BLOG Carl McTague mathematician, composer, photographer, fiddler

17 Dec 2014 | categories: Mathematics

# How to Recognize Generators for String Bordism

A key result of my paper:

unmentioned in the abstract or introduction—and likely of independent interest—is the following characteristic-number-theoretic criterion for a set to generate the String bordism ring (with 6 inverted).

Theorem 4. A set $S$ generates the ring $\pi_*\mathrm{MO}\langle8\rangle[1/6]$ if:

1. For each integer $n>1$, there is an element $M^{4n}$ of $S$ such that for any prime $p>3$: \begin{align*} \mathrm{ord}_p \big( \mathrm{s}_n[M^{4n}] \big) = \begin{cases} 1 & \text{if 2n=p^i-1 or 2n=p^i+p^j for some integers 0 \le i \le j} \\ 0 & \text{otherwise} \end{cases} \end{align*} where $\mathrm{s}_n[M]$ is the characteristic number corresponding to the $n$th power sum symmetric polynomial $\sum x_i^n$ in the Pontrjagin roots of the tangent bundle of $M$, and $\mathrm{ord}_p(-)$ its $p$-adic order, i.e. the heighest power of $p$ which divides it.
2. For each prime $p>3$ and each pair of integers $0<i<j$, there is an element $N^{2(p^i+p^j)}$ of $S$ such that: \begin{align*} \mathrm{s}_{(p^i+p^j)/2}[N^{2(p^i+p^j)}]&=0 \\ \mathrm{s}_{(p^i+1)/2,(p^j-1)/2}[N^{2(p^i+p^j)}] &\not\equiv 0 \mod p^2 \end{align*} where $\mathrm{s}_{m,n}[N]$ is the characteristic number corresponding to the symmetric polynomial $\sum_{i\ne j} x_i^mx_j^n$ in the Pontrjagin roots of the tangent bundle of $N$.

This is a consequence of Hovey’s calculation of:

The homotopy of $\mathrm{MString}$ and $\mathrm{MU}\langle6\rangle$ at large primes, Algebr. Geom. Topol. 8 (2008), 2401–2414.

which itself builds on Hopf-ring-theoretic work of Ravenel and Wilson. It should be compared with the analogous result for the oriented bordism ring (with 2 inverted):

Theorem (Novikov, cf. Stong p. 180). A sequence $\{M^{4n}\}_{n\ge1}$ generates the ring $\pi_*\mathrm{MSO}[1/2]$ if and only if:

• For any integer $n>0$ and any odd prime $p$: \begin{align*} \mathrm{ord}_p \big( \mathrm{s}_n[M^{4n}] \big) = \begin{cases} 1 & \text{if 2n=p^i-1 for some integer i>0} \\ 0 & \text{otherwise} \end{cases} \end{align*}

## References

13 Dec 2014 | categories: Photographs, Serpents, Archaelogy, Chronology

# After Laocoön [or Before?]

Little is known with certainty about the serpent—not even how many thousand years old it is. # References

[It struck me later how sim­i­lar my com­po­si­tion is to Edvard Munch’s in The Scream (1893).]

2 Dec 2014 | categories: Mathematics, Film, Crime

# Binomial Coefficients and Villainy Sherlock Holmes’s criminal archenemy Professor Moriarty was modeled on Hopkins mathematics professor Simon Newcomb (1835–1909).

“At the age of twenty-one [Moriarty] wrote a treatise upon the binomial theorem, which has had a European vogue,” and was “the celebrated author of The Dynamics of an Asteroid, a book which ascends to such rarefied heights of pure mathematics that it is said that there was no man in the scientific press capable of criticizing it” [from “The Final Problem” (1893) and The Valley of Fear (1914) by Arthur Conan Doyle].

At the age of nineteen Newcomb wrote an unpublished New Demonstration of the Binomial Theorem and was a celebrated expert on the dynamics of the solar system. He was also notoriously malicious. For example, he destroyed Charles Sanders Peirce’s career by secretly changing Hopkins president Daniel Coit Gilman’s mind about granting Peirce tenure, this after having been the favorite PhD student of Peirce’s father Benjamin Peirce at Harvard. (Newcomb was incidentally the maternal grandfather of Hassler Whitney.)

These and further similarities between Moriarty and Newcomb were first noted by Ronald Schorn in 1978, and the case for direct inspiration was convincingly made by Bradley Schaefer in 1993 (by arguing that Doyle would have known of Newcomb through close friend Alfred Drayson):

Sherlock Holmes and some astronomical connections, Journal of the British Astronomical Association, Vol. 103, No. 1 (1993), pp. 30–34.

The possibility of completing the square:

 Holmes — Moriarty ⋮ | CS Peirce — Newcomb

is suggested by the existence of the book:

Umberto Eco & Thomas A. Sebeok (eds.), The Sign of Three: Dupin, Holmes, Peirce, Indiana University Press (1984).

## Afterword

I got onto this after lecturing on Bernstein polynomials and Bézier curves. The (constructive) proof of the Stone-Weierstrass approximation theorem in that case requires some finesse with binomial coefficients. To lighten the mood I made an off-the-cuff remark about Moriarty’s young interest in binomial coefficients, and the idea that a genuine rival for Holmes could have no background but mathematics. After lecture I started digging through some references and was led to Newcomb.

I later pointed out that, although Holmes scoffs when in their first meeting Dr Watson makes the comparison, Holmes was himself modeled on C. Auguste Dupin, who first appeared in “The Murders in the Rue Morgue” (1841) by Edgar Allan Poe (†1849 in Baltimore). [And Watson himself, ironically, on Poe’s anonymous narrator.]

Having begun the course with an analysis of the word “analysis”:

From the Greek ana·lysis=“up”·“to loosen”=“to loosen up”. To be compared with “dissolve”→“solve”. But not to be confused with anal·ysis—“anal” coming from “anus”, the Latin for “ring”. Which is relevant since it leads to the mathematical word “annulus”. And which illustrates that etymology is nonassociative: (ana)·(lysis)≠(anal)·(ysis). Don’t you think it’s a little strange, though, that the Greek and Latin are not entirely unrelated?

I ended with the opening of “Rue Morgue”:

The mental features discoursed of as the analytical, are, in themselves, but little susceptible of analysis. We appreciate them only in their effects. We know of them, among other things, that they are always to their possessor, when inordinately possessed, a source of the liveliest enjoyment. As the strong man exults in his physical ability, delighting in such exercises as call his muscles into action, so glories the analyst in that moral activity which disentangles. He derives pleasure from even the most trivial occupations bringing his talent into play. He is fond of enigmas, of conundrums, of hieroglyphics; exhibiting in his solutions of each a degree of acumen which appears to the ordinary apprehension præternatural. His results, brought about by the very soul and essence of method, have, in truth, the whole air of intuition.

The faculty of re-solution is possibly much invigorated by mathematical study, and especially by that highest branch of it which, unjustly, and merely on account of its retrograde operations, has been called, as if par excellence, analysis. Yet to calculate is not in itself to analyse. A chess-player, for example, does the one without effort at the other. It follows that the game of chess, in its effects upon mental character, is greatly misunderstood. I am not now writing a treatise, but simply prefacing a somewhat peculiar narrative by observations very much at random…

[As an aside, I visited Thomas Browne’s monument in Norwich this summer. His Hydriotaphia, Urn Burial, or, a Discourse of the Sepulchral Urns lately found in Norfolk (1658)—from which Poe’s epigraph is taken—is otherworldly.]

Observe that the opening paragraph of Poe’s manuscript did not survive to the printed edition:

“It is not improbable that a few further steps in phrenological science will lead to a belief in the existence, if not to the actual discovery and location of an organ of analysis. [Moriarty, incidentally, had an associate named McTague in the 1899 play Sherlock Holmes, cowritten by Doyle and William Gillette. In fact, “Lightfoot” McTague appears in the 1916 silent film adaptation of the play, considered among the holy grails of lost films until a nitrate dupe negative was discovered mislabeled in the vaults of the Cinémathèque Française in 2014. For a far more cinematic appearance of a McTague in silent film, however, see Erich von Stroheim’s 1924 masterpiece film adaptation Greed of Frank Norris’s 1899 novel McTeague. [Is it a coincidence that McTeague and the Holmes play both date to 1899?] Seeing Greed at Eastman House inspired me to recreate one of its famous title cards—compare with the original and notice that the recreation even shakes like a projected film (the CSS for this relies on Bézier curves).]

27 Sep 2014 | categories: Chronology, Design

# Seminar Web Design

When I became an organizer of the Johns Hopkins Topology Seminar, I was asked to design a new webpage for it. A seminar webpage should:

1. Advertise future talks, especially the next one,
2. Allow invited speakers to easily identify available slots, and
3. Archive past talks.

My solution:

1. List all talks—both past and future—chronologically (oldest first), and
2. Automatically scroll down to the next talk when the page loads.

So to see the archive of past talks one scrolls up! I’ve not seen this approach before but think it’s quite natural. Additionally:

1. Animate the scrolling so visitors know to scroll up, and
2. Float the header above this whirling animation.
3. Separate past and future talks with a horizontal line, and
4. Align this line with the bottom edge of the header at the end of the animation.
5. Display all future speakers but hide their titles and abstracts until a week before they speak.

All this is achieved automatically with a few lines of jQuery code at the end of an easily maintained static webpage. The code parses the very same dates which humans read.

The page also automatically announces upcoming talks by email: A cron job (“seminarbot”) on the department server uses PhantomJS to fetch and execute an email-generating function reminderEmail() embedded in the webpage and drops the output into sendmail.

## References

20 Jul 2014 | categories: Mathematics

# Comments on a Paper of Karcher

Some comments on Hermann Karcher’s beautiful paper:

H. Karcher, Submersions via Projections, Geom. Dedicata 74 (1999), no. 3, pp. 249–260, MR1669359.

• Formulas (10), (11), (12) may be written: \begin{align*} R(X,Y)V &= -\overbrace{R(X,Y)\mathcal{H}\cdot V}^{\text{in HM}} + \overbrace{R^{\mathrm{V}}(X,Y)V-\big[\nabla_X\mathcal{H},\nabla_Y\mathcal{H}\big]V}^{\text{in VM}} \\ R(X,Y)H &= \phantom{-}\underbrace{R(X,Y)\mathcal{H}\cdot H}_{\text{in VM}} + \underbrace{R^{\mathrm{H}}(X,Y)H-\big[\nabla_X\mathcal{H},\nabla_Y\mathcal{H}\big] H}_{\text{in HM}} \end{align*}
• The last formula of (15) may be written: $$(\nabla_Xg_\epsilon)(Y,Z) = \epsilon(\epsilon-2)\cdot g(Y,\nabla_X\mathcal{V}\cdot Z)$$
• The equations (16) for the difference tensor $\Gamma^\epsilon(X,Y)=\nabla_XY-\nabla^\epsilon_XY$ extend to a single general formula: $$\Gamma^\epsilon(X,Y) = \epsilon(2-\epsilon)\cdot\big(\nabla_{\mathcal{H}X}\mathcal{H}\cdot\mathcal{V}Y +\nabla_{\mathcal{H}Y}\mathcal{H}\cdot\mathcal{V}X +\nabla_{\mathcal{V}X}\mathcal{H}\cdot\mathcal{V}Y\big)$$ The first and third terms combine easily but $\Gamma^\epsilon$—being the difference of torsion free connections—is symmetric, and this three-term formula showcases that symmetry—the third term being symmetric according to (63). Some useful special cases: \begin{align*} \Gamma^\epsilon(X,V) &= \epsilon(2-\epsilon)\cdot\nabla_X\mathcal{H}\cdot V \\ \Gamma^\epsilon(X,H) &= \epsilon(2-\epsilon)\cdot\nabla_H\mathcal{H}\cdot\mathcal{V}X \end{align*} Comparing with (9) leads to: \begin{align*} (\nabla^2\pi)(X,Y) &= D\pi\cdot\Gamma^1(X,Y) \end{align*}
• Some formulas worth recording: \begin{align*} \nabla_X^\epsilon\mathcal{H}\cdot H &= \nabla_X\mathcal{H}\cdot H \\ \nabla_X^\epsilon\mathcal{H}\cdot V &= (1-\epsilon)^2\cdot\nabla_X\mathcal{H}\cdot V \\ \big[\nabla^\epsilon_X\mathcal{H},\nabla^\epsilon_Y\mathcal{H}\big] &= (1-\epsilon)^2\cdot \big[\nabla_X\mathcal{H},\nabla_Y\mathcal{H}\big] \end{align*} Note that the second equation says that the second fundamental forms of the fibers go to zero as $\epsilon\to1$.

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