ODTÜ-BİLKENT Algebraic Geometry Seminar
(See all past talks
ordered according to speaker and
date)
2013 Spring Talks
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Abstract: This is going to be an informal talk on the dimension of the Fano variety of $k$-linear subspaces of projective hypersurfaces, with emphasis on the $k=1$ case. I will losely follow the contents of Chapter 7 and 8 of Eisenbud and Harris' to-be-published book Intersection Theory in Algebraic Geometry. |
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Abstract: We continue our leisurely paced learning seminar on Eisenbud and Harris' notes. I will start by reminding the definition of Chern classes as degeneracy cycles and continue with the calculation of the Chern classes of some interesting bundles. As an application I will talk about how these approaches are used to come up with the number 27, the number of lines on a smooth cubic surface in $\mathbb{P}^3$. Time permitting, I will also attempt to explain solutions to some of the keynote questions posed at the beginning of chapter 8. |
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Abstract: Discovery of the gauge-theoretic invariants (Donaldson's and later Seiberg-Witten's) brought a number of fundamental discoveries completely changing the landscape of Low-dimensional topology. I will review essentials of this theory tracing its later development (Ozsvath-Szabo theory) and focusing on the applications to algebraic geometry. |
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Abstract: After giving a general definition of Seiberg-Whitten invariants, their meaning in the case of Kahler surfaces will be explained. Some applications and developments will be discussed. |
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Abstract: After a short review of differential topological invariants of smooth manifolds, we will discuss some applications to algebraic surfaces. As an example I will discuss the complete intersection surfaces, presented by W. Ebeling (Invent. 1990), which form a pair of nondiffeomorphic but homeomorphic surfaces. |
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Abstract: Mid 90's, Broadhurst and Kreimer observed that multiple zeta values persist to appear in Feynman integral computations. Following this observation, Kontsevich proposed a conceptual explanation, that is, the loci of divergence in these integrals must be mixed Tate motives. In 2000, Belkale and Brosnan disproved this conjecture. In this talk, I will describe a way to correct Kontsevich's proposal and show that the regularized Feynman integrals in position space setting as well as their ambiguities are given in terms of periods of suitable configuration spaces, which are mixed Tate. Therefore, the integrals that are of our interest are indeed $\mathbb{Q}[1/2 \pi i]$-linear combinations of multiple zeta values. This talk is based on a joint work with M. Marcolli. |
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Abstract: In this two-part talk, we will define a moduli problem, and we will discuss the solutions in a number of well-known cases. We start by defining the moduli functor. Next, we show that the Grassmannian functor is represented by the Grassmann variety of linear subspaces of projective space. After discussing the Quot scheme in very general terms, we move to the construction of the moduli space of vector bundles of given rank and degree on an algebraic curve. |
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Abstract: In this two-part talk, we will define a moduli problem, and we will discuss the solutions in a number of well-known cases. We start by defining the moduli functor. Next, we show that the Grassmannian functor is represented by the Grassmann variety of linear subspaces of projective space. After discussing the Quot scheme in very general terms, we move to the construction of the moduli space of vector bundles of given rank and degree on an algebraic curve. |
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Abstract: Let $G=\langle g \rangle$ be a finite group generated by $g$. Given $h\in G$, the discrete logarithm problem (DLP) in $G$ with respect to the base $g$ is computing an integer $a$ such that $h=g^a$. The security of many cryptographic protocols relies on the intractability of DLP in the underlying group. Pollard's rho method is a general purpose algorithm to solve DLP in finite groups, and runs in fully-exponential expected time of $\sqrt{|G|}$. Some special purpose algorithms, such as index calculus method, can solve DLP in finite field groups in sub-exponential time. The lack of an efficient DLP solver for elliptic curve groups has been the main reason for elliptic curve based cryptography to shine compared to finite field based cryptography and the RSA cryptosystem. Recent results show that index calculus can be modified to solve ECDLP in certain settings faster than Pollard's rho algorithm. I will discuss recent developments in using index calculus method to solve ECDLP, and some restrictions of the method that motivate many open problems in the area. |
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Abstract: Paraphrasing A. Marin, we are "à la recherche de la géométrie algébrique perdue": a journey to forgotten algebraic geometry. Following Ethel I. Moody and taking her notes a bit further, I will discuss explicit equations (not just a formal construction in terms of some sheaves and their sections) describing the beautiful Bertini involution and related maps and curves. Should time permit, I will also say a few words justifying my interest in the subject: the Bertini involution can be used to produce explicit equations of the so-called maximizing plane sextics. In theory, all sextics that are still not understood can be handled in this way, but alas, sometimes Maple runs out of memory trying to solve the equations involved. |
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Abstract: We analyze the topological invariants of some specific Grassmannians, the Lie group $G_2$, and give some applications. This is a joint work with Selman Akbulut. |
Bilkent, 17 May 2013, Friday, 15:40
Emre Can Sertöz-[Humboldt] - Idea of the
Moduli Space of Curves
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Abstract: By considering Riemann surfaces from several different angles, we will see that there are many seemingly different ways to vary the complex structure on a surface, getting different Riemann surfaces. So we can ask "What is the most natural way to vary Riemann Surfaces?". This is what the moduli space construction answers, and we will talk about it. Also we will see why we need some extra structure on the moduli space besides the classical structures that come via a manifold (or a scheme). |
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) |