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1 = Topics in additive combinatorics and incidence geometry, fall 2015 =
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6 **Teacher:** [[Tuomas Orponen>>doc:mathstatHenkilokunta.Orponen, Tuomas]]
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8 **Scope:** 5 cr
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10 **Type:** Advanced studies
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12 **Teaching: **(% style="line-height: 1.4285715;" %)Weeks 36-42, Monday and Friday 14-16 in room B321. First lecture on Friday Sept. 4th.
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14 * (% style="color: rgb(153, 204, 0); color: rgb(0, 0, 255)" %)No lectures on Monday the 21st of September and Friday the 25th of September.
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16 **Topics:**
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19 * Freiman's theorem,
20 * Plünnecke-Ruzsa inequalities and Ruzsa's triangle inequality
21 * Roth's theorem on 3-term arithmetic progressions,
22 * Sum-product theory, including Solymosi's "4/3" theorem on the real line, and the Bourgain-Katz-Tao theorem in prime fields,
23 * The polynomial method in incidence geometry
24 * The Szemerédi-Trotter incidence theorem in the plane + applications
25 * The Kakeya problem in finite fields, and the Joints problem
26 * The multilinear Kakeya problem
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28 **Prerequisites: **The tools needed will be introduced on the go. There will be bits and pieces from Fourier analysis and algebraic geometry used, but most arguments are quite combinatorial and elementary. You can check out the draft lecture notes below to assess yourself.
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31 === {{toc maxLevel="4" minLevel="2" indent="20px"/}} ===
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33 == (% style="line-height: 1.5;" %)Course material (%%) ==
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35 The following lecture notes will be updated, amended – and certainly corrected – several times during the course:
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37 [[attach:OrponenLectureNotes.pdf]] (Last update: 5/11/2015, added Laura's notes on multilinear Kakeya)
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39 For additional reading, the most comprehensive text is
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41 T. Tao and V. Vu: Additive Combinatorics, Cambridge University Press (2006)
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43 For something freely available, you could try Ruzsa's lecture notes "Sumsets and structure", which you can easily find by googling.
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45 == [[Registration>>url:https://oodi-www.it.helsinki.fi/hy/opintjakstied.jsp?html=1&Tunniste=57072||shape="rect"]] ==
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47 No registration needed, just come to the first lecture.
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49 (% style="font-size: 20.0px;line-height: 1.5;" %)Exercises
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51 There will most likely be no official exercises. Instead, the students will be given individual topics – perhaps an article, or a book section – which they should study, write notes about, and present during the "exercise sessions", whose timetable will be discussed on the first lecture. This presentation will also be the means to get credit for the course (so if you don't need credit, you won't be required to prepare a presentation).
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53 (% style="font-size: 16.0px;font-weight: bold;line-height: 1.5;" %)Exercise classes
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78 == Course feedback ==
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80 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"]].