What is fairly new theorem one can teach (and prove) to an undergraduate student?

Question

Many students complain about how old the things in mathematics are. When students finish their undergraduate studies, there are usually not able to state results and prove them which were found after the 1950s (or even older). Of course, the notation has improved and maybe there are some nicer proof for theorems proven far back.

Are there examples of theorems (including proofs) which are as new as possible (at least newer than 1950/1960) and accessible to undergraduate students? Good students should be able to understand the meaning of the theorem and the tools which are needed for the proof should be available – the proof of Fermat’s Last Theorem would not fit into the examples here, although the statement is very clear.

Answer 1

There aren't many new theorems in old subjects that are accessible to undergraduates. However, as others have mentioned, there are many relatively new subjects within mathematics that are accessible. These include:

• Graph Theory, which really dates back to Dénes Kőnig's 1936 textbook. Many of the fundamental results were proven in the 1930's, but many others were proven several decades later. For example, the Max Flow Min Cut Theorem was first proven in 1956, and the definition and basic properties of treewidth were not introduced until 1976.

• Dynamical Systems picked up steam slowly, and many of the fundamental results were proven in the second half of the 20th century. For example, the theorem that period three implies chaos for a one-dimensional system was proven in 1975, though it was a special case of Sharkovskii's Theorem, which was proven in 1964 but not well known until later.

• Similarly, most of the fundamental results of Complex Dynamics and Fractal Geometry are quite new. The first pictures of the Mandelbrot set did not appear until 1978, and many important results are newer still.

• Game Theory is a relatively new subject. Nash equilibria were first defined in general by Nash in 1951, so most of the known results are newer than this.

• Knot Theory has been developing quickly in the last half of the 20th century. For example, the tabulation of knots with up to 11 crossing was not completed until 1974, and the Jones polynomial was not defined until 1984.

Answer 2

Chvátal's art gallery theorem, a visibility problem in computational geometry, was proposed and proved in the 1970s.

$\left\lfloor n/3 \right\rfloor$ guards are always sufficient and sometimes necessary to guard a simple polygon with $n$ vertices.

From Wikipedia:

Fisk (1978) proves the art gallery theorem as follows.

First, the polygon is triangulated (without adding extra vertices). The vertices of the polygon are then 3-colored in such a way that every triangle has all three colors. To find a 3-coloring, it is helpful to observe that the dual graph to the triangulation (the undirected graph having one vertex per triangle and one edge per pair of adjacent triangles) is a tree, for any cycle in the dual graph would form the boundary of a hole in the polygon, contrary to the assumption that it has no holes. Whenever there is more than one triangle, the dual graph (like any tree) must have a vertex with only one neighbor, corresponding to a triangle that is adjacent to other triangles along only one of its sides. The simpler polygon formed by removing this triangle has a 3-coloring by mathematical induction, and this coloring is easily extended to the one additional vertex of the removed triangle.

Once a 3-coloring is found, the vertices with any one color form a valid guard set, because every triangle of the polygon is guarded by its vertex with that color. Since the three colors partition the $n$ vertices of the polygon, the color with the fewest vertices forms a valid guard set with at most $\lfloor n/3\rfloor$ guards.

Answer 3

There are various results of this form, you can find some in a related MO question. But, I agree this is a problem that students often do not learn such results, or might not realize how recent some things are.

An example is Apéry's theorem (1979) establishing the irrationality of $\zeta(3)$, that is, the result that $\sum_{n=1}^{\infty} \frac{1}{n^3}$ is an irrational number.

What makes this result in my opinion especially well-suited for the intended purpose is that:

1. It fits together with things students should know anyway, namely $\sum_{n=1}^{\infty} \frac{1}{n^2} = \pi^2/6$, and possibly the evaluation of $\zeta(2k)$.

2. It is easy to motivated given the above that one seeks to understand the situation for odd values two.

3. The result touches on number theory and analysis.

4. There was continued activity around this problem since then and open questins remain, such as, already: is $\zeta(5)$ irrational?

For an overview see the link above and for an actual exposition of the proof written at that time see "A proof that Euler missed" by the van der Poorten; and meanwhile even more streamlined arguments exist.

Answer 4

I think that Bhargava's generalized factorial functions fit the bill perfectly. This construction, and the theorems he proves using them, are recent, topical, and completely undergraduate-accessible. Bhargava himself gives a good survey HERE, and there is a nice blog-post explanation of generalized factorials HERE.

Analogues of various number-theoretical results involving factorials, such as the fact that the gcd of elements in the image of a degree $k$ primitive polynomial with integer coefficients is divisible by $k!$, remain true when the integers are replaced by a subset $S$ of the integers and the factorial function is replaced by Bharagava's generalized factorial function $k!_S$. One of my students has been looking at these theorems, including the proofs, and I think that the level has proven to be just right for her to have understood everything and also to have learnt some useful mathematics.

Answer 5

The entire area of fractals was born after computers became easily available. Once the idea of fractals came up, it was easy to prove a lot of theorems about them that were only new because noone thought to pose them before.

The definition of the Mandelbrot set and its properties in particular are a good option for undergraduates.

See this paper for some sample theorems: homepages.warwick.ac.uk/~masdbl/dimension-total.pdf

Answer 6

I can think of at least these two good sources:

1. Number theory over function fields:

   Many undergraduates learn about rings, fields, the ring of integers of a function, and related topics in a second semester of algebra and/or a first semester of non-elementary number theory. Once you have these topics, you know all you need to learn about the zeta function over the field $\mathbb{F}_q(x)$. This would likely even solidify often-mistaken concepts about function fields and perhaps inspire some towards number theory or algebraic geometry.

   Resultwise, it's easy to prove the Riemann Hypothesis for this zeta function (which is nice and "cool"), and from that, the prime number theorem with optimal error term for function fields. Much of the material is approachable to undergrads, and it's all very recent - the best resource for inspiration would be Rosen's book, although this book would probably be a bit scary for the undergrads themselves. (That's the best part of course design, right? And why they want experts to digest material?).

2. Arithmetic/Complex Dynamics

   This field of arithmetic dynamics is less than 50 years old. The best resource would be Silverman's book, albeit with the same caveat as above. Most of the early proofs are extremely elementary, though recent, and have many connections to other areas of mathematics.

   Similarly, a lot of the material from Beardon's Iteration of Rational Functions is morally recent. It ties in nicely to a study of fractals, which is entirely new.

Answer 7

a neat theorem that highlights many emerging areas of research such as (un)decidability, cellular automata, Turing completeness, emergence, self-organization, computer science etc: the proof that Life is Turing complete, originally by Conway. Conways original version seems possibly unpublished however there is a version by Rendell that is quite intuitive and there are even youtube videos. the original proof seems to date to the ~1960s.

• This is a Turing Machine implemented in Conway's Game of Life Rendell
• A Turing Machine In Conway's Game Life pdf paper
• Game of life Turing machine youtube video

Answer 8

Are there examples of theorems (including proofs) which are as new as possible (at least newer than 1950/1960) and accessible to undergraduate students?

I understand that you're asking for historical proofs, but this idea that math is something that was done in the past might not be best ameliorated with examples from the recent past.

The easiest way to end up with new proofs and theorems is to help your students ask and answer their own mathematical questions. My favorite bit of writing on this sort of pedagogy is from this post:

http://christopherdanielson.wordpress.com/2012/10/12/the-hierarchy-of-hexagons/

I don't really see how this sort of questioning environment is compatible with a lecture-based course, though, so maybe this isn't helpful for your needs.

Answer 9

Here is a nice (basically linear algebra!) theorem from the representation theory of quivers proved by Gabriel:

Theorem. The number of indecomposable representations of a quiver is independent of the orientation of the edges of the underlying digraph. A connected graph gives rise to a quiver of finite type if and only if it is a simply laced Dynkin diagram.

Here is the proof due to Bernstein, Gelfand and Ponomarev which is usually discussed. Another very accessible exposition of this result may be found in these online notes (See §1.8 and Chapter 5).

Answer 10

Mason-Stothers Theorem, the polynomial analogue of ABC conjecture for integers.

Answer 11

Some famous theorems in mathematical statistics was proven arounf 1950. Some examples, the Rao-Blackwell theorem (see: https://en.wikipedia.org/wiki/Rao-Blackwell_Theorem) is interesting and accessible. Also Basus theorem should be, and that is even later:https://en.wikipedia.org/wiki/Basu_theorem That is from 1955.

More generally, a lot of results from distribution theory are new (distribution here in the probability sense)

Answer 12

And I've just recalled one more thing: Henstock–Kurzweil integral.

• New enough? Introduced in 1957. Check.
• Easy to understand? The definition is a slight variation on the Riemann integral, and the theory is only slightly more difficult (the difficulties being mostly technical), so (at least theoretically) it may be taught to first-year students. Check.
• Bonus points: interesting? Let's see: it's conceptually much simpler than the Lebesgue integral, though more powerful (it integrates all Lebesgue-integrable functions and then some – among others, all derivatives are H–K-integrable, and the Newton-Leibniz formula holds without exceptions); identical to Lebesgue integral when the latter is defined, hence can be used to define Lebesgue measure in an elementary way; and more. For me: check.
• Bonus points II: are there any open problems in the theory? Yes. For instance, the dual space of the space of H–K-integrable functions (with Alexiewicz norm) is known only in the case of functions on a bounded interval. The unbounded case is AFAIK terra incognita. Check.

Answer 13

In metric fixed point theory, you have Darbo (and Sadovski) theorems. They are generalizations of Schauder fixed point theorem, but instead of assuming that the set (or the mapping) is compact, one assumes that it maps sets into "more compact" sets (using some kind of a measure of noncompactness). (I don't have any reference with me at home now, but one classical source is Granas' and Dugundji's monograph on fixed point theory.) For instance, in Darbo's theorem, you assume that you have a continuous mapping $F$ on a non-empty bounded closed convex subset $C$ of a Banach space and a constant $k<1$ such that $\alpha(F(B))\le k\alpha(B)$ for each $B\subset C$, where $\alpha$ is the Kuratowski measure of noncompactness.

The notion of measure of noncompactness goes back to 1930s (Kuratowski was the one who introduced it, without relation to fixed point theory), Darbo's and Sadovski's theorems are (AFAIK) from 1950s (maybe a bit later).

On a related note, there are quite a few theorems (some fixed-point ones and some others) about hyperconvex metric spaces (a notion introduced in the 1930s by Aronszajn, but first appearing in a published paper in 1956) whose proofs employ simple tools from metric topology, but are highly nontrivial. A beautiful example is Baillon's theorems:

An intersection of a chain of bounded hyperconvex spaces is nonempty and hyperconvex.

Any nonexpansive (i.e., having Lipschitz constant $1$) mapping of a bounded hyperconvex space into itself has a fixed point.

Both can be proves using quite elementary methods (but rely heavily on the axiom of choice), but the proofs are far from trivial. Both were proved in a paper published in late 1980s.

(For more info on hyperconvex spaces, see e.g. http://en.wikipedia.org/wiki/Injective_metric_space and references therein, in particular the original paper of Aronszajn ana Panitchpakdi and the article by Espinola and Khamsi).

Answer 14

Theorem about geometric numerical integration.

Nowadays introductory ODE courses (naturally) usually take a dynamical systems point of view and asymptotic/geometric properties of the solutions play an ever increasing role.

I believe that an introductory numerical analysis course where time discretization of ODEs are treated should respect this and there are many easily accessible results in this direction that can be presented already on an undergraduate level.

The monograph by Hairer, Lubich and Wanner is hardly accessible for undergraduates but a good instructor should be able to extract results which can be presented. Most of these results are from the last four decades.

Answer 15

The whole field of NP completeness in computer science dates from around 1970. Some results are easy to understand (but the proofs tend to be gargantuan until some basic results are in).