Log24

Monday, April 2, 2018

Three Mother Cubes

Filed under: General,Geometry — Tags: — m759 @ 1:44 PM

From a Toronto Star video pictured here on April 1 three years ago:

The three connected cubes are labeled "Harmonic Analysis," 'Number Theory,"
and "Geometry."

Related cultural commentary from a review of the recent film "Justice League" —

"Now all they need is to resurrect Superman (Henry Cavill),
stop Steppenwolf from reuniting his three Mother Cubes
(sure, whatever) and wrap things up in under two cinematic
hours (God bless)."

The nineteenth-century German mathematician Felix Christian Klein
as Steppenwolf —

Volume I of a treatise by Klein is subtitled
"Arithmetic, Algebra, Analysis." This covers
two of the above three Toronto Star cubes.

Klein's Volume II is subtitled "Geometry."

An excerpt from that volume —

Further cultural commentary:  "Glitch" in this journal.

Saturday, March 24, 2018

Sure, Whatever.

Filed under: General,Geometry — Tags: — m759 @ 11:13 AM

The search for Langlands in the previous post
yields the following Toronto Star  illustration —

From a review of the recent film "Justice League" —

"Now all they need is to resurrect Superman (Henry Cavill),
stop Steppenwolf from reuniting his three Mother Cubes
(sure, whatever) and wrap things up in under two cinematic
hours (God bless)."

For other cubic adventures, see yesterday's post on A Piece of Justice 
and the block patterns in posts tagged Design Cube.

Friday, March 23, 2018

Reciprocity

Filed under: General,Geometry — Tags: — m759 @ 7:00 PM

Copy editing — From Wikipedia

"Copy editing (also copy-editing or copyediting, sometimes abbreviated ce)
is the process of reviewing and correcting written material to improve accuracy,
readability, and fitness for its purpose, and to ensure that it is free of error,
omission, inconsistency, and repetition. . . ."

An example of the need for copy editing:

Related material:  Langlands and Reciprocity in this  journal.

Friday, February 16, 2018

Two Kinds of Symmetry

Filed under: General,Geometry — m759 @ 11:29 PM

The Institute for Advanced Study (IAS) at Princeton in its Fall 2015 Letter 
revived "Beautiful Mathematics" as a title:

This ugly phrase was earlier used by Truman State University
professor Martin Erickson as a book title. See below. 

In the same IAS Fall 2015 Letter appear the following remarks
by Freeman Dyson —

". . . a special case of a much deeper connection that Ian Macdonald 
discovered between two kinds of symmetry which we call modular and affine.
The two kinds of symmetry were originally found in separate parts of science,
modular in pure mathematics and affine in physics. Modular symmetry is
displayed for everyone to see in the drawings of flying angels and devils
by the artist Maurits Escher. Escher understood the mathematics and got the
details right. Affine symmetry is displayed in the peculiar groupings of particles
created by physicists with high-energy accelerators. The mathematician
Robert Langlands was the first to conjecture a connection between these and
other kinds of symmetry. . . ." (Wikipedia link added.)

The adjective "modular"  might aptly be applied to . . .

The adjective "affine"  might aptly be applied to . . .

From 'Beautiful Mathematics,' by Martin Erickson, an excerpt on the Cullinane diamond theorem (with source not mentioned)

The geometry of the 4×4 square combines modular symmetry
(i.e., related to theta functions) with the affine symmetry above.

Hudson's 1905 discussion of modular symmetry (that of Rosenhain
tetrads and Göpel tetrads) in the 4×4 square used a parametrization
of that square by the digit 0 and the fifteen 2-subsets of a 6-set, but 
did not discuss the 4×4 square as an affine space.

For the connection of the 15 Kummer modular 2-subsets with the 16-
element affine space over the two-element Galois field GF(2), see my note
of May 26, 1986, "The 2-subsets of a 6-set are the points of a PG(3,2)" —

— and the affine structure in the 1979 AMS abstract
"Symmetry invariance in a diamond ring" —

For some historical background on the symmetry investigations by
Dyson and Macdonald, see Dyson's 1972 article "MIssed Opportunities."

For Macdonald's own  use of the words "modular" and "affine," see
Macdonald, I. G., "Affine Lie algebras and modular forms," 
Séminaire N. Bourbaki , Vol. 23 (1980-1981), Talk no. 577, pp. 258-276.

Wednesday, April 1, 2015

Math’s Big Lies

Filed under: General,Geometry — Tags: , — m759 @ 11:00 AM

Two mathematicians, Barry Mazur and Edward Frenkel,
have, for rhetorical effect, badly misrepresented the
history of some basic fields of mathematics. Mazur and
Frenkel like to emphasize the importance of new 
research by claiming that it connects fields that previously
had no known connection— when, in fact, the fields were
known to be connected since at least the nineteenth century.

For Mazur, see The Proof and the Lie; for Frenkel, see posts
tagged Frenkel-Metaphors.

See also a story and video on Robert Langlands from the
Toronto Star  on March 27, 2015:

"His conjectures are called functoriality and
reciprocity. They made it possible to link up
three branches of math: harmonic analysis,
number theory, and geometry. 

To mathematicians, this is mind-blowing stuff
because these branches have nothing to do
with each other."

For a much earlier link between these three fields, see the essay
"Why Pi Matters" published in The New Yorker  last month.

Sunday, March 29, 2015

Mathematics for Jews*

Filed under: General — Tags: , — m759 @ 11:00 PM

Headline at the Toronto Star  on Friday, March 27, 2015:

Robert Langlands: The Canadian
who reinvented mathematics

“He’s like a modern-day Einstein.”

Apparently, unlike God, Langlands würfelt .

* See also Blockheads  in this journal.

Sunday, February 23, 2014

Sunday School

Filed under: General — m759 @ 9:00 AM

Lang to Langlands

Lang —

“Elliptic functions parametrize elliptic curves, and the intermingling of the analytic and algebraic-arithmetic theory has been at the center of mathematics since the early part of the nineteenth century.”

— Serge Lang, preface to Elliptic Functions  (second edition, 1987)

Langlands

“The theory of modular functions and modular forms, defined on the upper half-plane H and subject to appropriate tranformation laws with respect to the group Gamma = SL(2, Z) of fractional linear transformations, is closely related to the theory of elliptic curves, because the family of all isomorphism classes of elliptic curves over C can be parametrized by the quotient Gamma\H. This is an important, although formal, relation that assures that this and related quotients have a natural structure as algebraic curves X over Q. The relation between these curves and elliptic curves predicted by the Taniyama-Weil conjecture is, on the other hand, far from formal.”

— Robert P. Langlands, review of Elliptic Curves , by Anthony W. Knapp. (The review appeared in Bulletin of the American Mathematical Society , January 1994.)

Thursday, December 5, 2013

Fields

Filed under: General,Geometry — Tags: , , — m759 @ 1:20 AM

Edward Frenkel recently claimed for Robert Langlands
the discovery of a link between two "totally different"
fields of mathematics— number theory and harmonic analysis.
He implied that before Langlandsno relationship between
these fields was known.

See his recent book, and his lecture at the Fields Institute
in Toronto on October 24, 2013.

Meanwhile, in this journal on that date, two math-related
quotations for Stephen King, author of Doctor Sleep

"Danvers is a town in Essex County, Massachusetts, 
United States, located on the Danvers River near the
northeastern coast of Massachusetts. Originally known
as Salem Village, the town is most widely known for its
association with the 1692 Salem witch trials. It is also
known for the Danvers State Hospital, one of the state's
19th-century psychiatric hospitals, which was located here." 

"The summer's gone and all the roses fallin' "

For those who prefer their mathematics presented as fact, not fiction—

(Click for a larger image.)

The arrows in the figure at the right are an attempt to say visually that 
the diamond theorem is related to various fields of mathematics.
There is no claim that prior to the theorem, these fields were not  related.

See also Scott Carnahan on arrow diagrams, and Mathematical Imagery.

Tuesday, November 26, 2013

Edward Frenkel, Your Order Is Ready.

Filed under: General — Tags: — m759 @ 11:00 AM

Backstory: Frenkel's Metaphors and Waitressing for Godot.

In a recent vulgarized presentation of the Langlands program,
Edward Frenkel implied that number theory and harmonic
analysis were, before Langlands came along, quite unrelated.

This is false.

"If we think of different fields of mathematics as continents,
then number theory would be like North America and
harmonic analysis like Europe." 

Edward Frenkel, Love and Math , 2013

For a discussion of pre-Langlands connections between 
these "continents," see

Ding!

"Fourier Analysis in Number Theory, my senior thesis, under the advisory of Patrick Gallagher.

This thesis contains no original research, but is instead a compilation of results from analytic
number theory that involve Fourier analysis. These include quadratic reciprocity (one of 200+
published proofs), Dirichlet's theorem on primes in arithmetic progression, and Weyl's criterion.
There is also a function field analogue of Fermat's Last Theorem. The presentation of the
material is completely self-contained."

Shanshan Ding, University of Pennsylvania graduate student

Monday, November 25, 2013

Pythagoras Wannabe*

Filed under: General,Geometry — Tags: — m759 @ 10:10 AM

A scholium on the link to Pythagoras
in this morning's previous post Figurate Numbers:

For related number mysticism, see Chapter 8, "Magic Numbers,"
in Love and Math: The Heart of Hidden Reality
by Edward Frenkel (Basic Books, Oct. 1, 2013).

(Click for clearer image.)

See also Frenkel's Metaphors in this journal. 

* The wannabe of the title is of course not Langlands, but Frenkel.

Sunday, November 24, 2013

Galois Groups and Harmonic Analysis

Filed under: General,Geometry — Tags: — m759 @ 9:29 AM

"In 1967, he [Langlands] came up with revolutionary
insights tying together the theory of Galois groups 
and another area of mathematics called harmonic
analysis. These two areas, which seem light years
apart
, turned out to be closely related."

— Edward Frenkel, Love and Math, 2013

"Class field theory expresses Galois groups of
abelian extensions of a number field F
in terms of harmonic analysis on the
multiplicative group of [a] locally compact
topological ring, the adèle ring, attached to F."

— Michael Harris in a description of a Princeton
    mathematics department talk of October 2012

Related material: a Saturday evening post.

See also Wikipedia on the history of class field theory.
For greater depth, see Tate's [1950] thesis and the book
Fourier Analysis on Number Fields .

Saturday, November 23, 2013

Light Years Apart?

Filed under: General,Geometry — Tags: — m759 @ 9:00 PM

From a recent attempt to vulgarize the Langlands program:

"Galois’ work is a great example of the power of a mathematical insight…. 

And then, 150 years later, Langlands took these ideas much farther. In 1967, he came up with revolutionary insights tying together the theory of Galois groups and another area of mathematics called harmonic analysis. These two areas, which seem light years apart, turned out to be closely related."

— Frenkel, Edward (2013-10-01).
     Love and Math: The Heart of Hidden Reality
     (p. 78, Basic Books, Kindle Edition) 

(Links to related Wikipedia articles have been added.)

 

Wikipedia on the Langlands program

The starting point of the program may be seen as Emil Artin's reciprocity law [1924-1930], which generalizes quadratic reciprocity. The Artin reciprocity law applies to a Galois extension of algebraic number fields whose Galois group is abelian, assigns L-functions to the one-dimensional representations of this Galois group; and states that these L-functions are identical to certain Dirichlet L-series or more general series (that is, certain analogues of the Riemann zeta function) constructed from Hecke characters. The precise correspondence between these different kinds of L-functions constitutes Artin's reciprocity law.

For non-abelian Galois groups and higher-dimensional representations of them, one can still define L-functions in a natural way: Artin L-functions.

The insight of Langlands was to find the proper generalization of Dirichlet L-functions, which would allow the formulation of Artin's statement in this more general setting.

 

From "An Elementary Introduction to the Langlands Program," by Stephen Gelbart (Bulletin of the American Mathematical Society, New Series , Vol. 10, No. 2, April 1984, pp. 177-219)

On page 194:

"The use of group representations in systematizing and resolving diverse mathematical problems is certainly not new, and the subject has been ably surveyed in several recent articles, notably [ Gross and Mackey ]. The reader is strongly urged to consult these articles, especially for their reformulation of harmonic analysis as a chapter in the theory of group representations.

In harmonic analysis, as well as in the theory of automorphic forms, the fundamental example of a (unitary) representation is the so-called 'right regular' representation of G….

Our interest here is in the role representation theory has played in the theory of automorphic forms.* We focus on two separate developments, both of which are eventually synthesized in the Langlands program, and both of which derive from the original contributions of Hecke already described."

Gross ]  K. I. Gross, On the evolution of non-commutative harmonic analysis . Amer. Math. Monthly 85 (1978), 525-548.

Mackey ]  G. Mackey, Harmonic analysis as the exploitation of symmetry—a historical survey . Bull. Amer. Math. Soc. (N.S.) 3 (1980), 543-698.

* A link to a related Math Overflow article has been added.

In 2011, Frenkel published a commentary in the A.M.S. Bulletin  
on Gelbart's Langlands article. The commentary, written for
a mathematically sophisticated audience, lacks the bold
(and misleading) "light years apart" rhetoric from his new book 
quoted above.

In the year the Gelbart article was published, Frenkel was
a senior in high school. The year was 1984.

For some remarks of my own that mention
that year, see a search for 1984 in this journal.

Monday, November 18, 2013

Teleportation Web?

Filed under: General,Geometry — Tags: — m759 @ 8:45 PM

"In this book, I will describe one of the biggest ideas
to come out of mathematics in the last fifty years:
the Langlands Program, considered by many as
the Grand Unified Theory of mathematics. It’s a
fascinating theory that weaves a web of tantalizing
connections between mathematical fields that
at first glance seem to be light years apart:
algebra, geometry, number theory, analysis
,
and quantum physics. If we think of those fields as
continents in the hidden world of mathematics, then
the Langlands Program is the ultimate teleportation
device, capable of getting us instantly from one of
them to another, and back."

— Edward Frenkel, excerpt from his new book
     in today's online New York Times  

The four areas of pure mathematics that Frenkel
names do not, of course, seem to be "light years
apart" to those familiar with the development of
mathematics in the nineteenth century.

Related material:  Sunday morning's post.

Sunday, November 17, 2013

The X-Men Tree

Filed under: General — Tags: — m759 @ 7:59 AM

Continued from November 12, 2013. A post on that date
showed the tree from Waiting for Godot  along with the two
X-Men patriarchs. See also last night's Chapel post,
which shows a more interesting tree—

A recent book on the Langlands program by Edward Frenkel
repeats a metaphor about building a bridge  between unrelated
worlds within mathematics. A review of the Frenkel book by
Marcus du Sautoy replaces the bridge  metaphor with a wormhole .
Some users of such metaphors seem to feel they are justified, 
for maximum rhetorical effect, in lying about the unrelatedness of
the worlds being connected. The connections they discuss are
surprising (see the Eichler function discussed by Frenkel and
du Sautoy), but the connections occur, at least in the case of
elliptic curves and modular forms, between areas of mathematics
long known to be, in less subtle ways, related. See remarks
from 2005 by Diamond and Shurman below.

Related material:

Wednesday, November 13, 2013

X-Code

Filed under: General,Geometry — m759 @ 8:13 PM

IMAGE- 'Station X,' a book on the Bletchley Park codebreakers

From the obituary of a Bletchley Park
codebreaker who reportedly died on
Armistice Day (Monday, Nov. 11)—

"The main flaw of the Enigma machine,
seen by the inventors as a security-enhancing
measure, was that it would never encipher
a letter as itself…."

Update of 9 PM ET Nov. 13—

"The rogue’s yarn that will run through much of
the material is the algebraic symmetry to which
the name of Galois is attached…."

— Robert P. Langlands,
     Institute for Advanced Study, Princeton

"All the turmoil, all the emotions of the scenes
have been digested by the mind into
a grave intellectual whole.  It is as though
Bach had written the 1812 Overture."

— Aldous Huxley, "The Best Picture," 1925

Friday, December 10, 2010

Cruel Star, Part II

Filed under: General,Geometry — Tags: — m759 @ 2:00 PM

Symmetry, Duality, and Cinema

— Title of a Paris conference held June 17, 2010

From that conference, Edward Frenkel on symmetry and duality

"Symmetry plays an important role in geometry, number theory, and quantum physics. I will discuss the links between these areas from the vantage point of the Langlands Program. In this context 'duality' means that the same theory, or category, may be described in two radically different ways. This leads to many surprising consequences."

Related material —

http://www.log24.com/log/pix10B/101210-CruelStarPartII.jpg

See also  "Black Swan" in this journal, Ingmar Bergman's production of Yukio Mishima's "Madame de Sade," and Duality and Symmetry, 2001.

This journal on the date of the Paris conference
had a post, "Nighttown," with some remarks about
the duality of darkness and light. Its conclusion—

"By groping toward the light we are made to realize
 how deep the darkness is around us."
  — Arthur Koestler, The Call Girls: A Tragi-Comedy,
      Random House, 1973, page 118

Saturday, December 4, 2010

Forgive Us Our Transgressions

Filed under: General — m759 @ 4:15 PM

Bulletin of the American Mathematical Society

"Recent Advances in the Langlands Program"

Author(s):  Edward Frenkel
Journal:     Bull. Amer. Math. Soc. 41 (2004), 151-184.
Posted:     January 8, 2004

Item in the references:

[La5] G. Laumon, La correspondance de Langlands sur les corps de fonctions  (d'après Laurent
La fforgue), Séminaire Bourbaki, Exp. No. 973, Preprint math.AG/0003131.

Correction—

http://www.log24.com/log/pix10B/101204-BourbakiNo873.jpg

Related material— Peter Woit 's post on Frenkel today—

"Math Research Institute, Art, Politics, Transgressive Sex and Geometric Langlands."

See also an item from a Google search on " 'nit-picking' + Bourbaki "—

White Cube — Jake & Dinos Chapman 

Fucking Hell is not, evidently, a realistic (much less nit-picking ) account of the ….
The following link enables you to pan virtually around the Bourbaki
www.whitecube.com/artists/chapman/texts/154/ – Cached

— as well as a search for "White Cube" in this journal.

Friday, September 17, 2010

The Galois Window

Filed under: General,Geometry — Tags: , — m759 @ 5:01 AM

Yesterday's excerpt from von Balthasar supplies some Catholic aesthetic background for Galois geometry.

That approach will appeal to few mathematicians, so here is another.

Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace  is a book by Leonard Mlodinow published in 2002.

More recently, Mlodinow is the co-author, with Stephen Hawking, of The Grand Design  (published on September 7, 2010).

A review of Mlodinow's book on geometry—

"This is a shallow book on deep matters, about which the author knows next to nothing."
— Robert P. Langlands, Notices of the American Mathematical Society,  May 2002

The Langlands remark is an apt introduction to Mlodinow's more recent work.

It also applies to Martin Gardner's comments on Galois in 2007 and, posthumously, in 2010.

For the latter, see a Google search done this morning—

http://www.log24.com/log/pix10B/100917-GardnerGalois.jpg

Here, for future reference, is a copy of the current Google cache of this journal's "paged=4" page.

Note the link at the bottom of the page in the May 5, 2010, post to Peter J. Cameron's web journal. Following the link, we find…

For n=4, there is only one factorisation, which we can write concisely as 12|34, 13|24, 14|23. Its automorphism group is the symmetric group S4, and acts as S3 on the set of three partitions, as we saw last time; the group of strong automorphisms is the Klein group.

This example generalises, by taking the factorisation to consist of the parallel classes of lines in an affine space over GF(2). The automorphism group is the affine group, and the group of strong automorphisms is its translation subgroup.

See also, in this  journal, Window and Window, continued (July 5 and 6, 2010).

Gardner scoffs at the importance of Galois's last letter —

"Galois had written several articles on group theory, and was
  merely annotating and correcting those earlier published papers."
Last Recreations, page 156

For refutations, see the Bulletin of the American Mathematical Society  in March 1899 and February 1909.

Saturday, April 3, 2010

Infinite Jest

Filed under: General,Geometry — Tags: — m759 @ 1:05 AM

"Democrats– in conclusion– Democrats in America
were put on earth to do one thing– Drag the
ignorant hillbilly half of this country into the next
century, which in their case is the 19th."

Bill Maher on March 26

Reply to Maher:

"Hell is other people."
— Jean-Paul Sartre

With a laugh track.

Related material:

Dragging Maher into the 18th  century–

From
N. H. Abel on Elliptic Functions:
Problems of Division and Reduction
,
by Henrik Kragh Sørensen —

Related material– Lemniscate to Langlands (2004)
and references to the lemniscate in
Galois Theory, by David A. Cox (Wiley-IEEE, 2004)

Saturday, February 16, 2008

Saturday February 16, 2008

Filed under: General,Geometry — m759 @ 9:29 AM
Bridges
Between Two Worlds


From the world of mathematics…


“… my advisor once told me, ‘If you ever find yourself drawing one of those meaningless diagrams with arrows connecting different areas of mathematics, it’s a good sign that you’re going senile.'”

— Scott Carnahan at Secret Blogging Seminar, December 14, 2007

Carnahan’s remark in context:

“About five years ago, Cheewhye Chin gave a great year-long seminar on Langlands correspondence for GLr over function fields…. In the beginning, he drew a diagram….

If we remove all of the explanatory text, the diagram looks like this:

CheeWhye Diagram

I was a bit hesitant to draw this, because my advisor once told me, ‘If you ever find yourself drawing one of those meaningless diagrams with arrows connecting different areas of mathematics, it’s a good sign that you’re going senile.’ Anyway, I’ll explain roughly how it works.

Langlands correspondence is a ‘bridge between two worlds,’ or more specifically, an assertion of a bijection….”

Compare and contrast the above…

… to the world of Rudolf Kaehr:

Rudolf Kaehr on 'Diamond Structuration'

The above reference to “diamond theory” is from Rudolf Kaehr‘s paper titled Double Cross Playing Diamonds.

Another bridge…

Carnahan’s advisor, referring to “meaningless diagrams with arrows connecting different areas of mathematics,” probably did not have in mind diagrams like the two above, but rather diagrams like the two below–

From the world of mathematics

Relationship of diamond theory to other fields

“A rough sketch of
how diamond theory is
related to some other
fields of mathematics”
— Steven H. Cullinane

… to the world of Rudolf Kaehr:

Relationship of PolyContextural Logic (PCL) to other fields

Related material:

For further details on
the “diamond theory” of
Cullinane, see

Finite Geometry of the
Square and Cube
.

For further details on
the “diamond theory” of
Kaehr, see

Rudy’s Diamond Strategies.

Those who prefer entertainment
may enjoy an excerpt
from Log24, October 2007:

“Do not let me hear
Of the wisdom
of old men,
but rather of
their folly”
 
Four Quartets   

Anthony Hopkins in 'Slipstream'

Anthony Hopkins
in the film
Slipstream

Anthony Hopkins  
in the film “Proof“–

Goddamnit, open
the goddamn book!
Read me the lines!

Thursday, January 31, 2008

Thursday January 31, 2008

Filed under: General — m759 @ 5:24 AM
From G. K. Chesterton,
The Black Virgin
 
As the black moon
of some divine eclipse,
As the black sun
of the Apocalypse,
As the black flower
that blessed Odysseus back
From witchcraft; and
he saw again the ships.

In all thy thousand images
we salute thee.

Earlier in the poem….
 
Clothed with the sun
or standing on the moon
Crowned with the stars
or single, a morning star,
Sunlight and moonlight
are thy luminous shadows,
Starlight and twilight
thy refractions are,
Lights and half-lights and
all lights turn about thee.

 
From Oct. 16, 2007,
date of death of Deborah Kerr:

"Harish, who was of a
spiritual, even religious, cast
and who liked to express himself in
metaphors, vivid and compelling,
did see, I believe, mathematics
as mediating between man and
what one can only call God."
R. P. Langlands

From a link of Jan. 17, 2008
Time and Eternity:

Abstract Symbols of Time and Eternity

Jean Simmons and Deborah Kerr in Black Narcissus
Jean Simmons (l.) and Deborah Kerr (r.)
in "Black Narcissus" (1947)

and from the next day,
Jan. 18, 2008:

… Todo lo sé por el lucero puro
que brilla en la diadema de la Muerte.

Rubén Darío,
born January 18, 1867

Related material:

Dark Lady and Bright Star,
Time and Eternity,
Damnation Morning

Happy birthday also to
the late John O'Hara.

Tuesday, October 16, 2007

Tuesday October 16, 2007

Filed under: General — m759 @ 10:00 AM
In memory of
Harish-Chandra,
who died at 60
on this date in 1983

  The image “http://www.log24.com/log/pix07A/071016-Harish-Chandra.jpg” cannot be displayed, because it contains errors.
Harish-Chandra in 1981
(Photo by Herman Landshof)

Recent Log24 entries have parodied the use of the phrase “deep beauty” as the title of the Oct. 3-4 physics symposium of that name, which was supported by a grant from the John Templeton Foundation and sponsored by the Department of Philosophy at Princeton University.
Such parody was in part suggested by the symposium’s sources of financial and academic support. This support had, in the view of some, the effect of linking the symposium’s topic, the mathematics of quantum theory, with both religion (the Templeton Foundation) and philosophy (a field sometimes associated in popular thought– though not at Princeton— with quantum mysticism.)

As a corrective to the previous parodies here, the following material on the mathematician Harish-Chandra may help to establish that there is, in fact, such a thing as “deep beauty”– if not in physics, religion, or philosophy, at least in pure mathematics.

MacTutor History of Mathematics:

“Harish-Chandra worked at the Institute of Advanced Study at Princeton from 1963. He was appointed IBM-von Neumann Professor in 1968.”

R. P. Langlands (pdf, undated, apparently from a 1983 memorial talk):

“Almost immediately upon his arrival in Princeton he began working at a ferocious pace, setting standards that the rest of us may emulate but never achieve. For us there is a welter of semi-simple groups: orthogonal groups, symplectic groups, unitary groups, exceptional groups; and in our frailty we are often forced to treat them separately. For him, or so it appeared because his methods were always completely general, there was a single group. This was one of the sources of beauty of the subject in his hands, and I once asked him how he achieved it. He replied, honestly I believe, that he could think no other way. It is certainly true that he was driven back upon the simplifying properties of special examples only in desperate need and always temporarily.”

“It is difficult to communicate the grandeur of Harish-Chandra’s achievements and I have not tried to do so. The theory he created still stands– if I may be excused a clumsy simile– like a Gothic cathedral, heavily buttressed below but, in spite of its great weight, light and soaring in its upper reaches, coming as close to heaven as mathematics can. Harish, who was of a spiritual, even religious, cast and who liked to express himself in metaphors, vivid and compelling, did see, I believe, mathematics as mediating between man and what one can only call God. Occasionally, on a stroll after a seminar, usually towards evening, he would express his feelings, his fine hands slightly upraised, his eyes intent on the distant sky; but he saw as his task not to bring men closer to God but God closer to men. For those who can understand his work and who accept that God has a mathematical side, he accomplished it.”

For deeper views of his work, see

  1. Rebecca A. Herb, “Harish-Chandra and His Work” (pdf), Bulletin of the American Mathematical Society, July 1991, and
  2. R. P. Langlands, “Harish-Chandra, 1923-1983” (pdf, 28 pp., Royal Society memoir, 1985)

Thursday, August 9, 2007

Thursday August 9, 2007

Filed under: General — m759 @ 12:00 PM
“Serious numbers  
will always be heard.”

— Paul Simon

(See St. Luke’s Day, 2005.)  


Bulletin of the American Mathematical Society
,
Volume 31, Number 1, July 1994, Pages 1-14

Selberg’s Conjectures
and Artin L-Functions
(pdf)

M. Ram Murty

Introduction

In its comprehensive form, an identity between an automorphic L-function and a “motivic” L-function is called a reciprocity law. The celebrated Artin reciprocity law is perhaps the fundamental example. The conjecture of Shimura-Taniyama that every elliptic curve over Q is “modular” is certainly the most intriguing reciprocity conjecture of our time. The “Himalayan peaks” that hold the secrets of these nonabelian reciprocity laws challenge humanity, and, with the visionary Langlands program, we have mapped out before us one means of ascent to those lofty peaks. The recent work of Wiles suggests that an important case (the semistable case) of the Shimura-Taniyama conjecture is on the horizon and perhaps this is another means of ascent. In either case, a long journey is predicted…. At the 1989 Amalfi meeting, Selberg [S] announced a series of conjectures which looks like another approach to the summit. Alas, neither path seems the easier climb….

[S] A. Selberg, Old and new
      conjectures and results
      about a class of Dirichlet series,
      Collected Papers, Volume II,
      Springer-Verlag, 1991, pp. 47-63.

Zentralblatt MATH Database
on the above Selberg paper:

“These are notes of lectures presented at the Amalfi Conference on Number Theory, 1989…. There are various stimulating conjectures (which are related to several other conjectures like the Sato-Tate conjecture, Langlands conjectures, Riemann conjecture…)…. Concluding remark of the author: ‘A more complete account with proofs is under preparation and will in time appear elsewhere.'”

Related material: Previous entry.

Friday, February 2, 2007

Friday February 2, 2007

Filed under: General — m759 @ 7:11 AM

The Night Watch

For Catholic Schools Week
(continued from last year)–

Last night’s Log24 Xanga
footprints from Poland:

Poland 2/2/07 1:29 AM
/446066083/item.html
2/20/06: The Past Revisited
(with link to online text of
Many Dimensions, by Charles Williams)

Poland 2/2/07 2:38 AM
/426273644/item.html
1/15/06 Inscape
(the mathematical concept, with
square and “star” diagrams)

Poland 2/2/07 3:30 AM
nextdate=2%252f8%252f20…
2/8/05 The Equation
(Russell Crowe as John Nash
with “star” diagram from a
Princeton lecture by Langlands)

Poland 2/2/07 4:31 AM
/524081776/item.html
8/29/06 Hollywood Birthday
(with link to online text of
Plato on the Human Paradox,
by a Fordham Jesuit)

Poland 2/2/07 4:43 AM
/524459252/item.html
8/30/06 Seven
(Harvard, the etymology of the
word “experience,” and the
Catholic funeral of a professor’s
23-year-old daughter)

Poland 2/2/07 4:56 AM
/409355167/item.html
12/19/05 Quarter to Three (cont.)

(remarks on permutation groups
for the birthday of Helmut Wielandt)

Poland 2/2/07 5:03 AM
/490604390/item.html
5/29/06 For JFK’s Birthday
(The Call Girls revisited)

Poland 2/2/07 5:32 AM
/522299668/item.html
8/24/06 Beginnings
(Nasar in The New Yorker and
T. S. Eliot in Log24, both on the 2006
Beijing String Theory conference)

Poland 2/2/07 5:46 AM
/447354678/item.html
2/22/06 In the Details
(Harvard’s president resigns,
with accompanying “rosebud”)

Friday, October 6, 2006

Friday October 6, 2006

Filed under: General — m759 @ 12:00 PM
A Visual Proof

The great mathematician
Robert P. Langlands
is 70 today.

In honor of his expository work–
notably, lectures at
The Institute for Advanced Study
on “The Practice of Mathematics
and a very acerbic review (pdf) of
a book called Euclid’s Window
here is a “Behold!” proof of
the Pythagorean theorem:

The image “http://www.log24.com/log/pix06A/Pythagorean_Theorem.jpg” cannot be displayed, because it contains errors.

The picture above is adapted from
 a sketch by Eves of a “dynamical”
proof suitable for animation.

The proof has been
 described by Alexander Bogomolny
as “a variation on” Euclid I.47.
Bogomolny says it is a proof
by “shearing and translation.”

It has, in fact, been animated.
The following version is
by Robert Foote:
The image “http://www.log24.com/log/pix06A/RobertFooteAnimation.gif” cannot be displayed, because it contains errors.

Thursday, June 23, 2005

Thursday June 23, 2005

Filed under: General,Geometry — m759 @ 3:00 PM

Mathematics and Metaphor

The current (June/July) issue of the Notices of the American Mathematical Society has two feature articles.  The first, on the vulgarizer Martin Gardner, was dealt with here in a June 19 entry, Darkness Visible.  The second is related to a letter of André Weil (pdf) that is in turn related to mathematician Barry Mazur’s attempt to rewrite mathematical history  and to vulgarize other people’s research by using metaphors drawn, it would seem, from the Weil letter.
 
A Mathematical Lie conjectures that Mazur’s revising of history was motivated by a desire to dramatize some arcane mathematics, the Taniyama conjecture, that deals with elliptic curves and modular forms, two areas of mathematics that have been known since the nineteenth century to be closely related.

Mazur led author Simon Singh to believe that these two areas of mathematics were, before Taniyama’s conjecture of 1955, completely unrelated — 

“Modular forms and elliptic equations live in completely different regions of the mathematical cosmos, and nobody would ever have believed that there was the remotest link between the two subjects.” — Simon Singh, Fermat’s Enigma, 1998 paperback, p. 182

This is false.  See Robert P. Langlands, review of Elliptic Curves, by Anthony W. Knapp, Bulletin of the American Mathematical Society, January 1994.

It now appears that Mazur’s claim was in part motivated by a desire to emulate the great mathematician André Weil’s manner of speaking; Mazur parrots Weil’s “bridge” and “Rosetta stone” metaphors —

From Peter Woit’s weblog, Feb. 10, 2005:

“The focus of Weil’s letter is the analogy between number fields and the field of algebraic functions of a complex variable. He describes his ideas about studying this analogy using a third, intermediate subject, that of function fields over a finite field, which he thinks of as a ‘bridge‘ or ‘Rosetta stone.'” 

In “A 1940 Letter of André Weil on Analogy in Mathematics,” (pdf), translated by Martin H. Krieger, Notices of the A.M.S., March 2005, Weil writes that

“The purely algebraic theory of algebraic functions in any arbitrary field of constants is not rich enough so that one might draw useful lessons from it. The ‘classical’ theory (that is, Riemannian) of algebraic functions over the field of constants of the complex numbers is infinitely richer; but on the one hand it is too much so, and in the mass of facts some real analogies become lost; and above all, it is too far from the theory of numbers. One would be totally obstructed if there were not a bridge between the two.  And just as God defeats the devil: this bridge exists; it is the theory of the field of algebraic functions over a finite field of constants….

On the other hand, between the function fields and the ‘Riemannian’ fields, the distance is not so large that a patient study would not teach us the art of passing from one to the other, and to profit in the study of the first from knowledge acquired about the second, and of the extremely powerful means offered to us, in the study of the latter, from the integral calculus and the theory of analytic functions. That is not to say that at best all will be easy; but one ends up by learning to see something there, although it is still somewhat confused. Intuition makes much of it; I mean by this the faculty of seeing a connection between things that in appearance are completely different; it does not fail to lead us astray quite often. Be that as it may, my work consists in deciphering a trilingual text {[cf. the Rosetta Stone]}; of each of the three columns I have only disparate fragments; I have some ideas about each of the three languages: but I know as well there are great differences in meaning from one column to another, for which nothing has prepared me in advance. In the several years I have worked at it, I have found little pieces of the dictionary. Sometimes I worked on one column, sometimes under another.”

Here is another statement of the Rosetta-stone metaphor, from Weil’s translator, Martin H.  Krieger, in the A.M.S. Notices of November 2004,  “Some of What Mathematicians Do” (pdf):

“Weil refers to three columns, in analogy with the Rosetta Stone’s three languages and their arrangement, and the task is to ‘learn to read Riemannian.’  Given an ability to read one column, can you find its translation in the other columns?  In the first column are Riemann’s transcendental results and, more generally, work in analysis and geometry.  In the second column is algebra, say polynomials with coefficients in the complex numbers or in a finite field. And in the third column is arithmetic or number theory and combinatorial properties.”

For greater clarity, see  Armand Borel (pdf) on Weil’s Rosetta stone, where the three columns are referred to as Riemannian (transcendental), Italian (“algebraico-geometric,” over finite fields), and arithmetic (i.e., number-theoretic).
 
From Fermat’s Enigma, by Simon Singh, Anchor paperback, Sept. 1998, pp. 190-191:

Barry Mazur: “On the one hand you have the elliptic world, and on the other you have the modular world.  Both these branches of mathematics had been studied intensively but separately…. Than along comes the Taniyama-Shimura conjecture, which is the grand surmise that there’s a bridge between these two completely different worlds.  Mathematicians love to build bridges.”

Simon Singh: “The value of mathematical bridges is enormous.  They enable communities of mathematicians who have been living on separate islands to exchange ideas and explore each other’s  creations…. The great potential of the Taniyama-Shimura conjecture was that it would connect two islands and allow them to speak to each other for the first time.  Barry Mazur thinks of the Taniyama-Shimura conjecture as a translating device similar to the Rosetta stone…. ‘It’s as if you know one language and this Rosetta stone is going to give you an intense understanding of the other language,’ says Mazur.  ‘But the Taniyama-Shimura conjecture is a Rosetta stone with a certain magical power.'”

If Mazur, who is scheduled to speak at a conference on Mathematics and Narrative this July, wants more material on stones with magical powers, he might consult The Blue Matrix and The Diamond Archetype.

Tuesday, February 8, 2005

Tuesday February 8, 2005

Filed under: General,Geometry — m759 @ 10:00 AM
New from the
Oscar-winning producer,
director, and screenwriter

of “A Beautiful Mind” –

The image “http://www.log24.com/log/pix05/050208-Crowe.jpg” cannot be displayed, because it contains errors.

With apologies to Dan Brown

“The Divine Proportion

is an irrational number and
the positive solution
of the quadratic equation

x2 – x – 1 = 0,

which is (1+Sqrt(5))/2,
about 1.618034.

The Greek letter ‘phi’
(see below for the symbol)
is sometimes used
to represent this number.”

The image “http://www.log24.com/log/pix05/050208-pentagon2.gif” cannot be displayed, because it contains errors.

Don Cohen  

For another approach to
the divine proportion, see

Best Picture.

“The rogue’s yarn that will run through much of the material is the algebraic symmetry to which the name of Galois is attached and which I wanted to introduce in as concrete and appealing a way as possible….

Apart from its intrinsic appeal, that is the reason for treating the construction of the pentagon, and our task today will be to acquire some feel for this construction.  It is not easy.”
 
— R. P. Langlands, 1999 lecture (pdf) at the Institute for Advanced Study, Princeton, in the spirit of Hermann Weyl

Friday, September 17, 2004

Friday September 17, 2004

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

God is in…
The Details

From an entry for Aug. 19, 2003 on
conciseness, simplicity, and objectivity:

Above: Dr. Harrison Pope, Harvard professor of psychiatry, demonstrates the use of the Wechsler Adult Intelligence Scale "block design" subtest.

Another Harvard psychiatrist, Armand Nicholi, is in the news lately with his book The Question of God: C.S. Lewis and Sigmund Freud Debate God, Love, Sex, and the Meaning of Life.

Pope

Nicholi

Old
Testament
Logos

New
Testament
Logos

For the meaning of the Old-Testament logos above, see the remarks of Plato on the immortality of the soul at

Cut-the-Knot.org.

For the meaning of the New-Testament logos above, see the remarks of R. P. Langlands at

The Institute for Advanced Study.

On Harvard and psychiatry: see

The Crimson Passion:
A Drama at Mardi Gras

(February 24, 2004)

This is a reductio ad absurdum of the Harvard philosophy so eloquently described by Alston Chase in his study of Harvard and the making of the Unabomber, Ted Kaczynski.  Kaczynski's time at Harvard overlapped slightly with mine, so I may have seen him in Cambridge at some point.  Chase writes that at Harvard, the Unabomber "absorbed the message of positivism, which demanded value-neutral reasoning and preached that (as Kaczynski would later express it in his journal) 'there is no logical justification for morality.'" I was less impressed by Harvard positivism, although I did benefit from a course in symbolic logic from Quine.  At that time– the early 60's– little remained at Harvard of what Robert Stone has called "our secret culture," that of the founding Puritans– exemplified by Cotton and Increase Mather.

From Robert Stone, A Flag for Sunrise:

"Our secret culture is as frivolous as a willow on a tombstone.  It's a wonderful thing– or it was.  It was strong and dreadful, it was majestic and ruthless.  It was a stranger to pity.  And it's not for sale, ladies and gentlemen."

Some traces of that culture:

A web page
in Australia:

A contemporary
Boston author:

Click on pictures for details.

A more appealing view of faith was offered by PBS on Wednesday night, the beginning of this year's High Holy Days:

Armand Nicholi: But how can you believe something that you don't think is true, I mean, certainly, an intelligent person can't embrace something that they don't think is true — that there's something about us that would object to that.

Jeremy Fraiberg: Well, the answer is, they probably do believe it's true.

Armand Nicholi: But how do they get there? See, that's why both Freud and Lewis was very interested in that one basic question. Is there an intelligence beyond the universe? And how do we answer that question? And how do we arrive at the answer of that question?

Michael Shermer: Well, in a way this is an empirical question, right? Either there is or there isn't.

Armand Nicholi: Exactly.

Michael Shermer: And either we can figure it out or we can't, and therefore, you just take the leap of faith or you don't.

Armand Nicholi: Yeah, now how can we figure it out?

Winifred Gallagher: I think something that was perhaps not as common in their day as is common now — this idea that we're acting as if belief and unbelief were two really radically black and white different things, and I think for most people, there's a very — it's a very fuzzy line, so that —

Margaret Klenck: It's always a struggle.

Winifred Gallagher: Rather than — I think there's some days I believe, and some days I don't believe so much, or maybe some days I don't believe at all.

Doug Holladay: Some hours.

Winifred Gallagher: It's a, it's a process. And I think for me the big developmental step in my spiritual life was that — in some way that I can't understand or explain that God is right here right now all the time, everywhere.

Armand Nicholi: How do you experience that?

Winifred Gallagher: I experience it through a glass darkly, I experience it in little bursts. I think my understanding of it is that it's, it's always true, and sometimes I can see it and sometimes I can't. Or sometimes I remember that it's true, and then everything is in Technicolor. And then most of the time it's not, and I have to go on faith until the next time I can perhaps see it again. I think of a divine reality, an ultimate reality, uh, would be my definition of God.

Winifred
Gallagher

Sangaku

Gallagher seemed to be the only participant in the PBS discussion that came close to the Montessori ideals of conciseness, simplicity, and objectivity.  Dr. Montessori intended these as ideals for teachers, but they seem also to be excellent religious values.  Just as the willow-tombstone seems suited to Geoffrey Hill's style, the Pythagorean sangaku pictured above seems appropriate to the admirable Gallagher.

Saturday, January 17, 2004

Saturday January 17, 2004

Filed under: General — m759 @ 12:00 PM

Math History

This morning’s web notes:

From Lemniscate to Langlands.

Monday, December 8, 2003

Monday December 8, 2003

Filed under: General — m759 @ 11:11 PM

Dream of Youth

Today is the feast day of

Saint Hermann Weyl.

In his honor, here are two links:

The Jugendtraum and

Langlands on the Jugendtraum.

Sunday, November 30, 2003

Sunday November 30, 2003

Filed under: General — m759 @ 3:27 PM

The Proof and the Lie

A mathematical lie has been circulating on the Internet.

It concerns the background of Wiles’s recent work on mathematics related to Fermat’s last theorem, which involves the earlier work of a mathematician named Taniyama.

This lie states that at the time of a conjecture by Taniyama in 1955, there was no known relationship between the two areas of mathematics known as “elliptic curves” and “modular forms.”

The lie, due to Harvard mathematician Barry Mazur, was broadcast in a TV program, “The Proof,” in October 1997 and repeated in a book based on the program and in a Scientific American article, “Fermat’s Last Stand,” by Simon Singh and Kenneth Ribet, in November 1997.

“… elliptic curves and modular forms… are from opposite ends of the mathematical spectrum, and had previously been studied in isolation.”

Site on Simon Singh’s 1997 book Fermat’s Last Theorem

“JOHN CONWAY: What the Taniyama-Shimura conjecture says, it says that every rational elliptic curve is modular, and that’s so hard to explain.

BARRY MAZUR: So, let me explain.  Over here, you have the elliptic world, the elliptic curves, these doughnuts.  And over here, you have the modular world, modular forms with their many, many symmetries.  The Shimura-Taniyama conjecture makes a bridge between these two worlds.  These worlds live on different planets.  It’s a bridge.  It’s more than a bridge; it’s really a dictionary, a dictionary where questions, intuitions, insights, theorems in the one world get translated to questions, intuitions in the other world.

KEN RIBET: I think that when Shimura and Taniyama first started talking about the relationship between elliptic curves and modular forms, people were very incredulous….”

Transcript of NOVA program, “The Proof,” October 1997

The lie spread to other popular accounts, such as the column of Ivars Peterson published by the Mathematical Association of America:

“Elliptic curves and modular forms are mathematically so different that mathematicians initially couldn’t believe that the two are related.”

Ivars Peterson, “Curving Beyond Fermat,” November 1999 

The lie has now contaminated university mathematics courses, as well as popular accounts:

“Elliptic curves and modular forms are completely separate topics in mathematics, and they had never before been studied together.”

Site on Fermat’s last theorem by undergraduate K. V. Binns

Authors like Singh who wrote about Wiles’s work despite their ignorance of higher mathematics should have consulted the excellent website of Charles Daney on Fermat’s last theorem.

A 1996 page in Daney’s site shows that Mazur, Ribet, Singh, and Peterson were wrong about the history of the known relationships between elliptic curves and modular forms.  Singh and Peterson knew no better, but there is no excuse for Mazur and Ribet.

Here is what Daney says:

“Returning to the j-invariant, it is the 1:1 map betweem isomorphism classes of elliptic curves and C*. But by the above it can also be viewed as a 1:1 map j:H/r -> C.  j is therefore an example of what is called a modular function. We’ll see a lot more of modular functions and the modular group. These facts, which have been known for a long time, are the first hints of the deep relationship between elliptic curves and modular functions.”

“Copyright © 1996 by Charles Daney,
All Rights Reserved.
Last updated: March 28, 1996″

Update of Dec. 2, 2003

For the relationship between modular functions and modular forms, see (for instance) Modular Form in Wikipedia.

Some other relevant quotations:

From J. S. Milne, Modular Functions and Modular Forms:

“The definition of modular form may seem strange, but we have seen that such functions arise naturally in the [nineteenth-century] theory of elliptic functions.”

The next quote, also in a nineteenth-century context, relates elliptic functions to elliptic curves.

From Elliptic Functions, a course syllabus:

“Elliptic functions parametrize elliptic curves.”

Putting the quotes together, we have yet another description of the close relationship, well known in the nineteenth century (long before Taniyama’s 1955 conjecture), between elliptic curves and modular forms.

Another quote from Milne, to summarize:

“From this [a discussion of nineteenth-century mathematics], one sees that arithmetic facts about elliptic curves correspond to arithmetic facts about special values of modular functions and modular forms.”

Serge Lang apparently agrees:

Elliptic functions parametrize elliptic curves, and the intermingling of the analytic and algebraic-arithmetic theory has been at the center of mathematics since the early part of the nineteenth century.”

Editorial description of Lang’s Elliptic Functions (second edition, 1987)

Update of Dec. 3, 2003

The theory of modular functions and modular forms, defined on the upper half-plane H and subject to appropriate tranformation laws with respect to the group Gamma = SL(2, Z) of fractional linear transformations, is closely related to the theory of elliptic curves, because the family of all isomorphism classes of elliptic curves over C can be parametrized by the quotient Gamma\H. This is an important, although formal, relation that assures that this and related quotients have a natural structure as algebraic curves X over Q. The relation between these curves and elliptic curves predicted by the Taniyama-Weil conjecture is, on the other hand, far from formal.”

Robert P. Langlands, review of Elliptic Curves, by Anthony W. Knapp.  (The review  appeared in Bulletin of the American Mathematical Society, January 1994.)

Monday, September 1, 2003

Monday September 1, 2003

Filed under: General — m759 @ 3:33 PM

The Unity of Mathematics,

or “Shema, Israel”

A conference to honor the 90th birthday (Sept. 2) of Israel Gelfand is currently underway in Cambridge, Massachusetts.

The following note from 2001 gives one view of the conference’s title topic, “The Unity of Mathematics.”

Reciprocity in 2001

by Steven H. Cullinane
(May 30, 2001)

From 2001: A Space Odyssey, by Arthur C. Clarke, New American Library, 1968:

The glimmering rectangular shape that had once seemed no more than a slab of crystal still floated before him….  It encapsulated yet unfathomed secrets of space and time, but some at least he now understood and was able to command.

How obvious — how necessary — was that mathematical ratio of its sides, the quadratic sequence 1: 4: 9!  And how naive to have imagined that the series ended at this point, in only three dimensions!

— Chapter 46, “Transformation”

From a review of Himmelfarb, by Michael Krüger, New York, George Braziller, 1994:

As a diffident, unsure young man, an inexperienced ethnologist, Richard was unable to travel through the Amazonian jungles unaided. His professor at Leipzig, a Nazi Party member (a bigot and a fool), suggested he recruit an experienced guide and companion, but warned him against collaborating with any Communists or Jews, since the objectivity of research would inevitably be tainted by such contact. Unfortunately, the only potential associate Richard can find in Sao Paulo is a man called Leo Himmelfarb, both a Communist (who fought in the Spanish Civil War) and a self-exiled Jew from Galicia, but someone who knows the forests intimately and can speak several of the native dialects.

“… Leo followed the principle of taking and giving, of learning and teaching, of listening and storytelling, in a word: of reciprocity, which I could not even imitate.”

… E. M. Forster famously advised his readers, “Only connect.” “Reciprocity” would be Michael Kruger’s succinct philosophy, with all that the word implies.

— William Boyd, New York Times Book Review, October 30, 1994

Reciprocity and Euler

Applying the above philosophy of reciprocity to the Arthur C. Clarke sequence

1, 4, 9, ….

we obtain the rather more interesting sequence
1/1, 1/4, 1/9, …..

This leads to the following problem (adapted from the St. Andrews biography of Euler):

Perhaps the result that brought Euler the most fame in his young days was his solution of what had become known as the Basel problem. This was to find a closed form for the sum of the infinite series

1/1 + 1/4 + 1/9 + 1/16 + 1/25 + …

— a problem which had defeated many of the top mathematicians including Jacob Bernoulli, Johann Bernoulli and Daniel Bernoulli. The problem had also been studied unsuccessfully by Leibniz, Stirling, de Moivre and others. Euler showed in 1735 that the series sums to (pi squared)/6. He generalized this series, now called zeta(2), to zeta functions of even numbers larger than two.

Related Reading

For four different proofs of Euler’s result, see the inexpensive paperback classic by Konrad Knopp, Theory and Application of Infinite Series (Dover Publications).

Related Websites

Evaluating Zeta(2), by Robin Chapman (PDF article) Fourteen proofs!

Zeta Functions for Undergraduates

The Riemann Zeta Function

Reciprocity Laws
Reciprocity Laws II

The Langlands Program

Recent Progress on the Langlands Conjectures

For more on
the theme of unity,
see

Monolithic Form
and
ART WARS.

Tuesday, August 19, 2003

Tuesday August 19, 2003

Filed under: General,Geometry — Tags: — m759 @ 5:23 PM

Intelligence Test

From my August 31, 2002, entry quoting Dr. Maria Montessori on conciseness, simplicity, and objectivity:

Above: Dr. Harrison Pope, Harvard professor of psychiatry, demonstrates the use of the Wechsler Adult Intelligence Scale "block design" subtest.

Another Harvard psychiatrist, Armand Nicholi, is in the news lately with his book The Question of God: C.S. Lewis and Sigmund Freud Debate God, Love, Sex, and the Meaning of Life

 

Pope

Nicholi

Old
Testament
Logos

New
Testament
Logos

For the meaning of the Old-Testament logos above, see the remarks of Plato on the immortality of the soul at

Cut-the-Knot.org.

For the meaning of the New-Testament logos above, see the remarks of R. P. Langlands at

The Institute for Advanced Study.

For the meaning of life, see

The Gospel According to Jill St. John,

whose birthday is today.

"Some sources credit her with an I.Q. of 162."
 

Thursday, March 13, 2003

Thursday March 13, 2003

Filed under: General — m759 @ 4:44 PM

ART WARS:

From The New Yorker, issue of March 17, 2003, Clive James on Aldous Huxley:

The Perennial Philosophy, his 1945 book compounding all the positive thoughts of West and East into a tutti-frutti of moral uplift, was the equivalent, in its day, of It Takes a Village: there was nothing in it to object to, and that, of course, was the objection.”

For a cultural artifact that is less questionably perennial, see Huxley’s story “Young Archimedes.”

Plato, Pythagoras, and
the diamond figure

Plato’s Diamond in the Meno
Plato as a precursor of Gerard Manley Hopkins’s “immortal diamond.” An illustration shows the ur-diamond figure.

Plato’s Diamond Revisited
Ivars Peterson’s Nov. 27, 2000 column “Square of the Hypotenuse” which discusses the diamond figure as used by Pythagoras (perhaps) and Plato. Other references to the use of Plato’s diamond in the proof of the Pythagorean theorem:

Huxley:

“… and he proceeded to prove the theorem of Pythagoras — not in Euclid’s way, but by the simpler and more satisfying method which was, in all probability, employed by Pythagoras himself….
‘You see,’ he said, ‘it seemed to me so beautiful….’
I nodded. ‘Yes, it’s very beautiful,’ I said — ‘it’s very beautiful indeed.'”
— Aldous Huxley, “Young Archimedes,” in Collected Short Stories, Harper, 1957, pp. 246 – 247

Heath:

Sir Thomas L. Heath, in his commentary on Euclid I.47, asks how Pythagoreans discovered the Pythagorean theorem and the irrationality of the diagonal of a unit square. His answer? Plato’s diamond.
(See Heath, Sir Thomas Little (1861-1940),
The thirteen books of Euclid’s Elements translated from the text of Heiberg with introduction and commentary. Three volumes. University Press, Cambridge, 1908. Second edition: University Press, Cambridge, 1925. Reprint: Dover Publications, New York, 1956.

Other sites on the alleged
“diamond” proof of Pythagoras

Colorful diagrams at Cut-the-Knot

Illustrated legend of the diamond proof

Babylonian version of the diamond proof

For further details of Huxley’s story, see

The Practice of Mathematics,

Part I, by Robert P. Langlands, from a lecture series at the Institute for Advanced Study, Princeton.

From the New Yorker Contributors page for St. Patrick’s Day, 2003:

Clive James (Books, p. 143) has a new collection, As of This Writing: The Essential Essays, 1968-2002, which will be published in June.”

See also my entry “The Boys from Uruguay” and the later entry “Lichtung!” on the Deutsche Schule Montevideo in Uruguay.

Tuesday, October 22, 2002

Tuesday October 22, 2002

Filed under: General,Geometry — m759 @ 1:16 AM

Introduction to
Harmonic Analysis

From Dr. Mac’s Cultural Calendar for Oct. 22:

  • The French actress Catherine Deneuve was born on this day in Paris in 1943….
  • The Beach Boys released the single “Good Vibrations” on this day in 1966.

“I hear the sound of a
   gentle word

On the wind that lifts
   her perfume
   through the air.”

— The Beach Boys

 
In honor of Deneuve and of George W. Mackey, author of the classic 156-page essay, “Harmonic analysis* as the exploitation of symmetry† — A historical survey” (Bulletin of the American Mathematical Society (New Series), Vol. 3, No. 1, Part 1 (July 1980), pp. 543-698), this site’s music is, for the time being, “Good Vibrations.”
 
For more on harmonic analysis, see “Group Representations and Harmonic Analysis from Euler to Langlands,” by Anthony W. Knapp, Part I and Part II.
 
* For “the simplest non-trivial model for harmonic analysis,” the Walsh functions, see F. Schipp et. al., Walsh Series: An Introduction to Dyadic Harmonic Analysis, Hilger, 1990. For Mackey’s “exploitation of symmetry” in this context, see my note Symmetry of Walsh Functions, and also the footnote below.
 
† “Now, it is no easy business defining what one means by the term conceptual…. I think we can say that the conceptual is usually expressible in terms of broad principles. A nice example of this comes in form of harmonic analysis, which is based on the idea, whose scope has been shown by George Mackey… to be immense, that many kinds of entity become easier to handle by decomposing them into components belonging to spaces invariant under specified symmetries.”
The importance of mathematical conceptualisation,
by David Corfield, Department of History and Philosophy of Science, University of Cambridge

Powered by WordPress