ODTÜ-BİLKENT Algebraic Geometry Seminar
(See all past talks
ordered according to speaker and
date)
2014 Fall Talks
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We start with two talks on the recent developments on
"Lines on Surfaces." After that we run a learning seminar on Dessins
d'enfants. We will mostly follow the following book: |
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Abstract: This is a joint project with I. Itenberg and S. Sertöz. I will discuss the recent developments in our never ending saga on lines in nonsingular projective quartic surfaces. In 1943, B. Segre proved that such a surface cannot contain more than 64 lines. (The champion, so-called Schur's quartic, has been known since 1882.) Even though a gap was discovered in Segre's proof (Rams, Schütt), the claim is still correct; moreover, it holds over any field of characteristic other than 2 or 3. (In characteristic 3, the right bound seems to be 112.) At the same time, it was conjectured by some people that not any number between 0 and 64 can occur as the number of lines in a quartic. We tried to attack the problem using the theory of K3-surfaces and arithmetic of lattices. Alas, a relatively simple reduction has lead us to an extremely difficult arithmetical problem. Nevertheless, the approach turned out quite fruitful: for the moment, we can show that there are but three quartics with more than 56 lines, the number of lines being 64 (Schur's quartic) or 60 (two others). Furthermore, we can prove that a real quartic cannot contain more than 56 real lines, and we have an example realizing this bound. We can also construct quartics with any number of lines in {0; : : : ; 52; 54; 56; 60; 64}, thus leaving only two values open. Conjecturally, we have a list of all quartics with more than 48 lines. (The threshold 48 is important in view of another theorem by Segre, concerning planar sections.) There are about two dozens of species, all but one 1-parameter family being projectively rigid. |
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Abstract: This is the second part of the previous talk. See the above abstract. |
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Abstract: With this talk we start our series of talks on "Girondo and Gonzalez-Diez, Introduction to Compact Riemann Surfaces and Dessins d'Enfants, London Mathematical Society Student Texts 79, Cambridge University Press, 2012." The first chapter is on Riemann surfaces with an emphasis on computable examples. |
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Abstract: We continue with the topology of Riemann surfaces. |
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Abstract: We will finish the first chapter on compact Riemann surfaces. The main topic this week will be function fields on Riemann surfaces. |
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Abstract: We will start by discussing the consequences of the Uniformization Theorem of compact Riemann surfaces and continue by discussing the groups which uniformize Riemann surfaces of genus greater than one. Expect lots of pictures. |
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Abstract: This week we start with hyperbolic geometry. |
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Abstract: We will continue with the fundamental group of compact Riemann surfaces and, time permitting, proceed with the existence of meromorphic functions on such surfaces. |
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Abstract: We will start talking about Fuchsian groups. |
ODTU, 28 November 2014, Friday, 15:40
Özgür Kişisel-[ODTÜ] -
Riemann surfaces and discrete groups - V
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Abstract: We will talk about automorphisms of Riemann surfaces. |
Bilkent, 5 December 2014, Friday, 15:40
Özgür Kişisel-[ODTÜ] -
Riemann surfaces and discrete groups - VI
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Abstract: We will talk about the moduli space of compact Riemann surfaces and conclude our discussion of chapter 2. |
ODTU, 12 December 2014, Wednesday, 15:40
Sinan Sertöz-[Bilkent] - Belyi's
Theorem-I
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Abstract: We will describe the content of what is known as Belyi's theorem and prove the hard part which is actually easier than the easy part! |
Bilkent, 19 December 2014, Friday, 15:40
Sinan Sertöz-[Bilkent] - Belyi's
Theorem-II
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Abstract: Last week we discussed the content of Belyi's theorem and worked out an example. So it is only this week that we start to prove the first part of Belyi's theorem: If a compact Riemann surface is defined over the field of algebraic numbers, then it has a meromorphic function which ramifies over exactly three points. This is know as the hard part, and the converse is known as the easy part even though the converse is more involved! |
ODTU, 26 December 2014, Tuesday, 15:40
Sinan Sertöz-[Bilkent] - Belyi's
Theorem-III
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Abstract: This week we will prove that if a compact Riemann surface admits a meromorphic function which ramifies over at most three points, then it is defined over the field of algebraic numbers. This was first proved by Weil in 1956. We will present a modern proof following Girondo and Gonzalez-Diez. |
ODTÜ talks are either at Hüseyin
Demir
Seminar room or at Gündüz İkeda seminar room at the Mathematics building of ODTÜ.
Bilkent talks are at room 141 of Faculty of Science A-building at
Bilkent.
| 2000-2001 Talks (1-28) | 2001 Fall Talks (29-42) | 2002 Spring Talks (43-54) | 2002 Fall Talks (55-66) |
| 2003 Spring Talks (67-79) | 2003 Fall Talks (80-90) | 2004 Spring Talks (91-99) | 2004 Fall Talks (100-111) |
| 2005 Spring Talks (112-121) | 2005 Fall Talks (122-133) | 2006 Spring Talks (134-145) | 2006 Fall Talks (146-157) |
| 2007 Spring Talks (158-168) | 2007 Fall Talks (169-178) | 2008 Spring Talks (179-189) | 2008 Fall Talks (190-204) |
| 2009 Spring Talks (205-217) | 2009 Fall Talks (218-226) | 2010 Spring Talks (227-238) | 2010 Fall Talks (239-248) |
| 2011 Spring Talks (249-260) | 2011 Fall Talks (261-272) | 2012 Spring Talks (273-283) | 2012 Fall Talks (284-296) |
| 2013 Spring Talks (297-308) | 2013 Fall Talks (309-319) | 2014 Spring Talks (320-334) |