Wiki source code of Funktioteoria III, syksy 2010

Last modified by saksman@helsinki_fi on 2024/03/27 10:09
Show last authors
1 = Function theory III, fall 2010 =
2
3 === What is going on? ===
4
5 Lectures have ended, thanks for participation !
6
7 Those who urgently need to get marks of the course during December, please come to meet me to discuss the essays you are writing.
8
9 === Exercises ===
10
11
12 \\
13
14 SUGGESTIONS FOR SOLUTIONS:
15
16 \\
17
18 === Lecture notes (scetches only) ===
19
20 \\
21
22 Rami Luisto has written an account of a self-contained proof (presented in the last lecture) of Jordan's curve theorem (with two proofs) and of the lemma of Janizewski, based only on results of FTII. The text is found below.
23 In turn, the text by Tomas Soto shows how the Caratheodory theorem of boundary extension for conformal maps (from the disc to Jordan domains), and as a simple consequence the Schoenfliess theorem, are proven by assuming Jordan theorem and Janitzewski's lemma.
24
25 [[Jordan's curve theorem >>attach:jordancurvethm_ft3.pdf]]
26 [[Schoenfliess theorem >>attach:ft3_schoenfliess(soto).pdf]]
27
28 === Content ===
29
30 We will cover several basic elements of function theory that have been not covered by Function Theory I-II.
31 Basic topic covered include modulus of curve families, boundary extensions of conformal maps, basic distortion estimates of univalent functions, hyperbolic and quasihyperbolic metrics with applications to conformal maps, modular groups and functions, the Picard theorems, harmonic measure, elliptic functions, and (if time allows) the uniformization theorem will be discussed.
32
33 === Prerequisities ===
34
35 Function theory II. Measure and integral. For the bits of topology, real or functional
36 analysis that will possibly be needed, careful references are given.
37
38 === Scope ===
39
40 10 op.
41
42 === Type ===
43
44 Advanced studies.
45
46 === Literature ===
47
48 Suitable sections from various books will be mentioned during the lectures.
49
50 === Lecturer ===
51
52 [[Eero Saksman>>doc:mathstatHenkilokunta.Saksman, Eero]]
53
54 === Lectures ===
55
56 Most of the weeks 36-42 and 44-50 Tu 10-12, We 9-12 C123 (during couple of weeks extra lectures on Fr 10-12 in C123).
57 Exercise classes are held only every second week, the answers should be returned (% style="text-decoration: underline;" %)in writing(%%) before the exercise class takes place.
58 If there are participants who do not know Finnish, the lectures will be held in English.
59
60 === Passing the course ===
61
62 The course can be passed by returning written exercises to the instructor before the instruction classes take place.
63 Also, most probably, one will be asked to write a small essay on a topic that will be chosen jointly with the lecturer.
64 It is also possible to pass the course by an oral examination (this sounds bad, I know, but the questions will be
65 easy and the examiner will help to answer them).
66
67 === [[Register>>url:https://oodi-www.it.helsinki.fi/hy/opintjakstied.jsp?html=1&Tunniste=57072||shape="rect"]] ===
68
69 Did you forget to register? [[What to do>>doc:mathstatOpiskelu.Kysymys4]].
70
71 === Exercise groups ===
72
73 Every second week, exact timing will be announced later on.
74
75 (% class="wrapped" %)
76 |=(((
77 Group
78 )))|=(((
79 Day
80 )))|=(((
81 Time
82 )))|=(((
83 Location
84 )))|=(((
85 Instructor
86 )))
87 |(((
88 1.
89 )))|(((
90 Fri
91 )))|(((
92 10-12
93 )))|(((
94 C123
95 )))|(((
96 Istvan Prause
97 )))
98
99 WELLCOME!
100
101 === Logbook ===
102
103 8.9 Definition of modulus of curve families, modulus of simplest configurations (rectangle, annulus), basic properties
104
105 10.9 Properties of modulus (continued). Continuous extension to the boundary (beginning)
106
107 14.9 Finite or local connectivity along the boundary. Continuous extension of conformal maps onto the boundary.
108
109 15.9 Caratheodory-Osgood theorem (homeomorphic extensions). Properties of Jordan domains and conformal maps between them.
110 Prime ends via modulus (beginning)
111
112 21.9 Prime ends via modulus (completion)
113
114 22.9 Area theorem of conformal maps. Koebe mapping. Koebe 1/4-theorem, Koebe estimate. Automorphisms of the unit disc.
115 Conformal invariance of the hyperbolic metric. Shortest hyperbolic distance, Non-Euclidean geometry.
116
117 28.9 Hyperbolic metric for simply connected domains, analytic functions as contraction of the hyperboli metric, kvasihyperbolic metric,
118 Whitney tilings, rigidity of conformla maps between simply connected domains.
119
120 29.9 Growth estimates for univalent functions on the unit disc. Convergence properties of conformal maps:
121 the Caratheodory kernel theorem on convergence (beginning).
122
123 1.10 Caratheodory kernel theorem on convergence (completion).
124
125 5.10 The modular group. Construction of the fundamental domain for the (mod 2) subgroup.
126
127 6.10 Fundamental domains continued. Construction of the modular function.
128
129 12.10 Construction and properties of the modular function (continued). Lifts of functions through the covering space.
130
131 13.10 Picard's little and great theorems. Schottky theorem. Fundamental group of the full modular group. Functions with one period.
132
133 19.10 Discrete period modules. Jacobi's theorem. Period parallegrams. Basic properties of elliptic functions: order and equidistribution.
134
135 20.10 Relation between sums of poles and zeros. Construction of Weierstrass p-function, and Weierstrass zeta- and eta-functions.
136 Representation of elliptic functions in terms of p and p'. Characterization via zeros and poles.
137
138 8.11 The differential equation of p-function. Every elliptic function satisfies an algebraic differential equation.
139 Weierstrass characterization of meromorphic functions admitting an algebraic addition theorem (beginning).
140
141 9.11 Weierstrass characterization of meromorphic functions admitting an algebraic addition theorem (completion). Mapping properties of elliptic integrals. Connection to modular functions. Construction of elliptic functions in terms of the Jacobi theta-functions.
142
143 16.11. Harmonic measure for intervals on the unit circle. Lindelöf's generalized maximum principle. Lindelöf's theorem on boundary limits of analytic functions.
144
145 17.11. Harmonic measure for Borel subsets of the unit circle. Caharcterization of harmonic functions that are Poisson extensions of boundary Borel measures. Non-tangential convergence. Domination of the non-tangential maximal function by the Hardy-Littlewood maximal function on the boundary.
146 Fatou's theorem for boundary limits of Poisson integrals.
147
148 23.11 Harmonic h^p-spaces, representation in terms of the radian limit function, norm equivalence.
149
150 24.11 Analytic Hardy spaces H^p, completeness, characterization in terms of the Fourier coefficients, Blaschke products, zeroes of Hardy functions satisfy the Blaschke condition.
151
152 1.12 Factroization theorem of hardy functions. Uniqueness via boundary values. Boundedness of th radial maximal function on H^p for p>0. Brothers Riesz theorem
153 on absolute continuity. Conformal maps onto Jordan domains with rectifiable boundary: (another) Brothers Riesz theorem.
154
155 7.12 Harmonic measure on Jordan domains with rectifiable boundary. Discussion of further topics in function theory.
156
157 8.12 (last lecture) the Jordan curve theorem (proven by Rami Luisto). Discussion of the Schoenfliess theorem.