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Thursday, December 5, 2013

Blackboard Jungle

Filed under: General,Geometry — Tags: , — m759 @ 11:07 AM

Continued from Field of Dreams, Jan. 20, 2013.

IMAGE- Richard Kiley in 'Blackboard Jungle,' with grids and broken records

That post mentioned the March 2011 AMS Notices ,
an issue on mathematics education.

In that issue was an interview with Abel Prize winner
John Tate done in Oslo on May 25, 2010, the day
he was awarded the prize. From the interview—

Research Contributions

Raussen and Skau: This brings us to the next
topic: Your Ph.D. thesis from 1950, when you were
twenty-five years old. It has been extensively cited
in the literature under the sobriquet “Tate’s thesis”.
Several mathematicians have described your thesis
as unsurpassable in conciseness and lucidity and as
representing a watershed in the study of number
fields. Could you tell us what was so novel and fruitful
in your thesis?

Tate: Well, first of all, it was not a new result, except
perhaps for some local aspects. The big global
theorem had been proved around 1920 by the
great German mathematician Erich Hecke, namely
‚Äčthe fact that all L -functions of number fields,
abelian -functions, generalizations of Dirichlet’s
L -functions, have an analytic continuation
throughout the plane with a functional equation
of the expected type. In the course of proving
it Hecke saw that his proof even applied to a new
kind of L -function, the so-called L -functions with
Grössencharacter. Artin suggested to me that one
might prove Hecke’s theorem using abstract
harmonic analysis on what is now called the adele
ring, treating all places of the field equally, instead
of using classical Fourier analysis at the archimedian 
places and finite Fourier analysis with congruences 
at the p -adic places as Hecke had done. I think I did
a good job —it might even have been lucid and
concise!—but in a way it was just a wonderful 
exercise to carry out this idea. And it was also in the
air. So often there is a time in mathematics for 
something to be done. My thesis is an example. 
Iwasawa would have done it had I not.

[For a different perspective on the highlighted areas of
mathematics, see recent remarks by Edward Frenkel.]

"So often there is a time in mathematics for something to be done."

— John Tate in Oslo on May 25, 2010.

See also this journal on May 25, 2010, as well as
Galois Groups and Harmonic Analysis on Nov. 24, 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 Langlands, no 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.

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