[Turkmath:7707] [FGE-Duyuru] Differential Geometry,May 27 - 29, 2011, TÜBİTAK - FEZA GÜRSEY INSTITUTE

Kursat Aker aker at gursey.gov.tr
10 Mayıs 2011 Sal 11:42:07 EEST


  Differential Geometry


      May 27 - 29, 2011

      TÜBİTAK - FEZA GÜRSEY INSTITUTE

*Speakers:*

    * Mustafa Kalafat, Middle East Technical University
    * Barış Coşkunüzer, Koc University
    * Tekin Dereli, Koc University
    * Cenap Özel, Abant İzzet Baysal University
    * İbrahim Ünal, Middle East Technical University North Cyprus Campus
    * F. Muazzez Şimşir, Middle East Technical University
    * Kadri Arslan, Uludag University
    * Bayram Tekin, Middle East Technical University

*Lectures:*

    * *Reidemeister torsion of Product Manifolds and Quantum
      Entanglement of Pure States with Schmidt Rank* /by/ Cenap Özel

      Using symplectic chain complex, a formula for the Reidemeis- ter
      torsion of product of oriented closed connected even dimensional
      mani- folds is presented. In applications, the formula is applied
      to Riemann sur- faces,Grassmannians, Projective spaces and
      manifolds of pure bipartite states with Schmidt ranks.

      *References:*

         1. M. F. Atiyah and R. Bott, The Yang-Mills Equations over
            Riemann Surfaces, Phil. Trans. R. Soc. London Series A, 308
            No. 1505 (1983), 523-615.
         2. J.M. Bismut, H. Gillet, and C. Soule, Analytic torsion and
            holomorphic determinant bundles I. Bott-Chern forms and
            analytic torsion, Comm. Math. Phys. 115 No 1 (1988), 49-78.
         3. J.M. Bismut and F. Labourie, Symplectic geometry and the
            Verlinde formulas, in: S.T. Yau (Ed.), Surveys in
            di®erential geometry. Vol. V. Di®erential geometry inspired
            by string theory. Boston, MA: International Press. Surv.
            Di®er. Geom., Suppl. J. Di®er. Geom. 5 (1999), 97-311.
         4. T.A. Chapman, Hilbert cube manifolds and the invariance of
            Whitehead torsion, Bull. Amer. Math. Soc. 79(1973), 52-56.
         5. T.A. Chapman, Topological invariance of Whitehead torsion,
            Amer. J. Math. 96(1974), 488-497.
         6. G. de Rham, Reidemeister's torsion invariant and rotation of
            Sn; in: Di®erential Analysis, Tata Institute and Oxford
            Univ. Press, 1964, 27-36.
         7. W. Franz, Uber die Torsion einer Uberdeckung, J. Reine
            Angew. Math. 173(1935), 245-254.
         8. J. Grabowski, G. Marmo, and M. Kus, , Geometry of quantum
            systems: density states and entanglemen, J.Phys. A 38(2005),
            10217-10244.
         9. J. Grabowski, M. Kus, and G. Marmo, Symmetries, group
            actions, and entanglement, Open Sys. and Information Dyn.
            13(2006), 343-362.
        10. P. Griffiths and J. Harris, Principles of Algebraic
            Geometry, John Willey Library Edition, 1994.
        11. A. Hatcher, Algebraic Topology, Cambridge University Press,
            2002.
        12. R.C. Kirby and L.C. Siebenmann, On triangulation of
            manifolds and Haupvermutung, Bull. Amer. Math. Soc.
            75(1969), 742-749.
        13. V.I. Man'ko, G. Marmo, E.C.G. Sudarshan, and F. Zaccaria,
            Di®erential geometry of density states and entanglement,
            Rep. Math. Phys. 55(2005), 405-422.
        14. J.P. May, A Concise Course in Algebraic Topology, The
            University of Chicago Press, 1999.
        15. J. Milnor, A duality theorem for Reidemeister Torsion, Ann.
            Math. (1962), 137-147.
        16. J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc.
            72(1966), 358-426.
        17. J. Milnor, Infinite cyclic covers, in: Topology of Manifolds
            in Michigan, 1967, 115-133.
        18. J. Porti, Torsion de Reidemeister pour les Varieties
            Hyperboliques, Mem. Amer. Math. Soc., 1997.
        19. K. Reidemeister, Homotopieringe und LinsenrÄaume, Abh. Math.
            Sem. Univ. Hamburg 11(1935), 102-109.
        20. Y. Sozen, On Reidemeister torsion of a symplectic complex,
            Osaka J. Math. 45(2008), 1-39.
        21. Y. Sozen, On Fubini-Study form and Reidemeister torsion,
            Topology and its Applications, 156 (2009), 951-955.
        22. Y. Sozen, A note on Reidemeister torsion and period matrix
            of Riemann surfaces, Math. Slovaca, 61 No. 1 (2011), 29-38.
        23. Y. Sozen, Symplectic Chain Complex, Reidemeister torsion,
            compact manifolds, submitted.
        24. E. Witten, On quantum gauge theories in two dimensions,
            Comm. Math. Phys. 141(1991), 153-209.
        25. Z. Yu, X. Jost-Li, Q. Li, J. Lv., and S. Fei, Differential
            Geometry of bipartiate quantum states, Rep. Math. Phys. 60
            No. 1 (2007), 125-133.
    * *Non-Abelian Magnetic Monopoles and Electric-Magnetic Duality*
      /by/ Tekin Dereli

      I will first present the U(1) gauge theory structure of Maxwell
      equations and discuss the definition of conserved charges. After a
      few comments on the Dirac monopoles I will give SU(2)
      Yang-Mills-Higgs theory and discuss the 't Hooft-Polyakov monopole
      and dyon solutions. Finally I will introduce the Montonen-Olive
      conjecture (1978) and the corresponding electric-magnetic duality
      that led in 1994 to the Seiberg-Witten construction of topological
      invariants.

      *References:*

         1. G. 't Hooft, Nucl.Phys.B79 (1974)276
         2. E.B.Bogomol'nyi, Sov.J.Nucl.Phys. 24(1976)449
         3. C.Montonen, D.Olive, Phys.Lett. B72(1977) 117
         4. N.Seiberg,E.Witten, Nucl.Phys.B426 (1994)19; ibid,
            B430(1994)485
    * *Foliations of Hyperbolic Space by Constant Mean Curvature
      Hypersurfaces* /by/ Barış Coşkunüzer

      In this talk, we will start with a survey of asymptotic Plateau
      problem. Then, we will show that the constant mean curvature
      surfaces in the hyperbolic 3-space spanning a star-shaped curve in
      the asymptotic sphere give a foliation of the hyperbolic 3-space.
      Then, we will talk about generalizations of these results in more
      general settings.

      *References:*

         1. M. Anderson, Complete minimal hypersurfaces in hyperbolic
            n-manifolds, Comment. Math. Helv. 58 (1983) 264-290.
         2. B. Coskunuzer, Foliations of Hyperbolic Space by Constant
            Mean Curvature Hypersurfaces, IMRN (2010) 1417-1431.
         3. B. Guan, and J. Spruck, Hypersurfaces of constant mean
            curvature in hyperbolic space with prescribed asymptotic
            boundary at infinity, Amer. J. Math. 122 (2000) 1039-1060.
         4. R. Hardt and F.H. Lin, Regularity at infinity for absolutely
            area minimizing hypersurfaces in hyperbolic space, Invent.
            Math. 88 (1987) 217-224.
         5. Y. Tonegawa, Existence and regularity of constant mean
            curvature hypersurfaces in hyperbolic space, Math. Z. 221
            (1996) 591--615.
    * *Algebraic Surfaces and their applications to Differential
      Geometry 1 & 2* /by/ Mustafa Kalafat

      We will talk about complex algebraic surfaces, Kodaira-Enriques
      Classification, and their applications to 4-dimensional Riemannian
      geometry. In particular we will talk about the Kodaira Dimension
      and its relationship with the Yamabe Invariant.

    * *Calibrated Geometries* /by/ Ibrahim Unal

      Calibrated Geometries are introduced by Harvey and Lawson in the
      foundational paper [1]. These are the geometries of minimal
      submanifolds which are determined by a form φ on a Riemannian
      manifold called calibration. I will talk about the well-known
      examples of calibrated submanifolds, especially coming from
      special holonomy, and their deformation spaces, time permitting.

    * *An Introduction to Potential Theory on Calibrated Manifolds* /by/
      İbrahim Ünal

      Recently, the notion of plurisubharmonic functions in calibrated
      geometries are introduced by Harvey and Lawson [2].These functions
      generalize the classical plurisubharmonic functions from complex
      geometry to calibrated manifolds. In this talk, I will give some
      information about these functions and their properties where the
      calibration is parallel.

      *References:*

         1. F. R. Harvey and H. B. Lawson, Jr, Calibrated geometries,
            Acta Mathematica 148 (1982), 47-157.
         2. F. R. Harvey and H. B. Lawson, Jr., An introduction to
            potential theory in calibrated geometry, Amer. J. Math. 131
            no. 4 (2009), 893-944. ArXiv:math.0710.3920.
         3. D. D. Joyce, Compact Manifolds with Special Holonomy, Oxford
            University Press, Oxford, 2000.
         4. I. Unal, Topology of Phi-Convex Domains in Calibrated
            Manifolds" ( to appear in"*Bull*. *Braz*. Math. Soc".)
    * *Non-divergence harmonic maps* /by/ F. Muazzez Şimşir

      We describe work on solutions of certain non-divergence type and
      therefore non-variational elliptic and parabolic systems on
      manifolds. These systems include Hermitian and affine harmonics
      which should become useful tools for studying Hermitian and affine
      manifolds, resp. A key point is that in addition to the standard
      condition of nonpositive image curvature that is well known and
      understood in the theory of ordinary harmonic maps (which arise
      from a variational problem), here we also need in addition a
      global topological condition to guarantee the existence of solutions.

      *References:*

         1. S.I. Al'ber, Spaces of mappings into a manifold with
            negative curvature, Sov. Math. Dokl. 9 (1967), 6--9.
         2. S.Y. Cheng and S.T. Yau, The real Monge-Amp\`ere equation
            and affine flat structures, Differential Geometry and
            Differential Equations, Proc. Beijing Symp. 1980, 339--370,
            1982.
         3. H.-Ch. Grunau and M.Kuhnel, On the existence of
            Hermitian-harmonic maps from complete Hermitian to complete
            Riemannian manifolds, Math. Zeit. 249 (2005), 297--325.
         4. J. Jost, Harmonic mappings between Riemannian manifolds,
            Canberra Univ. Press, 1984.
         5. J. Jost, Nonpositive curvature: Geometric and analytic
            aspects, Birkhauser, 1997.
         6. J. Jost, Riemannian geometry and geometric analysis, 5th
            ed., Springer, 2008.
         7. J. Jost, Harmonic mappings, L.Z. Ji et al. (editors),
            Handbook of Geometric Analysis, International Press, 2008,
            147--194.
         8. J. Jost and S.T. Yau, A nonlinear elliptic system for maps
            from Hermitian to Riemannian manifolds and rigidity theorems
            in Hermitian geometry, Acta Math. 170 (1993), 221--254.
         9. J. Jost and F.M. Simsir, Affine harmonic maps, Analysis 29
            (2009), 185--197.
        10. A. Milgram and P. Rosenbloom, Harmonic forms and heat
            conduction, I: Closed Riemannian manifolds, Proc. Nat. Acad.
            Sci. 37 (1951), 180--184.
        11. J. Milnor, On fundamental groups of complete affinely flat
            manifolds, Adv. Math. 25 (1977), 178--187.
        12. L. Ni, Hermitian harmonic maps from complete Hermitian to
            complete Riemannian manifolds, Math. Zeit. 232 (1999),
            331--355.
        13. W. von Wahl, Klassische Losbarkeit im Grosen fur
            nichtlineare parabolische Systeme und das Verhalten der
            Losungen fur t, Nachr. Akad. Wiss. Gottingen, II. Math. -
            Phys. Kl., 131--177, 1981.
        14. W. von Wahl, The continuity or stability method for
            nonlinear elliptic and parabolic equations and systems,
            Rend. Sem. Mat. Fis. Milano 62 (1992), 157--183.
    * *On translation-like covering transformations* /by/ F. Muazzez Şimşir

      The concept of "translation-like elements" of the group of
      covering transformations of a covering projection onto a compact
      space is defined. It is shown that the group of covering
      transformations of the universal covering projection of a compact
      Riemannian manifold with negative sectional curvatures admits no
      non-trivial translation-like elements.

      *References:*

         1. W. P. Byers, On a theorem of Preissmann , Proc. of the Amer.
            Math. Soc. 24 (1970), 50--51.
         2. P. Eberlein, Lattices in spaces of non-positive curvature,
            Annals of Math. 111 (1980), 435--476.
         3. M. Gromov, Almost flat manifolds, J. of Diff. Geo. 13
            (1978), 231--241.
         4. M. Gromov, Groups of polynomial growth and expanding maps,
            I. H. E. S. Publications mathematiques 53 (1981), 53--71.
         5. J. Tits, Appendix to M. Gromov, Groups of polynomial growth
            and expanding maps, I. H. E. S. Publications mathematiques
            53 (1981), 53--71., I. H. E. S. Publications mathematiques
            53 (1981) 74--78.
         6. A. Preissmann, Quelques propietes globales des espaces de
            Riemann, Comment. Math. Helvet. 15 (1943) 175--216.
    * *3D Shape modelling with superquadrics* /by/ Kadri Arslan

      The problem of recovering the shape of objects from unstructured
      3D data is important in many areas of computer graphics and
      computer vision, including robotics, medical images and the
      automatic construction of virtual environments. In last 30 years,
      much work has done focussed of finding suitable models for the
      recovery of objects from 3D data. This work has largely proposed
      the use of some form of parametric model, most commonly spherical
      product of two 2D curves. Quadrics are the simple type of
      spherical products. Superquadrics are solid models that can fairly
      simple parametrization of spherical product of two 2D curves. They
      represent a large variety of standard geometric solids, as well as
      smooth shapes in between. The superquadrics can be deformed by
      stretching, bending, tapering or twisting, to built complex objects.

      *References:*

         1. Arslan, K. Bulca B, Bayram B, et al., On Spherical Product
            Surfaces in E-3 International Conference on Cyberworlds (CW
            2009), SEP 07-11, 2009 Bradford, ENGLAND.
         2. T. Bhabhrawala, Shape Recovery from Medical Image Data Using
            Extended Superquadrics. MSc Thesis, State University of New
            York at Buffalo, December,2004.
         3. J. Gielis, J Beirinchx, and Bastianens, Superquadrics with
            rational and irrational symmetry, Symposium on solid
            modelling and Applications, 2003.
         4. A. Gupta, and R. Bajcsy, Surface and volumetric segmentation
            of range images using biquadrics and superquadrics. In Int'l
            Conf. Pattern Recognition, 1(1992). 158--162.
         5. A. Jaclic, A. Leonardis, and F. Solina, Segmentation and
            Recovery of Superquadrics. Kluwer Academic Publishers, Vol.
            20, 2000.
    * *Covariant Symplectic Structure and Conserved Charges of
      Topologically Massive Gravity* /by/ Bayram Tekin

      I will talk about the covariant symplectic structure of the
      Topologically Massive Gauge theory and Gravity. I will also
      present a compact expression for the conserved charges of generic
      spacetimes with Killing symmetries.
      This talk is based on : arXiv:1104.3404 [hep-th] by C. Nazaroglu,
      Y. Nutku and B. Tekin.

------------------------------------------------------------------------

*Accommodation* (including breakfast, lunch) will be provided by Feza 
Gürsey Institute *Student Hostel* for participants from outside İstanbul 
if desired.

*Travel funds* are *not* available for participants.

*Program* is also featured on the Istanbul Mathematical Agenda: 
http://www.google.com/calendar/embed?src=jdf754c331751cbt6q9vc281es%40group.calendar.google.com&ctz=Europe/Istanbul 
<http://www.google.com/calendar/embed?src=jdf754c331751cbt6q9vc281es%40group.calendar.google.com&ctz=Europe/Istanbul> 


All participants are encouraged to fill in the following *application 
form*. Filling in the form is essential for the TÜBİTAK - FEZA GÜRSEY 
INSTITUTE to provide the best service for all participants.

*Number of participants is limited to _30_ people.*

*Deadline:* May 22, 2011

*To Apply:* http://www.gursey.gov.tr/apps/app-frm-gen.php?id=diffgeo11

*Web site:* http://www.gursey.gov.tr/new/diffgeo11/

*Organizer:*
F. Muazzez Simsir

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Differential Geometry

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