保正版！随机矩阵、Frobenius特征值和单值性9787040534948高等教育出版社(美)尼古拉斯·M.卡茨,(美)彼得·萨纳克

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作者: (美)尼古拉斯·M.卡茨,(美)彼得·萨纳克
出版社: 高等教育出版社
ISBN: 9787040534948

出版时间: 2020-04
装帧: 精装
开本: 16开

作者: (美)尼古拉斯·M.卡茨,(美)彼得·萨纳克
出版社: 高等教育出版社

ISBN: 9787040534948
出版时间: 2020-04

装帧: 精装
开本: 16开

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目录
Introduction

Chapter 1.Statements of the Main Results

1.0.Measures attached to spacings of eigenvalues

1.1.Expected values of spacing measures

1.2.Existence, universality and discrepancy theorems for limits of expected values of spacing measures: the three main theorems

1.3.Interlude: A functorial property of Haar measure on compact groups

1.4.Application: Slight economies in proving Theorems 1.2.3 and 1.2.6

1.5.Application: An extension of Theorem 1.2.6

1.6.Corollaries of Theorem 1.5.3

1.7.Another generalization of Theorem 1.2.6

1.8.Appendix: Continuity properties of "the ith eigenvalue" as a function on U(N)

Chapter 2.Reformulation of the Main Results

2.0."Naive" versions of the spacing measures

2.1.Existence, universality and discrepancy theorems for limits of expected values of naive spacing measures: the main theorems bis

2.2.Deduction of Theorems 1.2.1, 1.2.3 and 1.2.6 from their bis versions

2.3.The combinatorics of spacings of finitely many points on a line: first discussion

2.4.The combinatorics of spacings of finitely many points on a line: second discussion

2.5.The combinatorics of spacings of finitely many points on a line: third discussion: variations on Sep(a)and Clump(a)

2.6.The combinatorics of spacings of finitely many points of a line: fourth discussion: another variation on Clump(a)

2.7.Relation to naive spacing measures on G(N): Int, Cor and TCor

2.8.Expected value measures via INT and COR and TCOR

2.9.The axiomatics of proving Theorem 2.1.3

2.10.Large N COR limits and formulas for limit measures

2.11.Appendix: Direct image properties of the spacing measures

Chapter 3.Reduction Steps in Proving the Main Theorems

3.0.The axiomatics of proving Theorems 2.1.3 and 2.1.5

3.1.A mild generalization of Theorem 2.1.5: the φ-version

3.2.M-grid discrepancy, L cutoff and dependence on the choice of coordinates

3.3.A weak form of Theorem 3.1.6

3.4.Conclusion of the axiomatic proof of Theorem 3.1.6

3.5.Making explicit the constants

Chapter 4.Test Functions

4.0.The classes T(n) and To(n) of test functions

4.1.The random variable Z[n, F, G(N)]on G(N) attached to a function F in T(n)

4.2.Estimates for the expectation E(Z[n, F, G(N)]) and variance Var(Z[n, F, G(N)])of Z[n, F, G(N)]on G(N)

Chapter 5.Haar Measure

5.0.The Weyl integration formula for the various G(N)

5.1.The K(x, y)version of the Weyl integration formula

5.2.The L(r, y) rewriting of the Weyl integration formula

5.3.Estimates for Ly(x, y)

5.4.The Lv(x, y) determinants in terms of the sine ratios Sv(x)

5.5.Case by case summary of explicit Weyl measure formulas via Sv

5.6.Unified summary of explicit Weyl measure formulas via Sy

5.7.Formulas for the expectation B(Z[n, F, G(N)])

5.8.Upper bound for E(Z[n, F, G(N)])

5.9.Interlude: The sin(rx)/rx kernel and its approximations

5.10.Large N limit of E(ZIn, F, G(N)]) via the sin(rx)/rx kernel

5.11.Upper bound for the variance

Chapter 6.Tail Estimates

6.0.Review: Operators of inite rank and their (reversed)characteristic polynomials

6.1.Integral operators of finite rank: a basic compatibility between spectral and Fredholm determinants

6.2.An integration formula

6.3.Integrals of determinants over G(N)as Fredholm determinants

6.4.A new spe case: O_(2N +1)

6.5.Interlude: A determinant-trace inequality

6.6.First application of the determinant-trace inequality

6.7.Application: Estimates for the numbers eigen(n, s, G(N))

6.8.Some curious identities among various eigen(n, s, G(N))

6.9.Normalized "nth eigenvalue" measures attached to G(N)

6.10.Interlude: Sharper upper bounds for eigen(0, s, SO(2N), for eigen(0, s, O_(2N +1)), and for eigen(0, s, U(N))

6.11.A more symmetric construction of the "nth eigenvalue" measures v(n, U(N))

……

内容摘要
本书主要论述了zeta和L函数之零点间距与大型紧典型群之随机元特征值间距之间的深层关系。这种称为Montgomery-Odlyzko定律的关系，对有限域上的zeta和L函数之宽类都成立。本书借鉴并描述了诸多不同的数学领域，从代数几何、模空间、单值性、等分布和Weil猜想，到关于紧典型群在维数趋于无穷的极限情况下的概率论，以及来自正交多项式和Fredholm行列式的相关技术。本书可供对有限域和局部域上的簇、zeta函数、极限理论和族结构感兴趣的研究生和科研人员阅读参考。
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