Last modified by iholopai@helsinki_fi on 2024/03/27 10:50
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1 {{panel}}
2 **Teacher:** [[Ilkka Holopainen>>doc:mathstatHenkilokunta.Holopainen, Ilkka]]
3
4 **Scope:** 10 cr
5
6 **Type:** Advanced studies
7
8 **Teaching: **Weeks 36-42 and 44-50, Tuesday 12-14 and Thursday 10-12 in room C124.** Note: No lectures (nor home work classes) during weeks 43 (25-26.10) and 44 (1-3.11).**
9
10 **Between 9.11 - 14.12 we will have extra classes/home work sessions on Wednesdays 12-14 in room B322.
11 **
12
13 **Topics:
14 **From the book //Geometric measure theory: A beginner's guide //by F. Morgan:
15
16 "//Geometric measure theory// could be described as differential geometry, generalized through measure theory to deal with maps and surfaces that are not necessary smooth, and applied to the calculus of variations."
17
18 The quotation above describes very well the goal of the course. This course gives an introduction to the theory of varifolds and currents that are kind of generalized surfaces. They have been used in many geometric variational problems, in particular, in connections with higher dimensional minimal surfaces.
19
20 Some topics that might be discussed:
21
22 //Review of measure theory//
23
24 (% style="list-style-type: square;" %)
25 * //measures and outer measures//
26 * //metric outer measures//
27 * //regularity of measures, Radon measures//
28 * //Hausdorff measure and dimension//
29 * //Riesz representation theorem
30 //
31
32 //Lipschitz mappings and rectifiable sets//
33
34 * //extension of Lipschitz functions
35 //
36 * //Rademacher's theorem
37 //
38 * //area and co-area formulae
39 //
40 * //rectifiable sets
41 //
42 * //approximate tangent space//
43 * //densities//
44 * //the structure theorem//
45
46 //Varifolds//
47
48 * //general varifolds//
49 * //rectifiable k-varifolds
50 //
51 * //first variation formula
52 //
53 * //monotonicity formula and its consequences
54 //
55 * //regularity theorem
56 \\//
57
58 //Currents//
59
60 * //k-vectors and k-covectors
61 //
62 * //differential forms
63 //
64 * //currents (definition and basic notions)
65 //
66 * //spaces of currents
67 //
68 * //slicing//
69 * //deformation theorem//
70 * //isoperimetric inequality//
71 * //rectifiability and compactness theorems//
72
73 //Mass (area) minimizing currents//
74
75 * //existence
76 //
77 * //properties of mass minimizing currents
78 //
79 * //regularity of mass minimizing currents
80 //
81
82 **Prerequisites:**
83
84 Good knowledge on measure and integration theory (courses like// Measure and integral //and //Real analysis I//). The course// Real analysis II //would be helpful but is not necessary.**
85 **
86 {{/panel}}
87
88 === {{toc maxLevel="4" minLevel="2" indent="20px"/}} ===
89
90 == News ==
91
92 * No lectures (nor home work classes) during weeks 43 (25-26.10) and 44 (1-3.11).
93
94 == Teaching schedule ==
95
96 Weeks 36-42 and 44-50, Tuesday 12-14 and Thursday 10-12 in room C124. **Note: **No lectures (nor home work classes) during weeks 43 (25-26.10) and 44 (1-3.11).
97
98 **Between 9.11 - 14.12 we will have extra classes/home work sessions on Wednesdays 12-14 in room B322.**
99
100 == Exams ==
101
102 The course can be passed by an (% class="external-link" %)[[exam>>url:https://wiki.helsinki.fi/display/mathstatOpiskelu/Yleistentit+2016-2017||shape="rect"]](%%) or preferably by giving presentations in the home work classes and writing essays.
103
104 How to pass the course by giving presentations? The plan is that a student gives two presentations (1-2 hours each) on a chosen topic and writes an essay on the topic (and, of course, follows the presentations of others).
105
106 == Course material ==
107
108 **Books**
109
110 L. Evans and R. Gariepy: (% style="color: rgb(0,128,0);" %) (%%)//Measure theory ans fine properties of functions, //CRC Press, 1992.
111
112 H. Federer: //Geometric Measure Theory//, Springer, 1969.
113
114 F. Lin and X. Yang: //Geometric Measure Theory: An Introduction//, International Press, 2002.
115
116 P. Mattila: //Geometry of sets and measures in Euclidean spaces//, Cambridge University Press,1995.
117
118 F. Morgan: //Geometric Measure Theory, A Beginner's Guide//, Academic Press, 1987. An[[ e-book>>url:https://www.terkko.helsinki.fi/booknavigator/geometric-measure-theory-a-beginners-guide||shape="rect"]] available for students of UH.
119
120 L. Simon: (% class="confluence-link" %)// //(%%)[[(% class="confluence-link" %)//Lectures on Geometric Measure Theory//>>url:https://maths-proceedings.anu.edu.au/CMAProcVol3/CMAProcVol3-Complete.pdf||shape="rect"]](%%), Australian National University, 1983. (% class="confluence-link" %)[[An updated version>>url:http://web.stanford.edu/class/math285/ts-gmt.pdf||shape="rect"]].
121
122
123
124 **Lecture notes**
125
126 P. Mattila: Currents and varifolds. Hand-written lecture notes, fall 2011.
127
128 I. Holopainen: [[Geometric measure theory>>attach:GMT.pdf]] (first 88 pages + appendix). Proof of the Deformation theorem, discussions on regularity results (stationary varifolds, mass minimizing currents) will be added later.
129
130 == [[Registration>>url:https://oodi-www.it.helsinki.fi/hy/opintjakstied.jsp?html=1&Tunniste=57266||shape="rect"]] ==
131
132
133 (% style="color: rgb(96,96,96);" %)Did you forget to register? (%%)[[What to do?>>url:https://wiki.helsinki.fi/display/mathstatOpiskelu/Kysymys4||style="text-decoration: underline;" shape="rect"]]
134
135 == Exercises ==
136
137 * [[Exercises 1>>attach:gmt16h1.pdf]] [[Solutions 1>>attach:gmt16r1.pdf]]
138 * [[Exercises 2>>attach:gmt16h2.pdf]] [[ Solutions 2>>attach:gmt16r2.pdf]]
139 * 28.9.2016, Presentation on the Riesz representaton theorem by Ville Marttila (see Appendix in the lecture notes)
140 * 5.10.2016, Presentation on weak convergence of measures and on the compactness of Radon measures by Janne Siipola (see Appendix in the lecture notes)
141 * 12.10.2016, [[Presentation on Rademacher's theorem>>attach:rademacher.pdf]] by Akseli Haarala
142 * 19.10.2016, [[Presentation on the area formula>>attach:area_formula.pdf]] by Krishnan Narayanan
143 * 8.11.2016 (Tuesday), [[Presentation on the co-area formula>>attach:co-area_formula.pdf]] by Otte Heinävaara
144 * 9.11.2016, Presentation on the Whitney extension by Valter Lillberg
145 * 9.11.2016, (% class="confluence-link" %) [[(% class="confluence-link" %)Presentation on submanifolds and mean curvature>>attach:Mean_curvatur.pdf]](%%) by Ville Karlsson
146 * 16.11.2016, Presentation on approximate tangent spaces and on rectifiable sets (part I) by Ville Marttila
147 * 16.11.2016, Presentation on approximate tangent spaces and on rectifiable sets (part II) by Janne Siipola
148 * 23.11.2016, Presentation on the [[monotonicity formula and isoperimetric inequality>>attach:Isoperimetric_inequality.pdf]] by Akseli Haarala
149 * 23.11.2016, Presentation on the rectifiability theorem and tangent cones (part I) by Krishan Narayanan
150 * 30.11.2016, Presentation on the rectifiability theorem and tangent cones (part II) by Otte Heinävaara
151 * 30.11.2016, Presentation on the regularity theory (part I) by Valter Lillberg
152 * 7.12.2016, Presentation on the regularity theory (part II) by Ville Karlsson
153
154 === Exercise classes ===
155
156 |=(((
157 Group
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159 Day
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161 Time
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163 Room
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165 Instructor
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168 1.
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170 Wednesday
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172 14-16
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174 C129
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176 Ilkka Holopainen
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178
179 == Course feedback ==
180
181 Course feedback can be given at any point during the course. Click [[here>>url:https://elomake.helsinki.fi/lomakkeet/11954/lomake.html||style="line-height: 1.4285;" shape="rect"]].