Log24

Thursday, April 15, 2021

A Pythagorean Letter*

Filed under: General — Tags: , — m759 @ 4:00 am

See other posts now tagged Yale Weekend.

That weekend, Sat. Nov. 23 — Sun. Nov. 24, 2013,
saw the death of Yale professor Sam See
in a New Haven Jail.

Related literary remarks:

Search  "Merve Emre" + "Sam See."

* Vide  Log24 references.

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 .

Logic for Jews*

The search for 1984 at the end of last evening’s post
suggests the following Sunday meditation.

My own contribution to this genre—

A triangle-decomposition result from 1984:

American Mathematical Monthly ,  June-July 1984, p. 382

MISCELLANEA, 129

Triangles are square

“Every triangle consists of n  congruent copies of itself”
is true if and only if  is a square. (The proof is trivial.)
— Steven H. Cullinane

The Orwell slogans are false. My own is not.

* The “for Jews” of the title applies to some readers of Edward Frenkel.

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.

Deo Gratias

Filed under: General — Tags: — m759 @ 12:00 pm

See also R. L. Burnside in this journal (Sept. 2, 2005).

Rabbi

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

“We’ll give the week-end to wisdom, to Weisheit, the rabbi….”

— Wallace Stevens in “Things of August” (see Storyville yesterday)

My choice for a rabbi would be George Steiner.

INTERVIEWER

You once referred to the “patience of apprehension” and “open-endedness of asking” which fiction can enact, and yet you have described your fictions as “allegories of argument, stagings of ideas.” Do you still consider them to be “stagings of ideas”?

GEORGE STEINER

Very much so. My writing of fiction comes under a very general heading of those teachers, critics, scholars who like to try their own hand once or twice in their lives.

The Paris Review, Winter 1995

For one such staging, see today’s earlier posts Chess and Frame Tale.

Frame Tale (continued)

Filed under: General — Tags: — m759 @ 10:30 am

See The X-Men Tree,  another tree,  and Trinity MOG.

Chess

Filed under: General — Tags: — m759 @ 9:29 am

Norwegian, 22, Takes World Chess Title

Quoted here on Thursday, the date of Kavli‘s death:

Herbert Mitgang’s New York Times 
obituary of Cleanth Brooks

“The New Critics advocated close reading of literary texts
and detailed analysis, concentrating on semantics, meter,
imagery, metaphor and symbol as well as references to
history, biography and cultural background.”

See also Steiner, Chess, and Death.

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