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

**** 2017 Spring Talks ****

 Learning seminar on K3 surfaces

1. Bilkent, 24 February 2017, Friday, 15:40

Ali Sinan Sertöz-[Bilkent] - On the moduli of K3 surfaces

 Abstract:  We will discuss the main line of ideas involved in the proofs of the Torelli theorems for K3 surfaces as outlined by Huybrechts in his recent book "Lectures on K3 Surfaces."

2. ODTÜ, 3 March 2017, Friday, 15:40

Ali Sinan Sertöz-[Bilkent- On the moduli of K3 surfaces-II

 Abstract:  This is going to be a continuation of last week's talk. In particular we will talk about the ideas involved around proving the Global Torelli Theorem for K3 surfaces. Most proofs will be referred to the literature but we will try to relate the concepts involved.

3. Bilkent, 10 March 2017, Friday, 15:40

Ali Ulaþ Özgür Kiþisel-[ODTU] - Tropical curves

 Abstract:  In this talk, we will discuss several approaches to defining tropical curves and the theory of linear systems on tropical curves.

4. ODTÜ, 17 March 2017, Friday, 15:40

Ali Ulaþ Özgür Kiþisel-[ODTU] - Tropical curves-II

 Abstract: In this talk, we will continue our discussion of several approaches to defining tropical curves and the theory of linear systems on tropical curves.

5. Bilkent, 24 March 2017, Friday, 15:40

Emre Coþkun-[ODTU] - The Beilinson spectral sequence

 Abstract: We overview the Beilinson spectral sequence and its applications in the construction of sheaves and vector bundles.

6. ODTÜ, 31 March 2017, Friday, 15:40

Alexander Degtyarev-[Bilkent] - Lines in polarized K3-surfaces

 Abstract: I will explain the proof of my conjectures (reported earlier in this seminar) on the maximal number of straight lines in sextic surfaces in $\mathbb{P}^4$, (42 lines) and octic surfaces/triquadrics in $\mathbb{P}^5$, (36 lines). I will also try to make it clear that the complexity of the problem decreases when the polarization grows. The asymptotic bound for K3-surfaces in large projective spaces is 24 lines, all constituting fiber components of an elliptic pencil.

7. Bilkent, 7 April 2017, Friday, 15:40

Mesut Þahin-[Hacettepe] - Lattice ideals and toric codes

 Abstract: I will briefly recall basics of toric varieties over finite fields and evaluation codes on them. Then, we will see that some vanishing ideals of subvarieties are lattice ideals. Using this, we characterize whether they are complete intersections or not. In the former case; dimension, length and regularity of the code will be understood easily.

8. Bilkent, 14 April 2017, Friday, 15:40

Nil Þahin-[Bilkent] - On Pseudo Symmetric Monomial Curves

 Abstract:  After giving basic definitions and concepts about symmetric and pseudo symmetric numerical semigroups, we will focus on 4-generated pseudo symmetric numerical semigroups/monomial curves. Determining the indispensable binomials of the defining ideal, we will give characterizations under which the tangent cone is Cohen-Macaulay. If time permits, determining minimal graded free resolutions of the tangent cones, we’ll show that “If the 4 generated pseudo symmetric numerical semigroup S is homogeneous and the corresponding tangent cone is Cohen Macaulay, then S is also Homogeneous type.

9. Bilkent, 21 April 2017, Friday, 15:40

Alexander Klyachko-[Bilkent] - Transformation of cyclic words into Lie elements

 Abstract:  Let $V$ be a complex vector space and $T(V)=\sum_{n=0}^\infty V^{\otimes n}$ be its tensor algebra.  We are primarily concerned with Lie subalgebra   $L(V)\subset T(V)$ generated by commutators of elements in $V$ and graded by degrees of the tensor components.   From practical point of view treating  Lie elements in terms of commutators is often awkward. Here we describe another approach that allows to write  Lie elements in terms of cyclic words. To wit, for every tensor component  $V^{\otimes n}$ define two operators :     c_n=\frac{1}{n} \sum_{k=0}^{n-1}\varepsilon^{-k}\tau^k,\qquad \ell_n=\frac{1}{n}\sum_{\sigma\in S_n} \varepsilon^{\text{maj}\,\sigma}\sigma   where $\varepsilon$ is a primitive root of unity of degree $n$, $\tau$ is $n$-cycle in symmetric group $S_n$ acting on $V^{\otimes n}$ by permutation of tensor factors. The majorization  index  $\text{maj}\,\sigma$ of permutation $\sigma$ is defined as follows $\operatorname{maj}\,\sigma =\sum_{\sigma(k)> \sigma(k+1)}k\;\mod n.$ The operators $\ell_n$ and $c_n$  satisfy the following equations $c_n\ell_n=\ell_n, \quad \ell_n c_n=c_n,\quad c_n^2=c_n,\quad \ell_n^2=\ell_n.$ Clearly, action of $c_n$ on a monomial in $V^{\otimes n}$ produces a cyclic word. It may be less straightforward that action of $\ell_n$ on a monomial gives Lie element    \ell_n(x_1,x_2,\ldots,x_n)=\frac{1}{n}\sum_I\varepsilon^{\text{maj}\, I} X_I    where  summation runs over all permutations $I=(i_1,i_2,\ldots, i_n)$ and $X_I=(x_{i_1},x_{i_2},\ldots ,x_{i_n})$.    It should be emphasised that cyclic permutation of arguments  in  $\ell_n(x_1,x_2,\ldots,x_n)$ adds only a phase factor equal to $n$-th root of unity.

10. Bilkent, 28 April 2017, Friday, 15:40

Özgün Ünlü-[Bilkent] - Semi-characteristic classes

 Abstract:  In this talk, I will first give basic definitions and theorems about semi-characteristic classes. Secondly, I will discuss some applications of semi-characteristic classes.

11. ODTÜ, 5 May 2017, Friday, 15:40

Çisem Güneþ Aktaþ-[Bilkent] - An Introduction to Nikulin's Theory of Discriminant Forms-I

 Abstract:  In this talk I will first recall some basic definitions and notions about lattices. Then I will introduce fundamentals of Nikulins's theory of discriminant forms. Finally, I will discuss some principal applications of this theory and give an example in the particular case of K3-lattice.

12. ODTU, 12 May 2017, Friday, 15:40

Çisem Güneþ Aktaþ-[Bilkent] - An Introduction to Nikulin's Theory of Discriminant Forms-II

 Abstract: This is the continuation of last week's talk.

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.

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