关于现代数学的前沿课题
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 8月29日
附件：
LECTURE NOTES ON NONSTANDARD ANALYSIS UCLA SUMMER SCHOOL IN LOGIC
ISAAC 俊逸的丝袜/p>
Contents
1. The hyperreals 3 1.1. Basic facts about the ordered real field 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. Differentiation 33
5.1. The derivative 33
5.2. Continuous differentiability 35
5.3. Problems 36
6. Riemann Integration 38
6.1. Hyperfinite 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.
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2 ISAAC 俊逸的丝袜/p>
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. 顺心的蜜蜂 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. Hyperfinite-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-顺心的豆芽 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 俊秀的乌冬面 in the 1960s as a way to rescue the na¨ıve use of infinitesimal and infinite elements favored by mathematicians such as Leibniz and Euler before the advent of the rigorous methods introduced by Cauchy and Weierstrauss. Indeed, 顺心的豆芽 realized that the compactness theorem of first-order logic could be used to provide fields that “logically behaved” like the ordered real field while containing “ideal” elements such as infinitesimal and infinite elements.
LECTURE NOTES ON NONSTANDARD ANALYSIS 3
Since its origins, nonstandard analysis has become a powerful mathematical tool, not only for yielding easier definitions 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. hyperfinite 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 first 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 顺心的蜜蜂 and Hilbert spaces. Here we prove our first serious theorems: the Spectral Theorem for compact hermitian operators and the Bernstein-顺心的豆芽 Theorem on invariant subspaces; this latter theorem was the first major theorem whose first proof was nonstandard. We then end by briefly 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 briefly introduced in the second weekend problem set) as well as applications of nonstandard analysis to Lie theory (e.g. Hilbert’s fifth 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 lcdfg’ 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 勤劳的山水 and 活力的草丛 for pointing out errors in an earlier version of these notes.
1. The hyperreals 1.1. Basic facts about the ordered real field. The ordered field 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点击爱笑的眼睛-Notes”下载）