Seminar on Geometry


Seminar on Geometry Summer 2015, ODTÜ, Ankara Time / Location: (Monday), Wednesday and Fridays 2:30 / M-214

Schedule of talks


TIME	SPEAKER	TITLE
Jun 5 Fri, 11:00	Eyüp Yalçınkaya	h-principle and φ-free embeddings in calibrated manifolds 1
3:30		Special submanifolds of the Page space 1
Jun 8 Mon, 2:00		Special submanifolds of the Page space 2
3:30		Special submanifolds of the Page space 3
Jun 10 Wed, 2:00		Special submanifolds of the Page space 4
3:30		Special submanifolds of the Page space 5
Jun 12 Fri, 2:00	Şahin Koçak	Metrik Geometri ve Filogenetik Ağaçlar 10. Ankara Matematik Günleri , M-13
3:30		Special submanifolds of the Page space 6
Jun 15-19		ACM Bundles on Algebraic Varieties
Jun 17 Wed, 2:00		Scalar curvature of complex surfaces 1
Jun 19 Fri, 2:00	Aleksandra Borówka	Quaternionic Geometry 1
3:30		h-principle and φ-free embeddings in calibrated manifolds 2
Jun 22 Mon, 2:00	Özgür Kelekçi	Conical Singularities 1
3:30		Scalar curvature of complex surfaces 2
Jun 24 Wed, 2:00		Quaternionic Geometry 2
Jun 26 Fri, 2:00		Quaternionic Geometry 3
3:30		h-principle and φ-free embeddings in calibrated manifolds 3
Jun 29 Mon, 2:00	Özgür Kelekçi	Conical Singularities 2
3:30		Scalar curvature of complex surfaces 3
Jul 1 Wed, 2:00		Quaternionic Geometry 4
3:30		Scalar curvature of complex surfaces 4
Jul 3 Fri, 2:00		Quaternionic Geometry 5
3:30		h-principle and φ-free embeddings in calibrated manifolds 4
Jul 8 Wed, 2:00		Conical Singularities 3
3:30		Scalar curvature of complex surfaces 5
Jul 10 Fri, 2:00		Quaternionic Geometry 6
3:30		h-principle and φ-free embeddings in calibrated manifolds 5
Jul 13 Mon, 2:00		Scalar curvature of complex surfaces 6
3:30		Coassociative-free immersions 1
Jul 15 Wed, 12:00	Time Changed	Coassociative-free immersions 2
2:00	Aleksandra Borówka	Hyperkähler Geometry 1
Jul 17 Fri	Holiday	Ramazan Bayramı (1.gün)
Jul 22 Wed, 2:00		Coassociative-free immersions 3
3:30		Coassociative-free immersions 4
Jul 24 Fri, 2:00		Hyperkähler Geometry 2
3:30		Coassociative-free immersions 5
Jul 27-31		No Seminar this week
Aug 5 Wed, 2:00		Characteristic Classes of Bundles 1 See also: Gazi University
Aug 6 Thu, 15:40	Ken Richardson (TCU-USA)	Generalizing the Gauss-Bonnet Theorem to the foliation setting
Aug 7 Fri, 2:00		Hyperkähler Geometry 3
3:30		Characteristic Classes of Bundles 2
Aug 10-14	No Seminar	Invariants in Low Dimensional Geometry Gazi University, Aug 10-14
Aug 17 Mon, 2:00		Hyperkähler Geometry 4
3:30		Characteristic Classes of Bundles 3
Aug 19 Wed, 2:00		Characteristic Classes of Bundles 4
Aug 21 Fri, 2:00		Hyperkähler Geometry 5
3:30		Characteristic Classes of Bundles 5
Aug 24 Mon, 2:00		Characteristic Classes of Bundles 6
3:30		Characteristic Classes of Bundles 7
Aug 26 Wed, 2:00		Hyperkähler Geometry 6
3:30		Characteristic Classes of Bundles 8
Sep 2 Wed, 2:00		Hyperkähler Geometry 7
3:30		Characteristic Classes of Bundles 9
Sep 4 Fri, 2:00	Paweł Borówka	Hyperelliptic curves on abelian surfaces
3:30		Characteristic Classes of Bundles 10
Sep 7-9		28. Ulusal Matematik Sempozyumu Akdeniz Üniversitesi, Antalya
Sep 11 Fri, 2:00		Conformal structures on tori
Sep 16 Wed, 2:00		Conformal structures on tori 2
3:30		Conformal structures on tori 3
Sep 18 Fri, 2:30		Quaternionic Geometry 7
4:00		Quaternionic Geometry 8
Sep 21-25	Kurban Bayramı (1. gün 24 Eyl)	German National Meeting

Abstracts/Notlar

Quaternion-Kähler manifolds: Lectures by Aleksandra Borówka
In this seminar series we will present results of a joint project with David Calderbank. The main result concerning quaternionic geometry is a generalization of the Feix-Kaledin construction of hyperkähler metrics in quaternionic geometry. The result is obtained using twistor theory. We will also present a similar construction, where we obtain a minitwitor space of an asymptotically hyperbolic Einstein-Weyl 3-fold.
Reference 1 : A. Borówka - Twistor constructions of quaternionic manifolds and asymptotically hyperbolic Einstein-Weyl spaces.
PhD thesis, University of Bath, UK, 2014.
Reference 2 : A. Borówka - Twistor construction of asymptotically hyperbolic Einstein-Weyl spaces.
Differential Geometry and its Applications., 35, (September 2014), 224-241.

Twistor 1: Introduction to Quaternionic manifolds.

Twistor 2: Introduction to Einstein-Weyl manifolds, parabolic structures.

Twistor 3: Introduction to Twistor theory.

Twistor 4: Generalized Feix-Kaledin construction 1.

Twistor 5: Generalized Feix-Kaledin construction 2.

Twistor 6: Generalized Feix-Kaledin construction - examplex and properties.

Twistor 7: Construction of minitwistor spaces of asymptotically hyperbolic Einstein-Weyl manifolds 1.

Twistor 8: Construction of minitwistor spaces of asymptotically hyperbolic Einstein-Weyl manifolds 2.

Hyperkähler manifolds: Lectures by Aleksandra Borówka
In this seminar series we will introduce and discuss ideas around the following papers:
Reference 1 : M. Kalafat, J. Sawon - Hyperkähler manifolds with circle actions and the Gibbons-Hawking Ansatz.
Preprint available at Arxiv math.DG/0910.0672.
Reference 2 : R. Bielawski - Complete hyper-Kahler 4n-manifolds with a local tri-Hamiltonian Rn-action.
Math. Ann. 314 (1999), no. 3, 505-528.

Hyperkähler 1: Introduction.

Hyperkähler 2: Gibbons-Hawking ansatz.

On the scalar curvature of complex surfaces
In this seminar series we will be working on the following paper:

Reference: C. LeBrun - On the Scalar Curvature of Complex Surfaces.
Geom. Func. An. 5 (1995) 619--628.

OSCCS 1: Introduction. Seiberg-Witten equations.

OSCCS 2: Chambers. Seiberg-Witten invariants.

Special submanifolds of the Page space
In this seminar series we will be introducing the Page metric. Also introduce an efficient coordinate system on it. Then analyse some submanifolds which make these coordinates partially constant. An introduction to the subject can be found at:

Reference: M. Kalafat, C. Koca - On the curvature of Einstein-Hermitian surfaces.

Subpage 1: Introduction. Topology of the space.

Subpage 2: Euler coordinates. Special surfaces of constant coordinates.

Subpage 3: Computing the connection 1-forms.

Subpage 4: Curvature 2-forms.

Lectures on Conformal Structures on Riemann Surfaces
In this seminar series we will give elementary talks on moduli space of conformal classes on elliptic curves. Reference:

J.Jost - Compact Riemann Surfaces. Springer-Verlag.

Tori 1: Teichmüller space and Moduli space of conformal structures. Meromorphic functions.

Tori 2-3: Weierstrass p-function. Embedding into CP_2. Elliptic integrals.

An introduction to Conifold Singularities
In this seminar series we will present an introduction to conifolds and their singularities. We will also comment on two different ways of repairing these singularities.

References: 1.P. Candelas and X.C. de la Ossa, ‘Comments on Conifolds’ Nucl. Phys., B 342 (1990), 246-268. 2.P. Candelas, P.S. Green and T. Hübsch’ Rolling Among Calabi-Yau Vacua’,Nucl. Phys. B330 (1990) 49–102. 3.Y-M. Chan, ‘Calabi–Yau and Special Lagrangian 3-folds with conical singularities and their desingularizations’, PhD Thesis , Oxford University (2005) 4.K. Becker, M. Becker and J. H. Schwarz, "String Theory and M-Theory, A Modern Introduction", Chapter 9 and 10 (2007).

Conical 1: Conifolds and conifold singularities.

Conical 2: The Deformed Conifold.

Conical 3: The Resolved Conifold.

h-principle and φ-free embeddings in calibrated manifolds
In this seminar series we will be working on the following paper:

Reference: İ. Ünal - h-Principle and φ-free embeddings in calibrated manifolds.
International Journal of Mathematics Vol. 26, No. 5 (2015).

h-principle 1: Introduction.

h-principle 2: Embeddings.

Characteristic Classes of Bundles
In this seminar series we will be working on the chapter 4 of the following book:

Reference: Bott, Raoul; Tu, Loring W. - Differential forms in algebraic topology.
Graduate Texts in Mathematics, 82. Springer-Verlag, New York-Berlin, 1982. xiv+331 pp. ISBN: 0-387-90613-4.

Bundles 1: Chern classes of wedge products, symmetric products and tensor products.

Bundles 2: Splitting principle.

Bundles 3: Euler class of a complex line bundle. (Section 6).

Bundles 4: Blow up process. Sphere as the unit sphere bundle of the universal bundle. (Section 20)

Bundles 5: Projectivization of a vector bundle.