[Math] Quirky, non-rigorous, yet inspiring, literature in mathematics

booksreference-requestsoft-question

In contrast with such lucid, pedagogical, inspiring books such as Visualizing Complex Analysis by Needham and Introduction to Applied Mathematics by Strang, I've had the pleasure of coming across the non-rigorous, thought-provoking/stimulating, somewhat quirky works of Heaviside on operational calculus and divergent series; of Ramanujan on series, in particular, his use of his master theorem/formula (as explicated by Hardy); and the relatively unknown posthumous notes of Bernard Friedman on distributions and symbolic/operational calculus Lectures on Applications-Oriented Mathematics.

These works just blindside you. You think, "What the hey?" and slowly they grow on you and you start to understand them after further research using other texts, translating the terminology/concepts, and working out details. You're left with a deeper understanding and appreciation of the originality and applicability of the work–much like Hardy professed the day after he received Ramanujan's letter, no doubt.

(Friedman's Applied Mathematics, in contrast, is of a very different nature and an immediately enlightening intro to its topics.)

Any similar experiences with other mathematical works?

(This question is not research-level per se and may be more appropriate for MSE, but certainly the works cited have inspired and continue to inspire much advanced research into the related topics, and the question falls in a similar category to MO-Q1 and the MO-Qs that pop up in the Related section of that question).

Best Answer

Andrew Granville and Jennifer Granville's "Prime Suspects" is a comic book which despite being phrased as a murder mystery does a surprisingly lucid job connecting ideas involving primes with ideas involving permutations. This paper Andrew Granville outlining the basic idea is pretty non-rigorous but fascinating.

Related Solutions

[Math] How To Present Mathematics To Non-Mathematicians

I have given talks about mathematics to non-mathematicians, for example to a bunch of marketing people. Supplemental: to see an example of a talk of mine that was given to a general audience, see my TEDx talk "Zeroes" (with supplemental material). The talk lasted 15 minutes and it took me about two weeks to prepare.

In my experience the following points are worth noting:

If the audience does not understand you it is all in vain.
You should interact with your audience. Ask them questions, talk to them. A lecture is a boring thing.
Pick one thing and explain it well. The audience will understand that in 10 minutes you cannot explain all of math. The audience will not like you if you rush through a number of things and you don't explain any one of them well. So an introductory sentence of the form "Math is a vast area with many uses, but in these 10 minutes let me show you just one cool idea that mathematicians have come up." is perfectly ok.
A proof of something that seems obvious does not appeal to people. For example, the proof of Kepler's conjecture about sphere packing is a bad example because most people won't see what the fuss is all about. So Kepler's conjecture would be a bad example.
You are not talking to mathematicians. You are not allowed to have definitions, theorems or proofs. You are not allowed to compute anything.
Pictures are your friend. Use lots of pictures whenever possible.
You need not talk about your own work, but pick something you know well.
Do not pick examples that always appear in popular science (Fermat's Last Theorem, the Kepler conjecture, the bridges of Koenigsberg, any of the 1 million dollar problems). Pick something interesting but not widely known.

Here are some ideas I used in the past. I started with a story or an intriguing idea, and ended by explaining which branch of mathematics deals with such ideas. Do not start by saying things like "an important branch of mathematics is geometry, let me show you why". Geometry is obviously not important since all of mathematics has zero importance for your audience. But they like cool ideas. So let them know that math is about cool ideas.

To explain what topology and modern geometry are about, you can talk about the Lebesgue covering dimension. Our universe is three-dimensional. But how can we find this out? Suppose you wake up in the morning and say "what's the dimension of the universe today?" You walk into your bathroom and look at the tiles. There is a point where three of them meet and you say to yourself "yup, the universe is still three-dimensional". Find some tiles in the classroom and show people how always at least three of them meet. Talk about how four of them could also meet, but at least three of them will always meet in a point. In a different universe, say in a plane, the tiles would really be segments and so only two of them would meet. Draw this on a board. Show slides of honeycombs in which three honeycomb cells meet. Show roof tilings in which thee tiles meet, etc. Ask the audience to imagine what happens in four dimensions: what do floor tiles in a bathroom look like there? They must be like our bricks. What is a chunk of space for us is just a wall for them. So if we have a big pile of bricks stacked together, how many will meet at a point? At least four (this will require some help from you)!
To explain knot theory, start by stating that we live in a three-dimensional space because otherwise we could not tie the shoelaces. It is a theorem of topology that knots only exist in three dimensions. You proceed as follows. First you explain that in one or two dimensions you can't make a knot because the shoelace can't cross itself. It can only be a circle. In three dimensions you can have a knot, obviously. In four dimensions every knot can be untied as follows. Imagine the that the fourth dimension is the color of the rope. If two points of the rope are of different color they can cross each other. That is not cheating because in the fourth dimension (color) they're different. So take a knot and color it with the colors of the rainbow so that each point is a different color. Now you can untie the knot simply by pulling it apart in any which way. Crossing points will always be of different colors. Show pictures of knots. Show pictures of knots colored in the color of the rainbow.
Explain infinity in terms of ordinal numbers (cardinals are no good for explaining infinity because people can't imagine $\aleph_1$ and $2^{\aleph_0}$). An ordinal number is like a queue of people who are waiting at a counter (pick an example that everyone hates, in Slovenia this might be a long queue at the local state office). A really, really long queue contains infinitely many people. We can imagine that an infinite queue 1, 2, 3, 4, ... is processed only after the world ends. Discuss the following question: suppose there are already infinitely many people waiting and one more person arrives. Is the queue longer? Some will say yes, some will say no. Then say that an infinite row of the form 1, 2, 3, 4, ... with one extra person at the end is like waiting until the end of the world, and then one more day after that. Now more people will agree that the extra person really does make the queue longer. At this point you can introduce $\omega$ as an ordinal and say that $\omega + 1$ is larger than $\omega$. Invite the audience to invent longer queues. As they do, write down the corresponding ordinals. They will invent $\omega + n$, possibly $\omega + \omega$. Someone will invent $\omega + \omega + \omega + \ldots$, you say this is a bit imprecise and suggest that we write $\omega \cdot \omega$ instead. You are at $\omega^2$. Go on as far as your audience can take it (usually somewhere below $\epsilon_0$). Pictures: embed countable ordinals on the real line to show infinite queues of infinite queues of infinite queues...

Noncommutative Geometry – Significance of Non-Commutative Geometry in Mathematics

$\DeclareMathOperator\coker{coker}$I think I'm in a pretty good position to answer this question because I am a graduate student working in noncommutative geometry who entered the subject a little bit skeptical about its relevance to the rest of mathematics. To this day I sometimes find it hard to get excited about purely "noncommutative" results, but the subject has its tentacles in so many other areas that I never get bored.

Before saying anything further, I need to say a few words about the Atiyah–Singer index theorem. This theorem asserts that if $D$ is an elliptic differential operator on a manifold $M$ then its Fredholm index $\dim(\ker(D)) - \dim(\coker(D))$ can be computed by integrating certain characteristic classes of $M$. Non-trivial corollaries (obtained by "plugging in" well chosen differential operators) include the generalized Gauss–Bonnet formula, the Hirzebruch signature theorem, and the Hirzebruch–Riemann–Roch formula. It was quickly realized (first by Atiyah, I think) that the proof of the theorem can be viewed as a statement about the Poincaré duality pairing between topological K-theory and its associated homology theory (these days called K-homology).

I wasn't around, but I'm told that people were very excited about Atiyah and Singer's achievement (understandably so!). People quickly began trying to generalize and strengthen the theorem, and my claim is that noncommutative geometry is the area of mathematics that emerged from these attempts. Saying that marginalizes the other important reasons for developing the subject, but I think it was Connes' main motivation and in any event it is a convenient oversimplification for a MO answer. It also helps me answer your first question by playing to my personal biases: when I was choosing an area of research I told my adviser that I was interested in learning more about that Atiyah–Singer index theorem and I was led inexorably toward the tools of noncommutative geometry.

The origin of the relationship between NCG and Atiyah–Singer lies in equivariant index theory. Atiyah and Singer realized from the start that if $M$ admits an action by a compact Lie group $G$ and $D$ is invariant under the group action then it is better to think of the index of $D$ as a virtual representation of $G$ (i.e. an element of the $G$-equivariant K-theory of a point) rather than as an integer. If $G$ is not compact then this doesn't really work, but the noncommutative geometers realized that $D$ does have an index in the K-theory of the reduced group C$^\ast$-algebra $C_r^\ast(G)$. Indeed, to a noncommutative geometer equivariant index theory is all about a map $K_\ast(M) \to K_\ast(C_r^\ast(M)$ where $K_\ast(M)$ is the K-homology of $M$; in the case where $M$ is the universal classifying space of $G$, Baum and Connes conjectured that this map is an isomorphism. Proving this conjecture for more and more groups and understanding its consequences motivates a great deal of the development of NCG to this day.

The conjecture is interesting in its own right if you already care about index theory, but even if you don't injectivity of the Baum–Connes map implies the Novikov conjecture (see Alain Valette's answer) and surjectivity is related to the Borel conjecture. It has numerous other applications, for example to the theory of positive scalar curvature obstructions in Riemannian geometry or to the Kadison–Kaplansky conjecture in functional analysis (which would follow from surjectivity). Recently there has been a lot of interest in connections between the Baum–Connes conjecture and representation theory; the Baum–Aubert–Plyman conjecture in $p$-adic representation theory has its origins in these sorts of considerations.

Much of the rest of NCG can also be traced back to index theory. Kasparov's KK-theory arose as a way to understand maps and pairings between K-theory and K-homology, motivated in part by index theory. Connes' work on noncommutative measure theory arose from his work on index theory for measurable foliations (with applications to dynamical systems). Cyclic (co)homology was invented in part to gain access in a noncommutative setting to the Chern character map from K-theory to cohomology which translates the K-theoretic formulation of the index theorem into a cohomological formula. Connes' theory of spectral triples and noncommutative Riemannian geometry is based on the theory of Dirac operators which was invented by Atiyah and Singer to prove the index theorem. I guess my point with all of this is that all the esoteric machinery of NCG seems less artificial when viewed through the lens of index theory.

Best Answer

Related Solutions

[Math] How To Present Mathematics To Non-Mathematicians

Noncommutative Geometry – Significance of Non-Commutative Geometry in Mathematics

Related Question