Last modified by pankka@helsinki_fi on 2024/03/27 10:05
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1 = Introduction to differential forms, spring 2011 =
2
3 === Lecturer ===
4
5 [[Pekka Pankka>>doc:mathstatHenkilokunta.Pankka, Pekka]]
6
7 === Scope ===
8
9 10 cu.
10
11 === Type ===
12
13 Advanced studies
14
15 === Description ===
16
17 Differential forms give a coordinate free formalism for multivariable calculus and a point of view to modern analysis. This formalism can also be used
18 to understand basic concepts in algebraic topology as homology and cohomology. In this course we discuss how familiar methods from calculus give rise
19 to more abstract concepts in topology.
20
21 Background in algebraic topology or in differential geometry is not necessary.
22
23 === Prerequisites ===
24
25 Real analysis and Topology II (or equivalent).
26
27 === Lectures ===
28
29 Weeks 3-9 and 11-18, Tuesday 16-18 and Thursday 10-12 in room B321. ((% style="color: rgb(255,0,0);" %)Note:(%%) (% style="color: rgb(0,0,0);" %)Change in Tuesdays lecture time.)
30
31 Easter holiday 21.-27.4.
32
33 === Methods of passing the course ===
34
35 The course can be passed by returning written solutions to weekly problem sets. Solutions are graded with scale 0-6. Problems discussed in the exercise sessions give extra credit. Course can be also passed by an exam. Contact the lecturer for details.
36
37 === Topics (preliminary list) ===
38
39 * Multilinear algebra
40 * Vector fields and differential forms
41 * Simplicial complexes
42 * Integration
43 * Homology and de Rham cohomology
44 * Methods of calculation in cohomology
45 * Degree theory
46 * Classical applications of cohomology
47
48 === Bibliography ===
49
50 * Tu: An Introduction to Manifolds
51 * Madsen - Tornehave: From Calculus to Cohomology
52 * Bott - Tu: Differential Forms in Algebraic Topology
53
54 === [[Registration>>url:https://oodi-www.it.helsinki.fi/hy/opintjakstied.jsp?html=1&Tunniste=57072||shape="rect"]] ===
55
56 Did you forget to register? [[What to do>>doc:mathstatOpiskelu.Kysymys4]].
57
58 === Lecture notes ===
59
60 [[Introduction and 1-forms >>attach:IDF_notes_Intro.PDF]]
61 [[Multilinear algebra>>attach:IDF_alternating_algebra.PDF]]
62 [[From Orientation to Integration>>attach:IDF_notes_From_Orientation_to_Integration.PDF]]
63 [[Exterior derivative>>attach:IDF_notes_Exterior_derivative.PDF]]
64 [[Chains, Forms, and Integration>>attach:IDF_notes_Chains_and_Stokes.PDF]]
65 [[de Rham cohomology: Poincaré's lemma>>attach:IDF_notes_Poincare_lemma.PDF]]
66 [[de Rham cohomology: Exact sequences>>attach:IDF_notes_Exact_Sequences.PDF]]
67 [[de Rham cohomology: Homotopy methods>>attach:IDF_notes_ThmA.PDF]]
68 [[de Rham cohomology: Applications>>attach:IDF_notes_Applications.PDF]]
69 [[Manifolds I>>attach:IDF_notes_Manifolds.PDF]]
70 [[Integration on manifolds>>attach:IDF_notes_Integration_II.PDF]]
71
72 === Exercise sets ===
73
74 [[Problem set 1>>attach:IDFexer01.pdf]] [[Solutions to Problem set 1>>attach:problems1.pdf]]
75 [[Problem set 2>>attach:IDFexer02.pdf]] [[Solutions to Problem Set 2>>attach:problems2.pdf]]
76 [[Problem set 3>>attach:IDFexer03.pdf]] [[Solutions to Problem Set 3>>attach:problems3.pdf]]
77 [[Problem set 4>>attach:IDFexer04.pdf]] [[Solutions to Problem Set 4>>attach:DOC001.PDF]]
78 [[Problem set 5>>attach:IDFexer05.pdf]] [[Solutions to Problem Set 5>>attach:problems5.pdf]]
79 [[Problem set 6>>attach:IDFexer06.pdf]] [[Solutions to Problem Set 6>>attach:problems6.pdf]]
80 [[Problem set 7>>attach:IDFexer07.pdf]] [[Solutions to Problem Set 7>>attach:problems7.pdf]]
81 [[Problem set 8>>attach:IDFexer08.pdf]] [[Solutions to Problem Set 8>>attach:problems8.pdf]]
82 [[Problem set 9>>attach:IDFexer09.pdf]] [[Solutions to Problem Set 9>>attach:problems9.pdf]]
83 [[Problem set 10>>attach:IDFexer10.pdf]] [[Solutions to Problem Set 10>>attach:problems10.pdf]]
84 [[Problem set 11>>attach:IDFexer11.pdf]]
85 [[Problem set 12>>attach:IDFexer12.pdf]][[Solutions to Problem Set 12>>attach:IDF_Ex12_Solutions.PDF]]
86
87 === Exercise groups ===
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90 Group
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92 Day
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98 Instructor
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101 1.
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103 Wed
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105 14-16
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107 C124
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109 Jan Cristina
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