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Henry.L
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Henry.L
Henry.L
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It has been recently mentioned by a speaker (his talk is completely not relevant to random matrix theory/RMT though) that modern statistics, especially random matrices theory, will help solving some number theoretic problems.

I was quite intrigued and ask for more explanation but the speaker himself said he did not understand the philosophy well enough but point me to a few keywords to search for, one among which is "Katz-Sarnak philosophy".

After a few searches, I figured out that all sources seem to point to [KS]. The major results in [KS] seems like saying that the class of general linear (compact) groups have the same n-level correlations. While some researchers [Kowalski][Miller] do mention that they applied KS philosophy, but what they have done seems very different...So for number theorists and experts on elliptic curves:

(1) When you refer to "Katz-Sarnak philosophy", what kind of thinking/technique do you actually mean?

(2) Is there a formalism/explanation of this philosophy in language of RMT? I asked this because it might be helpful to understand it in this perspective (at least to a probabilist).

Any inputs are highly appreciated.

Reference

[KS]Katz, Nicholas M., and Peter Sarnak, eds. Random matrices, Frobenius eigenvalues, and monodromy. Vol. 45. American Mathematical Soc., 1999.Google books

[Kowalski]http://blogs.ethz.ch/kowalski/2008/07/30/finding-life-beyond-the-central-limit-theorem/

[Miller]Miller, Steven J. "One-and two-level densities for rational families of elliptic curves: evidence for the underlying group symmetries." Compositio Mathematica 140.4 (2004): 952-992.

It has been recently mentioned by a speaker (his talk is completely not relevant to random matrix theory/RMT though) that modern statistics, especially random matrices theory, will help solving some number theoretic problems.

I was quite intrigued and ask for more explanation but the speaker himself said he did not understand the philosophy well enough but point me to a few keywords to search for, one among which is "Katz-Sarnak philosophy".

After a few searches, I figured out that all sources seem to point to [KS]. The major results in [KS] seems like saying that the class of general linear (compact) groups have the same n-level correlations. While some researchers [Kowalski][Miller] do mention that they applied KS philosophy, but what they have done seems very different...So for number theorists and experts on elliptic curves:

(1) When you refer to "Katz-Sarnak philosophy", what kind of thinking/technique do you actually mean?

(2) Is there a formalism/explanation of this philosophy in language of RMT? I asked this because it might be helpful to understand it in this perspective (at least to a probabilist).

Reference

[KS]Katz, Nicholas M., and Peter Sarnak, eds. Random matrices, Frobenius eigenvalues, and monodromy. Vol. 45. American Mathematical Soc., 1999.Google books

[Kowalski]http://blogs.ethz.ch/kowalski/2008/07/30/finding-life-beyond-the-central-limit-theorem/

[Miller]Miller, Steven J. "One-and two-level densities for rational families of elliptic curves: evidence for the underlying group symmetries." Compositio Mathematica 140.4 (2004): 952-992.

It has been recently mentioned by a speaker (his talk is completely not relevant to random matrix theory/RMT though) that modern statistics, especially random matrices theory, will help solving some number theoretic problems.

I was quite intrigued and ask for more explanation but the speaker himself said he did not understand the philosophy well enough but point me to a few keywords to search for, one among which is "Katz-Sarnak philosophy".

After a few searches, I figured out that all sources seem to point to [KS]. The major results in [KS] seems like saying that the class of general linear (compact) groups have the same n-level correlations. While some researchers [Kowalski][Miller] do mention that they applied KS philosophy, but what they have done seems very different...So for number theorists and experts on elliptic curves:

(1) When you refer to "Katz-Sarnak philosophy", what kind of thinking/technique do you actually mean?

(2) Is there a formalism/explanation of this philosophy in language of RMT? I asked this because it might be helpful to understand it in this perspective (at least to a probabilist).

Any inputs are highly appreciated.

Reference

[KS]Katz, Nicholas M., and Peter Sarnak, eds. Random matrices, Frobenius eigenvalues, and monodromy. Vol. 45. American Mathematical Soc., 1999.Google books

[Kowalski]http://blogs.ethz.ch/kowalski/2008/07/30/finding-life-beyond-the-central-limit-theorem/

[Miller]Miller, Steven J. "One-and two-level densities for rational families of elliptic curves: evidence for the underlying group symmetries." Compositio Mathematica 140.4 (2004): 952-992.

Source Link
Henry.L
Henry.L
  • 8.1k
  • 8
  • 48
  • 74

What is the Katz-Sarnak philosophy?

It has been recently mentioned by a speaker (his talk is completely not relevant to random matrix theory/RMT though) that modern statistics, especially random matrices theory, will help solving some number theoretic problems.

I was quite intrigued and ask for more explanation but the speaker himself said he did not understand the philosophy well enough but point me to a few keywords to search for, one among which is "Katz-Sarnak philosophy".

After a few searches, I figured out that all sources seem to point to [KS]. The major results in [KS] seems like saying that the class of general linear (compact) groups have the same n-level correlations. While some researchers [Kowalski][Miller] do mention that they applied KS philosophy, but what they have done seems very different...So for number theorists and experts on elliptic curves:

(1) When you refer to "Katz-Sarnak philosophy", what kind of thinking/technique do you actually mean?

(2) Is there a formalism/explanation of this philosophy in language of RMT? I asked this because it might be helpful to understand it in this perspective (at least to a probabilist).

Reference

[KS]Katz, Nicholas M., and Peter Sarnak, eds. Random matrices, Frobenius eigenvalues, and monodromy. Vol. 45. American Mathematical Soc., 1999.Google books

[Kowalski]http://blogs.ethz.ch/kowalski/2008/07/30/finding-life-beyond-the-central-limit-theorem/

[Miller]Miller, Steven J. "One-and two-level densities for rational families of elliptic curves: evidence for the underlying group symmetries." Compositio Mathematica 140.4 (2004): 952-992.