# Modern Mathematical Achievements Accessible to Undergraduates

While there is tremendous progress happening in mathematics, most of it is just accessible to specialists. In many cases, the proofs of great results are both long and use difficult techniques. Even most research topologists would not be able to understand the proof of the virtually fibering conjecture or the Kervaire problem, to name just two recent breakthroughs in topology, without spending months on it.

But there are some exceptions from this rule. As a topologist, I think here mostly about knot theory:

• The Jones and HOMFLY polynomials. While the Jones polynomial was first discussed from a more complicated context, a rather simple combinatorial description was found. These polynomials help to distinguish many knots.
• The recent proof by Pardon that knots can be arbitrarily distorted. This might be not as important as the Jones polynomial, but quite remarkably Pardon was still an undergrad then!

Or an example from number theory are the 15- and 290-theorems: If a positive definite integer valued qudadratic form represents the first 290 natural numbers, it represents every natural number. If the matrix associated to the quadratic form has integral entries, even the first 15 natural numbers are enough. [Due to Conway, Schneeberger, Bhargava and Hanke.] [Edit: As mentioned by Henry Cohn, only the 15-theorem has a proof accessible to undergrads.]

My question is now the following:

What other major achievements in mathematics of the last 30 years are there which are accessible to undergraduates (including the proofs)?

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PRIMES is in P? –  cardinal May 5 '13 at 19:15
Regarding the 290-theorem, the proof is not accessible to undergraduates. (As written, it uses the Ramanujan conjecture for weight 2 cusp forms. There might be an undergraduate-accessible proof, but the one Bhargava and Hanke wrote up isn't it.) For the 15-theorem, I don't recall anything nearly as high-powered, so I think you could teach it to undergraduates, but you'd have to spend some time teaching them about genera of quadratic forms first. –  Henry Cohn May 5 '13 at 23:24
Could you clarify the time frame you have in mind? A single lecture, a week, or a month? –  Alex R. May 6 '13 at 0:11
Is it fair to characterize the use of the Jones Polynomial to distinguish knot as tremendous progress in mathematics? My impression is that its primary source of interest is in its deeper significance. –  Jonah Sinick May 6 '13 at 3:57
@Jonah: I think, if one is too picky about the notion of tremendous, there won't be any examples...But I think, making significant progress on the 100-year old problem of classifiying knots is already pretty good. –  Lennart Meier May 6 '13 at 14:44

Primes are in P. The proof is indeed accessible, see for example the article "Primes are in P: A breakthrough for "Everyman", http://www.ams.org/notices/200305/fea-bornemann.pdf‎. The idea is really simple, based on the observation, that, if the natural numbers $a$ and $n$ are relatively prime, then $n$ is prime if and only if $$(x − a)^n \equiv (x^n − a) \mod n$$ in the ring of polynomials $\mathbb{Z}[x]$. Of course, more precise results concerning complexity are not so easy.

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Not only is it accessible to undergraduates, two of the authors were themseleves undergraduates when they found it. –  Chandan Singh Dalawat Aug 28 '13 at 1:17

Google PageRank algorithm is remarkable mathematics we experience daily. It's even taught in some undergraduate classes. Here is the link to one such class

http://www.math.harvard.edu/~knill/teaching/math19b_2011/index.html

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It is not difficult to think of examples where the content, statement, and import of a result are accessible, but the proof is not accessible. Hales' resolution of the sphere-packing Kepler conjecture can certainly be appreciated, and the proof outline and technique are accessible, but it would be a stretch to say that the proof is "accessible to undergraduates" (or to anyone, for that matter!).

I have had some success explaining to undergraduates the proof of the Bellows Conjecture: that the volume of any flexible polyhedron is constant (and so cannot serve as a bellows). The 1995 proof by Idzhad Sabitov (for genus-zero polyhedra) uses a grand generalization of Francesca's 15th-century formula for the volume of a tetrahedron as a function of its six edge lengths. Sabitov showed that the volume of a polyhedron can be expressed as a root of a polynomial whose coefficients are polynomials in its edge lengths, a remarkable result. (The polynomial is already degree-16 for an octahedron.) Because the edge lengths of a flexing polyhedron are constant, the polynomial is fixed and can only change discretely by jumping from one root to another. But this contradicts what should be a continuous volume change under a continuous flex.

Steffen's 14-triangle, 9-vertex flexible polyhedron (Fig.23.9 in Geometric Folding Algorithms).

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Correction: Sabitov's result is that the volume of a polyhedron can be expressed as a root of a polynomial whose coefficients are polynomials in the edge lengths. –  John Pardon May 7 '13 at 1:16
Thank you for the correction! –  Joseph O'Rourke May 7 '13 at 1:41
Alexander Gaifullin generalized it to arbitrary dimensions: arxiv.org/abs/1210.5408 "Generalization of Sabitov's Theorem to Polyhedra of Arbitrary Dimensions" –  Alexander Chervov Jun 27 '13 at 12:53

The existence of a gömböc. From Wikipedia:

A gömböc (pronounced [ˈɡømbøts] in Hungarian, sometimes spelled gomboc and pronounced GOM-bock in English) is a convex three-dimensional homogeneous body which, when resting on a flat surface, has just one stable and one unstable point of equilibrium. Its existence was conjectured by Russian mathematician Vladimir Arnold in 1995 and proven in 2006 by Hungarian scientists Gábor Domokos and Péter Várkonyi.

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There are all sorts of assumptions behind the question, and answers, such as that the solution of famous problems is the test of progress in mathematics. At my first international conference in 1964 I met Stanislaw Ulam, and he mentioned to me: "A young person may think the most ambitious thing to do is to tackle some famous problem or conjecture. But that might distract that person from developing the kind of mathematics most appropriate to them."

In the 1980s I gave a talk to teachers and children on "How mathematics gets into knots" and mentioned prime knots and prime numbers. After the talk, a teacher came up to me and said: "That is the first time anyone in my career has used the word "analogy" in relation to mathematics." I find that tragic! We have to be careful not to fall into: "They ask for bread and we give them stones."

The notion of fractal is well known to the public, but how many mathematics courses give a simple account of the Hausdorff metric, and let students see some of the mathematics behind the fractal notion.

Other scientists would like to know what is new in mathematics, in terms of concepts and ideas. My talk to a Conference on Theoretical Neuroscience in 2003, was well received. it included an email analogy for colimits, as well as ideas on higher dimensional algebra. One participant told me: " That was the first time I had heard a seminar by a mathematician which made any sense!" So this time I managed to get it right!

First year main math courses at University should contain something which excites the imagination. (I am told Physics courses usually have something on current research.) That Euclidean geometry has been out of most syllabi makes this harder, especially to get over the idea of proof. Is it too harsh to say that courses on "Proof" are about how to write clear proofs of boring things? It is good to show proofs of otherwise not so believable things.

In the 20th century, a main contributor to the unity of mathematics has been Category Theory; I feel this is a high order mathematical achievement! See the article Analogy and Comparison for a discussion of analogy in this context.

A simple talk to the first year on cubes of dimension $0$ to $5$, and how to count faces of various dimensions, awakened the interest of a student, who later went on to a PhD. (This was also a talk I have given to 13 year olds: they end up by counting the $2$-dimensional faces of a $5$-dimensional cube.)

There is also a lot to say about the contribution of mathematics over the millennia to science and culture. See an article on Mathematics in Context.

Edit: January 12, 2014 : I would like to add that since the general public are often familiar with the words fractal and chaos, it is sad if undergraduates in mathematics are not given some idea of the rigorous mathematics behind these notions, in particular for fractals that is the notion of Hausdorff metric. I have given a light hearted course on this in a second year course on analysis, without proving the completeness theorem, but explaining what it means, and with exercises on calculating the Hausdorff distance between subsets of the plane. I also asked them to do a short project on "The importance of fractals" using the web to get evidence, and also encouraged use of fractal computer programs.

The notion of "chaos" is also important in view of the financial situation, and the everyday notion of weather and climate!

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@Ronnie: who is "Saul Ulam?" Do you mean "Stan-the-Man?" –  John Klein May 6 '13 at 18:19
@John: thanks John. Corrected. The conference was in Syracuse, Sicily, in honour of Archimedes, and Sierpinski and Ulam stayed at the posh hotel; but Stan came down regularly to chat with the rest! –  Ronnie Brown May 6 '13 at 19:58
Can you say something about the email analogy for colimits? (If it was once at that link, it is no longer...) –  Cam McLeman Aug 19 '13 at 15:09
@CamMcLeman: Thanks for drawing this to my attention! It is now there. My idea is that this intuition gives a possible mathematical framework for dispersed communication, which must be how the brain works. (Except that emails while dispersed in pieces are so only from one server, usually. I was thinking about statements/proofs of van Kampen type theorems.) –  Ronnie Brown Aug 24 '13 at 16:17

Dvir's proof of the finite field Kakeya conjecture in 2008 surely should count as a modern achievement. This problem was considered to be extremely hard and even though it originated as a toy model of the Euclidean Kakeya problem, it had become an important problem unto itself. Moreover, it was the first and dramatic example of the application of the "algebraic method" in these sort of geometric combinatorics problems, which was later extended to the Euclidean setting (albeit in a highly nontrivial way, and not to the Euclidean Kakeya problem), most significantly in the solution of the Erdös distance set conjecture by Guth and Katz.

The proof is fairly short and elementary and certainly accessible to undergraduates with the right background, see Terry Tao's notes.

Added by PLC: Yes, this a very nice proof to show to undergraduates, especially those who know about the Chevalley-Warning Theorem. (A little searching on this site and elsewhere will reveal that Chevalley-Warning is one of my very favorite results in undergraduate number theory.) I was extremely taken by Dvir's proof when it came out and wrote up a treatment here. And I agree: if you're trying to convince someone that there is really something to this "polynomial method" business, I think it would be hard to do better than this beautiful result.

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Rivoal's proof of the irrationality of infinitely many $\zeta(2n+1)$.

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So the proof of this result is really that accessible? Can you provide a link? –  rem May 26 '13 at 13:01
See the undergraduate textbook by Pierre Colmez : editions.polytechnique.fr/?afficherfiche=168 –  Chandan Singh Dalawat Aug 28 '13 at 1:20
At least Apéry's irrationality of zeta(3) is accessible, though not quite in the 30 year window. Or at least Beukers's reformulation. –  Ben Wieland Dec 19 '13 at 4:00

The entire material taught in an undergraduate course of quantum information at math/comp sci departments.

If I should pick one specific theorem, maybe I'd go with the fact that encoding $\vert0\rangle$ and $\vert1\rangle$ as

$$\vert0\rangle \rightarrow \frac{(\vert000\rangle+\vert111\rangle)(\vert000\rangle+\vert111\rangle)(\vert000\rangle+\vert111\rangle)}{2\sqrt{2}}$$ and $$\vert1\rangle \rightarrow \frac{(\vert000\rangle-\vert111\rangle)(\vert000\rangle-\vert111\rangle)(\vert000\rangle-\vert111\rangle)}{2\sqrt{2}}$$ respectively allows for correcting an arbitrary quantum error on one qubit, which was proved by Peter Shor.

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There are many examples in Alon and Spencer's The Probabilistic Method. One such example is the Cheeger inequality for graphs. Unfortunately for this thread, the canonical proofs which use the probabilistic method were originally written over 30 years ago.

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Yeah, I also wanted to pick a pretty awesome classical result or two from that book or Additive Combinatorics by Tao and Vu. Alas, the basic probabilistic techniques like the first moment method are actually quite old. I checked when Lovász local lemma first appeared, and, of course, it was 1975... It would've been such a nice example if it were 8 years younger... I'd love to know recent and equally accessible problem-solving techniques that experts in this field think are as significant. –  Yuichiro Fujiwara May 6 '13 at 0:57
How about Moser's recent constructive proof of the Lovasz Local Lemma? It's the first which actually gives an effective algorithm for finding the object that Lovasz proved existed, and it's accessible for undergrads. –  Peter Shor May 6 '13 at 1:07

It's hard to tell what the OP means by "achievement", and I'm slightly worried he is referring to something thunderous and game-changing: a longstanding conjecture proven, a new theory suddenly opening up a whole new world, a simplification obsoleting lots of mathematics, etc.. Meanwhile, mathematics is progressing in many places at its regular pace without such abrupt changes. Much of this progress is accessible at undergrad level.

Maybe the neatest new development in combinatorics is the theory of abelian networks: arXiv:0801.3306, arXiv:0010241, arXiv:0608360 and many others. Some of these things originally started as problems in high-school math contests, though the new interest has been triggered only when Deepak Dhar introduced sandpiles as a physical model.

Coding theory has developed greatly in the last 30 years (turbo, polar, space-time), though this is where my knowledge ends.

Quasisymmetric functions have seen a lot of progress. $\mathbf{QSym}$ over $\mathbf{Sym}$ has a stable basis, and $\mathbf{QSym}$ is a free polynomial algebra are two discoveries that come into my mind.

An alternative understanding of the Robinson-Schensted correspondence in terms of growth diagrams has emerged in the 1990s, in the works of Fomin and Roby and later van Leeuwen.

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@darij: I agree with the tenor of this answer. I have read a book on Quantum Physics with the disclaimer that it cannot mention the thousands of physicists who had contributed to the advancement of the area under discussion. The advancement of mathematics does also require a broad front, and this general advance needs to be brought to the attention of students. How to do this successfully, in the context of an examination system, is not so clear. A feature of our Maths in Context course was the wide variety of project topics chosen by the students, many not on our lists. –  Ronnie Brown May 6 '13 at 21:57
Surely, an achievement does not need to be thunderous. But a thunderous achievement with proofs that are easy to understand is especially surprising and noteworthy. –  Lennart Meier May 7 '13 at 2:22
Personally, I am attracted by those advances which widen perspectives without being too technical. So I am fond of the Lawvere advertisement for the topos of directed graphs, in which the logic is non Boolean, as can be easily explained, to undergraduates and to scientists. –  Ronnie Brown May 7 '13 at 14:13
@Ronnie: any good source on that? (The topos of graphs I mean.) –  darij grinberg May 7 '13 at 15:47
@darij: such a result would be in the book by Mac Lane and Moerdijk. More generally, the topos of presheaves on a small category $C$ is Boolean iff $C$ is a groupoid. For the specific case of directed graphs, it would suffice to find a subgraph of a graph which has no subgraph complement. For example, a vertex as a subgraph of the graph consisting of that vertex and a single loop has no complement. –  Todd Trimble May 26 '13 at 15:58

I would suggest Furstenberg's proof of Szemeredi's theorem. Of course, it has to be cleaned up a bit removing Rochlin's theorem, fibrations, ergodic theorem itself, the transfinite induction, and leaving just Radon-Nikodym (conditional expectations) and elementary functional analysis in the Hilbert spaces (a bounded sequence has a weakly convergent subsequence) before you present it to the students. This cleaning requires some effort, but the result is pretty neat. It is also one of the instances where using "actual infinity" is advantageous to explicit epsilonics not only phylosophically, but technically as well.

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$26824404^4+153656394^4+187967604^4=206156734^4$ (Elkies, 1988). This is not such an interesting result by itself but it can be used to tell a nice story.

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This cannot possibly be correct because it doesn't add up modulo 10. But yes, Elkies has results in a similar flavor. –  Abhishek Parab May 11 '13 at 1:26
If you google on "206156734 elkies" (for example), you find that many of the synopses, including the one for Elkies's original paper, state the result as 26824404 + 153656394 + 187967604 = 206156734. The final "4" in each number is meant as the exponent. I recommend leaving the answer here unedited, partly to see if it propagates forward as 268244044 + 1536563944 + 1879676044 = 2061567344 and partly because it tells its own nice story about the value of simple checks. (Even without checking mod 10, one might why Elkies didn't factor out a common 2 to get a smaller set of numbers.) –  Barry Cipra Jun 27 '13 at 12:23

Alon's Combinatorial Nullstellensatz is an algebraic technique developed in the 1990's which has many applications in number theory, combinatorics and graph theory.

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Good point! I can't really say that the NSS technique is new, since more or less similar arguments have been made using finite/divided differences long ago (unfortunately I don't have good references). But some applications of it certainly are new, and e. g. Christian Reiher's proof of the Kemnitz conjecture (Ramanujan J (2007) 13, pp. 333-337) is well accessible to undergrads and well-versed highschoolers. –  darij grinberg May 6 '13 at 18:04
The link in this answer invalid: I get a 'not permitted' message. Couls you provide another one? –  rem May 26 '13 at 13:06
I've updated the link: tau.ac.il/~nogaa/PDFS/null2.pdf should work. –  Thomas Kalinowski May 26 '13 at 13:36
Thanks! It works. –  rem May 28 '13 at 20:56
@darijgrinberg: I think Christian Reiher was himself an undergrad or well-versed highschooler when he came up with that proof. –  Omar Antolín-Camarena Jun 27 '13 at 18:55

Some theorems whose proofs are very simple from mathematical viewpoint (they use elementary probability theory and geometry), but with huge impact to quantum mechanics:

1. Bell's theorem 1, 2.
2. Kochen-Specker's theorem 3, 4.
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I'll elaborate on a line from Darij Grinberg's answer.

The Shannon capacity says how much information you can transmit with high probability over a channel when each bit is unreliable. Shannon proved that random codes approach the theoretical maximum. However, when there is no structure in the code, it is hard to figure out which code word is closest to the received message.

Turbo codes from 1993 onward are a breakthrough which can be presented to undergraduates. These are families of codes which approach the Shannon capacity of noisy channels and which have practical iterative decoding algorithms.

Initially, the effectiveness of the decoders was a mystery. One helpful perspective is to recognize the decoding as an example of (loopy) belief propagation in a Bayesian network, which is an intuitive algorithm. This problem can also be described as trying to solve a crossword puzzle by alternately trying to improve your solution by studying the (ambiguous) down clues and the (ambiguous) across clues.

Turbo codes are actually used in wireless networks such as for mobile phones.

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Are turbo codes, LDPC codes, or polar codes the best choice for presentation to undergraduates? I think they're all possible, although you probably need a couple of hours to do it well. Turbo codes seem to have the disadvantage that you need more background; in particular, you need to go over convolutional codes first. Does anybody have any comments from experience? –  Peter Shor May 10 '13 at 12:23

If you would allow a 43 year-old mathematical achievement, Yuri Matiyasevich's solution of Hilbert's tenth problem is accessible even to high school students.

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The disproving of Borsuk's conjecture (http://en.wikipedia.org/wiki/Borsuk%27s_conjecture) by Jeff Kahn and Gil Kalai (http://arxiv.org/abs/math.MG/9307229)

In general combinatorial geometry is often quite accessible as proofs if they have been discovered can be short and elegant.

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The matrix formalism of self-similar groups makes the theory (and numerous examples - beginning with the groups of intermediate growth) perfectly accessible for undergrads.

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The existence and uniqueness of self-similar set and measures for iterated function systems (you find in all books on fractal) is a good example. Students will like it, cause You may provide many nice pictures. I hope this counts also the result is due to Hutchinson, J. "Fractals and Self-Similarity." Indiana Univ. J. Math. 30, 713-747, 1981, which is 32 years ago.

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Cryptographic Restults like e.g. the Diffie-Hellman method should be quite accessible.

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I was able to explain the following corollary of Gromov's theorem to the undergraduates. It took few lectures, but the material brakes nicely into interesting parts. This could be considered as a student-friendly introduction to h-principle

There is a length preserving map from unit sphere to the plane.

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do you have the lecture notes of the proof of the corollary you mentioned? Thanks! –  math Oct 18 '13 at 8:41
@mathandzen, a preliminary version is here math.psu.edu/petrunin/papers/mass/euclid2alexandrov.pdf –  Anton Petrunin Oct 18 '13 at 22:09
Thank you. It looks accessible to me also. :-) –  math Oct 19 '13 at 7:39

This kind of achievements rarely happens in Analysis, but here is a recent example: Herbert Stahl's proof of the BMV Conjecture. A version of the proof accessible to undergraduates (who had an undergraduate complex variables course) can be found here:

www.math.purdue.edu/~eremenko/dvi/bmv.pdf The recent Russian version is even further simplified:

www.math.purdue.edu/~eremenko/dvi/talk2.pdf

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I could not find the file talk2.pdf. –  Dietrich Burde Jun 27 '13 at 12:33
I fixed this, thanks, Dietrich. But it is in Russian. –  Alexandre Eremenko Jun 30 '13 at 15:15

The recent solution of the Kadison-Singer problem, by Marcus, Spielman and Srivastava (see http://arxiv.org/abs/1306.3969) involves linear algebra, elementary probability theory, and some calculus in several variables. There is a nice exposition on T. Tao's blog: http://terrytao.wordpress.com/2013/11/04/real-stable-polynomials-and-the-kadison-singer-problem/#more-7109

(Actually what is elementary is the deduction by MSS of Weaver's $(KS_2)$-conjecture. To see that Weaver's conjecture is equivalent to the Kadison-Singer problem, you need some basic $C^*$-algebra theory).

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The ideas behind Gödel's theorem are accessible in highschool, and the proof can be completely explained if we admit the most techniical part (that Demonstration and Substitution in a formula can be completely encoded in First Order Logic).

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But that is a very old result, from the 30s. –  The User May 26 '13 at 13:41
ah sorry I didn't see that it was required to be new –  Denis May 27 '13 at 8:16