Mathematics Site - J.S. Milne (since 1996)

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1960s 70s 80s 90s 00s 10s

Etale Cohomology. • Hodge Cycles, Motives, and Shimura Varieties • Arithmetic Duality Theorems • Automorphic Forms, Shimura Varieties, and L-functions • Elliptic curves • Algebraic groups.

Group Theory • Fields and Galois Theory • Algebraic Geometry • Algebraic Number Theory • Modular Functions and Modular Forms • Elliptic Curves • Abelian Varieties • Lectures on Etale Cohomology • Class Field Theory • Algebraic Groups, Lie Groups, and their Arithmetic Subgroups • Complex Multiplication

Commutative Algebra • Motives • Shimura Varieties • Tannakian Categories • Work of Tate

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William Kingdon Clifford (of Clifford algebras).

- Nov 24,2013. A proof of the Barsotti-Chevalley theorem on algebraic groups here
- Nov 13,2013. Motivic complexes and special values of zeta functions (with NR) here
- Oct 16,2013. The
*p*-cohomology of algebraic varieties and special values of zeta functions (with NR) here

- September, 2017.
*Algebraic groups: The theory of group schemes of finite type over a field.*My page; CUP page - April, 2017.
*Etale Cohomology*is available in paperback PUP

- December 20, 2015. New version of
*Algebraic Groups*iAG - August 24, 2014. New version of
*Algebraic Geometry*AG - May 5, 2013. New version of
*Lie Algebras, Algebraic Groups, ...*LAG

- September 5, 2016. Review of the Collected Works of John Tate
- July 2, 2015. The Riemann hypothesis over finite fields: from Weil to the present day.
- August 1, 2014. New version of
*Commutative Algebra*. CA - March 18, 2012. The Work of John Tate For Abel prize volume.

- October 2016. More notes of courses by Tate from the 1960s.
- July 2016. Notes from Séminaire Heidelberg-Strasbourg 1965-66 (Groupes Algébriques)
- Jan. 2016. Letter of Deligne on Tannakian categories.
- Sept. 2013. Proceedings of the 1955 Tokyo-Nikko Conference on Algebraic Number Theory
- Jan. 2010.
*Complete*notes from the famous 1964 Woods Hole conference.

Narasimhan (NAMS 2010, Sept, p.955).

*A mathematician is so rare an animal
that he deserves to be preserved,
be it only on the score of curiosity.*

Simone Weil (as quoted on p.396 of the Pétrement biography).

*Mathematical laws may be discovered by man, but they are not created by man, nor even by God. That two
plus two equals four is not a decree of God that He is free to change into two plus two equals three, or five. I perceive the
mathematical laws as being part of the very nature of God--a tiny part, certainly, the most superficial in some
sense, and the only part accessible to reason alone.*

Grothendieck, La Clef des Songes (via Juan Antonio Navarro Gonzalez)

*But then I felt possessed by an aura of inspiration that allowed me
to improvise credible answers and miraculous lucky guesses. Except in mathematics, which not even God could make me understand.*

Gabriel García Márquez, Living to Tell the Tale, p204.

T.S. Eliot

*Physics is very interesting. There are many, many interesting theorems. Unfortunately,
there are no definitions.*

David Kazhdan.

*Early on I noticed that mathematicians live in a world inaccessible to common mortals
... They are a special breed possessed by an intense cerebral life;
simultaneously living on two distinct levels of consciousness,
they are at once present and able to carry on normally
and yet are immersed in the abstractions that form the core of their lives. *

Françoise Ulam (Stanislaw Ulam's wife).

I came into the room, which was half dark, and presently spotted Lord Kelvin in the
audience and realised that I was in for trouble at the last part of my speech dealing with
the age of the earth, where my views conflicted with his. To my relief, Kelvin fell fast
asleep, but as I came to the important point, I saw the old bird sit up, open an eye and
cock a baleful glance at me! Then a sudden inspiration came, and I said Lord Kelvin had
limited the age of the earth, *provided no new source (of energy) was discovered.* That
prophetic utterance refers to what we are now considering tonight, radium! Behold! the old
boy beamed upon me.

Ernest Rutherford

*Every mathematician worthy of the name has experienced, if only rarely,
the state of lucid exaltation in which one thought
succeeds another as if miraculously, and
in which the unconscious (however one interprets that word) seems
to play a role.*

André Weil

*... it is impossible to explain honestly the beauties of the laws of
nature in a way that people can feel, without their having some deep
understanding of mathematics. I am sorry, but this seems to be the
case.*

Richard Feynman
er...

*In the broad light of day mathematicians check their equations and
their proofs, leaving no stone unturned in their search for rigour.
But, at night, under the full moon, they dream, they float among the
stars and wonder at the miracle of the heavens. They are inspired.
Without dreams there is no art, no mathematics, no life.*

Atiyah (NAMS Jan 2010 p.8).

*I still say to myself when I am depressed, and find myself forced to
listen to pompous and tiresome people, "Well, I have done one thing
you could never have have done, and that is to have collaborated with
both Littlewood and Ramanujan on something like equal terms."*

Hardy, Apology, Sect. 29.

*The problem with the global village is all the
global-village idiots.*

Attributed to Sidney Coleman in arXiv:1108.2700.

Paul Krugman, blog April 8 2011.

Bill Clinton (Sept. 6, 2012).

*Do not work within two hours of a substantial meal; blood cannot be in
two places at once.*

J.E. Littlewood, in Littlewood's miscellany, p199.

*Certainly the best times were when I was alone with mathematics, free of ambition and pretense, and indifferent to the world.*

Langlands, in Mathematicians: An Outer View of the Inner World, p142.

*Mathematics has been for me, not only a profession, but also my
preferred hobby. ... Again and again I have been guided by a sense of
the architecture of this edifice, to which we continue to add new
wings and new floors while renovating the parts already constructed,
into feeling that certain problems had priority as opening new
perspectives or establishing a new foundation for future
constructions. This is the professional point of view, but happily
these problems were those that attracted me the most. In other
instances I was not guided by such motives, being attracted only by
curiosity, by the need to know the answer to an enigma, without
reference to its importance in a general context.* Borel, Œuvres
IV, p376.

*Even God wouldn't get a grant today because somebody on the committee
would say, oh those were very interesting experiments (creating the universe),
but they've never been repeated. And then someone else would say, yes and he
did it a long time ago; what's he done recently? And a third would say,
to top it all, he published it all in an un-refereed journal (The Bible).*

Sydney Brenner.

*It is while doing mathematical research that one truly comes to see the beauty of mathematics. It faces you in those moments when the underlying simplicity of a question appears and its meaningless complications can be forgotten. In those moments a piece of a colossal logical structure is illuminated, and some of the meaning hidden in the nature of things is finally revealed.
*

Ruelle, The Mathematician's Brain, p130.

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