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

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With abstracts
 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.

Course Notes
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

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

   Documents by other mathematicians.

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Personal stuff: cv, photo, etc.
Tips for authors.
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I had been to school most all the time ... and could say the multiplication table up to six times seven is thirty-five, and I don't reckon I could ever get any further than that if I was to live forever. I don't take no stock in mathematics, anyway.
        Huckleberry Finn.

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Near the end of the lecture, the speaker said that he would conclude the proof with some hand-waving. Cartan obviously did not approve. He turned to me and said: "Now I understand why Indian Gods have so many hands; they want to give proofs in n-dimensions."
        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.

My excuse for not having written the book is that there have been a great many other people, better equipped in many ways than I, who have not written it either.
         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.

Arithmetic has a well-known liberal bias. [or worse; perhaps much worse]
         Paul Krugman, blog April 8 2011.

Now, people ask me all the time how we got four surplus budgets in a row. What new ideas did we bring to Washington? I always give a one-word answer: Arithmetic. (Sustained cheers, applause.)
         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.

The danger to society is not merely that it should believe wrong things, though that is great enough; but that it should become credulous, and lose the habit of testing things and inquiring into them; for then it must sink back into savagery.
        William Kingdon Clifford (of Clifford algebras).


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