ODTÜ-BÝLKENT Algebraic Geometry Seminar

2000 Fall Talks

1. Bilkent, 29 Sept 2000
Ali Sinan Sertöz, Cauchy's formula and applications

2. Bilkent, 6 Oct 2000
Ali Sinan Sertöz, Cauchy's formula and applications II

3. ODTÜ, 13 Oct 2000
Özgür Kiþisel, Weierstrass theorems and corollaries

4. ODTÜ, 20 Oct 2000
Yýldýray Ozan, Nulstellensatz

5. Bilkent, 27 Oct 2000
Yýldýray Ozan, Complex manifolds

6. Bilkent, 3 Nov 2000
Yýldýray Ozan, Complex manifolds II

7. ODTÜ, 10 Nov 2000
Yýldýray Ozan, Submanifolds and subvarieties

8. ODTÜ, 17 Nov 2000
Feza Arslan, deRham and Dolbeault cohomology

9. Bilkent, 24 Nov 2000
Hurþit Önsiper, Calculus on complex manifolds

10. Bilkent, 1 Dec 2000
Hurþit Önsiper, Sheaves and cohomology

11. ODTÜ, 8 Dec 2000
Hurþit Önsiper, Sheaves and cohomology II

12. ODTÜ, 15 Dec 2000
Ali Sinan Sertöz, Cohomology

13. Bilkent, 22 Dec 2000
Ali Sinan Sertöz, Some cohomology calculations

14. Bilkent, 12 Jan 2001
Yýldýray Ozan, Topology of manifolds

15. ODTÜ, 19 Jan 2001
Yýldýray Ozan, Topology of manifolds II

2001 Spring Talks

1. Bilkent, 9 Feb 2001
Ali Sinan Sertöz, Vector Bundles, connections and curvature

2. Bilkent, 16 Feb 2001
Ali Sinan Sertöz, Vector Bundles, connections and curvature II

3. ODTÜ, 23 Feb 2001
Özgür Kiþisel, Harmonic theory on compact complex manifolds

4. ODTÜ, 2 Mar 2001
Özgür Kiþisel, Kahler manifolds

5. Bilkent, 16 Mar 2001
Özgür Kiþisel, Lefschetz decomposition

6. Bilkent, 23 Mar 2001
Özgür Kiþisel, Lefschetz decomposition II

7. ODTÜ, 30 Mar 2001
Ali Sinan Sertöz, Divisors and line bundles

8. ODTÜ, 6 Apr 2001
Yýldýray Ozan, Chern classes of line bundles

9. Bilkent, 13 Apr 2001
Yýldýray Ozan, Chern classes of line bundles II

10. Bilkent, 20 Apr 2001

11. ODTÜ, 27 Apr 2001
Özgür Kiþisel, Kodaira vanishing theorem

12. ODTÜ, 4 May 2001
Yýldýray Ozan, Lefschetz theorem on hyperplane sections

13. Bilkent, 11 May 2001
Yýldýray Ozan, Lefschetz theorem on (1,1)-classes

2001 Fall Talks:

1. Bilkent,  27 Sept 2001
Alexander Degtyarev- Hodge Theory; A Fresh Look

2. ODTÜ,  5 Oct 2001
Ali Sinan Sertöz- On some K3 covers of Enriques surfaces

3. ODTÜ, 12 Oct 2001
Ali Sinan Sertöz- On some K3 covers of Enriques surfaces II

4. Bilkent, 19 Oct 2001
Ali Sinan Sertöz- Smooth curves on algebraic K3 surfaces

5. Bilkent, 26 Oct 2001 Friday 16:00
Alexander Klyachko- Curves with big number of rational points

6. ODTÜ, 2 Nov 2001 Friday 16:00
Yýldýray Ozan-On relative topology of a real algebraic set in its complexification

7. ODTÜ, 9 Nov 2001 Friday 16:00
Hurþit Önsiper-On the moduli spaces of some special surfaces of general type

8. Bilkent, 16 Nov 2001 Friday 16:05
Feza Arslan-Hilbert functions and Arf rings

9. Bilkent, 23 Nov 2001 Friday 16:05
Serguei Stepanov-Codes on fibre products of Artin-Schreier curves

10. ODTÜ, 30 Nov 2001 Friday 16:00
Ergün Yalçýn- The variety of the cohomology ring of a finite group

11. ODTÜ, 7 Dec 2001 Friday 16:00
Ersan Akyýldýz-Group actions and cohomology of homogeneous spaces

12. Bilkent, 14 Dec 2001 Friday 16:05
Özgür Kiþisel- Integrable systems and quantum cohomology

13. Bilkent, 21 Dec 2001 Friday 16:05
Meral Tosun-Tyurina components

14. Bilkent, 28 Dec 2001 Friday 16:05
Sergey Finashin- Invariants of 3+1 dimensional manifolds after Ozvath and Szabo

2002 Spring Talks

1. ODTÜ, 1 March 2002 Friday 15:40
Ersan Akyýldýz- Group actions and cohomology of homogeneous spaces-II

2. ODTÜ,  8 March 2002 Friday 15:40
Ersan Akyýldýz- Group actions and cohomology of homogeneous spaces-III

3. ODTÜ, 15 March 2002 Friday 15:40
Hurþit Önsiper- Some pathologies in characteristic p

4. Bilkent, 22 March 2002 Friday 15:40
Ergün Yalçýn-Steenrod closed ideals in polynomial rings over Fp

5. Bilkent, 29 March 2002 Friday 15:40
Feza Arslan- Cayley-Bacharach Theorems

6. ODTÜ, 5 April 2002 Friday 15:40
Hurþit Önsiper- Cayley-Bacharach property revisited

7. ODTÜ, 12 April 2002 Friday 15:40
Özgür Kiþisel- Quantum Cohomology

8. Bilkent, 19 April 2002 Friday 16:40
Alexander Klyachko- Branched coverings of torus and holomorphic differentials

9. Bilkent, 26 April 2002 Friday 15:40
Yýldýray Ozan- Algebraic and Hamiltonian circle actions

10. ODTÜ, 3 May 2002 Friday 15:40
Sergey Finashin- Some Real Algebraic Aspects of Quantum Cohomology and Mirror Symmetry

11. ODTÜ, 10 May 2002 Friday 15:40
Lucian Badescu-An introduction to the algebro-geometric aspect of Hilbert's 14th problem (according to Zariski)
12. Bilkent, 17 May 2002 Friday 15:40
Alexander Klyachko- Branched coverings of torus and holomorphic differentials-II, Counting coverings

2002 Fall Talks

1. Bilkent, 11 October 2002 Friday 15:40
Alexander Klyachko- Spectral Problems and Geometry (ICM 2002 Beijing Talk)

2. ODTÜ, 18 October 2002 Friday 15:40
Hurþit Önsiper- Vanishing Theorems

3. ODTÜ, 25 October 2002 Friday 15:40
Turgut Önder- Equivariant Almost Complex Substructures on Manifolds-an update and the techniques involved

4. Bilkent, 1 November 2002 Friday 15:40
Yusuf Civan- Topology of  Torus Actions

5. Bilkent, 8 November 2002 Friday 15:40
Yýldýray Ozan- Real Algebraic Differential Forms in Real Algebraic Geometry

6. ODTÜ, 15 November 2002 Friday 15:40
Ebru Keyman- Braids with Multiple Points

7. ODTÜ, 22 November 2002 Friday 15:40
Burak Özbaðcý- Topology of Stein Domains

8. Bilkent, 29 November 2002 Friday 15:40
Huishi Li- Some Noncommutative Quadric Algebras

9. Bilkent, 13 December 2002 Friday 15:40
Emrah Çakçak- Subfields of the Function Field of the Deligne-Lusztig Curve of Ree Type

10. ODTÜ, 20 December 2002 Friday 15:40
Ersan Akyýldýz- On the Factorization of Poincare Polynomials

11. ODTÜ, 27 December 2002 Friday 15:40
Ali Öztürk- Real Abelian Varieties with Many Line Bundles (after Prieto and Kollar)

12. Bilkent, 3 January 2003 Friday 15:40
Alexander Degtyarev- On Symmetric K3 Surfaces

2003 Spring Talks

1. Bilkent, 21 February 2003 Friday 15:40
Özgür Kiþisel- Introduction to Hodge Theory

2. ODTÜ, 28 February 2003 Friday 15:40
Yýldýray Ozan-Fundamental Structures in Hodge Theory

3. ODTÜ, 7 March 2003 Friday 15:40
Yýldýray Ozan- More Fundamental Structures in Hodge Theory

4. Bilkent, 14 March 2003 Friday 15:40
Yýldýray Ozan- Lefschetz Decomposition

5. Bilkent, 21 March 2003 Friday 15:40
Ali Sinan Sertöz- Spectral Sequences

6. ODTÜ, 28 March 2003 Friday 15:40
Ali Sinan Sertöz- Hodge Structures

7. Bilkent, 4 April 2003 Friday 15:40
Anthony Scholl- Recent Progress in the Arithmetic of Elliptic Curves

8. Bilkent, 11 April 2003 friday 15:40
Alexander Degtyarev- Hodge Theory

9. Bilkent, 18 April 2003 Friday 15:40
Alexander Degtyarev- Deligne Cohomology

10. ODTÜ, 25 April 2003 Friday 15:40
Ali Sinan Sertöz- Variation of Hodge Structure within a Flat Family

11. ODTÜ, 2 May 2003 Friday 15:40
Ali Sinan Sertöz- Griffiths Periodicity

12. ODTÜ, 9 May 2003 Friday 15:40
Ali Özgür Kiþisel-Infinitesimal Period Relations

13. Bilkent, 16 May 2003 Friday 15:40
Ali Özgür Kiþisel-Infinitesimal Period Relations-II

2003 Fall Talks

1. ODTÜ, 3 October 2003 Friday 15:40
Ali Sinan Sertöz-  Residues and sheaf cohomology

2. Bilkent, 10 October  2003 Friday 15:40
Hurþit Önsiper-The Abel-Jacobi mapping

3. ODTÜ, 17 October 2003 Friday 15:40
Hurþit Önsiper-Properties of the Abel-Jacobi mapping

4. Bilkent, 24 October 2003 Friday 15:40
Alexander Degtyarev- The Inversion Theorem for Generic Hypersurfaces

5. ODTÜ, 31 October 2003 Friday 15:40
Ali Ulaþ Özgür Kiþisel- Lefschetz Pencils and Normal Functions

6. Bilkent, 7 November 2003 Friday 15:40
Ali Ulaþ Özgür Kiþisel- Lefschetz Pencils for the Working Algebraic Geometer

7. ODTÜ, 14 November 2003 Friday 15:40
Yýldýray Ozan- Topology of Certain Singular Hypersurfaces

8. Bilkent, 5 December 2003 friday 15:40
Yýldýray Ozan- Topology of Certain Singular Hypersurfaces-II

9. ODTÜ, 12 December 2003 Friday 15:40
Ali Sinan Sertöz-  Intermediate Jacobians

10. Bilkent, 19 December 2003 Friday 15:40
Ali Ulaþ Özgür Kiþisel-The Infinitesimal Abel-Jacobi Mapping for Hypersurfaces-I

11. ODTÜ, 26 December 2003 Friday 15:40
Ali Ulaþ Özgür Kiþisel-The Infinitesimal Abel-Jacobi Mapping for Hypersurfaces-II

2004 Spring Talks

1. Bilkent, 5 March 2004 Friday 15:40
Alp Bassa-  Introduction to Tropical Algebraic Geometry

2. ODTÜ, 12 March 2004 Friday 15:40
Ali Ulaþ Özgür Kiþisel-Work of Mikhalkin

3. Bilkent, 19 March 2004 Friday 15:40
Çaðatay Kutluhan-Countýng Curves via Lattice Paths in Polygons

4. ODTÜ, 26 March 2004 Friday 15:40
Ýnan Utku Türkmen-Enumeration of Tropical Curves in R2

5. Bilkent, 2 April 2004 Friday 15:40
Mustafa Topkara-Relating the enumeration of tropical curves to counting lattice paths

6. ODTÜ, 9 April 2004 Friday 15:40
Burcu Baran-Amoebas of algebraic varieties and tropical geometry

7. Bilkent, 30 April 2004 Friday 15:40
Hakan Güntürkün-Non-Archimedean Amoebas

8. ODTÜ, 7 May  2004 Friday 15:40
Ali Ulaþ Özgür Kiþisel-Connection between classical and tropical geometries

9. Bilkent, 14 May 2004 Friday 15:40
Yýldýray Ozan-Counting Holomorphic Curves by Tropical Curves

2004 Fall Talks

1. ODTÜ, 1 October 2004 Friday 15:40
Hurþit Önsiper- Hodge Theory and Algebraic Cycles-I

2. ODTÜ, 8 October 2004 Friday 15:40
Hurþit Önsiper- Hodge Theory and Algebraic Cycles-II

3. Bilkent, 15 October 2004 Friday 14:40
Alexander Klyachko-Schubert Calculus and Quantum Marginal Problem

4. ODTÜ, 22 October 2004 Friday 15:40
Yýldýray Ozan-Action of the homology of the diffeomorphism group

5. Bilkent, 5 November 2004 Friday 15:40
Ali Ulaþ Özgür Kiþisel-On parts of Kontsevich's proof of Witten's conjecture

6. ODTÜ, 12 November  2004 Friday 15:40
Ali Ulaþ Özgür Kiþisel-On parts of Kontsevich's proof of Witten's conjecture, II

7. Bilkent, 19 November 2004 Friday 15:40
Alexander Degtyarev- Dessins d'enfants, trigonal curves and elliptic surfaces, I

8. ODTÜ, 26 November 2004 Friday 15:40
Alexander Degtyarev- Dessins d'enfants, trigonal curves and elliptic surfaces, II

9. Bilkent, 3 December 2004 Friday 15:40
Ergün Yalçýn-Varieties of Modules and a Theorem of Jon Carlson

10. ODTÜ, 10 December 2004 Friday 15:40

11. Bilkent, 17 December 2004 Friday 15:40
Turgut Önder- Foliations on 4-manifolds and minimal genus of embedded surfaces

12. ODTÜ, 24 December 2004 Friday 15:40
Ferruh Özbudak- Improvements on Tsfasman-Vladut-Zink Bound

2005 Spring Talks

1. Bilkent, 18 February 2005 Friday 15:40
Ali Sinan Sertöz-  Zero cycles on surfaces

2. ODTÜ, 25 February 2005 Friday 15:40
Hurþit Önsiper- 0-cycles on open surfaces, I

3. Bilkent, 4 March 2005 Friday 15:40
Hurþit Önsiper- 0-cycles on open surfaces, II

4. ODTÜ, 11 March 2005 Friday 15:40
Yýldýray Ozan- Curves on threefolds and the intermediate Jacobians

5. Bilkent, 18 March 2005 Friday 15:40
Yýldýray Ozan- Curves on threefolds and the intermediate Jacobians, II

6. Bilkent, 25 March 2005 Friday 15:40
Alexander Degtyarev- K-theoretic and cohomological methods

7. Bilkent, 1 April  2005 Friday 15:40
Alexander Degtyarev- K-theoretic and cohomological methods-II

8. ODTÜ, 22 April 2005 Friday 15:40
Alexander Klyachko- Etale Cohomology

9. ODTÜ, 29 April 2005 Friday 15:40
Hurþit Önsiper-  Vengeance of Arithmetic, I

10. ODTÜ, 6 May 2005 Friday 15:40
Hurþit Önsiper- Vengeance of Arithmetic, II

2005 Fall Talks

1. Bilkent, 23 September 2005 Friday 15:40
Ali Sinan Sertöz-On the anti-invariant lattice of some K3 surfaces (joint with C. Koca)

2. ODTÜ, 7 October 2005 Friday 15:40
Ali Ulaþ Özgür Kiþisel - Roitman's theorem

3. Bilkent, 14 October 2005 Friday 15:40
Yýldýray Ozan-Roitman's theorem II

4. ODTÜ, 21 October 2005 Friday 15:40
Ali Ulaþ Özgür Kiþisel - Roitman's theorem III

5. ODTÜ,  11 November 2005 Friday 15:40
Müfit Sezer - Noether numbers for cyclic groups of prime order

6. ODTÜ, 18 November 2005 Friday 15:40
Selma Altýnok - Toric varieties associated with weighted graphs

7. Bilkent, 25 November 2005 Friday 15:40
Mesut Þahin- Roitman's theorem on complete intersections

8. Bilkent, 2 December 2005 Friday 15:40
Ali Sinan Sertöz- Introduction to pure motives

9. ODTÜ, 9 December 2005 Friday 15:40
Ali Sinan Sertöz- Introduction to pure motives-II

10. Bilkent, 16 December 2005 Friday 15:40
Alexander Klyachko- Cayley hyperdeterminants

11. ODTÜ,  23 December 2005 Friday 15:40
Ali Sinan Sertöz- Introduction to pure motives-III

12. Bilkent,  30 December 2005 Friday 14:40
Kazým Büyükboduk-Trivial Zeros, Kolyvagin Systems and Main Conjectures of Iwasawa Theory

2006 Spring Talks

1. ODTÜ,  17 February 2006 Friday 15:40
Ali Sinan Sertöz- The mean Jacobian of a threefold (après Tyurin)

2. Bilkent, 24 February 2006 Friday 15:40
Ali Sinan Sertöz- The cylinder map

3. ODTÜ,  3 March 2006  Friday 15:40
Ýnan Utku Türkmen- The Griffiths component

4. Bilkent,  10 March 2006 Friday 15:40
Ali Ulaþ Özgür Kiþisel- Abelian varieties, theta function and the Riemann theorem

5. Bilkent, 17 March 2006  Friday 15:40
Alexander Degtyarev- On deformations of singular sextics

6. ODTÜ, 24 March 2006  Friday 15:40
Ali Sinan Sertöz- Geometry of the cubic

7. ODTÜ, 31 March 2006  Friday 15:40
Ali Sinan Sertöz- Geometry of the cubic II

8. Bilkent, 7 April 2006  Friday 15:40
Ýnan Utku Türkmen- Lines on a cubic hypersurface

9. ODTÜ, 14 April 2006  Friday 15:40
Ali Sinan Sertöz-
The cubic threefold

10. Bilkent, 21 April 2006  Friday 15:40
Anthony J. Scholl- Moduli of curves of genus three

11. ODTÜ, 28 April 2006  Friday 15:40
Ali Ulaþ Özgür Kiþisel-
Spectral curves

12. Bilkent, 5 May 2006  Friday 15:40
Alexander Klyachko-Poncelet porism and the numerical range of matrices

2006 Fall Talks

1. Bilkent, 29 September 2006 Friday 15:40
Mesut Þahin- On some monomial curves that are set theoretic complete intersections

2. ODTÜ, 6 October 2006 Friday 15:40
Feza Arslan-Arf rings and index of regularity

3. Bilkent, 13 October 2006 Friday 15:40
Engin Özkan-Compatible (Ga,Gm ) actions on a toric surface

4. ODTÜ, 3 November 2006 Friday 15:40
Ali Ulaþ Özgür Kiþisel- An additive group action on the hyperquot scheme

5. Bilkent, 10 November 2006 Friday 15:40
Franz Lemmermeyer-Values of polynomials over F_p

6. ODTÜ, 17 November 2006 Friday 15:40
Alexander Klyachko-Kahler-Einstein metric on toric varieties

7. Bilkent, 24 November 2006 Friday 15:40
Ergün Yalçýn-The top Stiefel-Whitney class of an augmented regular representation

8. ODTÜ, 1 December 2006 Friday 15:40
Alexander Degtyarev-On total reality of meromorphic functions

9. Bilkent, 8 December 2006 Friday 15:40
Yýldýray Ozan-Contact Topology : An introduction and some examples

10. ODTÜ, 15 December 2006 Friday 15:40
Piotr Pragacz-Positivity of Schur function expansions of Thom polynomials

11. Bilkent, 22 December 2006 Friday 15:40
Marcel Morales-On the Nash problem for arcs on a singular surface

12. ODTÜ, 29 December 2006 Friday 15:40
Mustafa Kalafat-Scalar curvature and connected sums of self-dual 4-manifolds

2007 Spring Talks

1. Bilkent, 23 February 2007 Friday 15:40
Ergün Yalçýn- Serre's theorem in group cohomology

2. ODTÜ, 2 March 2007 Friday 15:40
Ali Sinan Sertöz- Preliminaries on K3 surfaces

3. Bilkent, 9 March 2007 Friday 15:40
Ali Sinan Sertöz- The Fermat quartic

4. ODTÜ, 16 March 2007 Friday 15:40
Ali Sinan Sertöz- Rational curves on an Enriques surface, apres Namikawa

5. Bilkent, 23 March 2007 Friday 15:40
Mustafa Devrim Kaba-Cycles, zeta functions and Tate conjectures

6. ODTÜ,  30 March 2007 Friday 15:40
Mustafa Devrim Kaba-Tate conjectures for some fibered surfaces

7. Bilkent, 6 April 2007 Friday 15:40
Alexander Degtyarev-Oka's conjecture on irreducible plane sextics

8. Bilkent, 20 April 2007 Friday 15:40
Ali Ulaþ Özgür Kiþisel- Rational curves on K3 surfaces

9. Bilkent, 4 May 2007 Friday 15:40
Alexander Klyachko-Invariants and covariants of skew symmetric forms

10. ODTÜ, 11 May 2007 Friday 15:40
Ali Ulaþ Özgür Kiþisel- Rational curves on K3 surfaces-II

11. Bilkent, 18 May  2007 Friday 15:40
Mesut Þahin- Extending STCI property of monomial curves

2007 Fall Talks

1. Bilkent, 19 October 2007 Friday 15:40
Mesut Þahin - Extending certain properties of monomial curves

2. ODTÜ, 26 October 2007 Friday 15:40
Feza Arslan - Standard bases and Cohen-Macaulayness

3. Bilkent, 2 November 2007 Friday 15:40
Pýnar Mete - Hilbert function via Semigroup gluing

4. ODTÜ,  9 November 2007 Friday 15:40
Müfit Sezer-Grobner basis of ideals generated by minors of matrices of indeterminates

5. Bilkent, 16 November 2007 Friday 15:40
Müfit Sezer-Grobner basis of ideals generated by minors of matrices of indeterminates-II

6. ODTÜ, 23 November 2007 Friday 15:40
Özgür Ünlü-On Buchsbaum-Eisenbud-Horrocks conjecture

7. Bilkent, 30 November 2007 Friday 15:40
Alexander Degtyarev-On irreducible sextics with non-abelian fundamental group

8. ODTÜ, 7 December  2007 Friday 15:40
Engin Özkan-Normality of  Toric Orbit Closures in G/P

9. Bilkent, 14 December  2007 Friday 15:40
Alexander Klyachko-Hurwitz numbers and moduli spaces

10. ODTÜ, 28 December  2007 Friday 15:40
Yýldýray Ozan-Luttinger surgery along Lagrangian submanifolds

2008 Spring Talks

1. Bilkent, 22 February 2008 Friday 15:40
Ali Ulaþ Özgür Kiþisel - Introduction to Spectral Curves

2. ODTÜ, 29 February 2008 Friday 15:40
Ali Sinan Sertöz- General Information on theta functions

3. Bilkent, 7 March 2008 Friday 15:40
Ali Sinan Sertöz- Theta functions on Abelian tori

4. ODTÜ, 21 March 2008 Friday 15:40
Ali Sinan Sertöz- Theta functions on Riemann surfaces

5. Bilkent, 28 March 2008 Friday 15:40
Ali Sinan Sertöz- Abel map and the Jacobi inversion theorem

6. ODTÜ, 4 April 2008 Friday 15:40
Ali Ulaþ Özgür Kiþisel - Applications to non-linear equations

7. Bilkent, 11 April 2008 Friday 15:40
Ali Ulaþ Özgür Kiþisel - Applications to non-linear equations-II

8. ODTÜ, 18 April 2008 Friday 15:40
Ali Ulaþ Özgür Kiþisel - Applications to non-linear equations-III

9. ODTÜ, 2 May 2008 Friday 15:40
Pýnar Topaloðlu Mete-Minimal systems of generators of toric varieties

10. Bilkent, 9 May 2008 Friday 15:40
Müfit Sezer-Separating invariants

11. ODTÜ,  16 May 2008 Friday 15:40
Alexander Klyachko-Pauli principle revisited

2008 Fall Talks

1. Bilkent, 19 September 2008 Friday 15:40
Ýnan Utku Türkmen-Higher indecomposable Chow cycles and Hodge-D conjecture

2. ODTÜ, 26 September 2008 Friday 15:40
Ali Sinan Sertöz- Correspondences apres Manin

3. Bilkent, 10 October 2008 Friday 15:40
Alexander Degtyarev-On the number of solutions of quadratic equations

4. ODTÜ, 17 October 2008 Friday 15:40
Ali Sinan Sertöz- Motifs apres Manin

5. Bilkent, 24 October 2008 Friday 15:40
Mehmetcik Pamuk-4-Manifolds with Free Fundamental Group

6. ODTÜ, 31 October 2008 Friday 15:40
Alexander Degtyarev-Singular plane sextics via dessins d'enfants

7. Bilkent, 7 November 2008 Friday 15:40
Müfit Sezer- Toric ideals and partition identities

8. ODTÜ, 14 November 2008 Friday 15:40
Mustafa Devrim Kaba- Manin's identity principle

9. Bilkent, 21 November 2008 Friday 15:40
Mustafa Devrim Kaba- Motives of curves and surfaces

10. ODTÜ, 28 November 2008 Friday 15:40
Sema Salur- Applications of Calibrations: Mirror Dualities

11. Bilkent, 5 December 2008 Friday 15:40
Alexander Klyachko- Quantum mechanics and Poncelet porism

12. ODTÜ, 19 December 2008 Friday 15:40
Ergün Yalçýn- A problem in commutative algebra related to group actions

13. Bilkent, 26 December 2008 Friday 15:40
Özgün Ünlü- Steenrod Operations and Transformation Groups

14. Bilkent, 22 January 2009 Thursday 15:40
James D. Lewis, Biextensions associated to algebraic cycles, I

15. Bilkent, 23 January 2009 Friday 15:40
James D. Lewis, Biextensions associated to algebraic cycles, II

2009 Spring Talks

1. Bilkent, 20 February 2009 Friday 15:40
Ali Sinan Sertöz--What is wrong with the proof of the Hodge conjecture?

 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!

2. ODTÜ, 27 February 2009 Friday 15:40
Ali Sinan Sertöz--Hodge conjecture; is it still open?

 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.

3. ODTÜ, 6 March 2009 Friday 15:40
Deniz Kutluay- Knot groups

 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.

4. Bilkent, 13 March 2009 Friday 15:40
Deniz Kutluay- Fox calculus

 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.

5. Bilkent, 20 March2009 Friday 15:40
Mesut Þahin-Toric ideals of simple surface singularities

 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.

6. ODTÜ, 27 March 2009 Friday 15:40
Münevver Çelik-Calculating Alexander polynomials

 Abstract: We will demonstrate different methods of calculating the Alexander polynomial on several examples.

7. Bilkent, 3 April 2009 Friday 15:40
Alexander Degtyarev-Towards the generalized Shapiro and Shapiro conjecture

 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 --> P1  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>d2/3+O(d). In the talk, we improve the above inequality to g>d2/4+O(d).

8. ODTÜ, 10 April 2009 Friday 15:40
Yýldýray Ozan-J-holomorphic curves in the study of symplectomorphism groups of symplectic 4-manifolds

 Abstract:  In this talk, after I describe algebraic automorphisms group of P1xP1, 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 J-holomorphic curves in symplectic topology and how they are employed in this work.

9. Bilkent, 17 April 2009 Friday 15:40
Alexander Degtyarev-Real elliptic surfaces and real elliptic curves of type I (joint w/I. Itenberg)

 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 (quasi-simplicity) in the case of rational base. (The results are partially interlaced with those by V. Zvonilov.) As a by-product, we obtain a criterion for a trigonal curve of type I to be isotopic to a maximally inflected one.

10. ODTÜ, 24 April 2009 Friday 15:40
Ýnan Utku Türkmen-A brief introduction to higher Chow groups

 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 Hodge-D conjecture) for product of two general elliptic curves.

11. Bilkent, 4 May 2009 Monday 15:40 -- Note the unusual date
Fatma Altunbulak Aksu-Varieties of modules and a filtration theorem

 Abstract: The variety of a finitely generated kG-module 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.

12. ODTÜ, 8 May 2009 Friday 15:40
Ali Sinan Sertöz-Preliminaries on motifs

 Abstract: We will outline the construction of pure motifs, concentrating on the Chow-Kunneth decomposition. Time permiting we intend to describe the transcendental part of the motif of a surface. This is an informal introductory talk.

13. Bilkent, 15 May 2009 Friday 15:40
Muhammed Uludað-The Universal Arithmetic Curve

 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" galois-invariant subsystem (not a subcategory) of genus-0 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 non-negatively 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 galois-invariant genus-0 subcategories of non-galois coverings in C. The talk is intended to be informal, relaxed and audience friendly.

2009 Fall Talks

1. Bilkent, 2 October 2009 Friday 15:40
Aslý Güçlükan--Vector Bundles and their classification

 Abstract:  The aim of this talk is to give the necessary background material on vector bundles to introduce the topological K-theory. We also explain the classification theorem for vector bundles. This talk is accessible to graduate students at any level.

2. Bilkent, 9 October 2009 Friday, 15:40
Aslý Güçlükan--Introduction to topological K-theory
 Abstract:  Last week we  discussed the basic properties of vector bundles over a compact base space X to introduce the topological K-theory. The set of isomorphism classes of vector bundles on X forms a commutative monoid. The idea of K-theory of X is the completion of this monoid to a ring. In this talk, we will discuss basic concepts in K-theory.

3. ODTU, 16 October 2009 Friday, 15:40
Yýldýray Ozan- On Bott periodicity theorem
 Abstract:  This is going to be an introductory talk to Bott's periodicity theorem.

4. Bilkent, 23 October 2009 Friday, 15:40
Ýnan Utku Türkmen- Introduction to Algebraic K-Theory
 Abstract:  This is going to be a introductory talk to algebraic K-theory. I will introduce algebraic K-theory 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 K-theories.

5. ODTU, 6 November 2009 Friday, 15:40
Ýnan Utku Türkmen- Introduction to Algebraic K-Theory II: K1 of rings
 Abstract:  In this introductory talk we will define K1 of rings and discuss their basic properties.

6. Bilkent, 13 November 2009 Friday, 15:40
Ergün Yalçýn- Wall's finiteness obstruction and its generalizations
 Abstract:  As one of the topological applications of algebraic K-theory, 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 CW-complex. Then, I will discuss the orbit category version of Wall's finiteness obstruction.

7. ODTU, 4 December 2009 Friday, 15:40
Ali Sinan Sertöz- K0 and K1 of categories
 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 Riemann-Roch theorem but that may be left to the next talk if time runs up.

8. Bilkent, 11 December 2009 Friday, 15:40
Özgün Ünlü- Spheres which are H-spaces
 Abstract:  We will talk about the proof of the well-known fact that an n-dimensional sphere is an H-space if and only if n=0, 1, 3, or 7.

9. Bilkent, 18 December 2009 Friday, 15:40
Ýzzet Coþkun-Birational geometry of moduli spaces
 Abstract:  The Kontsevich moduli space of stable maps is the central object in Gromov-Witten 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

1. Bilkent, 19 February 2010, Friday, 15:40
Ali Sinan Sertöz-[Bilkent University]-Grothendieck-Riemann-Roch Theorem

 Abstract:  We will conclude last term's seminar on K-theory with an application to algebraic geometry by developing Grothendieck's Riemann-Roch theorem. The talk will be expository and will be accessible even to those who do not remember much of last semester's talks!

2. ODTU, 26 February 2010 Friday, 15:40
Deniz Kutluay-[Bilkent University]-Jones Polynomial

 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.

3. 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).

4. ODTU, 12 March 2010 Friday, 15:40
Ýnan 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.

5. 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.

6. 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 G-bundle over the prismatic star. This is a joint work with Johan L. Dupont.

7. ODTU, 9 April 2010 Friday, 15:40
Yýldýray Ozan-[ODTU]- Algebraic K-theory 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 K-theory is used to study regular maps between real algebraic varieties. Namely, I will talk about the results of Loday and Bochnak-Kucharz which mainly show that regular maps between certain products of spherees are all null-homotopic. For example, Loday showed that any regular map from S1 x S1 to S2 is homotopically trivial, where Sk is the unit sphere in Rk+1.

8. ODTU, 16 April 2010 Friday, 15:40
Ali Kemal Uncu-[TOBB ETU]-  Modular symbols on congruence subgroups of SL2(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 SL2(Z). We will work with the Riemann surface related to the congruence subgroup of SL2(Z), define modular symbols and give the relation between modular symbols and Fourier coefficients of modular forms.

9. ODTU, 30 April 2010 Friday, 15:40
Ergün Yalçýn-[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.

10. 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.

11. ODTU, 14 May 2010 Friday, 15:40
Hamza Yeþilyurt-[Bilkent University]-Rogers-Ramanujan Functions

 Abstract:  We present several identities for the Rogers-Ramanujan functions along with their partition theoretic interpretations and conclude with our recent work on such identities.

12.

13. 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

1. Bilkent, 1 October 2010, Friday, 15:40

Alexander Degtyarev-[Bilkent University] - The Alexander module of a trigonal curve

 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 Fp ) 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.

2. ODTU, 8 October 2010 Friday, 15:40

Ömer Küçüksakallý-[ODTU] - Frey Curves and Fermat's Last theorem

 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 Taniyama-Shimura conjecture) implies Fermat's Last Theorem. A central role in Ribet's proof is played by elliptic curves introduced by Frey.

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On 13-15 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.
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3. Bilkent, 22 October 2010 Friday, 15:40

Christophe Eyral-[Aarhus University] - A short introduction to Lefschetz theory on the topology of algebraic varieties

 Abstract:

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29 October is Republic Day, a national day for Turkey. No talks!
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4. Bilkent, 5 November 2010 Friday, 16:00

Muhammed Uludað-[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.

5. ODTU, 12 November 2010 Friday, 15:40

Ýnan 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 non-trivial 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; CH3ind(E1x E2 x E1 x E2). I will review the results in literature related with this group and sketch an alternative proof for non-triviality of this group using a constructive method.

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16-19 November is a religious holiday in Turkey. No talks!
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6. ODTU, 26 November 2010 Friday, 15:40

Mehmetcik Pamuk-[ODTU] - s-cobordism classification of 4-manifolds

 Abstract: In this talk we are going to show how one can use the group of homotopy self-equivalences of a 4-manifold together with the modified surgery of Matthias Kreck to give an s-cobordism classification of topological 4-manifolds. We will work with certain fundamental groups and give s-cobordism classification in terms of standard invariants.

7. Bilkent, 3 December 2010 Friday, 15:40

Ergün Yalçýn-[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.

8. ODTU, 10 December 2010 Friday, 15:40

Mustafa Kalafat-[University of Wisconsin at Madison and ODTU] -
Hyperkahler manifolds with circle actions and the Gibbons-Hawking Ansatz

 Abstract: We show that a complete simply-connected hyperkahlerian 4-manifold with an isometric triholomorphic circle action is obtained from the Gibbons-Hawking ansatz with some suitable harmonic function.

9. Bilkent, 17 December 2010 Friday, 15:40

Kürþat 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).   Multiplicative generators for centers of integral symmetric groups were first identified by Farahat and Higman, which were later shown to be elementary symmetric polynomials in the celebrated Young-Jucys-Murphy elements by Jucys.  In this talk, we will present a set of multiplicative generators for the Hecke ring of (S(2n), H(n)), affirming a conjecture of Matsumoto, who showed these elements are elementary symmetric polynomials evaluated at odd Young-Jucys-Murphy elements after projecting from the integral group ring of S(2n) to the Hecke ring of (S(2n),H(n)). This is a joint work with Mahir Bilen Can, Tulane University.  * Reference: http://front.math.ucdavis.edu/1009.5373

10. 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!

11. -----------------------------------------------------------------------------------------------------------------------
31 December  afternoon is no time to hold seminars on this planet! No talks!
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2011 Spring Talks

1. Bilkent, 18 February 2011, Friday, 15:40

Ali Sinan Sertöz-[Bilkent University] - Basics of Spectral Sequences

 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!

2. ODTU, 25 February 2011 Friday, 15:40

Ali Sinan Sertöz-[Bilkent University] - Basics of Spectral Sequences II

 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.

3. 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.

-

4. 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.

5. Bilkent, 18 March 2011 Friday, 15:40

Alexander Degtyarev-[Bilkent University] - Leray-Serre 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.

6. ODTU, 1 April 2011 Friday, 15:40

Alexander Degtyarev-[Bilkent University] - Leray-Serre 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.

7. Bilkent, 8 April 2011 Friday, 15:40

Ergün Yalçýn- [Bilkent University] -The Lyndon-Hochschild-Serre spectral sequence

 Abstract:   Let G be a group and H be a normal subgroup of G. Then there is a spectral sequence, called LHS-spectral 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 HLS-spectral 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.

8. ODTU, 15 April 2011 Friday, 15:40

Ergün Yalçýn-[
Bilkent University] - Calculating with the LHS-spectral sequence

 Abstract: Let G be a group and H be a normal subgroup of G. There is a spectral sequence, called LHS-spectral 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 LHS-spectral 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.

9. Bilkent, 22 April 2011 Friday, 14:35 (Notice the new time for this talk)

Özgün Ünlü-[Bilkent University] -Atiyah-Hirzebruch spectral sequence

 Abstract: Let X be a CW complex and h be a generalized cohomology theory. Atiyah-Hirzebruch spectral sequence relates the generalized cohomology groups h_*(X) with ordinary cohomology groups with coefficients in the generalized cohomology of a point.

10. ODTU, 29 April 2011 Friday, 15:40

Yýldýray 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.

11. 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 Atiyah-Hirzebruch spectral sequence.

12. ODTU, 13 May 2011 Friday, 15:40

Mehmetcik Pamuk-[ODTU] - An Application of Atiyah-Hirzebruch Spectral Sequence

 Abstract:

13.

2011 Fall Talks

1. Bilkent, 7 October  2011, Friday, 15:40

Alexander Degtyarev-[Bilkent University] - Products of two Dehn twists and real Lefschetz fibrations

 Abstract:  (joint with Nermin Salepci, Université de Lyon)  An object repeatedly occurring in algebraic geometry is a fibration with singular fibers. If the base is a topological disk and the number of singular fibers is finite, the topology of such a fibration can adequately be described by the so called monodromy factorization of the monodromy at infinity (boundary of the disk), regarded up to a certain collection of moves, called Hurwitz moves and, possibly, global conjugation. We consider the simplest nontrivial case: factorizations into product of two Artin generators in the braid group B_3 (equivalently, two Dehn twists in the mapping class group of a torus). Even here, the results obtained are quite unexpected; considering the known examples, the general case (even in B_3) is very far from its complete understanding.    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 semi-simplicity statement for real trigonal M-curves 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.

2. ODTU, 14 October  2011, Friday, 15:40

Alexander Degtyarev-[Bilkent University] - Lines on quartic surfaces

 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 P3 is merely a K3-surface equipped with a polarization of degree 4. Thus, I will give a gentle introduction to theory of K3-surfaces: the period space, the global Torelli theorem and surjectivity of the period map, and the implications of the Riemann--Roch 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.

3. Bilkent, 21 October  2011, Friday, 15:40

Richard Gonzales-[Boðazici University] - KM theory of rationally smooth group embeddings.

 Abstract:  Let G be a reductive group. A GxG-variety 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 GxG-orbits. 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 three-fold. 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: Brion, M. ''The behaviour at infinity of the Bruhat decomposition''. Comment. Math. Helv. 73, pp. 137-174 (1998). Gonzales, R. ''GKM theory of rationally smooth group embeddings''. PhD thesis (2011).http://ir.lib.uwo.ca/etd/216/  Goresky, M., Kottwitz, R., MacPherson, R. ''Equivariant Cohomology, Koszul duality, and the localization theorem''. Invent. math. 131, pp. 25-83 (1998). Renner, L. '' The H-polynomial of an Irreducible Representation''. Journal of Algebra 332, pp. 159-186 (2011).  PS: The speaker is supported under TUBITAK ISBAP Grant  107T897 -Matematik Ýþbirliði Aðý: Cebir ve Uygulamalarý.

The afternoon of 28 October is a National Holiday.

4. Bilkent, 4 November  2011, Friday, 15:40

Ali Sinan Sertöz-[Bilkent University] - An overview of counting lines on algebraic surfaces

 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 Bayramý.

5. ODTU, 18 November  2011, Friday, 15:40

Mehmetcik Pamuk-[ODTU] - Surgery Method of Classifying Manifolds

 Abstract:  The surgery method of classifying manifolds seeks to answer the following question: Given a homotopy equivalence of m-dimensional 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.

6. Bilkent, 25 November  2011, Friday, 15:40

Selma A. Bhupal-[Hacettepe University] - On Piecewise Polynomial Functions and their Dimension

 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  Rn  and their dimension.

7. ODTU, 2 December  2011, Friday, 15:40

Ahmet Beyaz-[ODTU] - Genus Zero Gromov-Witten Invariants

 Abstract:  In this talk, we review the genus zero Gromov-Witten 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 Gromov-Witten invariants to distinguish the symplectic structures on a smooth 6-manifold is restricted in a certain sense.

8. Bilkent, 9 December  2011, Friday, 15:40

Mustafa Kalafat-[ODTU] - Geometric Invariant Theory and Einstein-Weyl Geometry

 Abstract:  We give a survey of Geometric Invariant Theory for Toric Varieties, and present an application to the Einstein-Weyl 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.

9. ODTU, 16 December   2011, Friday, 15:40

Ferruh Özbudak-[ODTU] - Finite number of Kummer cover and curves with many points

 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 construct expicit examples of fibre products of Kummer covers with many rational rational points, which includes a record and two new entries of the current table of the manypoints". This is a report on a joint work with Burcu Gulmez Temur.

10. Bilkent, 23 December  2011, Friday, 15:40

Mesut Þahin-[Karatekin University] - On Toric Codes

 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.

11. ODTU, 30 December  2011, Friday, 15:40

Alexander Klyachko-[Bilkent University] - Toric and Flag varieties

 Abstract:  In the talk I will discuss the structure of toric variety XG equal to closure of a generic orbit of a maximal torus of a simple group G in its flag variety FG, the respective restriction map H*(FG)-->H*(XG) together with some applications.

12. Bilkent, 6 January  2012, Friday, 15:40

Çetin Ürtiþ-[TOBB] - Sphere Packings, Lattices and Theta Functions

 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 face-centered 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 Atiyah-Singer Index Theorem,  Springer Lecture Notes in Mathematics No: 638.

1. Bilkent, 2 March  2012, Friday, 15:40

Ali Sinan Sertöz-[Bilkent University] - Atiyah-Singer Index Theorem - Preliminaries

 Abstract:  Preliminaries will be discussed; mostly characteristic classes.

2. ODTU, 9 March  2012, Friday, 15:40

Ali Sinan Sertöz-[Bilkent University] - Atiyah-Singer Index Theorem - Motivation and Statement

 Abstract:  I will talk about the motivation for the index theorem and discuss the individual terms in the statement of the theorem.

3. Bilkent, 16 March  2012, Friday, 15:40

Ýnan Utku Türkmen-[Bilkent University] - The de Rham and Dolbeault operators

 Abstract:  We examine the consequences of applying the Atiyah-Singer Index Theorem to de Rham and Dolbeault operators.

4. ODTU, 23 March  2012, Friday, 15:40

Mustafa Kalafat-[ODTU] - The Hodge operator

 Abstract:  In this talk we will demonstrate that the application of the Atiyah-Singer Index Theorem to Hodge operator yields the Hirzebruch signature theorem.

5. Bilkent, 30 March   2012, Friday, 15:40

Yýldýray Ozan-[ODTU] - The Dirac operator

 Abstract:  In this talk we will discuss the application of the Atiyah-Singer Index Theorem to Dirac operator.

6. ODTU, 6 April  2012, Friday, 15:40

Turgut Onder-[ODTU] - The ring K(X)

 Abstract:  In this talk we give a brief description of the ring K(X) of  stable vector bundles over X.

7. Bilkent, 13 April   2012, Friday, 15:40

Turgut Onder-[ODTU] - The ring K(X), II

 Abstract:  In this talk we continue to give a brief description of the ring K(X) of  stable vector bundles over X.

8. Bilkent, 20 April  2012, Friday, 15:40

Asli Guclukan Ilhan-[Bilkent] - The topological index B

 Abstract:  In this talk we will elaborate on the topological index B as covered in Shanahan's boook.

9. ODTU, 27 April  2012, Friday, 15:40

Mehmetcik Pamuk-[ODTU] - Pseudodifferential operators

 Abstract:  In this talk we will discuss pseudodifferential operators and their suitable generalizations as discussed in Shanahan's book.

10. Bilkent, 4 May  2012, Friday, 15:40

Özgün Ünlü-[Bilkent] - Construction of the index homomorphism

 Abstract:  In this talk we will discuss the construction of the index homomorphism as given in Shanahan's book.

11. Bilkent, 11 May  2012, Friday, 15:40

Ali Sinan Sertöz-[Bilkent University] - Proof of the index theorem

 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

1. Bilkent, 21 September  2012, Friday, 15:40

Alexander Degtyarev-[Bilkent] - On lines on smooth quartics

 Abstract:  (a never ending joint project with I. Itenberg and S. Sertoz) It is a common understanding that, thanks to the global Torelli theorem and the surjectivity of the period map, any reasonable question concerning the topology and geometry of $K3$-surfaces can be reduced to a certain arithmetical problem. We tried to apply this ideology to the study of the possible configurations of straight lines on a nonsingular quartic surface in $\mathbb{P}^3$. According to C. Segre, a nonsingular quartic in $\mathbb{P}^3$ may contain at most 64 lines, and one explicit example of a surface with exactly 64 lines is known. The original proof, using classical algebraic geometry in the Italian school style, is very complicated. We managed to reprove Segre's result using the contemporary arithmetical approach. In addition, we prove that, up to projective equivalence, a nonsingular quartic with 64 lines is unique. Furthermore, we show that a real nonsingular quartic may contain at most 56 real lines and, conjecturally, such a quartic is also unique (although the latter statement is not quite definite yet).  Alas, the proof is transparent but heavily computer aided, the principal achievement being a stage at which my laptop can handle it in finite time (although a human still cannot).

2. ODTU, 28 September   2012, Friday, 15:40

Ali Sinan Sertöz-[Bilkent] - The Chow ring of $\mathbb{G}(1,3)$

 Abstract:  I will start with Chapter 2 of Eisenbud-Harris 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 $\mathbb{P}^3$.

3. Bilkent, 5 October  2012, Friday, 15:40

Ali Sinan Sertöz-[Bilkent- The Chow ring of $\mathbb{G}(1,3)$, part II

 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.

4. ODTU, 12 October  2012, Friday, 15:40

Ali Sinan Sertöz-[Bilkent- The Chow ring of $\mathbb{G}(1,3)$, part III

 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.

5. Bilkent, 19 October   2012, Friday, 15:40

Ali Sinan Sertöz-[Bilkent- The Chow ring of $\mathbb{G}(1,3)$, part IV-last!

 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!

6. ODTU, 2 November  2012, Friday, 15:40

Tolga Karayayla-[ODTU] - Schubert calculus, Chow ring of Grassmannians, part I

 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.

7. Bilkent, 9 November  2012, Friday, 15:40

Tolga Karayayla-[ODTU] - Schubert calculus, Chow ring of Grassmannians, part II

 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.

8. ODTU, 16 November  2012, Friday, 15:40

Müfit Sezer-[Bilkent] - Invariants of the Klein four group in chracteristic two

 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)

9. Býlkent, 23 November  2012, Friday, 15:40

Emre Þen-[Bilkent] - Chow Groups of Rational Equivalence Classes of Cycles

 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:  "Let $L,Q\subset \mathbb{P}^3$ be a line and a nonsingular conic in $\mathbb{P}^3$. Is $\left( \mathbb{P}^3\setminus L\right)\cong\left( \mathbb{P}^3\setminus Q\right)$ as schemes?" (ref. page 139)

10. Bilkent, 30 November  2012, Friday, 15:40

Özgür Kiþisel-[ODTU] - Tropical Intersections

 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.

11. ODTU, 14 December  2012, Friday, 15:40

Tolga Karayayla-[ODTU] - Schubert calculus, Chow ring of Grassmannians, part III

 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.

12. ODTU, 21 December  2012, Friday, 15:40

Nil Þahin-[ODTU] - Singularity Theory and Arf Rings

 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.

13. Bilkent, 28 December  2012, Friday, 15:40

Mustafa Kalafat-[Tunceli] - Einstein-Hermitian 4-Manifolds of Positive Bisectional Curvature

 Abstract:  We show that a compact complex surface together with an Einstein-Hermitian metric of positive holomorphic bisectional curvature is biholomorphically isometric to the complex projective plane with its Fubini-Study metric up to rescaling.  (Joint work with C.Koca.)

2013 Spring Talks

1. ODTU, 1 March 2013, Friday, 15:40

Ali Sinan Sertöz-[Bilkent] - Lines on hypersurfaces

 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.

2. Bilkent, 8 March 2013, Friday, 15:40

Ali Sinan Sertöz-[Bilkent] - Chern classes as degeneracy cycles and some applications

 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.

3. ODTU, 15 March 2013, Friday, 15:40

Sergey Finashin-[METU] - Invariants of complex algebraic surfaces via differential topology

 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.

4. Bilkent, 22 March 2013, Friday, 15:40

Sergey Finashin-[METU] - Invariants of complex algebraic surfaces via differential topology, II

 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.

5. ODTU, 29 March 2013, Friday, 15:40

Yýldýray Ozan-[METU] - Some applications of Differential Topological Invariants to Algebraic Surfaces

 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.

6. Býlkent, 5 April 2013, Friday, 15:40

Özgür Ceyhan-[Luxembourg] - Feynman integrals as periods in configuration spaces

(This talk is organized in collaboration with Bilkent Department of Mathematics.)

 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.

7. ODTU, 12 April 2013, Friday, 15:40

Emre Coþkun-[METU] - An Introduction to Moduli Problems

 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.

8. Bilkent, 19 April 2013, Friday, 15:40

Emre Coþkun-[METU] - An Introduction to Moduli Problems, II

 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.

9. ODTU, 26 April 2013, Friday, 15:40

Koray Karabina-[Bilkent] - Elliptic Curve Discrete Logarithm Problem

 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.

10. Bilkent, 3 May 2013, Friday, 15:40

Alexander Degtyarev-[Bilkent] - On the Bertini involution

 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.

11. ODTU, 10 May 2013, Friday, 15:40

Mustafa Kalafat-[Tunceli] - Topology of $G_2$ manifolds

 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.

12. 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

1. ODTU, 4 October 2013, Friday, 15:40

Mesut Þahin-[Karatekin] - Affine toric varieties, cones, lattices, semigroup rings, toric ideals

 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.

2. Bilkent, 11 October 2013, Friday, 15:40

Mesut Þahin-[Karatekin] - Fans, toric variety of a fan via gluing affine toric varieties, Orbit-Cone correspondence

 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 Bayramý.

3. ODTU, 25 October 2013, Friday, 15:40

Mustafa Kalafat-[Tunceli] - Examples. Blow ups. Resolution of Singularities. Torus action. Orbits. Divisors.

 Abstract: We will revise the material on toric varieties with emphasis on examples and introduce some new concepts as time permits.

4. Bilkent, 1 November 2013, Friday, 15:40

Mustafa Kalafat-[Tunceli]- Blow ups. Resolution of Singularities. Torus action. Orbits. Divisors - I

 Abstract:  We will continue to discuss the material in Brasselet's exposition  "Geometry of toric varieties", sections 5 and 6, as time permits.

5. ODTU, 8 November 2013, Friday, 15:40

Mustafa Kalafat-[Tunceli]- Blow ups. Resolution of Singularities. Torus action. Orbits. Divisors - II

 Abstract:  We will complete our discussion of the material in Brasselet's exposition  "Geometry of toric varieties", sections 5 and 6.

6. Býlkent, 15 November 2013, Friday, 15:40

Mustafa Kalafat-[Tunceli]- Blow ups. Resolution of Singularities. Torus action. Orbits. Divisors - III

 Abstract:  We will complete our discussion with more examples.

7. ODTU, 29 November 2013, Friday, 15:40

Alexander Degtyarev-[Bilkent] - Resolutions of singularities, Viro’s patchworking, and tropical geometry

 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.

8. Bilkent, 6 December 2013, Friday, 15:40

Mesut Þahin-[Karatekin] - Projective toric varieties

 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 $X_A$.

9. ODTU, 13 December 2013, Friday, 15:40

Alexander Degtyarev-[Bilkent] - Viro's patchworking

 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.

10. Bilkent, 20 December 2013, Friday, 15:40

Mesut Þahin-[Karatekin] - Coordinate ring of a toric variety I

 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.

11. ODTU, 27 December 2013, Friday, 15:40

Mesut Þahin-[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, Springer-Verlag, 2006. (Grundlehren der mathematischen Wissenschaften, no. 334)

1. Bilkent, 28 February 2014, Friday, 15:40

Mesut Þahin-[Karatekin] - Toric Varieties I

 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.

2. ODTU, 7 March 2014, Friday, 15:40

Mesut Þahin-[Karatekin] - Toric Varieties II

 Abstract: We will continue with divisors and sheaves on toric varieties. Reference is chapter 4 of Cox, Little and Schenck.

3. Bilkent, 14 March 2014, Friday, 15:40

Mesut Þahin-[Karatekin] - Toric Varieties III

 Abstract:  We will talk about quasicoherent sheaves on the normal toric variety which come from multigraded modules over its Cox ring.

4. ODTU, 21 March 2014, Friday, 15:40

Mesut Þahin-[Karatekin] - Toric Varieties IV

 Abstract:  We will talk about The Toric Ideal-Variety Correspondence from Cox-Little-Schenck's book Toric Varieties,  see in particular page 220.

5. Býlkent, 28 March 2014, Friday, 15:40

Mesut Þahin-[Karatekin] - Toric ideal-subscheme correspondence

 Abstract:  .We will talk about the correspondence between closed subschemes in a normal toric variety and B-saturated homogeneous ideals in its Cox ring.

6. ODTU, 4 April 2014 Friday, 15:40

Mesut Þahin-[Karatekin] - Multigraded Hilbert functions and toric complete intersection codes 1

 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. This is a joint work with Ivan Soprunov of Cleveland State University.

7. Bilkent, 11 April 2014, Friday, 15:40

Mesut Þahin-[Karatekin] - Multigraded Hilbert functions and toric complete intersection codes 2

 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. This is a joint work with Ivan Soprunov of Cleveland State University.

8. ODTU, 18 April 2014, Friday, 15:40

Emre Coskun-[ODTÜ] - Deformation Theory 1

 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.

9. Bilkent, 25 April 2014, Friday, 15:40

Emre Coskun-[ODTÜ] - Deformation Theory 2

 Abstract:   Last week we defined the R-module ExA(R,I). This week we will continue from there and talk about the extensions of schemes.

10. 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.

11. 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.

12. Bilkent, 21 May 2014, Wednesday, 15:40

Caner Koca-[Vanderbild] - The Monge-Ampere 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 non-linear second order elliptic PDE known as the complex Monge-Ampere equation. An immediate consequence of the conjecture is the existence of Kähler-Einstein metrics on compact Kähler manifolds with vanishing first Chern class (better known as Calabi-Yau 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.

13. 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.

14. 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 Fubini-Study metric on the complex projective plane.

15. Bilkent, 3 June 2014, Tuesday, 15:40

Caner Koca-[Vanderbilt] - Extremal Kähler Metrics and Bach-Maxwell 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 Kähler-Einstein 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 conformally-invariant version of the classical Einstein-Maxwell equations, known as the Bach-Maxwell 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: Girondo and Gonzalez-Diez, Introduction to Compact Riemann Surfaces and Dessins d'Enfants, London Mathematical Society Student Texts 79, Cambridge University Press, 2012.

1. Bilkent, 26 September 2014, Friday, 15:40

Alexander Degtyarev-[Bilkent] - Lines on surfaces - I

 Abstract:   This is a joint project with I. Itenberg and S. Sertöz. I will discuss the recent developments in our never ending saga on lines in nonsingular projective quartic surfaces. In 1943, B. Segre proved that such a surface cannot contain more than 64 lines. (The champion, so-called Schur's quartic, has been known since 1882.) Even though a gap was discovered in Segre's proof (Rams, Schütt), the claim is still correct; moreover, it holds over any field of characteristic other than 2 or 3. (In characteristic 3, the right bound seems to be 112.) At the same time, it was conjectured by some people that not any number between 0 and 64 can occur as the number of lines in a quartic. We tried to attack the problem using the theory of K3-surfaces and arithmetic of lattices. Alas, a relatively simple reduction has lead us to an extremely difficult arithmetical problem. Nevertheless, the approach turned out quite fruitful: for the moment, we can show that there are but three quartics with more than 56 lines, the number of lines being 64 (Schur's quartic) or 60 (two others). Furthermore, we can prove that a real quartic cannot contain more than 56 real lines, and we have an example realizing this bound. We can also construct quartics with any number of lines in {0; : : : ; 52; 54; 56; 60; 64}, thus leaving only two values open. Conjecturally, we have a list of all quartics with more than 48 lines. (The threshold 48 is important in view of another theorem by Segre, concerning planar sections.) There are about two dozens of species, all but one 1-parameter family being projectively rigid.

2. ODTU, 2 October 2014, Thursday, 15:40

Alexander Degtyarev-[Bilkent] - Lines on surfaces - II

 Abstract:  This is the second part of the previous talk. See the above abstract.

3. Bilkent, 10 October 2014, Friday, 15:40

Emre Coþkun-[ODTÜ] - Compact Riemann surfaces and algebraic curves-I

 Abstract:  With this talk we start our series of talks on "Girondo and Gonzalez-Diez, Introduction to Compact Riemann Surfaces and Dessins d'Enfants, London Mathematical Society Student Texts 79, Cambridge University Press, 2012." The first chapter is on Riemann surfaces with an emphasis on computable examples.

4. ODTU, 17 October  2014, Friday, 15:40

Emre Coþkun-[ODTÜ] - Compact Riemann surfaces and algebraic curves-II

 Abstract:  We continue with the topology of  Riemann surfaces.

5. Býlkent, 24 October 2014, Friday, 15:40

Emre Coþkun-[ODTÜ] - Compact Riemann surfaces and algebraic curves-III

 Abstract:  We will finish the first chapter on compact Riemann surfaces. The main topic this week will be function fields on Riemann surfaces.

6. ODTU, 31 October 2014 Friday, 15:40

Özgür Kiþisel-[ODTÜ] - Riemann surfaces and discrete groups - I

 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.

7. Bilkent, 7 November, Friday, 15:40

Özgür Kiþisel-[ODTÜ] - Riemann surfaces and discrete groups - II

8. ODTU, 14 November 2014, Friday, 15:40

Özgür Kiþisel-[ODTÜ] - Riemann surfaces and discrete groups - III

 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.

9. Bilkent, 21 November 2014, Friday, 15:40

Özgür Kiþisel-[ODTÜ] - Riemann surfaces and discrete groups - IV

 Abstract:   We will start talking about Fuchsian groups.

10. ODTU, 28 November 2014, Friday, 15:40

Özgür Kiþisel-[ODTÜ] - Riemann surfaces and discrete groups - V

 Abstract:  We will talk about automorphisms of Riemann surfaces.

11. Bilkent, 5 December 2014, Friday, 15:40

Özgür Kiþisel-[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.

12. ODTU, 12 December 2014, Wednesday, 15:40

Sinan Sertöz-[Bilkent] - Belyi's Theorem-I

 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!

13. Bilkent, 19 December 2014, Friday, 15:40

Sinan Sertöz-[Bilkent] - Belyi's Theorem-II

 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!

14. ODTU, 26 December 2014, Tuesday, 15:40

Sinan Sertöz-[Bilkent] - Belyi's Theorem-III

 Abstract:  This week we will prove that if a compact Riemann surface admits a meromorphic function which ramifies over at most  three points, then it is defined over the field of algebraic numbers. This was first proved by Weil in 1956. We will present a modern proof following Girondo and Gonzalez-Diez.

ODTÜ-BÝLKENT 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: Girondo and Gonzalez-Diez, Introduction to Compact Riemann Surfaces and Dessins d'Enfants, London Mathematical Society Student Texts 79, Cambridge University Press, 2012.

1. Bilkent, 13 February 2015, Friday, 15:40

Davide Cesare Veniani-[Leibniz University of Hanover] - Lines on K3 quartic surfaces

 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.

2. ODTÜ, 20 February 2015, Friday, 15:40

 Ferruh Özbudak-[ODTÜ] - Perfect nonlinear and quadratic maps on finite fields and some connections to finite semifields, algebraic curves and cryptography

 Abstract:    Let K be a finite field with q elements, where q is odd. Let E3 and E2 be extensions of K of index 3 and 2. We show that all perfect nonlinear K-quadratic maps from E3 to E2 are extended affine equivalent (and also CCZ-equivalent). These notions are naturally connected to finite semifields (and to finite projective planes) and to certain important functions in cryptography. The proof is based on Bezout's Theorem of algebraic curves. We also give a related non-extendability result.

3. Bilkent, 27 February 2015, Friday, 15:40

Ali Sinan Sertöz-[Bilkent] - Belyi's Theorem-IV

 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 S is defined over the algebraic numbers Q¯ if and only if the orbit of S under the action of the Galois group Gal(C/Q) is finite.

4. ODTÜ, 6 March 2015, Friday, 15:40

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.  The talk will be aimed at students who took a first course in algebraic geometry.

5. Bilkent, 13 March 2015, Friday, 15:40

Davide Cesare Veniani-[Leibniz University of Hanover] -
An introduction to elliptic fibrations - part II: Mordell-Weil group and torsion sections

 Abstract:    Given an elliptic surface, the set of sections of its fibration forms a group called the Mordell-Weil 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.  The talk will be aimed at students who took a first course in algebraic geometry.

6. ODTÜ, 20 March 2015, Friday, 15:40

Ali Sinan Sertöz-[Bilkent] - Belyi's Theorem-V

 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 surfaceS is defined over the algebraic number field if and only if the orbit of S under the Galois group Gal(C/Q)  contains only finitely many isomorphism classes of Riemann surfaces. Once this is established, we will show that having a Belyi map for S leads to the finiteness of the isomorphism classes in {Sσ}σ∈Gal(C/Q). This will conclude our study of  the first three chapters of Girondo and Gonzalez-Diez's book.

7. Bilkent, 27 March 2015, Friday, 15:40

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.

8. ODTÜ, 3 April 2015, Friday, 15:40

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.

9. Bilkent, 10 April 2015, Friday, 15:40

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.

10. ODTÜ, 17 April 2015, Friday, 15:40

Ali Sinan Sertöz-[Bilkent] - From Belyi pairs to dessins

 Abstract:   We will talk about obtaining a dessin from a Belyi function.

11. Bilkent,  24 April 2015, Friday, 15:40

Alexander Klyachko-[Bilkent] - Exceptional Belyi coverings

 Abstract:   (This is a joint project with Cemile Kürkoðlu.)  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 Q or its quadratic extension.  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 g>1  found none of degree 18 or less. Absence of exceptional hyperbolic coverings is a mystery we couldn’t explain.

12. ODTÜ, 8 May 2015, Friday, 15:40

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.

ODTÜ-BÝLKENT Algebraic Geometry Seminar

**** 2015 Fall Talks ****

1. Bilkent, 2 October 2015, Friday, 15:40

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, so-called 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.  We tried to attack the problem using the theory of K3$K3$-surfaces and arithmetic of lattices. This relatively simple reduction has lead us to an extremely difficult arithmetical problem. Nevertheless, the approach turned out quite fruitful: for the moment, we have a complete classification of smooth quartics containing more than 52 lines. As an immediate consequence of this classification, we have the following:   -- an alternative proof of Segre's bound 64;   -- Shur's quartic is the only one with 64 lines;   -- a real quartic may contain at most 56 real lines;   -- a real quartic with 56 real lines is also unique;   -- the number of lines takes values {0,...,52,54,56,60,64}. Conjecturally, we have a complete list of all quartics with more than 48 lines; there are about two dozens of species, most projectively rigid. I will discuss methods used in the proof and a few problems that are still open, e.g., the minimal fields of definition, triangle-free configurations, lines in singular quartics, etc. This subject is a joint work in progress with Ilia Itenberg and Sinan Sertöz.

2. ODTÜ, 9 October 2015, Friday, 15:40

Ali Sinan Sertöz-[Bilkent] - The basic theory of elliptic surfaces-I

 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.

3. Bilkent, 16 October 2015, Friday, 15:40

Ali Sinan Sertöz-[Bilkent] - The basic theory of elliptic surfaces-II

 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 P3${\mathbb{P}}^{3}$.

4. ODTÜ, 23 October 2015, Friday, 15:40

Ergün Yalçýn-[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 G-actions on a closed manifold homotopy equivalent to a sphere, (B) finite G-homotopy representations (as defined by tom Dieck), and (C) finite G-CW complexes homotopy equivalent to a sphere. These three settings generalize the basic models arising from unit spheres S(V) in orthogonal or unitary G-representations. 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 Z/p×Z/p$\mathbb{Z}/p×\mathbb{Z}/p$, for any prime p$p$). This is joint work with Ian Hambleton.

5. Bilkent, 6 November 2015, Friday, 15:40

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.

6. ODTÜ, 13 November, Friday, 15:40

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.

7. Bilkent, 20 November 2015, Friday, 15:40

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.

8. ODTÜ, 27 November 2015, Friday, 15:40 (Joint with METU Mathematics Seminars)

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.

9. Bilkent, 4 December 2015, Friday, 15:40

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. This is joint work with Maurice Rojas and Grigoris Paouris.

10. ODTÜ, 11  December  2015, Friday, 15:40

Ali Ulaþ Özgür Kiþisel-[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.

11. Bilkent,  18 December 2015, Friday, 15:40

Mesut Þahin-[Hacettepe] - On Pseudo Symmetric Monomial Curves

 Abstract:   In this talk, we introduce monomial curves, toric ideals and monomial algebras associated to 4-generated 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 Cohen-Macaulay in which case Sally's conjecture will be true.  Joint with Nil Þahin of Bilkent University.  Supported by Tubitak No:114F094.

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

**** 2016 Spring Talks ****

 The theme of this term is Topology of Algebraic Curves, by Alex Degtyarev De Gruyter, 2012 The following is a tentative distribution of the talks. Changes will be done to suit our mathematical pleasures!

1. ODTÜ, 26 February 2016, Friday, 15:40

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.)

2. Bilkent, 4 March 2016, Friday, 15:40

Alexander Degtyarev-[Bilkent] - Skeletons-II

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

3. ODTÜ, 11 March 2016, Friday, 15:40

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.

4. Bilkent, 18 March 2016, Friday, 15:40

Ali Sinan Sertöz-[Bilkent] - Elliptic Surfaces and Weierstrass theory

 Abstract: We will talk about the Weierstrass theory and the j-invariant of elliptic surfaces.

5. ODTÜ, 25 March 2016, Friday, 15:40

Alexander Degtyarev-[Bilkent] - Trigonal curves and monodromy

 Abstract:  We will discuss the simple analytic (Calculus 101) properties of the j$j$-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 j$j$-invariant.

6. Bilkent, 1 April 2016, Friday, 15:40

Alexander Degtyarev-[Bilkent] - Trigonal curves and monodromy - II

 Abstract: We will discuss the fundamental groups, braid monodromy, Zariski--van Kampen theorem, and the relation between the braid monodromy, dessins, and the j$j$-invariant, implying that the monodromy group is one of genus zero and imposing strong restrictions on the fundamental group. Another application of this ideology is the concept of universal (for a given fundamental group) trigonal curve.

ODTU, 8 April 2016, Friday, 15:40
Cancelled due to the memorial meeting for Tosun Terzioðlu who passed away only six weeks ago.
The meeting will be at ODTU
Mathematics Department Cahit Arf Amphitheater, starting at 13:30.

7. Bilkent, 15 April 2016, Friday, 15:40

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 so-called Oka conjecture.

ODTÜ, 22 April 2016, Friday, 15:40
Cancelled in favour of 4th Cemal Koç Algebra Days
at METU

8. Bilkent, 29 April 2016, Friday, 15:40

Alexander Degtyarev-[Bilkent] - Trigonal curves and monodromy: further applications

 Abstract: As yet another application of the relation between the braid monodromy and j$j$-invariant, we will derive certain universal bounds for the metabelian invariants of the fundamental group of a trigonal curve.

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

**** 2016 Fall Talks ****

 Learning seminar on K3 surfaces

1. Bilkent, 7 October 2016, Friday, 15:40

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 K3$K3$-surfaces of small degree. Starting with such classical results as Schur's quartic and Segre's bound (proved by Rams and Schütt) of 64$64$ 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. There are limitless opportunities in extending and generalizing these results. First, one can switch from C$\mathbb{C}$ to an algebraically closed field of characteristic p>0$p>0$. Here, of course, most interesting are the so-called (Shioda) supersingular surfaces. I will discuss the properties of (quasi-)elliptic pencils on such surfaces, culminating in the classification of large configurations of lines for p=2,3$p=2,3$. Alternatively, one may consider non-closed fields such as R$\mathbb{R}$ or Q$\mathbb{Q}$. For the former, the sharp bound is 56$56$ real lines in a real quartic; for the latter, the current bound is 52$52$, and the best known example has 46$46$ 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 Φ4:={z40+z41+z42+z43=0},${\mathrm{\Phi }}_{4}:=\left\{{z}_{0}^{4}+{z}_{1}^{4}+{z}_{2}^{4}+{z}_{3}^{4}=0\right\},$ 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 K3$K3$-surfaces of small discriminant, arriving at a remarkable alternative characterisation of Schur's quartic---the champion carrying 64$64$ lines: it is also the (only) smooth quartic of the smallest possible discriminant, which is 48$48$. Going even further, we can study other projective models of small degree; counting lines in these models, we arrive at the following conjectures:  a smooth sextic curve in P2${\mathbb{P}}^{2}$ has at most 72$72$ tritangents; a smooth sextic surface in P4${\mathbb{P}}^{4}$ has at most 42$42$ lines; a smooth octic surface in P5${\mathbb{P}}^{5}$ has at most 36$36$ lines. These conjectures are still wide open; I only have but a few examples. A few other sporadic problems may be mentioned in the talks: growth of the number of smooth models, hyperelliptic models, Mukai groups, explicit equations, lines in singular quartics (including the current champion with 52$52$ lines), etc. I hope to conclude with a brief account of the tools used in the proofs (the global Torelli theorem and surjectivity of the period map, both over C$\mathbb{C}$ and over p>0$p>0$, elliptic and quasi-elliptic pencils, arithmetic of integral lattices and Nikulin's theory, Niemeier lattices, etc.), raising the audience's interest in a semester long learning seminar.

2. ODTÜ, 14 October 2016, Friday, 15:40

Alexander Degtyarev-[Bilkent - Lines in K3 surfaces-II

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

3. Bilkent, 21 October 2016, Friday, 15:40

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".

**** No talks are scheduled for 28 October 2016 Friday *****

4. ODTÜ, 4 November   2016, Friday, 15:40

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.

5. Bilkent, 11 November 2016, Friday, 15:40

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.

6. ODTÜ, 18 November 2016, Friday, 15:40

Çisem Güneþ-[Bilkent] - Classification of simple quartics up to equisingular deformation-I

 Abstract: In this talk we discuss the problem of classifying complex non-special 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 Miranda-Morrison's theory computing the genus groups. We give a complete description of equisingular strata of non-special simple quartics. First we recall fundamentals of Nikulin's theory of discriminant forms and lattice extensions and give a brief introduction to Miranda-Morrison's theory and recast some of their results in a form more suitable for our computations. Then we recall the notion of abstract homological types and arithmetical reduction of classification problem. Finally we give ideas of the proof of our principal result.

7. Bilkent, 25 November 2016, Friday, 15:40

Çisem Güneþ-[Bilkent] - Classification of simple quartics up to equisingular deformation-II

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

8. ODTU, 2 December 2016, Friday, 15:40

Oðuzhan 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 X$X$ determine, in most cases, if X$X$ covers an Enriques surface or not. We will summarize what is known about this problem and talk about the missing case when ρ=19$\rho =19$.

9. Bilkent, 9 December 2016, Friday, 15:40

Alexander Degtyarev-[Bilkent] - Projective models of $K3$-surfaces

 Abstract: Now, that we know everything about abstract K3$K3$-surfaces, I will try to take a closer look at Saint-Donat's seminal paper∗${}^{\ast }$ and share my findings. This paper is the foundation for all arithmetical reductions of geometric problems about projective K3$K3$-surfaces: it gives the conditions for an algebraic class to be (very) ample, i.e., to define a map from the K3$K3$-surface to a projective space, serving as the hyperplane section. If time permits, we will also discuss various properties of the maps obtained in this way: whether they are embeddings, the degree of the image, the generators of the defining ideal, etc. (∗$\ast$) Saint-Donat, B.  Projective models of K−3 surfaces,  Amer. J. Math.  96  (1974), 602--639.

10. ODTU, 16 December 2016, Friday, 15:40

Ali Ulaþ Özgür Kiþisel - [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.

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 P4${\mathbb{P}}^{4}$, (42 lines) and octic surfaces/triquadrics in P5${\mathbb{P}}^{5}$, (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$V$ be a complex vector space and T(V)=∑∞n=0V⊗n$T\left(V\right)=\sum _{n=0}^{\mathrm{\infty }}{V}^{\otimes n}$ be its tensor algebra.  We are primarily concerned with Lie subalgebra   L(V)⊂T(V)$L\left(V\right)\subset T\left(V\right)$generated by commutators of elements in V$V$ 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⊗n${V}^{\otimes n}$ define two operators :   cn=1n∑k=0n−1ε−kτk,ℓn=1n∑σ∈Snεmajσσ${c}_{n}=\frac{1}{n}\sum _{k=0}^{n-1}{\epsilon }^{-k}{\tau }^{k},\phantom{\rule{2em}{0ex}}{\ell }_{n}=\frac{1}{n}\sum _{\sigma \in {S}_{n}}{\epsilon }^{\text{maj}\phantom{\rule{thinmathspace}{0ex}}\sigma }\sigma$ where ε$\epsilon$ is a primitive root of unity of degree n$n$, τ$\tau$ is n$n$-cycle in symmetric group Sn${S}_{n}$ acting on V⊗n${V}^{\otimes n}$ by permutation of tensor factors. The majorization  index  majσ$\text{maj}\phantom{\rule{thinmathspace}{0ex}}\sigma$ of permutation σ$\sigma$ is defined as follows  majσ=∑σ(k)>σ(k+1)kmodn.$\mathrm{maj}\phantom{\rule{thinmathspace}{0ex}}\sigma =\sum _{\sigma \left(k\right)>\sigma \left(k+1\right)}k\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{1em}{0ex}}\mathrm{mod}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}n.$ The operators