a natural affinity for. – Harry Gindi Dec 7 '09 at 10:15
IMHO there is already too much unmotivated analysis in undergrad education. I rarely meet undergrads who think categories are the pinnacle of math, but I do meet more than enough undergrads who think that two-lines bounds involving epsilons, deltas, absolute values (as commonly seen in stochastics, diff. equations and asymptotics) are the pinnacle of maths. – darij grinberg Mar 15 '10 at 11:06
@Harry: I used to think so too. But then I saw analytic combinatorics, 'done right', and I learned to really enjoy it, while I hated my 2nd year combinatorics class. – Jacques Carette May 20 '10 at 2:31
@Jacques, which book did you use? or did you go from lecture notes? It always seems that combinatorics is almost never taught in an understandable way... – Michael Hoffman May 20 '10 at 21:14

I would love to see more differential geometry offered in undergrad. The course I envision would start with a review of vector calculus, move to studying hypersurfaces in $\mathbb{R}^n$, and then move into a study of manifolds. You could tie all these subjects together via the Fundamental Theorem of Curves, the Fundamental Theorem of Hypersurfaces, and the Fundamental Theorem of Riemannian Geometry. I feel such a course would help bridge the gap between undergrad and grad school; simultaneously reviewing the key ideas of calculus at a high level while also giving a solid foundation from which to study manifolds in grad school.

Here are some topics that could be covered:

• Curves in $\mathbb{R}^n$
• $k$-frames and curvature, leading to the Fundamental Theorem of Curves

• Hypersurfaces in $\mathbb{R}^n$

• Tangent spaces and curvature
• First/second fundamental forms
• Moving frames, Christoffel Symbols, Gauss Equations, Codazzi-Mainardi Equations.
• Fundamental Theorem of Hypersurfaces, a word on curvature tensors

• (Real) Manifolds, charts, multiple definitions of tangent vectors

• Mention Lie Bracket and Lie Algebras (after doing derivations for tangent vectors)
• Affine Connection, leading to the Fundamental Theorem of Riemannian Geometry

P.S. I am not a differential geometer. I study homotopy theory. I just thought a course like this was missing from the curriculum. Any feedback on other topics that could be covered or tangents that could be mentioned leading off from this material would be welcome.

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