# What parts of a pure mathematics undergraduate curriculum have been discovered since $1964?$

What parts of an undergraduate curriculum in pure mathematics have been discovered since, say, $1964?$ (I'm choosing this because it's $50$ years ago). Pure mathematics textbooks from before $1964$ seem to contain everything in pure maths that is taught to undergraduates nowadays.

I would like to disallow applications, so I want to exclude new discoveries in theoretical physics or computer science. For example I would class cryptography as an application. I'm much more interested in finding out what (if any) fundamental shifts there have been in pure mathematics at the undergraduate level.

One reason I am asking is my suspicion is that there is very little or nothing which mathematics undergraduates learn which has been discovered since the $1960s$, or even possibly earlier. Am I wrong?

• A first year graduate course is still teaching material from the late 1800s to early 1900s. – Matt Samuel Dec 27 '14 at 3:17
• To elaborate more, though, the way it is taught is not the way it was discovered historically. Newton studied elliptic curves in the 1600s while Cauchy defined the basic notion of a limit decades later, and Galois theory came before many things they teach you before you get to Galois theory. – Matt Samuel Dec 27 '14 at 3:23
• Seidenberg’s $1959$ proof of the Erdős-Szekeres theorem is only five years too early, and I taught it in a sophomore-level discrete math course more than five years ago as an application of the pigeonhole principle. (The theorem itself is from $1935$, though.) – Brian M. Scott Dec 27 '14 at 3:31
• Most undergraduate math curricula will refer to newer results like the classification of finite simple groups without going into all of the details of the proofs. Also it isn't always that the content has changed since 1964 but the emphasis and the methods of delivery definitely change (which is easy to see by looking at some very old books and some new books). – user171177 Dec 27 '14 at 3:46
• In the spirit of Matt Samuel's comment, I think the "main parts" of an undergraduate curriculum that are new are the presentations rather than actual "results," emphasizing more modern ideas (e.g., functorial properties). This is not insignificant. – Kimball Dec 27 '14 at 6:59

Except for top university mathematics programs which have truly gifted undergraduates in them-such as Harvard, Yale or the University of Chicago-I seriously doubt undergraduates are exposed to truly modern breakthroughs in mathematics in any significant manner. Indeed, it's rare for first year graduate courses to contain any of this material in large doses!

This question reminds me of an old story my friend and undergraduate mentor Nick Metas used to tell me. When he was a graduate student at MIT in the early 1960's,he had a fellow graduate student who was top of his class as an undergraduate and published several papers before graduating. When he got to MIT, he refused to attend classes, feeling such "textbook work" was beneath him." This is all dead mathematics-I want to study living mathematics! Stop wasting my time with stuff from before World War I!" As a result, he had some really bizarre holes in his training. For example, he understood basic notions of algebraic geometry and category theory, but he didn't understand what the limit of a complex function was. As a result, not only did he fail his qualifying exams, his own presented research suffered greatly-he was always playing catch-up. Eventually, he dropped out and Nick never heard from him again. He always tells his students this story in order to make them understand something fundamental about mathematics-it's a subject that builds vertically, from the most basic foundations upward to not only more sophisticated results, but from the oldest to the most recent results.

This is why I think undergraduates simply can't be exposed to "recent" results-it takes until they're at least first year graduate students for a wide enough conceptual foundations to be erected in them to even begin to understand these concepts.

• Sorry but I do wonder if that is a true story, it sounds like the "dead white men" stuff. Hard to imagine someone in maths graduate school saying "I don't need no Pythagoras's theorem, I only use category theory to solve my right-angled triangle problems". – Suzu Hirose Dec 27 '14 at 8:38
• The reputation of Harvard, etc., is earned through their graduate programs. It's unfair to extend that reputation to their undergraduate programs, which are not substantially different than the typical undergraduate math program in the U.S. There are truly gifted students all over the place. And there are engaged teachers all over the place too, some of whom find ways to insert more modern mathematics into their senior level courses. – alex.jordan Dec 27 '14 at 16:49
• @abel There is nothing substantially different about their program---the collection of undergraduate courses they offer. See course catalogs and course descriptions. When this answer says "except for schools like A, B, and C", I disagree that that clause needs to be here. Undergrads seeing more recent math is not limited to any class of school, and better Putnam scores are not evidence of students seeing more recent math, nor even of a better program. Better Putnam scores can also be explained by having a good Putnam training regimen or having a reputation that draws in talented students. – alex.jordan Dec 27 '14 at 17:25
• @abel You say "take a look at their sumer programs for undergrads". I invite you to actually do that. It sounds like by "look at what most other undergrads do in the summer" that you are not aware of what actually goes on all over the place with math majors in summer. – alex.jordan Dec 27 '14 at 17:35
• If they really have sumer programs, they are learning some very old mathematics, indeed. storyofmathematics.com/sumerian.html – Gerry Myerson Dec 27 '14 at 21:31

A lot of the important basic results of complexity theory postdate 1964. The most important example that comes to mind is that the formulation of NP-completeness didn't occur until the seventies; this includes the Cook-Levin theorem, which states that SAT is NP-complete (1971) and the identification of NP-completeness as something that was important and common to many natural computational problems (Karp 1972). These results certainly appear in an undergraduate course on computability and complexity.

Ladner's theorem, which would at least be mentioned in an undergraduate course, was proved in 1975.

(In contrast, the basic results of computability theory date to Turing, Post, Church, and Gödel in the 1930s.)

• I would say that complexity theory itself dates from 1971, and in particular all results in complexity theory are post 1971. – Yuval Filmus Dec 27 '14 at 8:57
• This is all true but is it a common component of undergrad mathematics curricula? – David Richerby Dec 27 '14 at 17:00
• It was when I was an undergraduate. My mathematics coursework had a CS elective that could be filled with C&C, which was commonly attended by many 3rd and 4th-year undergraduate math majors. NP-completeness is treated in the common undergraduate texts by Sipser and by Hopcroft and Ullman. – MJD Dec 27 '14 at 17:15
• @yuval Complexity analysis of algorithms certainly predates 1964. – MJD Dec 27 '14 at 17:16
• @Tyler Skiena The Algorithm Design Manual (2009) refers numerous times to ‘the boolean satisfiability problem’, called ‘satisfiability’ for short; for example “Cook's theorem proves that satisfiability is as hard as any problem in NP.” (p.343); “the hardness of 3-SAT implies that satisfiability is hard.” (p.329; compare my earlier comment) A Google Book search will demonstrate that your claim that “specialists never refer to plain satisfiability” is mistaken. Both 3-SAT and SAT are NP-complete. – MJD Dec 31 '14 at 4:03

Graph theory is a relatively recent subject. Although the Seven Bridges of Königsberg problem dates back as far as 1736, one of the first accepted textbook on the subject was published by Harary in 1969, thirty years after Kőnig's work.

Many theorems were stated and proved in the last fifty years or so. For instance, the strong perfect graph theorem was conjectured in 1961 and was not proved until 2006 by Chudnovsky and al. The statement and the proof of the weaker version are often taught at the undergraduate level.

• Combinatorics in general is a good place to look (esp. books which put lots of references, like Stanley's Enumerative Combinatorics) – Batman Dec 27 '14 at 4:37
• There's 230 years difference there. – Mark Hurd Dec 30 '14 at 0:43
• @MarkHurd: I think iHubble is referring to Dénes Kőnig and not to Euler. – rrufai Dec 30 '14 at 19:14

A lot of things connected to what is conventionally called the chaos theory. For example, it is quite accessible for undergraduates to prove that period three implies chaos.

(Li and Yorke published their paper in 1975. However, a more general result was given by Sharkovsky in 1964 (reprinted here), which still fits your question).

I taught students how to compute the Homfly polynomial in an undergrad topology course. This is an invariant for distinguishing inequivalent knots and links, and only goes back to 1985.

• Many textbooks on Knot Theory and related areas include such material and more recent stuff as well. The material is relatively accessible without a huge amount of preliminary work, yet looks at recent ideas. – Tim Porter Oct 16 '17 at 11:05

I taught undergraduates various primality testing and factorization algorithms (Pollard rho, quadratic sieve, elliptic curve methods) that were news in the 1970s and/or 1980s.

• What level of class would this be, final year students or first year students? – Suzu Hirose Dec 27 '14 at 8:14
• I am an undergraduate but I took a class on the mentioned topics. I am a final year undergraduate but there were also some third year undergraduates with me. A lot of the students were from the computer science stream. – Asvin Dec 27 '14 at 8:33
• It was a 3rd year class --- that's the final year, in Australia. – Gerry Myerson Dec 27 '14 at 21:33
• We did Pollard’s rho method in second year (Germany), in a course focused on getting the computer do advanced computations. (Kind of a math-centered programming course, if you will, but still a math course, including the theoretical discussion. No complexity analysis or anything like that, though.) – Christopher Creutzig Dec 29 '14 at 16:28
• Quite a number of algorithms in number theory are relatively new and are easy for undergrads to understand. RSA dates back to the 70s but undergirds our modern public key encryption as does the Diffie-Hellman key exchange, used in SSL and TSL protocols. Is RSA or Diffie-Hellman "pure" math? Probably not, but the recognition that simple number theory concepts (available to any undergrad) could have significant applications to communications -- that IS new. – KenWSmith Dec 30 '14 at 19:33

Many of the things taught in an undergraduate Algorithms course were only discovered in the last 50 years (and nearly all in the last 100). For instance, A*, the foundation of all modern graph-searching algorithms, was discovered in 1968.

Another subject at the intersection of Math and Comp Sci which has been mostly developed in the last 50 years is Cryptography. For example, RSA (the popular public-key encryption algorithm, which relies on simple modular arithmetic) is taught early on in undergraduate Cryptography, yet was only created in the late 1970's.

• It's true that these are new things but also I think it's rather a statement of the obvious to say that most computer-related or physics-related stuff is fairly new. I'd be more interested in whether, for example, a new theorem in topology, or geometry, or complex analysis, is taught to undergraduates. Apologies for writing a confusing question. – Suzu Hirose Dec 27 '14 at 8:31

In CAGD (computer-aided geometric design), pretty much everything was developed after 1964. The fundamental concepts of Bézier curves and surfaces were developed by Bézier and de Casteljau in the early 1960's, and NURBS curves and surfaces didn't appear until Versprille's thesis in 1975.

CAGD is basically the study of the mathematics of free-form shapes, as used in design, engineering, manufacturing, entertainment, etc. Without it, you couldn't produce a modern car, airplane, ship, or even a high-end golf club. I expect some would argue that it's not really mathematics, though it's often taught in math departments, in both undergraduate and graduate classes. I don't suppose many people would consider it be "pure" mathematics, though it involves lots of (fairly old) algebraic and differential geometry. Personally, I do consider it to be pure/applied mathematics; the computer implementations are typically performed just to confirm that the mathematics is working correctly (and to make money, of course).

There is a bibliography of CAGD and related fields here. If you filter to isolate the works written prior to 1964, you'll see that there's not much.

• This is really applied mathematics,although certainly no one would question its' importance. I just don't think it qualifies for what the OP is shooting for here. – Mathemagician1234 Dec 27 '14 at 6:03
• @Mathemagician1234 -- Sure. I said in my answer that some people would not consider CAGD to be "pure" mathematics. But I thought that some of the other answers were very narrow, refering to various specific theorems that seemed pretty obscure, to me (which means I never heard of them). So, I wanted to include something big and broad. Anyway, we can let the OP decide whether or not the answer is useful. – bubba Dec 27 '14 at 6:40

I nominate Carleson's theorem from 1962 (only two years too early), which is a must-mention (probably without a proof) in any Fourier analysis course now.

Note:it's pointed out that the year here is incorrect. The theorem was published in 1966.

• Wikipedia refers Carleson's 1966 paper which would make this within the 50 years period. Carleson is still alive. – Jeppe Stig Nielsen Dec 29 '14 at 11:04

A ton of the most important theorems in graph theory are a decade away from the mark (Turáns theorem, Brook's theorem,)

Robertson-Seymour theorem has less than $40$ years. If you look for graph theory results you'll probably find a ton in that time which are taught in undergrad graph theory courses.

I'm not sure if my answer qualifies, but here it is: as a second year student I took a course in mathematical logic, which included modal logic. Some of it predates the 60's, but many proofs are quite recent. Unfortunately, I'm not able to remember which proofs exactly, but given that Kripke only began to publish in the 60's, I'd imagine there ought to be more than one.

Also, if you are rather interested in what could be taught rather than what is actually taught, then, during my first year I was reading about cellular automata. The content was very easy to understand, so, I'd imagine if it was taught, that wouldn't pose a problem for students.

• Modal logic preceded Kripke by a decade or so, although his work really legitimized it and made it a popular area of research. And don't forget,Kripke's original researches were really motivated by problems in philosophy rather then mathematics.At least,I think so. I'll have to ask Saul next time I attend one of his seminars.......... – Mathemagician1234 Dec 28 '14 at 22:38
• @Mathemagician1234 Yes, I'm actually doing a math+philosophy curriculum. And yes, I envy you (just a little bit) for being able to attend Kripke's seminars :P – wvxvw Dec 31 '14 at 19:33

I'm really surprised that noone has mentioned Apery's 1978 proof of the irrationality of $\zeta (3)$, which should be (and most of the time, is) mentioned in every book that touches this function.

• What's more, this proof - and even some of the 'magic' behind its sequences - is accessible to a skilled undergraduate and I wouldn't be surprised if it could be taught in a 400-level combinatorics class or the like. – Steven Stadnicki Jan 1 '15 at 16:34

There are some not so old developments in the geometry of Banach spaces. They are typically mentioned in the functional analysis course.

Namely, in 1972 Per Enflo presented a famous example of a separable Banach space without Schauder basis. This space doesn't have an approximation property (i.e. there are compact operators in it which are not limits of finite dimensional operators).

In 1971 Lindenstrauss and Tzafriri proved that any Banach space not isomorphic to a Hilbert space has a closed subspace which cannot be complemented. In particular, you cannot have a continuous projection onto any closed subspace you like.

• Can you point out at which university these particular developments are included into the undergraduate program? – Artem Dec 28 '14 at 1:40
• @Artem The relevant links are asc.tuwien.ac.at/~funkana/downloads_general/bac_zeh.pdf and math.stanford.edu/~lecomte/Complemented_Subspaces.pdf. Also, I mention these results when talking to good students at Belarusian State University (3d or 4th year). The answers are understandable after an introductory course in functional analysis. Though, I doubt that these theorems are often mentioned by name in programs. I guess, more vague "Schauder basis in Banach space" or "Projection in Banach space" are used. – Yauhen Radyna Dec 28 '14 at 3:01

Solitons were discovered in 1965, and might appear in an undergratuade PDE course. (This may or may not count as "too applied" for this question, but basically it is a purely mathematical discovery.)

Just a list of some theorems I have learned as an undergraduate student.

(1) $$(U^*)$$ Morely's Categoricity Theorem: 1965; Model Theory -- This was technically a graduate course, but I believe all the students were undergrads.

(2) $$(U)$$ Ax-Grothendieck Theorem: 1968; Model Theory (or at least that's the context we learned the proof in).

(3) $$(U)$$ Mayer–Vietoris sequence; First appeared in print in 1952; Differential Manifolds.

(4) $$(U)$$ Representation Theory applied to Random Walks - This was engineered by Persi Diaconis, who began this line of work in 1974. This definately counts.

(5) $$(R)$$ Cell Decomposition Theorem/O-minimality is preserved under elementary equivalence; 1984-5: I learned this in a Reading and Research Course, so this might not count.

I will add to this list when I think of more results.

Key: $$U \equiv$$ Undergraduate Course; $$U^* \equiv$$ Graduate Course with mostly undergraduates; $$R \equiv$$ Reading and Research Course.

I have seen Lie superalgebras and groups to be taught in a ugrad course (somewhere - can't find the place now). Most of the foundations for Lie superalgebras were laid in the 70s by B Kostant.

Correct me if not true but graded rings and other graded structures popped out in the 50s (not before) and they can't be accused of being applied.

Since the dividing line between 'pure' and 'applied' is pretty arbitrary, optimization may or may not qualify. But there's plenty of very recent results there that are covered in introductory courses. Dantzig didn't publish the simplex algorithm until 1947, the KKT conditions was published in the fifties and the entire field was extremely under-developed until computers came along.

This is an old debate on whether or to what extent the undergraduate syllabus should be related to research, or reflect, and stimulate discussion on, the nature of mathematics, as understood by professionals. J.E.Littlewood in his book "A Mathematicians Miscellany" claimed his excellent result at Cambridge was little relevant to research. Many, including me, have found the transition from undergraduate to postgraduate a culture shock. I was fortunate to find eventually that my way into research was in writing, and then getting a good problem.

Some have even found that their undergraduate degree "taught me to hate mathematics" or that "the difficulty and inaccessability of the courses left me and my friends scarred". See also a little article Carpentry.

Texts in algebraic topology often ignore relevant work on groupoids developed since 1967: see this mathoverflow discussion.

Another area neglected is fractals; I believe every undergraduate should be given some information on the Hausdorff metric and its application to fractals, since the latter have had wide publicity, and the course on this can be lots of fun with wide applications, provided it emphasises in the first instance ideas rather than proofs.

On the other hand, undergraduate courses often fail to give any background or context to the study of our subject. So the linked article on The Methodology of Mathematics has just been revised and republished. There are more articles on my Popularisation and Teaching page.

I confess to have been shocked that our students usually did not know why the angle in semicircle is a right angle, and so did some Euclidean Geometry for a special course, introducing proof as having the purpose of showing something surprising and capable of development, so showing the intent of a theorem.

As elementary number theory is still considered to be a pure mathematics course, much has entered this field which is currently being applied. In 1964, there was no Diffie-Hellman nor RSA public key cryptography; nor were elliptic curves being used for digital signatures, key agreements, or to generate $$$$random'' numbers; nor were computers an integral tool in cryptography. There were no sieve methods as far as I know, being taught at the undergraduate level, for they had yet to be invented. And besides, the main focus of cryptography in 1964 was on encryption. All that has changed---semiprimes, academically speaking, are now in vogue; a solution to the discrete logarithm problem ranks much higher (I dare say) than it did a half century ago; and even the amateur is trying to grasp at the notion of what quantum computing is all about.