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If they are connected, can anyone give a link?

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- Thread starter pivoxa15
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If they are connected, can anyone give a link?

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mathman

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mathman

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Number theory was studied by the ancient Greeks (Euclid et al). Measure theory development is around 100 years old.

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There is also algebraic number theory, but algebra and number theory are related as soon as you start learning about them.

I assume you are referring to ring theory in algebra.

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quasar987

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http://en.wikipedia.org/wiki/Haar_measure

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No. Field theory in particular over finite fields and rational numbers. And Galois theory are used a lot in number theory. That is probably the largest area of algebra that is used in number theory. And p-adic analysis.I assume you are referring to ring theory in algebra.

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No. Field theory in particular over finite fields and rational numbers. And Galois theory are used a lot in number theory. That is probably the largest area of algebra that is used in number theory. And p-adic analysis.

All fields are rings, but not conversely.

You said, '...algebra and number theory are related as soon as you start learning about them...'

I assume one does not learn Galois theory as the first exposure to algebra. One does rings and group theory before moving on to Galois theory which uses a combination of them.

- #10

mathwonk

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read andre weils book, basic number theory, where you will see haar measure treated as one of the first topics (page 3).

(in fact this appears in the references for quasar's link)

or see fourier analysis on number fields, by ramakrishnan.

thus to say the two subjects have nothing to do with one another is to me incorrect.

John Tate

From Wikipedia, the free encyclopedia

• Ten things you may not know about Wikipedia •

For other people with similar names, see John Tate (disambiguation).

John Torrence Tate, born March 13, 1925 in Minneapolis, Minnesota, is an American mathematician, distinguished for many fundamental contributions in algebraic number theory and related areas in algebraic geometry. He wrote a Ph.D. at Princeton in 1950 as a student of Emil Artin, was at Harvard University 1954-1990, and is now at the University of Texas at Austin.

Tate's thesis, on the analytic properties of the class of L-functions introduced by Erich Hecke, is one of the relatively few such dissertations that have become a by-word. In it the methods, novel for that time, of Fourier analysis on groups of adeles, were worked out to recover Hecke's results.

Subsequently Tate worked with Emil Artin to give a treatment of class field theory based on cohomology of groups, explaining the content as the Galois cohomology of idele classes, and introduced Tate cohomology groups. In the following decades Tate extended the reach of Galois cohomology: Poitou-Tate duality, abelian varieties, the Tate-Shafarevich group, and relations with algebraic K-theory.

He made a number of individual and important contributions to p-adic theory: the Lubin-Tate local theory of complex multiplication of formal groups; rigid analytic spaces; the 'Tate curve' parametrisation for certain p-adic elliptic curves; p-divisible (Tate-Barsotti) groups. Many of his results were not immediately published and were written up by Jean-Pierre Serre. They collaborated on a major published paper on good reduction of abelian varieties.

The Tate conjectures are the equivalent for étale cohomology of the Hodge conjecture. They relate to the Galois action on the l-adic cohomology of an algebraic variety, identifying a space of 'Tate cycles' (the fixed cycles for a suitably Tate-twisted action) that conjecturally picks out the algebraic cycles. A special case of the conjectures, which are open in the general case, was involved in the proof of the Mordell conjecture by Gerd Faltings.

Tate has had a profound influence on the development of number theory through his role as a PhD advisor. His students include Joe Buhler, Benedict Gross, Robert Kottwitz, James Milne, Carl Pomerance, Ken Ribet and Joseph H. Silverman.

He was awarded a Wolf Prize in Mathematics in 2002/3.

[edit]See also

Sato-Tate conjecture

Sato-Tate measure

Tate module

Néron–Tate height

[edit]Selected publications

J. Tate, Fourier analysis in number fields and Hecke's zeta functions (Tate's 1950 thesis), reprinted in Algebraic Number Theory by J. W. S. Cassels, A. Frohlich ISBN 0-12-163251-2

(in fact this appears in the references for quasar's link)

or see fourier analysis on number fields, by ramakrishnan.

thus to say the two subjects have nothing to do with one another is to me incorrect.

John Tate

From Wikipedia, the free encyclopedia

• Ten things you may not know about Wikipedia •

For other people with similar names, see John Tate (disambiguation).

John Torrence Tate, born March 13, 1925 in Minneapolis, Minnesota, is an American mathematician, distinguished for many fundamental contributions in algebraic number theory and related areas in algebraic geometry. He wrote a Ph.D. at Princeton in 1950 as a student of Emil Artin, was at Harvard University 1954-1990, and is now at the University of Texas at Austin.

Tate's thesis, on the analytic properties of the class of L-functions introduced by Erich Hecke, is one of the relatively few such dissertations that have become a by-word. In it the methods, novel for that time, of Fourier analysis on groups of adeles, were worked out to recover Hecke's results.

Subsequently Tate worked with Emil Artin to give a treatment of class field theory based on cohomology of groups, explaining the content as the Galois cohomology of idele classes, and introduced Tate cohomology groups. In the following decades Tate extended the reach of Galois cohomology: Poitou-Tate duality, abelian varieties, the Tate-Shafarevich group, and relations with algebraic K-theory.

He made a number of individual and important contributions to p-adic theory: the Lubin-Tate local theory of complex multiplication of formal groups; rigid analytic spaces; the 'Tate curve' parametrisation for certain p-adic elliptic curves; p-divisible (Tate-Barsotti) groups. Many of his results were not immediately published and were written up by Jean-Pierre Serre. They collaborated on a major published paper on good reduction of abelian varieties.

The Tate conjectures are the equivalent for étale cohomology of the Hodge conjecture. They relate to the Galois action on the l-adic cohomology of an algebraic variety, identifying a space of 'Tate cycles' (the fixed cycles for a suitably Tate-twisted action) that conjecturally picks out the algebraic cycles. A special case of the conjectures, which are open in the general case, was involved in the proof of the Mordell conjecture by Gerd Faltings.

Tate has had a profound influence on the development of number theory through his role as a PhD advisor. His students include Joe Buhler, Benedict Gross, Robert Kottwitz, James Milne, Carl Pomerance, Ken Ribet and Joseph H. Silverman.

He was awarded a Wolf Prize in Mathematics in 2002/3.

[edit]See also

Sato-Tate conjecture

Sato-Tate measure

Tate module

Néron–Tate height

[edit]Selected publications

J. Tate, Fourier analysis in number fields and Hecke's zeta functions (Tate's 1950 thesis), reprinted in Algebraic Number Theory by J. W. S. Cassels, A. Frohlich ISBN 0-12-163251-2

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- #11

Office_Shredder

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All fields are rings, but not conversely.

You said, '...algebra and number theory are related as soon as you start learning about them...'

I assume one does not learn Galois theory as the first exposure to algebra. One does rings and group theory before moving on to Galois theory which uses a combination of them.

You can do number theory using rings and groups. For example, the group of integers 1,...p-1 with multiplication mod p can be used to show for all m, m^p-1=1 mod p, a trivial result once you talk about the order of a group. I'm sure there are other things I can't remember off the top of my head

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mathwonk

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here is talk title from an international conference on probability nd number theory:

"Asymptotic probability measures of Dedekind zeta-functions of non-Galois fields."

see:

http://www.math.tohoku.ac.jp/~hattori/pnt5.htm [Broken]

"Asymptotic probability measures of Dedekind zeta-functions of non-Galois fields."

see:

http://www.math.tohoku.ac.jp/~hattori/pnt5.htm [Broken]

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- #13

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If they are connected, can anyone give a link?

These two fields are actually quite intimately connected. Statements like those of mathman at the beginning of the thread are completely against the spirit of mathematics. I suggest that if you had this impression about the various fields of mathematics, you read Terence Tao's essay http://arxiv.org/pdf/math/0702396"

Now, on to measure theory and number theory! So, the first thing to note is that people often associate measure theory with real analysis because the first thing anyone did with it was generalize the definition of integration, and most of the early and famous applications of measure theory were of purely analytic, or geometric-analytic, interest. But measure theory is simply concerned with

So, let me briefly mention analytic number theory. Basically, measure theory is essential to the foundation of modern analysis. Have you heard of the Fourier transform? L^2 functions? These concepts are central to analytic number theory, and they are founded on measure-theoretic principles. But I don't think this really answers pivoxa's question. This isn't really an overlap between the fields. This is more like measure theory providing some rigor and clarity to things that were pretty much already there.

Therefore, let's consider a more interesting situation. http://en.wikipedia.org/wiki/Ergodic_theory" [Broken] is a sub-field of measure theory; it's essentially the study of measure-preserving transformations on probability spaces. This seems like analysis, probability theory, whatever. Right? So you might be a bit surprised to hear that some of the best known results about additive structure of the integers can so far be proven

In 1975 (?), Hillel Furstenberg opened the door to these developments with his ergodic-theoretic proof of Szemeredi's theorem, which states that any subset of the integers with positive density contains arithmetic progressions of arbitrary length. More recently (and more sensationally), Ben Green and Terence Tao combined the concepts from the four independent proofs of Szemeredi's theorem (Szemeredi's combinatorial proof, Furstenberg's ergodic proof, Gowers' Fourier-analytic proof, and Gowers' hypergraph proof) and were able to show that the primes contain arithmetic progressions of arbitrary length. This monumental achievement, which was a marvelous unification of four seemingly different fields of mathematics (graph theory, Fourier analysis, additive number theory, ergodic theory) was part of the reason for Terry Tao's Fields Medal.

You asked for links.

http://projecteuclid.org/DPubS/Repo...=body&id=pdf_1&handle=euclid.bams/1183549768"

http://terrytao.wordpress.com/2007/07/10/norm-convergence-of-multiple-ergodic-averages-for-commuting-transformations/" [Broken]

http://www.math.ucla.edu/~tao/254a.1.03w/" [Broken]

If you want other examples of the relationship between measure theory and number theory, let me know and I'll post more...

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- #14

Chris Hillman

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Welcome to PF, Xevarion!

And hurrah, it's a a joy to see such a hip newbie I just [post=1416125]mentioned[/post] the Szemeredi theorem a few months ago in another thread. And of course we want to see more examples!

BTW, for those who want to run to their local math library, good for you; some relevant undergraduate textbooks are:

G. J. O. Jameson,

Mark Pollicott and Michiko Yuri,

Two superb graduate level textbooks which offer discussions of the Szemeredi theorem in the context of graph theory are

Bela Bolobas,

Reinhard Diestel,

Note that one could not find two graduate level textbooks with fewer mathematical prerequisites, so I encourage adventurous readers of all backgrounds to take a look if they have the opportunity.

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O.K. :)And of course we want to see more examples!

Another thing I learned about recently is one of Melvyn Nathanson's pet problems. I think it's very interesting. Here's the idea: if A is a set of integers, define

A + A = {a1 + a2 | a1, a2 in A} (the sum set)

A - A = {a1 - a2 | a1, a2 in A} (the difference set)

Since a1 - a2 is not the same as a2 - a1, but a1+a2 = a2+a1, one would expect that A-A is bigger than A+A. Perhaps surprisingly, one can actually find sets with a larger sum set than difference set. For example, A = {0, 2, 3, 4, 7, 11, 12, 14} has sum set 0-28 missing just 1, 20, and 27; but A-A is -14 to 14 missing +-6, +-13.

So, now that we know these weird sets exist, we might at least hope that there are not many of them. This is where measure theory comes in. How do we define "not many"? We'd like a statement of the following type:

"The measure of the collection of sets A with more sums than differences and with A subset of [1, N] goes to 0 and N goes to infinity."

What measure do we use? Apparently, somebody proved that if you use the uniform measure (which gives the same measure to any subset of [1, N]), actually some positive fraction of sets have more sums than differences! But you can put other measures on those collections of sets (possibly more natural ones, although less simple). And in some of those other measures, there are actually "few" such "bad" sets, as we'd hope.

----

Another thing that's worth pointing out is that Furstenberg proved Szemeredi's theorem by a kind of transference principle. He found a purely measure-theoretic statement which implied the number theory result. The amazing thing is that the two theorems are actually equivalent. That is, the number theoretic result also implies the seemingly more general ergodic theorem!

----

If I think of anything else worth mentioning, I'll post it here too. Unfortunately I've been spending a lot of time on arithmetic combinatorics lately so that's all that comes to mind right now!

- #16

Chris Hillman

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[EDIT: rambling class p rapids "stream of conciousness" largely removed]

Many ergodic theorists have been interested in Penrose tilings and the theory of aperiodic tiling spaces generally. One place where this theory intersects at least two of the topics you mentioned (Fourier analysis and difference sets) is in the application of Meyer sets; see R. V. Moody, "Meyer Sets and Their Duals",

I am thinking of the example you gave as a "patch" from a "tiling" with prototiles of length 1,2,4, namely

oo*o****oooo*o

In the rambling first version of this post, I seized upon translation invariants aspects of your comments and ignored everything else. This prompts me to inquire whether one can arbitrage sum sets versus difference sets?

I've always been intrigued by Moody's paper but AFAIK this point of view has not been followed up, and it occurs to me that AFAIK Szemeredi's theorem has not been applied directly to tiling theory.

I think that I'm trying to suggest that it might be suggestive to try to interpret additive phenomena in the integers using some of the language of tiling theory, which might suggest some interesting problems. In addition, while IMO the general theory of tiling has not yet appeared, I expect it should provide a scheme for founding mathematics upon tilings rather than upon sets. If so from this POV it would not be surprising that "additive phenomena" can describe seemingly unrelated phenomenon.

The discovery of Penrose tilings gave rise to a great deal of interest in how unexpectedly rigid long range order can result from simple local rules. One could say that Szemeredi theorem concerns unexpectedly unavoidable order of a kind. So there does seem to be at least a vague spiritual connection.

Incidently, Bombieri, who worked on analytic topics related to some of the topics you mentioned, was one of those who became intrigued by Penrose tilings!

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- #17

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Unfortunately I don't actually know that much about tiling, so I am not sure exactly what kind of application you want to make here. I can at least tell you a few relevant things:

1) There is a generalization of Szemeredi's theorem to Z^n. However as far as I know the only proof of this right now is ergodic in nature, and therefore there are no bounds.

2) People also do try to prove results of this nature about thin subsets of the reals. So if you're actually interested in difference sets of some collection of vectors in R^n, there should be some theory too. But I don't think it is as rich or well-studied as in the cases of Z, Z/nZ, or finite fields. In particular, I don't know much about what's known in R^n for n>1.

3) Somehow from a measure-theoretic perspective, R^n (n > 1) suddenly gets complicated. They contain too much stuff. You can have integer-dimensional sets that are nice, but don't look that much like copies of R^k. Possibly if you had some set and you wanted to impose some additive structure on it, that would force it to be contained in some proper (affine) subspace... I dunno :-o

edit: by the way, you might be interested in reading about some of the recent results of Bourgain, Gamburd, and Sarnak about lattices and orbits. E.g. http://www.sciencedirect.com/scienc...1082852&md5=3881f0053692a3bbc99e47aec129fc07" (sorry I don't have a better link)

1) There is a generalization of Szemeredi's theorem to Z^n. However as far as I know the only proof of this right now is ergodic in nature, and therefore there are no bounds.

2) People also do try to prove results of this nature about thin subsets of the reals. So if you're actually interested in difference sets of some collection of vectors in R^n, there should be some theory too. But I don't think it is as rich or well-studied as in the cases of Z, Z/nZ, or finite fields. In particular, I don't know much about what's known in R^n for n>1.

3) Somehow from a measure-theoretic perspective, R^n (n > 1) suddenly gets complicated. They contain too much stuff. You can have integer-dimensional sets that are nice, but don't look that much like copies of R^k. Possibly if you had some set and you wanted to impose some additive structure on it, that would force it to be contained in some proper (affine) subspace... I dunno :-o

edit: by the way, you might be interested in reading about some of the recent results of Bourgain, Gamburd, and Sarnak about lattices and orbits. E.g. http://www.sciencedirect.com/scienc...1082852&md5=3881f0053692a3bbc99e47aec129fc07" (sorry I don't have a better link)

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- #18

Chris Hillman

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I am not sure exactly what kind of application you want to make here.

Neither am I (that's why I wrote "mumble, mumble"; I was thinking aloud).

1) There is a generalization of Szemeredi's theorem to Z^n. However as far as I know the only proof of this right now is ergodic in nature, and therefore there are no bounds.

I think that is worth elaborating on (better you than me!).

by the way, you might be interested in reading about some of the recent results of Bourgain, Gamburd, and Sarnak about lattices and orbits. E.g. http://www.sciencedirect.com/scienc...1082852&md5=3881f0053692a3bbc99e47aec129fc07" (sorry I don't have a better link)

Regarding expanders, I have "read" Davidov, Sarnak, Valette,

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- #19

morphism

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Really? I find that hard to believe after reading his book on functional analysis.Bela Bolobas,Modern Graph Theory, GTM 184, Springer, 1998 (one of the best books ever published in any subject!)

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