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1 = Geometric measure theory and singular integrals, spring 2017 =
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6 **Teachers:** [[Henri Martikainen>>url:http://wiki.helsinki.fi/display/mathstatHenkilokunta/Henri+Martikainen||shape="rect"]] and [[Tuomas Orponen>>doc:mathstatHenkilokunta.Orponen, Tuomas]]
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8 **Scope:** 10 cr
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10 **Type:** Advanced studies
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12 **Teaching:**
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14 Weeks 3-9 and 11-18, Tuesday 14-16 in room C122 and Friday 10-12 in room C124. Two hours of exercise classes per week.
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16 Easter holiday 13.-19.4.
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18 **Topics: **The course investigates the connection between the geometry of planar sets and the boundedness of the Cauchy transform (a singular integral operator in the plane). Specific topics may include:
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21 * Basic concepts of the theory of singular integral operators; in particular the Cauchy transform.
22 * Curvature and rectifiability of sets and measures in the plane.
23 * Boundedness of the Cauchy transform on curves.
24 * Analytic capacity and rectifiability.
25 * Characterising rectifiability via P. Jones' beta-numbers; connection to curvature.
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27 **Prerequisites: **(% style="color: rgb(96,96,96);" %)Basic knowledge on measure theory, Lebesgue integration and //L^^p^^-//spaces as covered e.g. in the courses "Mitta ja integraali" and "Reaalianalyysi I".
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30 === {{toc maxLevel="4" minLevel="2" indent="20px"/}} ===
31
32 == News ==
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34 * The first exercise set is now available. These are for the exercise session on Wednesday, 1 February.
35 * The list of topics covered on the lectures of the first part of the course: [[LOG>>attach:GMTSIO_LOG.pdf]].
36 * The second set of exercises is now available. These are for the exercise session on Wednesday, 15 February.
37 * Draft lecture notes for the second half of the course are now available.
38 * The third set of exercises is now available. These are for the exercise session on Wednesday, 1 March.
39 * The first part of the course is over: **no lecture on Friday, 3 March.** The second part of the course by Tuomas starts on Tuesday, 14 March.
40 * The fourth set of exercises is now available; session on March 29.
41 * Ella's presentation on Tuesday, Mar 21. A normal lecture on Wednesday, Mar 22.
42 * The fifth set of exercises is now available; session on April 12.
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44 == Topics for presentations ==
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46 Below are a few suggestions for the presentations. If you're interested in one of them, contact either one of the lecturers (e.g. after a lecture), and we'll discuss the details and the schedule. The schedule is tentative.
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49 * Geometry of 1-rectifiable sets (reserved by Ville Marttila, 15.3.)
50 * Analytic capacity (as in the books of Tolsa and Mattila, Chapter 1 and Chapter 19) (Reserved by Janne Siipola, 26.4.)
51 * Tangent measures (as in the book of Mattila, Chapter 14) (Reserved by Hans Groeniger, 5.4.)
52 * The traveling salesman theorem (as in the book of Bishop-Peres, Chapter 8).  (Reserved by Ella Tamir, 22.3.).
53 * A short, complex-analytic proof of the L2-boundedness of the Cauchy transform on curves, as in a paper of Coifman-Jones-Semmes (Two elementary proofs of the L2 boundedness of the Cauchy transform on Lipschitz curves, J. AMS Vol. 2, No. 3 (1989), 553–564 (reserved by Juuso Nyyssönen, 3.5.)
54 * [[Lecture notes for the second half of the course>>attach:OrponenURLectureNotes.pdf]]
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56 * (% style="font-size: 20.0px;" %)Course material
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64 1. (((
65 X. Tolsa, Analytic capacity, the Cauchy transform, and non-homogeneous Calderón–Zygmund theory, Progress in Mathematics, Vol. 307, Birkhäuser Verlag, Basel, 2014.
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67 1. P. Mattila: Geometry of Sets and Measures in Euclidean Spaces: Fractals and rectifiability, Cambridge University Press (1995).
68 1. (% style="color: rgb(96,96,96);" %)C. J. Bishop and Y. Peres, Fractal Sets in Probability and Analysis, Cambridge University Press (2015).
69 1. [[Lecture notes for the second half of the course>>attach:OrponenURLectureNotes.pdf]]
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74 == [[Registration>>url:https://weboodi.helsinki.fi/hy/opintjakstied.jsp?html=1&Tunniste=57072||shape="rect"]] ==
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76 (% style="color: rgb(96,96,96);" %)No registration required; come to the first lecture.
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78 == Exercises ==
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80 You should complete at least 60% of the exercises.
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82 === Assignments ===
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84 * [[Set 1>>attach:GMTSIO_Exercises_1.pdf]] [[Solution to Exercise 5>>attach:GMTSIO, Ex. Set 1, solutions.pdf]]
85 * [[Set 2>>attach:GMTSIO_Exercises_2.pdf]]
86 * [[Set 3>>attach:GMTSIO_Exercises_3.pdf]] [[Hints for Set 3>>attach:GMTSIO_Exercises_3-HINTS.pdf]]
87 * [[Set 4>>attach:GMTSIOEx4.pdf]]
88 * [[Set 5>>attach:GMTSIOEx5.pdf]]
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90 === Exercise classes ===
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94 Group
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102 Instructor
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105 1.
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107 Wednesday
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109 10-12
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111 C122
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116 == Course feedback ==
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118 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"]].