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关于现代数学的前沿课题

(2019-08-29 08:19:26)

关于现代数学的前沿课题

   今年7月,国家4部委发出通知,要求全国高校及科研院所设立“基础数学中心”。    

   新学年即将开始。我们可以设想,“基础数学中心”不是空集合。他们干什呢?

   他们对于现代数学的前沿课题一定有兴趣,不愿意甘当数学守旧派。

   请见本文附件文章。

袁萌  陈启清  829

附件:

LECTURE NOTES ON NONSTANDARD ANALYSIS UCLA SUMMER SCHOOL IN LOGIC

ISAAC GOLDBRING

Contents

1. The hyperreals 3 1.1. Basic facts about the ordered real eld 3

1.2. The nonstandard extension 4

1.3. Arithmetic in the hyperreals 5

1.4. The structure of N 7

1.5. More practice with transfer 8

1.6. Problems 9

2. Logical formalisms for nonstandard extensions 10

2.1. Approach 1: The compactness theorem 11

2.2. Approach 2: The ultrapower construction 12

2.3. Problems 16

3. Sequences and series 17

3.1. First results about sequences 17

 

3.2. Cluster points 19

 

3.3. Series 21

3.4. Problems 22

4. Continuity 23

4.1. First results about continuity 23

4.2. Uniform continuity 25

4.3. Sequences of functions 27

4.4. Problems 30

5. Dierentiation 33

 

5.1. The derivative 33

5.2. Continuous dierentiability 35

 

5.3. Problems 36

6. Riemann Integration 38

6.1. Hypernite Riemann sums and integrability 38

6.2. The Peano Existence Theorem 41

6.3. Problems 43

7. Weekend Problem Set #1 44

8. Many-sorted and Higher-Type Structures 47

8.1. Many-sorted structures 47

Date: November 10, 2014.

1

2 ISAAC GOLDBRING

8.2. Higher-type sorts 48

8.3. Saturation 51

 

8.4. Useful nonstandard principles 53

8.5. Recap: the nonstandard setting 54

8.6. Problems 54

9. Metric Space Topology 55

9.1. Open and closed sets, compactness, completeness 55

9.2. More about continuity 63

9.3. Compact maps 64

9.4. Problems 65

10. Banach Spaces 67

10.1. Normed spaces 67

10.2. Bounded linear maps 68

10.3. Finite-dimensional spaces and compact linear maps 69

10.4. Problems 71

11. Hilbert Spaces 73 11.1. Inner product spaces 73

11.2. Orthonormal bases and `2 75

11.3. Orthogonal projections 79

11.4. Hypernite-dimensional subspaces 82

 

11.5. Problems 83

12. Weekend Problem Set #2 85

13. The Spectral Theorem for compact hermitian operators 88

13.1. Problems 93

14. The Bernstein-Robinson Theorem 94

15. Measure Theory 101

15.1. General measure theory 101

 

15.2. Loeb measure 102

15.3. Product measure 103

15.4. Integration 104 15.5. Conditional expectation 104

15.6. Problems 105 16. Szemer´edi Regularity Lemma 106

16.1. Problems 108 References 110

Nonstandard analysis was invented by Abraham Robinson in the 1960s as a way to rescue the na¨ve use of innitesimal and innite elements favored by mathematicians such as Leibniz and Euler before the advent of the rigorous methods introduced by Cauchy and Weierstrauss. Indeed, Robinson realized that the compactness theorem of rst-order logic could be used to provide elds that “logically behaved” like the ordered real eld while containing “ideal” elements such as innitesimal and innite elements.

LECTURE NOTES ON NONSTANDARD ANALYSIS 3

Since its origins, nonstandard analysis has become a powerful mathematical tool, not only for yielding easier denitions for standard concepts and providing slick proofs of well-known mathematical theorems, but for also providing mathematicians with amazing new tools to prove theorems, e.g. hypernite approximation. In addition, by providing useful mathematical heuristics a precise language to be discussed, many mathematical ideas have been elucidated greatly. In these notes, we try and cover a wide spectrum of applications of nonstandard methods. In the rst part of these notes, we explain what a nonstandard extension is and we use it to reprove some basic facts from calculus. We then broaden our nonstandard framework to handle more sophisticated mathematical situations and begin studying metric space topology. We then enter functional analysis by discussing Banach and Hilbert spaces. Here we prove our rst serious theorems: the Spectral Theorem for compact hermitian operators and the Bernstein-Robinson Theorem on invariant subspaces; this latter theorem was the rst major theorem whose rst proof was nonstandard. We then end by briey discussing Loeb measure and using it to give a slick proof of an important combinatorial result, the Szemer´edi Regularity Lemma. Due to time limitations, there are many beautiful subjects I had to skip. In particular, I had to omit the nonstandard hull construction (although this is briey introduced in the second weekend problem set) as well as applications of nonstandard analysis to Lie theory (e.g. Hilbert’s fth problem), geometric group theory (e.g. asymptotic cones), and commutative algebra (e.g. bounds in the theory of polynomial rings). Wehaveborrowedmuchofourpresentationfromtwomainsources: Goldblatt’s fantastic book [2] and Davis’ concise [1]. Occasionally, I have borrowed some ideas from Henson’s [3]. The material on Szemer´edi’s Regularity Lemma and the Furstenberg Correspondence come from Terence Tao’s blog. I would like to thank Bruno De Mendonca Braga and Jonathan Wolf for pointing out errors in an earlier version of these notes.

1. The hyperreals 1.1. Basic facts about the ordered real eld. The ordered eld of real numbers is the structure (R;+,·,0,1,<). We recall some basic properties:… …(全文很长,请见“无穷小微积分”网站,di点击“NSA-Notes”下载)


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