ODTÜ
-BÝLKENT Algebraic Geometry Seminar
(See all past talks
ordered according to speaker and date)

**** 2019 Spring Talks ****

1. Bilkent, 22 February 2019, Friday, 15:40

Ali Sinan Sertöz-[Bilkent] - Arf Rings I

 Abstract: The aim of these two talks is to discuss the background and the content of Arf's 1946 paper on the multiplicity sequence of an algebraic curve branch. I will start by giving the geometric and algebraic descriptions of a singular branch for a curve, describe its multiplicity sequence obtained until it is resolved by blow up operations. Du Val defines some geometrically significant steps of the resolution process and shows that if the multiplicity sums up to those points are known then the whole multiplicity sequence can be recovered by a simple algorithm. However all this information must be encoded at the very beginning in the local ring of the branch. The problem is then to decipher this data. This week I will mostly describe the background and explain what is involved in actually finding these numbers. Arf's original article "Une interpretation algebrique de la suite des ordres de multiplicite d'une branche algebrique", together with my English translation can be found on: http://sertoz.bilkent.edu.tr/arf.htm

2. ODTÜ, 1 March 2019, Friday, 15:40

Ali Sinan Sertöz-[Bilkent] - Arf Rings II

 Abstract: I will first describe the structure of the local ring of a singular branch and explain how the blow up process affects it. Then I will describe, aprés Arf, how the multiplicity sequence can be recovered, not from this ring but from a slightly larger and nicer ring which is now known as the Arf ring. The process of finding this nicer ring is known as the Arf closure of this ring. Finally I will explain how Arf answered Du Val's question of reading off the multiplicity sequence from the local ring.

3. Bilkent, 8 March 2019, Friday, 15:40

Alexander Degtyarev-[Bilkent] - Tritangents to sextic curves via Niemeier lattices

 Abstract: I will address the following conjecture (and some refinements thereof): “A smooth plane curve of degree 6 has at most 72 tritangents.” After a brief introduction to the subject and a survey of the known results for the other polarized K3-surfaces, I will explain why the traditional approach does not work and suggest a new one, using the embedding of the Néron—Severi lattice of a K3-surface to an appropriate Niemeier lattice. I will also discuss the pros and contras of several versions of this approach and report the partial results obtained so far.

4. ODTÜ, 15 March 2019, Friday, 15:40

Alexander Degtyarev-[Bilkent] - Positivity and sums of squares of real polynomials

 Abstract: I will discuss the vast area of research (in which I am not an expert) related to Hilbert's 17th problem, namely, positivity of real polynomials vs. their representation as sums of squares (SOS). As is well known, "most" PSD (positive semi definite) forms in more than two variables are not SOS of polynomials, although they are SOS of rational function. I will consider a few simplest classical counterexamples, and then I will outline the construction part of our recent paper (in collaboration with Erwan Brugallé, Ilia Itenberg, and Frédéric Mangolte). Thinking that we were dealing with Hilbert's 16th problem (widely understood, i.e., topology of real algebraic varieties), we constructed real plane algebraic curves with large finite numbers of real points. These curves provide new lower bounds on the denominators needed to represent a PSD ternary form as a SOS of rational functions.

********  No talk on 22 March 2019 due to Spring Break  *****

5. Bilkent, 29 March 2019, Friday, 15:40

Yýldýray Ozan-[ODTÜ] - Equivariant Cohomology and Localization after Anton Alekseev

 Abstract: We will try to present the notes by Anton Alekseev on Equivariant Localization, mainly focusing on $S^1$-actions. First, we will introduce Stationary Phase Method. Then we will define equivariant $S^1$-cohomology and present a proof of the localization theorem suggested by E. Witten.  If time permits, finally we will end by the Duistermaat-Heckman formula and its proof.

6. ODTU, 5 April 2019, Friday, 15:40

Muhammed Uludað-[Galatasaray] - TBA

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7. Bilkent, 12 April 2019, Friday, 15:40

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8. ODTÜ, 19 April 2019, Friday, 15:40

Halil Ýbrahim Karakaþ-[Baþkent] - Arf Numerical Semigroups

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9. Bilkent, 26 April 2019, Friday, 15:40

Mesut Þahin-[Hacettepe] - TBA

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10. ODTÜ, 3 May 2019, Friday, 15:40

Tolga Karayayla-[ODTÜ] - TBA

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11. Bilkent, 10 May 2019, Friday, 15:40

Nil Þahin-[Bilkent] - TBA

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12. ODTÜ, 17 May 2019, Friday, 15:40

Yýldýray Ozan-[ODTÜ] - TBA

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ODTÜ talks are either at Hüseyin Demir Seminar room or at Gündüz Ýkeda 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) 2014 Fall Talks (335-348) 2015 Spring Talks (349-360) 2015 Fall Talks (361-371) 2016 Spring Talks (372-379) 2016 Fall Talks (380-389) 2017 Spring Talks (390-401) 2017 Fall Talks (402-413) 2018 Spring Talks (414-425) 2018 Fall Talks (426-434)