ODTÜBİLKENT Algebraic Geometry Seminar
2000 Fall Talks
2001 Spring Talks
2001 Fall Talks:
2002 Spring Talks
2002 Fall Talks
2003 Spring Talks
2003 Fall Talks
2004 Spring Talks
2004 Fall Talks
2005 Spring Talks
2005 Fall Talks
2006 Spring Talks
2006 Fall Talks
2007 Spring Talks
2007 Fall Talks
2008 Spring Talks
2008 Fall Talks
2009 Spring Talks
Abstract: Last year a 6 page proof of Hodge conjecture was deposited into the arXives. Later a 7 page revision was posted, see arXiv:0808.1402 This paper uses only the material found in chapter 0 of Griffiths and Harris' Principles of Algebraic Geometry. In this talk we will review this introductory material for the graduate students and then present the arguments of the alleged proof and ask the audience to find the error! 
Abstract: Last week we mentioned a subtle gap in the alleged proof of Hodge conjecture in arXiv:0808.1402. This week we will mention an irrecoverable gap in the proof and then give an informal survey of what is know in the Hodge conjecture front. 
Abstract: We will give an old constructive method to find the presentation of the knot group which is a knot invariant and we will finish with some illustrations. 
Abstract: There is a method of finding the group presentation of a tame knot. However, it is not an easy task to distinguish groups given their presentations, even in particular examples. Therefore, one needs to find presentation invariants. We shall first consider the Alexander matrix and elementary ideals of a given finite presentation in a general setup then restrict our attention to knot groups and get knot polynomials which happen to be knot invariants of trivial distinguishability. 
Abstract: We will present a class of toric varieties with exceptional properties. These are toric varieties corresponding to rational singularities of DE type. We show that their toric ideals have a minimal generating set which is also a Groebner basis consisting of large number of binomials of degree at most 4. 
Abstract: We will demonstrate different methods of calculating the Alexander polynomial on several examples. 
Abstract: We deal with the following generalized version of the Shapiro and Shapiro total reality conjecture: given a real curve C of genus g and a regular map C > P^{1} of degree d whose all critical points are distinct and real (in C), the map itself is real up to a Mőbius transformation in the target. The generalization was suggested by B. and M. Shapiro in about 2005, after the original conjecture was proved, and it was shown that the statement does hold for g>d^{2}/3+O(d). In the talk, we improve the above inequality to g>d^{2}/4+O(d). 
Abstract: In this talk, after I describe algebraic automorphisms group of P^{1}xP^{1}, I will consider the analogous problem in the category of symplectic topology. I will present some results comparing them with the results in the study of volume preserving diffeomorphisms group. In the remaining time, I will talk about the main technique used in the proof, so called the theory of Jholomorphic curves in symplectic topology and how they are employed in this work. 
Abstract: We attempt to study/classify real Jacobian elliptic surfaces of type I or, equivalently, separating real trigonal curves in geometrically ruled surfaces. (On the way, we extend the notions of type I and being separating to make them more suitable for elliptic surfaces.) We reduce the problem to a simple graph theoretical question and, as a result, obtain a characterization and complete classification (quasisimplicity) in the case of rational base. (The results are partially interlaced with those by V. Zvonilov.) As a byproduct, we obtain a criterion for a trigonal curve of type I to be isotopic to a maximally inflected one. 
Abstract: I will talk about the fundemantal concepts in the study of Higher Chow groups, historical background and main research subjects in this field in relation with classical Hodge Theory. I will demonstrate some of these concepts and methods by discussing in a "genaralization" of Hodge conjecture (so called HodgeD conjecture) for product of two general elliptic curves. 
Abstract: The variety of a finitely generated kGmodule is a closed homogeneous subvariety of the maximal ideal spectrum of the cohomology ring of a finite group G with coefficients in an algebraically closed field k of characteristic p>0. I will give some basic definitions and properties of varieties in group cohomology. Then I will present some results on filtration of modules related to varieties. 
Abstract: We will outline the construction of pure motifs, concentrating on the ChowKunneth decomposition. Time permiting we intend to describe the transcendental part of the motif of a surface. This is an informal introductory talk. 
Abstract: I will discuss the limit space F of the category of coverings C of the "modular interval" as a deformation retract of the universal arithmetic curve, which is by (my) definition nothing but the punctured solenoid S of Penner. The space F has the advantage of being compact, unlike S. A subcategory of C can be interpreted as ribbon graphs, supplied with an extra structure that provides the appropriate morphisms for the category C. After a brief discussion of the mapping class groupoid of F, and the action of the Absolute Galois Group on F, I will turn into a certain "hypergeometric" galoisinvariant subsystem (not a subcategory) of genus0 coverings in C. One may define, albeit via an artificial construction, the "hypergeometric solenoid" as the limit of the natural completion of this subsystem to a subcategory. Each covering in the hypergeometric system corresponds to a nonnegatively curved triangulation of a punctured sphere with flat (euclidean) triangles. The hypergeometric system is related to plane crystallography. Along the way, I will also discuss some other natural solenoids, defined as limits of certain galoisinvariant genus0 subcategories of nongalois coverings in C. The talk is intended to be informal, relaxed and audience friendly. 
2009 Fall Talks
Abstract: The aim of this talk is to give the necessary background material on vector bundles to introduce the topological Ktheory. We also explain the classification theorem for vector bundles. This talk is accessible to graduate students at any level. 
Abstract: Last week we discussed the basic properties of vector bundles over a compact base space X to introduce the topological Ktheory. The set of isomorphism classes of vector bundles on X forms a commutative monoid. The idea of Ktheory of X is the completion of this monoid to a ring. In this talk, we will discuss basic concepts in Ktheory. 
Abstract: This is going to be an introductory talk to Bott's periodicity theorem. 
Abstract: This is going to be a introductory talk to algebraic Ktheory. I will introduce algebraic Ktheory and discuss some basic properties of it. I will give the sketch of the proof of Swan'a theorem, which gives us the relation between topological and algebraic Ktheories. 
Abstract: In this introductory talk we will define K_{1} of rings and discuss their basic properties. 
Abstract: As one of the topological applications of algebraic Ktheory, I will introduce Wall's finiteness obstruction which is defined as the obstruction for a finitely dominated space to be homotopy equivalent to a finite CWcomplex. Then, I will discuss the orbit category version of Wall's finiteness obstruction. 
Abstract: Following Rosenberg, we will describe the K theory of certain categories and talk about conditions under which we can use a more `reasonable' collection of modules instead of projective modules and still get the same K theory. This will eventually be applied to discuss Grothendieck's RiemannRoch theorem but that may be left to the next talk if time runs up. 
Abstract: We will talk about the proof of the wellknown fact that an ndimensional sphere is an Hspace if and only if n=0, 1, 3, or 7. 
Abstract: The Kontsevich moduli space of stable maps is the central object in GromovWitten theory. In this talk, I will discuss its birational geometry and describe how to run Mori's program on small degree examples. I will focus on a few concrete examples.This is joint work with Dawei Chen and builds on joint work with Joe Harris and Jason Starr. 
2010 Spring Talks
Abstract: We will conclude last term's seminar on Ktheory with an application to algebraic geometry by developing Grothendieck's RiemannRoch theorem. The talk will be expository and will be accessible even to those who do not remember much of last semester's talks! 
Abstract: In 1984, V. Jones introduced a new (polynomial) knot invariant by using an operator algebra. Later, it became clear that this polynomial can be obtained by several different methods. We will pick a simple approach, namely defining it by means of the slightly different Kauffman bracket polynomial. We will then consider Jones polynomials of alternating links. In the remaining time, we will finish with the proofs of Tait's conjectures (due to K. Murasugi) by using Jones Polynomial. 
Bilkent, 5 March 2010 Friday, 15:40
Deniz Kutluay[Bilkent University]Tait's
Conjectures
Abstract: P.G. Tait conjectured, in 1898, that a reduced alternating diagram of a knot achieves the minimum possible number of crossings for that knot (1), and writhe of such diagrams of the same knot is the same (2). We will first give K. Murasugi's proof to (1) which involves usage of Jones polynomial. We will then use the idea of taking parallels of diagrams (due to R.A. Stong) to prove (2). 
ODTU, 12 March 2010 Friday, 15:40
İnan Türkmen[Bilkent University] Detecting
Indecomposable Higher Chow Cycles
Abstract: Spencer Bloch defined the higher Chow in mid 80's as a "natural" extension of classical Chow groups and analysed basic properties of these groups in terms of maps to Deligne Cohomology, named regulators. There is a subgroup of higher Chow groups, group of indecomposables, of special interest. In this talk I will introduce two different methods to detect indecomposables; regulator indecomposability and filtrations on arithmetic Hodge structures. 
Bilkent, 19 March 2010 Friday, 15:40
Alexander Degtyarev[Bilkent University] Dihedral
covers of trigonal curves
Abstract: We classify irreducible trigonal curves in Hirzebruch surfaces that admit a dihedral cover and study geometric properties of such curves. In particular, we prove an analog of Oka's conjecture stating that an irreducible trigonal curve admits an S_3 cover if and only if it is of torus type. 
Bilkent, 26 March 2010 Friday, 15:40
Bedia Akyar[Dokuz Eylul University] Prismatic
sets in topology and geometry
Abstract: We study prismatic sets analogously to simplicial sets except that realization involves prisms. In particular, I will mention the examples; the prismatic subdivision of a simplicial set S and the prismatic star of S. Both have the same homotopy type as S. Moreover, I will give the role of prismatic sets in lattice gauge theory, that is, for a Lie group G and a set of parallel transport functions defining the transition over faces of the simplices, we define a classifying map from the prismatic star to a prismatic version of the classifying space of G. In turn this defines a Gbundle over the prismatic star. This is a joint work with Johan L. Dupont. 
ODTU, 9 April 2010 Friday, 15:40
Yıldıray Ozan[ODTU] Algebraic
Ktheory in the study of regular maps in real algebraic
geometry
Abstract: After introducing some preliminary material about real algebraic varieties I will try to summarize how algebraic Ktheory is used to study regular maps between real algebraic varieties. Namely, I will talk about the results of Loday and BochnakKucharz which mainly show that regular maps between certain products of spherees are all nullhomotopic. For example, Loday showed that any regular map from S^{1} x S^{1} to S^{2} is homotopically trivial, where S^{k} is the unit sphere in R^{k+1}. 
ODTU, 16 April 2010 Friday, 15:40
Ali Kemal Uncu[TOBB ETU] Modular
symbols on congruence subgroups of SL_{2}(Z)
Abstract: The talk will be about finding the Fourier coefficients of a modular form of the given even weight on a congruence subgroup of SL_{2}(Z). We will work with the Riemann surface related to the congruence subgroup of SL_{2}(Z), define modular symbols and give the relation between modular symbols and Fourier coefficients of modular forms. 
ODTU, 30 April 2010 Friday, 15:40
Ergün Yalçın[Bilkent University]Koszul
Resolutions and the Lie Algebra Cohomology
Abstract: Cohomology of a Lie algebra is defined both as the cohomology of its universal algebra and via a Koszul resolution. I will introduce both of the definitions and discuss their equivalence. Then, I will show how the Lie algebra cohomology appears in the integral cohomology calculation of a group extension. 
Bilkent, 7 May 2010 Friday, 15:40
Özgün Ünlü[Bilkent University] Homologically
trivial group actions on products of spheres
Abstract: In this talk, I will discuss some constructions of free group actions on products of spheres with trivial action on homology. 
ODTU, 14 May 2010 Friday, 15:40
Hamza Yeşilyurt[Bilkent University]RogersRamanujan
Functions
Abstract: We present several identities for the RogersRamanujan functions along with their partition theoretic interpretations and conclude with our recent work on such identities. 
Bilkent, 28 May 2010 Friday, 15:40
Mutsuo Oka[Tokyo University of Science]Polar
weighted homogeneous polynomials and mixed Brieskorn
singularity
Abstract: 
2010 Fall Talks
Abstract: The Alexander module of an algebraic curve is a certain purely algebraic invariant of the fundamental group of (the complement of) the curve. Introduced by Zariski and developed by Libgober, it is still a subject of intensive research. We will describe the Alexander modules and Alexander polynomials (both over Q and over finite fields F_{p }) of a special class of curves, the so called generalized trigonal curves. The rational case is closed completely; in the case of characteristic p>0, a few points remain open. (Conjecturally, all polynomials that can appear are indeed listed.) Unlike most known divisibility theorems, which rely upon the degree and the types of the singularities of the curve, our bounds are universal: essentially, the Alexander module of a trigonal curve can take but a finitely many values. 
Abstract: The curious history of Fermat's Last Theorem starts with Fermat's famous marginal commentary. The quest for the solution of this problem has created theories which affect all of mathematics. In this seminar, we will talk about Ribet's theorem which states that modularity theorem (previously known as TaniyamaShimura conjecture) implies Fermat's Last Theorem. A central role in Ribet's proof is played by elliptic curves introduced by Frey. 

On 1315 October, we are having an Algebra and Number
Theory Symposium in
honor of Prof Mehpare Bilhan's retirement.
There will be no Algebraic Geometry talk this week.

Bilkent, 22 October 2010 Friday, 15:40
Christophe Eyral[Aarhus University] 
A short introduction to
Lefschetz theory on the topology of algebraic
varieties
Abstract: 

29 October is Republic Day, a national day for Turkey. No
talks!

Bilkent, 5 November 2010 Friday, 16:00
Muhammed Uludağ[Galatasaray University] 
The Groupoid of Orientation Twists
Abstract: This is an essay to define a higher modular groupoid. The usual modular groupoid of triangulation flips admits ideal triangulations of surfaces of fixed genus and punctures as objects and flips as morphisms. The higher groupoid of orientation twists admits usual modular groupoids as its objects. 
ODTU, 12 November 2010 Friday, 15:40
İnan Utku Türkmen[Bilkent University]  An
Indecomposable Cycle on Self Product of Sufficiently
General
Product of Two Elliptic Curves
Abstract: The group of indecomposables is too complicated to compute in general and the results in literature are cenrered around proving that this group is nontrivial or in certain cases finitely generated. In this talk I will focus on the group of indeconposables of self product of sufficiently general product of two elliptic curves, namely; CH^{3}_{ind}(E_{1}x E_{2} x E_{1} x E_{2}). I will review the results in literature related with this group and sketch an alternative proof for nontriviality of this group using a constructive method. 

1619 November is a religious holiday in Turkey. No talks!

ODTU, 26 November 2010 Friday, 15:40
Mehmetcik Pamuk[ODTU]  scobordism
classification of 4manifolds
Abstract: In this talk we are going to show how one can use the group of homotopy selfequivalences of a 4manifold together with the modified surgery of Matthias Kreck to give an scobordism classification of topological 4manifolds. We will work with certain fundamental groups and give scobordism classification in terms of standard invariants. 
Bilkent, 3 December 2010 Friday, 15:40
Ergün Yalçın[Bilkent University] 
Productive elements in group cohomology
Abstract: I will give the definition of a productive element in group cohomology and describe a new approach to productive elements using Dold's Postnikov decomposition theory for projective chain complexes. The motivation for studying productive elements comes from multiple complexes which is an important construction for studying varieties of modules in modular representation theory. 
ODTU, 10 December 2010 Friday, 15:40
Mustafa Kalafat[University of Wisconsin at
Madison and ODTU] 
Hyperkahler manifolds with circle actions and the
GibbonsHawking Ansatz
Abstract: We show that a complete simplyconnected hyperkahlerian 4manifold with an isometric triholomorphic circle action is obtained from the GibbonsHawking ansatz with some suitable harmonic function. 
Bilkent, 17 December 2010 Friday, 15:40
Kürşat Aker[Feza Gürsey]  Multiplicative
Generators for the Hecke ring of the Gelfand Pair (S(2n),
H(n))
Abstract: For
a given positive integer n, Gelfand pair (S(2n),
H(n)) resembles the symmetric group S(n) in
numerous ways. Here, H(n) is a hyperoctahedral
subgroup of the symmetric group S(2n). In this
talk, we will exhibit a new similarity between the
Hecke ring of the pair (S(2n), H(n)) and the
center of the integral group ring of S(n). 
ODTU, 24 December 2010 Friday, 15:40
Ali Sinan Sertöz[Bilkent]  Counting
the number of lines on algebraic surfaces
Abstract: This is mostly an expository talk on the problem of counting the number of lines on an algebraic surface. The problem is to respect the rigidity of the line as opposed to accepting all rational curves as lines. Surprisingly some of the work done by Segre has not yet been matched by contemporary techniques. We will summarize what is known and speculate about what can be known! 

31 December afternoon is no time to hold seminars on
this planet! No talks!

2011 Spring Talks
Abstract: This term we plan to go over the interesting parts of J. McCleary's book, The User's Guide to Spectral Sequences (2nd Edition, 2001). I will begin with some motivation and basic definitions. This may take a few weeks after which many people promised to talk about the wonderful spectral sequences they have met! 
Abstract: We are continuing with J. McCleary's book, The User's Guide to Spectral Sequences (2nd Edition, 2001). I will repeat the basic definitions and work on some simple examples. 
Bilkent, 4 March 2011 Friday, 15:40
Ali Sinan Sertöz[Bilkent University]  Basics of
Spectral Sequences III
Abstract: We are continuing with J. McCleary's book, The User's Guide to Spectral Sequences (2nd Edition, 2001). I will start with the second chapter and describe two situations where spectral sequences arise. 

ODTU, 11 March 2011 Friday, 15:40
Ali Sinan Sertöz[Bilkent University]  Basics of Spectral Sequences IV
Abstract: We are continuing with J. McCleary's book, The User's Guide to Spectral Sequences (2nd Edition, 2001). I will summarize the third chapter and discuss convergence of spectral sequences. 
Bilkent, 18 March 2011 Friday, 15:40
Alexander Degtyarev[Bilkent University]  LeraySerre
spectral sequence I
Abstract: We start exploring the geometric application of the machinery of spectral sequence. As the simplest examples, we consider the spectral sequence(s) of a filtered topological space (as a straightforward generalization of the exact sequence of a pair) and the Serre spectral sequence of a simple fibration. 
ODTU, 1 April 2011 Friday, 15:40
Alexander Degtyarev[Bilkent
University]  LeraySerre
spectral sequence II
Abstract: We will continue exploring the immediate consequences and applications of the Serre spectral sequence. Then we will switch to the Leray spectral sequence, which will be derived as a special case of one of the hypercohomology spectral sequences; in particular, we will show that the Leray (and hence Serre) spectral sequences are natural and retain the multiplicative structure, facts that are not immediately obvious from Serre's construction via skeletons. 
Bilkent, 8 April 2011 Friday, 15:40
Ergün Yalçın [Bilkent University]
The LyndonHochschildSerre spectral sequence
Abstract: Let G be a group and H be a normal subgroup of G. Then there is a spectral sequence, called LHSspectral sequence, which converges to the cohomology of G and whose E_2 term can be expressed in terms of cohomology of H and G/H. I will show how the HLSspectral sequence can be constructed as a spectral sequence of a double complex and then I will illustrate its usage by doing some group cohomology calculations using it. 
ODTU, 15 April 2011 Friday, 15:40
Ergün Yalçın[Bilkent University]  Calculating
with the LHSspectral sequence
Abstract: Let G be a group and H be a normal subgroup of G. There is a spectral sequence, called LHSspectral sequence, which converges to the cohomology of G and whose E_2 term can be expressed in terms of cohomology of H and G/H. In last week's seminar, I showed how the LHSspectral sequence can be constructed as a spectral sequence of a double complex. This week I will show how this spectral sequence is used to do group cohomology calculations. I plan to bring enough examples to illustrate different situations that one faces while doing calculations with spectral sequences. 
Bilkent, 22 April 2011 Friday, 14:35 (Notice the new time for this talk)
Özgün Ünlü[Bilkent University] AtiyahHirzebruch spectral sequence
Abstract: Let X be a CW complex and h be a generalized cohomology theory. AtiyahHirzebruch spectral sequence relates the generalized cohomology groups h_*(X) with ordinary cohomology groups with coefficients in the generalized cohomology of a point. 
ODTU, 29 April 2011 Friday, 15:40
Yıldıray Ozan[ODTU]  On Cohomology of the Hamiltonian Gorups
Abstract: Homotopy properties of the group of Hamiltonian diffeomorphisms of symplectic manifolds are far richer than those of the diffeomorphism groups. Abrue, Anjos, Kedra, McDuff ve Reznikov are some of the authors who contributed to the theory. In this talk, I will explain basics of the theory and try to present sample arguments. 
Bilkent, 6 May 2011 Friday, 15:40
Mehmet Akif Erdal[Bilkent University]  James Spectral Sequence
Abstract: We will construct the James spectral sequence which is a variant of AtiyahHirzebruch spectral sequence. 
ODTU, 13 May 2011 Friday, 15:40
Mehmetcik Pamuk[ODTU]  An Application
of AtiyahHirzebruch Spectral Sequence
Abstract: 
2011 Fall Talks
Abstract: (joint
with Nermin Salepci, Université de Lyon) Trivial as it seems, this simplest case has a number of geometric applications. As a first one, we prove that any maximal real elliptic Lefschetz fibration over the sphere is algebraic. Other applications include the semisimplicity statement for real trigonal Mcurves in Hirzebruch surfaces. (One may try to speculate that products of two Dehn twists are still `tame' precisely because they are related to maximal geometric objects.) The principal tool is a description of subgroups of the modular group in terms of a certain class of Grothendieck's dessins d'enfants, followed by high school geometry. 
Abstract: The purpose of this expository talk is to lay a basis for Sinan's forthcoming account of our joint project. Recall that a quartic surface in P^{3} is merely a K3surface equipped with a polarization of degree 4. Thus, I will give a gentle introduction to theory of K3surfaces: the period space, the global Torelli theorem and surjectivity of the period map, and the implications of the RiemannRoch theorem. I will explain how the problem of counting lines on a quartic can be reduced to a purely arithmetical question and, should time permit, give a brief account of the results obtained so far, viz. a more or less explicit description of the Picard group of the champion quartic. 
Abstract: Let G be a reductive group. A GxGvariety X is called an embedding of G if X is normal, projective, and contains G as an open dense orbit. Regular compactifications and standard embeddings are the main source of examples. In the former case, they are smooth varieties, and their equivariant cohomology has been explicitely described by Brion using GKM theory. His description relies on the associated torus embedding and the structure of the GxGorbits. In contrast, standard embeddings constitute a much larger class of embeddings than the smooth ones, and their equivariant cohomology was, just until recently, only understood in some cases. Based on results of Renner, standard embeddings were known to come equipped with a canonical cell decomposition, given in terms of underlying monoid data. The purpose of this talk is threefold. First, I will give an overview of the theory of group embeddings, putting more emphasis on Renner's approach, and describe the structure of the so called rational cells. Secondly, I will explain how such cellular decompositions lead to a further application of GKM theory to the study of standard embeddings. Finally, I provide a complete description of the equivariant cohomology of any rationally smooth standard embedding. The major results of this talk are part of the speaker's PhD thesis. References: PS: The speaker is supported under TUBITAK ISBAP Grant 107T897 Matematik İşbirliği Ağı: Cebir ve Uygulamaları. 
The afternoon of 28 October is a
National Holiday.
Abstract: I will wrap up my recent investigations on lines on surfaces with a view towards settling some problems jointly with Degtyarev. 
There is no talk on 11 November
2011 due to Kurban Bayramı.
Abstract: The surgery method of classifying manifolds seeks to answer the following question: Given a homotopy equivalence of mdimensional manifolds f: M > N, is f homotopic to a diffeomorphism ? The surgery theory developed by Browder, Novikov, Sullivan and Wall in the 1960’s provides a systematic solution to this problem. My talk will aim to be a friendly introduction to the basic concepts of the surgery theory. 
Abstract: Splines or piecewise polynomial functions are used most commonly to approximate functions, especially by numerical analysts for approximating solutions to differential equation. Most recently, splines have also played an important role in computer graphics. That’s why it is of interest to study spline spaces. In this talk, we will discuss analyzing the piecewise functions with a specified degree of smoothness on polyhedral subdivision of region on R^{n} and their dimension. 
Abstract: In this talk, we review the genus zero GromovWitten invariants by first defining them in a brief way and then applying them in examples of dimension four and six. We also prove that the use of genus zero GromovWitten invariants to distinguish the symplectic structures on a smooth 6manifold is restricted in a certain sense. 
Abstract: We give a survey of Geometric Invariant Theory for Toric Varieties, and present an application to the EinsteinWeyl Geometry. We compute the image of the Minitwistor space of the Honda metrics as a categorical quotient according to the most efficient linearization. The result is the complex weighted projective space CP_(1,1,2). We also find and classify all possible quotients. 
Abstract: We
study the fibre products of a finite number of
Kummer covers of the projective line over finite
fields. We determine the number of rational points
of the fibre product under certain conditions. We
also 
Abstract: Toric codes are some evaluation codes obtained by projective toric varieties corresponding to convex lattice polytopes. We will explain how their basic parameters are related to the torus and the number of lattice points of the polytope and introduce certain generalizations. We will also review some recent results about the minimum distance. 
Abstract: In the talk I will discuss the structure of toric variety X_{G} equal to closure of a generic orbit of a maximal torus of a simple group G in its flag variety F_{G}, the respective restriction map H*(F_{G})>H*(X_{G}) together with some applications. 
Abstract: How should greengrocers most efficiently stack their oranges? How about pennies on a tabletop or atoms of a single element in a crystal? More than 400 years ago Kepler conjectured that the most efficient way is the facecentered cubic packing which is well known for greengrocers nowadays. Just recently a "proof" (referees are 99% are certain) for Kepler's conjecture is given. In this talk we will give a brief history of the conjecture and related problems. By considering the problem in higher dimensions we will illustrate some special cases and their applications to different areas of mathematics. In particular, the connection between lattices and theta functions will be discussed. 
2012 Spring Talks
This semester we are going
to run a learning
seminar on Patrick
Shanahan's book, The AtiyahSinger Index Theorem, Springer Lecture Notes in Mathematics No: 638. 
Abstract: Preliminaries will be discussed; mostly characteristic classes. 
Abstract: I will talk about the motivation for the index theorem and discuss the individual terms in the statement of the theorem. 
Abstract: We examine the consequences of applying the AtiyahSinger Index Theorem to de Rham and Dolbeault operators. 
Abstract: In this talk we will demonstrate that the application of the AtiyahSinger Index Theorem to Hodge operator yields the Hirzebruch signature theorem. 
Abstract: In this talk we will discuss the application of the AtiyahSinger Index Theorem to Dirac operator. 
Abstract: In this talk we give a brief description of the ring K(X) of stable vector bundles over X. 
Abstract: In this talk we continue to give a brief description of the ring K(X) of stable vector bundles over X. 
Abstract: In this talk we will elaborate on the topological index B as covered in Shanahan's boook. 
Abstract: In this talk we will discuss pseudodifferential operators and their suitable generalizations as discussed in Shanahan's book. 
Abstract: In this talk we will discuss the construction of the index homomorphism as given in Shanahan's book. 
Abstract: In this talk we will discuss the main ideas surrounding the proof of the index theorem as given in Shanahan's book. 
2012 Fall Talks
This semester we are going
to run a learning seminar on intersection theory.
We will loosely follow the notes 3264 & All That Intersection Theory in Algebraic Geometry by David Eisenbud and Joe Harris Here is a copy of these notes to save you some Googling. Research talks from other parts of geometry will not be excluded from our program 
Abstract:
(a never ending joint project
with I. Itenberg and S. Sertoz) 
Abstract: I will start with Chapter 2 of EisenbudHarris notes and after a brief introduction I will describe the Chow ring of $\mathbb{G}(1,3)$, with a view toward counting the number of lines which meet four general lines in $\mathbb{P}^3$. 
Abstract: I will continue to explore the geometry of Grassmannians, after which I will start discussing the Chow ring of $\mathbb{G}(1,3)$. I hope to have time to talk about the number of lines meeting four general lines in space. 
Abstract: I will start by describing the Chow ring of $\mathbb{G}(1,3)$ and then attack the "Keynote Questions" quoted at the beginning of the chapter. 
Abstract: I will complete the multiplication table of the Chow ring of $\mathbb{G}(1,3)$ and then attack the "Keynote Questions" quoted at the beginning of the chapter. Rain or shine, I will finish my talk series this week! 
Abstract: We generalize the discussion on the intersection theory on G(1,3) given in the previous seminars to the Grassmanian variety G(k,n). We are going to define the Schubert cells and cycles and discuss their properties known as Schubert Calculus. We will determine the Chow Ring A(G(k,n)) and mention some applications to intersection theory problems. 
Abstract: We generalize the discussion on the intersection theory on G(1,3) given in the previous seminars to the Grassmanian variety G(k,n). We are going to define the Schubert cells and cycles and discuss their properties known as Schubert Calculus. We will determine the Chow Ring A(G(k,n)) and mention some applications to intersection theory problems. 
Abstract: We consider an indecomposable representation of the Klein four group over a field of characteristic two and compute a generating set for the corresponding invariant ring up to a localization. We also obtain a homogeneous system of parameters consisting of twisted norms and show that the ideal generated by positive degree invariants is a complete intersection. (joint with J. Shank) 
Abstract:
First we start with defining
rational equivalence between two cycles. Then we
define the chow group as a group of rational
equivalence classes. Then we will present essential
theorems and propositions which are developed at the
fourth chapter (D. Eisenbud and J. Harris, All That
Intersection Theory in Algebraic Geometry) to solve
the keynote question b: 
Abstract: After an introductory discussion of tropical varieties, I intend to talk about tropical intersections and in particular the tropical Grassmannian. 
7 December 2012, Friday
This week's seminar is cancelled due to the traffic of
Docent juries taking place this week.
Abstract: We generalize the discussion on the intersection theory on G(1,3) given in the previous seminars to the Grassmanian variety G(k,n). We are going to define the Schubert cells and cycles and discuss their properties known as Schubert Calculus. We will determine the Chow Ring A(G(k,n)) and mention some applications to intersection theory problems. 
Abstract: Arf Closure of a local ring corresponding to a curve branch, which carries a lot of information about the branch, is an important object of study, and both Arf rings and Arf semigroups are being studied by many mathematicians, but there is not an implementable fast algorithm for constructing the Arf closure. The main aim of this work is to give an easily implementable fast algorithm for constructing the Arf closure of a given local ring. The speed of the algorithm is a result of the fact that the algorithm avoids computing the semigroup of the local ring. Moreover, in doing this, we give a bound for the conductor of the semigroup of the Arf Closure without computing the Arf Closure by using the theory of plane branches. We also give an exposition of plane algebroid curves and present the SINGULAR library written by us to compute the invariants of plane algebroid curves. 
Abstract:
We show that a compact complex
surface together with an EinsteinHermitian metric
of positive holomorphic bisectional curvature is
biholomorphically isometric to the complex
projective plane with its FubiniStudy metric up to
rescaling. 
2013 Spring Talks
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' tobepublished book Intersection Theory in Algebraic Geometry. 
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. 
Abstract: Discovery of the gaugetheoretic invariants (Donaldson's and later SeibergWitten's) brought a number of fundamental discoveries completely changing the landscape of Lowdimensional topology. I will review essentials of this theory tracing its later development (OzsvathSzabo theory) and focusing on the applications to algebraic geometry. 
Abstract: After giving a general definition of SeibergWhitten invariants, their meaning in the case of Kahler surfaces will be explained. Some applications and developments will be discussed. 
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. 
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. 
Abstract: In this twopart talk, we will define a moduli problem, and we will discuss the solutions in a number of wellknown 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. 
Abstract: In this twopart talk, we will define a moduli problem, and we will discuss the solutions in a number of wellknown 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. 
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 fullyexponential expected time of $\sqrt{G}$. Some special purpose algorithms, such as index calculus method, can solve DLP in finite field groups in subexponential 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. 
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 socalled 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. 
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
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). 
2013 Fall Talks
Abstract: In this talk, we give the classical definition of a toric variety involving the torus action and provide examples to illustrate it. We introduce two important lattices that play important roles in the theory of algebraic tori and demonstrate how they arise naturally in the toric case. Finally, we introduce affine toric varieties determined by strongly convex rational cones. 
Abstract: In this talk, we introduce fans and the (abstract) toric variety determined by a fan via gluing affine toric varieties defined by the cones in the fan. We include some examples and conclude with the correspondence between orbits of the torus action and the cones in the fan. 
18 October is Kurban Bayramı.
Abstract: We will revise the material on toric varieties with emphasis on examples and introduce some new concepts as time permits. 
Abstract: We will continue to discuss the material in Brasselet's exposition "Geometry of toric varieties", sections 5 and 6, as time permits. 
Abstract: We will complete our discussion of the material in Brasselet's exposition "Geometry of toric varieties", sections 5 and 6. 
Abstract: We will complete our discussion with more examples. 
2124
Nov 2013 Japanese Turkish Joint Geometry Meeting,
Galatasaray University, İstanbul
Abstract: In this very introductory talk I will try to discuss the interplay between such concepts as embedded toric resolutions of singularities via Newton polygons, Viro’s combinatorial patchworking, and tropical geometry. 
Abstract: We start with the definition of normal, very ample and smooth polytopes. We next define the projective toric variety $X_A$ determined by a finite set $A$ of lattice points. When $A$ is the lattice points of a polytope $P$ we demonstrate that $X_A$ reflects the properties of $P$ best if $P$ is very ample. We also define the normal fan of $P$ and discuss the relation between the corresponding "abstract" variety $X_P$ and the embedded variety $X_A$. 
Abstract: This is a continuation of my previous talk. After a brief introduction to Hilbert’s 16$^{\rm th}$ problem, I will try to outline the basic ideas underlying Viro’s method of patchworking real algebraic varieties. 
Abstract: The aim of this talk is to introduce the so called homogeneous coordinate ring of a normal toric variety. We will see how Chow group of Weil divisors turn this ring into a graded ring. Finally we show that every normal toric variety is a categorical quotient. 
ODTU, 27 December 2013, Friday, 15:40
Mesut Şahin[Karatekin]
 Coordinate ring of a toric variety
II
Abstract: After the promised example of "bad" quotient, I will review the correspondence between subschemes of a normal toric variety and multigraded ideals of its homogeneous coordinate ring. 
2014 Spring Talks
The first half of this semester is devoted
to toric varieties. The speaker will be mostly MESUT
SAHIN. The basic source will be the book: Toric Varieties, Cox, Little and Schenck, Graduate studies in mathematics vol 124, American Mathematical Society, 2011. 
The second half of this semester will be
devoted to deformation theory. The speaker for this topic
will be exclusively EMRE
COSKUN. He will follow the book: Deformations of Algebraic Schemes, Edoardo Sernesi, SpringerVerlag, 2006. (Grundlehren der mathematischen Wissenschaften, no. 334) 
Abstract: After recalling briefly basics of sheaf of a divisor on a normal variety, we will concentrate on the toric case. In particular, we give an explicit description of global sections of the sheaf of a torus invariant divisor. 
Abstract: We will continue with divisors and sheaves on toric varieties. Reference is chapter 4 of Cox, Little and Schenck. 
Abstract: We will talk about quasicoherent sheaves on the normal toric variety which come from multigraded modules over its Cox ring. 
Abstract: We will talk about The Toric IdealVariety Correspondence from CoxLittleSchenck's book Toric Varieties, see in particular page 220. 
Abstract: .We will talk about the correspondence between closed subschemes in a normal toric variety and Bsaturated homogeneous ideals in its Cox ring. 
Abstract: We
will talk about how multigraded Hilbert functions
can be used to compute dimensions of toric codes and
list some basic properties of multigraded Hilbert
functions. 
Abstract: We
will give a nice formula for the dimension of toric
complete intersection codes. We also give a bound on
the multigraded regularity of a zero dimensional
complete intersection subscheme of a projective
simplicial toric variety. The latter is important to
eliminate trivial codes. 
Abstract: In this series of lectures, we will develop deformation theory of functors of Artin rings. After discussing extensions of algebras over a fixed base ring, we will develop the theory of functors of Artin rings. These occur as 'local' versions of various moduli problems, and can give information about the local structure (e.g. smoothness, dimension) of moduli spaces near a point. We apply the theory to concrete examples of moduli problems, such as invertible sheaves on a variety, Hilbert schemes and Quot schemes. 
Abstract: Last
week we defined the 
ODTU, 9 May 2014, Friday, 15:40
Emre Coskun[ODTÜ]  Deformation Theory 3
Abstract: This week we will start formal deformation theory. This will be the content of chapter 2 in Sernesi's book. 
Bilkent, 16 May 2014, Friday, 15:40
Emre Coskun[ODTÜ]  Deformation Theory 4
Abstract: Last time we discussed briefly Schlessinger's theorem. We will continue from there. 
Bilkent, 21 May 2014, Wednesday,
15:40
Caner Koca[Vanderbild]  The MongeAmpere Equations and Yau's Proof of
the Calabi Conjecture
Abstract: The resolution of Calabi's Conjecture by S.T. Yau in 1977 is considered to be one of the crowning achievements in mathematics in 20th century. Although the statement of the conjecture is very geometric, Yau's proof involves solving a nonlinear second order elliptic PDE known as the complex MongeAmpere equation. An immediate consequence of the conjecture is the existence of KählerEinstein metrics on compact Kähler manifolds with vanishing first Chern class (better known as CalabiYau Manifolds). In this expository talk, I will start with the basic definitions and facts from geometry to understand the statement of the conjecture, then I will show how to turn it into a PDE problem, and finally I will highlight the important steps in Yau's proof. 
ODTU, 23 May 2014, Friday, 15:40
Emre Coskun[ODTÜ]  Deformation Theory 5
Abstract: We will discuss the closing remarks of deformation theory for this semester. 
Bilkent, 27 May 2014, Tuesday,
15:40
Caner Koca[Vanderbilt]  Einstein's Equations on Compact Complex
Surfaces
Abstract: After a brief review of Einstein's Equations in General Relativity and Riemannian Geometry, I will talk about one of my results: The only positively curved Hermitian solution to Einstein's Equations (in vacuo) is the FubiniStudy metric on the complex projective plane. 
Bilkent, 3 June 2014, Tuesday,
15:40
Caner Koca[Vanderbilt]  Extremal Kähler Metrics and BachMaxwell
Equations
Abstract: Extremal Kähler metrics are introduced by Calabi in 1982 as part of the quest for finding "canonical" Riemannian metrics on compact complex manifolds. Examples of such metrics include the KählerEinstein metrics, or more generally, Kähler metrics with constant scalar curvature. In this talk, I will start with an expository discussion on extremal metrics. Then I will show that, in dimension 4, these metrics satisfy a conformallyinvariant version of the classical EinsteinMaxwell equations, known as the BachMaxwell equations, and thereby are related to physics (conformal gravity) in a surprising and mysterious way. 
2014 Fall Talks
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: 
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, socalled 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 K3surfaces 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 1parameter family being projectively rigid. 
Abstract: This is the second part of the previous talk. See the above abstract. 
Abstract: With this talk we start our series of talks on "Girondo and GonzalezDiez, 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. 
Abstract: We continue with the topology of Riemann surfaces. 
Abstract: We will finish the first chapter on compact Riemann surfaces. The main topic this week will be function fields on Riemann surfaces. 
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. 
Abstract: This week we start with hyperbolic geometry. 
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. 
Abstract: We will start talking about Fuchsian groups. 
ODTU, 28 November 2014, Friday, 15:40
Özgür Kişisel[ODTÜ]  Riemann surfaces and discrete groups  V
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
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 TheoremI
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 TheoremII
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 TheoremIII
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 GonzalezDiez. 
ODTÜBİLKENT Algebraic
Geometry Seminar
(See all past talks ordered according
to speaker and date)
2015 Spring Talks
We will mainly continue our
learning seminar on Dessins
d'enfants. We follow the following book: 
Abstract: Counting lines on surfaces of fixed degree in projective space is a topic in algebraic geometry with a long history. The fact that on every smooth cubic there are exactly 27 lines, combined in a highly symmetrical way, was already known by 19th century geometers. In 1943 Beniamino Segre stated correctly that the maximum number of lines on a smooth quartic surface over an algebraically closed field of characteristic zero is 64, but his proof was wrong. It has been corrected in 2013 by Slawomir Rams and Matthias Schütt using techniques unknown to Segre, such as the theory of elliptic fibrations. The talk will focus on the generalization of these techniques to quartics admitting isolated ADE singularities. 
Ferruh Özbudak[ODTÜ]   Perfect nonlinear and quadratic maps on finite fields and some connections to finite semifields, algebraic curves and cryptography 
Abstract: Let 
Ali Sinan Sertöz[Bilkent]  Belyi's TheoremIV
Abstract: In
the previous talks, the proof of Belyi's theorem was
completed modulo a finiteness criterion. In this
talk we will prove that criterion. Namely, we
will prove that a compact Riemann surface 
Davide Cesare Veniani[Leibniz
University of Hanover] 
An introduction to elliptic fibrations 
part I: Singular Fibres
Abstract: The
theory of elliptic fibrations is an important tool
in the study of algebraic and complex surfaces. The
talk will focus on Kodaira's classification of
possible singular fibres. I will construct some
examples of rational and K3 elliptic surfaces to
illustrate the theory, coming from pencils of plane
cubics and lines on quartic surfaces. 
Davide Cesare Veniani[Leibniz
University of Hanover] 
An introduction to elliptic fibrations  part II:
MordellWeil group and torsion sections
Abstract: Given
an elliptic surface, the set of sections of its
fibration forms a group called the MordellWeil
group. After recalling the main concepts from part
I, I will expose the main properties of this group,
with a special focus on torsion sections. I will
give two constructions on quartic surfaces which
appear naturally in the study of the enumerative
geometry of lines, where torsion sections play a
prominent role. 
Ali Sinan Sertöz[Bilkent]  Belyi's TheoremV
Abstract: This
is the last talk in our series of talks on Belyi's
theorem. In this talk I will outline the proof of
the fact that a compact Riemann surface 
Ali Sinan Sertöz[Bilkent]  Exit Belyi, enter dessins d'enfants
Abstract: This week I will first clarify some of the conceptual details of the proof of Belyi's theorem that were left on faith last week. After that we will start talking about dessins d'enfants. 
Ali Sinan Sertöz[Bilkent]  From dessins d'enfants to Belyi pairs
Abstract: We will describe the process of obtaining a Belyi pair starting from a dessin d'enfant. 
Ali Sinan Sertöz[Bilkent]  Calculating the Belyi function associated to a dessin
Abstract: I will go over the calculation of the Belyi pair corresponding to a particular dessin given in the book, see Example 4.21. Time permitting, I will briefly talk about constructing a dessin from a Belyi pair. 
Ali Sinan Sertöz[Bilkent]  From Belyi pairs to dessins
Abstract: We will talk about obtaining a dessin from a Belyi function. 
Alexander Klyachko[Bilkent]  Exceptional Belyi coverings
Abstract: (This is a joint project with Cemile Kürkoğlu.) Exceptional covering is a connected Belyi coverings uniquely determined by its ramification scheme. Well known examples are cyclic, dihedral, and Chebyshev coverings. We add to this list a new infinite series of rational exceptional coverings together with the respective Belyi functions. We shortly discuss the minimal
field of definition of a rational exceptional
covering and show that it is either Existing theories give no upper bound on degree of the field of definition of an exceptional covering of genus 1. It is an open question whether the number of such coverings is finite or infinite. Maple search for an exceptional
covering of 
Alexander Degtyarev[Bilkent]  Dessins d'enfants and topology of algebraic curves
Abstract: I will give a brief introduction into the very fruitful interplay between Grothendieck's dessins d'enfants, subgroups of the modular group, and topology and geometry of trigonal curves/elliptic surfaces/Lefschetz fibrations. 
ODTÜBİLKENT Algebraic Geometry
Seminar
**** 2015 Fall Talks ****

Alexander Degtyarev[Bilkent]  Lines
on smooth quartics
Abstract: In
1943, B. Segre proved that a smooth quartic surface in
the complex projective space cannot contain more than
64 lines. (The champion, socalled Schur's quartic,
has been known since 1882.) Even though a gap was
discovered in Segre's proof (Rams, Schütt, 2015), 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 that not any number
between 0 and 64 can occur as the number of lines in a
quartic. 
Ali Sinan Sertöz[Bilkent]  The
basic theory of elliptic surfacesI
Abstract: This term we will be running a learnin seminar on elliptic surfaces with a view toward "lines on quartic surfaces". We will be mainly following Miranda's classical notes but other sources will not be excluded. 
Ali Sinan Sertöz[Bilkent]  The basic theory of elliptic surfacesII
Abstract: We continue our learning seminar talk on elliptic surfaces. We will also mention how this topic shows up in the search for lines on quartic surfaces in . 
Ergün Yalçın[Bilkent]  Group actions on spheres with rank one isotropy
Abstract: Actions of finite groups on spheres can be studied in various different geometrical settings, such as (A) smooth Gactions on a closed manifold homotopy equivalent to a sphere, (B) finite Ghomotopy representations (as defined by tom Dieck), and (C) finite GCW complexes homotopy equivalent to a sphere. These three settings generalize the basic models arising from unit spheres S(V) in orthogonal or unitary Grepresentations. In the talk, I will discuss the group theoretic constraints imposed by assuming that the actions have rank 1 isotropy (meaning that the isotropy subgroups of G do not contain , for any prime ). This is joint work with Ian Hambleton. 
Özgün Ünlü[Bilkent]  Free
group actions on products of spheres
Abstract: In this talk we will discuss the problem of finding group theoretic conditions that characterizes the finite groups which can act freely on a given product of spheres. The study of this problem breaks up into two aspects: (1) Find group theoretic restrictions on finite groups that can act freely on the given product. (2) Construct explicit free actions of finite groups on the given product. I will give a quick overview of the first aspect of this topic. Then I will discuss some recently employed methods of constructing such actions. 
Recep Özkan[ODTÜ] Concrete sheaves and continuous spaces
Abstract: This
is a talk from the speaker's recent dissertation.
After he summarizes the historical background and the
recent developments in the field he will motivate his
dissertation problems. Time permitting he will talk
about the ideas behind the proof of his main theorem. 
Cem Tezer[ODTÜ]  Anosov
diffeomorphisms : Revisiting an old idea
Abstract: Introduced by D. V. Anosov as the discrete time analogue of geodesic flows on Riemann manifolds of negative sectional curvature, Anosov diffeomorphisms constitute one of the leitmotivs of contemporary abstract dynamics. It is conjectured that these diffeomorphisms occur on very exceptional homogeneous spaces. The speaker will delineate the basic facts and briefly mention his own recent work towards settling this conjecture. 
Haydar Göral[Université Lyon 1]  Primality via Height Bound
Abstract: Height functions are of fundamental importance in Diophantine geometry. In this talk, we obtain height bounds for polynomial ring over the field of algebraic numbers. This enables us to test the primality of an ideal. Our approach is via nonstandard methods, so the mentioned bounds will be ineffective. We also explain the tools from nonstandard analysis. 
Alperen Ergür[Texas A&M]  Tropical Varieties for Exponential Sums
Abstract: We
define a variant of tropical varieties for exponential
sums. These polyhedral complexes can be used to
approximate, within an explicit distance bound, the
real parts of complex zeroes of exponential sums. We
also discuss the algorithmic efficiency of tropical
varieties in relation to the computational hardness of
algebraic sets. Our proof involves techniques from
basic complex analysis, inequalities and some recent
probabilistic estimates on projections that might be
of interest to analyst. 
Ali Ulaş Özgür Kişisel[ODTÜ] Moduli space of elliptic curves
Abstract: The aim of this talk is to view the moduli space of elliptic curves in different contexts. After briefly discussing the classical setting, we will see how it can be viewed as an orbifold and as an algebraic stack. 
Mesut Şahin[Hacettepe]  On
Pseudo Symmetric Monomial Curves
Abstract: In this talk,
we introduce monomial curves, toric ideals and
monomial algebras associated to 4generated pseudo
symmetric numerical semigroups. We give a
characterization of indispensable binomials of
these toric ideals, and of these monomial algebras
to have strongly indispensable minimal graded free
resolutions. We also discuss when the tangent
cones of these monomial curves at the origin are
CohenMacaulay in which case Sally's conjecture
will be true. 
ODTÜBİLKENT Algebraic Geometry
Seminar
(See all past talks ordered according
to speaker and date)
**** 2016 Spring Talks ****
The theme of this term is 
Alexander Degtyarev[Bilkent]  Skeletons
Abstract: This is
section 1.2. In a sense it is the heart of the
book: It explains how boring algebra can be
translated into the intuitive language of
pictures. (Of course, then it turns out that
pictures are not so easy, either, but that’s
another story.) 
Alexander Degtyarev[Bilkent]  SkeletonsII
Abstract: This
talk is a continuation of the previous week's talk. 
Ali Sinan Sertöz[Bilkent]
 Elliptic Surfaces
Abstract: We
will give an introduction to the concepts of elliptic
surfaces. We will mainly follow the order of Section
3.2 of the book. 
Ali Sinan Sertöz[Bilkent]
 Elliptic Surfaces and Weierstrass theory
Abstract: We will talk
about the Weierstrass theory and the jinvariant
of elliptic surfaces. 
Alexander Degtyarev[Bilkent]  Trigonal
curves and monodromy
Abstract: We will
discuss the simple analytic (Calculus 101)
properties of the invariant
and the way how it affects the singular fibers.
Then, we will start the discussion of trigonal
curves, fundamental groups, the braid monodromy,
and its relation to the invariant. 
Alexander Degtyarev[Bilkent]  Trigonal curves and monodromy  II
Abstract: We will
discuss the fundamental groups, braid monodromy,
Zariskivan Kampen theorem, and the relation
between the braid monodromy, dessins, and the invariant,
implying that the monodromy group is one of genus
zero and imposing strong restrictions on the
fundamental group. 
Alexander Degtyarev[Bilkent]  Trigonal
curves and monodromy  III
Abstract: We continue the description of the braid monodromy of a trigonal curve and its relation to the dessin. The principal result is the fact that the monodromy group is a subgroup of genus zero. As an immediate application, we will discuss the dihedral coverings ramified at trigonal curves (equivalently, torsion of the Mordell—Weil group of an elliptic surface) and a trigonal curve version of the socalled Oka conjecture. 
ODTÜ, 22 April 2016, Friday, 15:40
Cancelled in favour of 4th Cemal Koç
Algebra Days at METU
Alexander Degtyarev[Bilkent]  Trigonal
curves and monodromy: further applications
Abstract: As yet another
application of the relation between the braid
monodromy and invariant,
we will derive certain universal bounds for the
metabelian invariants of the fundamental group of
a trigonal curve. 
ODTÜBİLKENT
Algebraic Geometry Seminar
(See all past talks ordered according
to speaker and date)
**** 2016 Fall Talks ****

Alexander Degtyarev[Bilkent]  Lines
in K3 surfaces
Abstract: The unifying
theme of this series of talks is the classical
problem of counting lines in the projective models
of surfaces
of small degree. Starting with such classical
results as Schur's quartic and Segre's bound
(proved by Rams and Schütt) of lines
in a nonsingular quartic, I will discuss briefly
our recent contribution (with I. Itenberg and A.
S. Sertöz), i.e., the complete classification of
nonsingular quartics with many lines. Most quartics found (in an
implicit way) in our work are ``new'', attracting
the attention of experts in the field (Rams,
Schütt, Shimada, Shioda, Veniani). For example,
one of them turned out an alternative nonsingular
quartic model of the famous Fermat surface raising
the natural question if there are other such
models. An extensive search (Shimada, Shioda)
returned no results, and we show that, although
there are over a thousand singular models,
only two models
are smooth! Taking this line of research slightly
further, one can classify all smooth quartic
models of singular surfaces
of small discriminant, arriving at a remarkable
alternative characterisation of Schur's
quarticthe champion carrying lines:
it is also the (only) smooth quartic of the
smallest possible discriminant, which is .
Going even further, we can study other projective
models of small degree; counting lines in these
models, we arrive at the following conjectures:
These conjectures are still wide
open; I only have but a few examples. 
Alexander Degtyarev[Bilkent]  Lines in K3
surfacesII
Abstract: This
is the continuation of last week's talk. 
Ali Sinan Sertöz[Bilkent]  Introduction
to complex K3 surfaces
Abstract: We will start
reviewing and explaining as the case might be some
introductory concepts in K3 surface theory. The
level will be introductory so it is a good
opportunity so jump on the "wagon". 
Ali Sinan Sertöz[Bilkent]  K3
surfaces and lattices
Abstract:We will introduce some basic concepts of lattice theory that are used to understand K3 surfaces with a view towards Torelli type theorems. 
Ali Sinan Sertöz[Bilkent] K3 lattice of a K3 surface
Abstract: We will
continue our series on K3 surfaces by examining
the cohomology of K3 surfaces and finding out how
this cohomology structure characterizes the
surface. 
Abstract: In this talk
we discuss the problem of classifying complex
nonspecial simple quartics up to equisingular
deformation by reducing the problem to an
arithmetical problem about lattices. On this
arithmetical side, after applying Nikulin's
existence theorem, our computation based on the
MirandaMorrison's theory computing the genus
groups. We give a complete description of
equisingular strata of nonspecial simple
quartics. Finally we give ideas of the proof
of our principal result. 
Çisem Güneş[Bilkent]  Classification
of simple quartics up to equisingular deformationII
Abstract: This is the
continuation of last week's talk. 
Oğuzhan Yörük[Bilkent]  Which
K3 surfaces of Picard rank 19 cover an Enriques surface?
Abstract: The
parities of the entries of the transcendental
lattice of a K3 surface determine,
in most cases, if covers an
Enriques surface or not. We will summarize what is
known about this problem and talk about the missing
case when . 
Alexander Degtyarev[Bilkent]  Projective
models of surfaces
Abstract: Now, that we
know everything about abstract surfaces,
I will try to take a closer look at SaintDonat's
seminal paper and
share my findings. This 
Ali Ulaş Özgür Kişisel  [ODTÜ]  Arithmetic
of K3 surfaces
Abstract: I'll try to
outline some of the results in the survey paper of
M. Schütt with the same title. 
ODTÜBİLKENT
Algebraic Geometry Seminar
(See all past talks ordered
according to speaker and date)
**** 2017 Spring Talks ****

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." 
Ali Sinan Sertöz[Bilkent]  On
the moduli of K3 surfacesII
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. 
Ali Ulaş Özgür Kişisel[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. 
Ali Ulaş Özgür Kişisel[ODTU]  Tropical curvesII
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. 
Emre Coşkun[ODTU]  The Beilinson spectral sequence
Abstract: We overview
the Beilinson spectral sequence and its
applications in the construction of sheaves and
vector bundles. 
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 ,
(42 lines) and octic surfaces/triquadrics in ,
(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
K3surfaces in large projective spaces is 24
lines, all constituting fiber components of an
elliptic pencil. 
Mesut Şahin[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. 
Nil Şahin[Bilkent]  On Pseudo Symmetric Monomial Curves
Abstract: After giving
basic definitions and concepts about symmetric and
pseudo symmetric numerical semigroups, we will
focus on 4generated pseudo symmetric numerical
semigroups/monomial curves. Determining the
indispensable binomials of the defining ideal, we
will give characterizations under which the
tangent cone is CohenMacaulay. 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. 
Alexander Klyachko[Bilkent]  Transformation of cyclic words into Lie elements
Abstract: Let be
a complex vector space and be
its tensor algebra. We are primarily
concerned with Lie subalgebra generated
by commutators of elements in and
graded by degrees of the tensor components. where is a primitive root of unity of degree , is cycle in symmetric group acting on by permutation of tensor factors. The majorization index of permutation is defined as follows The operators 