[Turkmath:7741] Differential Geometry May, 27 - 29, 2011 TÜBİTAK - FEZA GÜRSEY INSTITUTE
Kursat Aker
aker at gursey.gov.tr
26 Mayıs 2011 Per 17:42:37 EEST
Differential Geometry May 27 - 29, 2011
TÜBİTAK - FEZA GÜRSEY INSTITUTE
*Speakers:*
- Mustafa Kalafat, Middle East Technical University
- Barış Coşkunüzer, Koç University
- Tekin Dereli, Koç University
- Cenap Özel, Abant İzzet Baysal University
- F. Muazzez Şimşir, 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.
- *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.
------------------------------
*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
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
Algebraic Combinatorics June 5 - 18, 2011
TÜBİTAK - FEZA GÜRSEY INSTITUTE
*Speakers:*
- Alain Lascoux, IGM, Université de Marne-la-Vallée
- Piotr Pragacz, Institute of Mathematics of Polish Academy of Sciences
*Lectures:*
- *Combinatorial Operators on Polynomials, Schubert and Macdonald
Polynomials* *by* Alain Lascoux
*References:*
1. A. Lascoux, Symmetric Functions and Combinatorial Operators on
Polynomials, CBMS/AMS Lectures Notes 99, Providence (2003). (No ACE,
Exercises)
*Versions Available Online:*
- Symmetric Functions, Nankai University, October-November
2001.<http://www.gursey.gov.tr/new/co1106/Lascoux/ALCoursSf2.pdf>(ACE,
Exercises)
- Operators on Polynomials, ACE Summer School, July
2004.<http://www.gursey.gov.tr/new/co1106/Lascoux/Luminy.pdf>(ACE,
Exercises)
- Symmetric Functions and Combinatorial Operators on Polynomials,
North-Carolina, June
2001.<http://www.gursey.gov.tr/new/co1106/Lascoux/CbmsTout.pdf>(No
ACE, No Exercises)
2. A. Lascoux, *Schubert and Macdonald Polynomials, a parallel*(preprint:
http://igm.univ-mlv.fr/~al/ARTICLES/Dummies.pdf )
- *Posititivity in Global Singularity Theory* *by* Piotr Pragacz
The pioneering papers of Griffiths and Fulton and Lazarsfeld investigated
numerical positivity related to ample vector bundles in differential and
algebraic geometry. Their various variants are nowadays widely investigated
in algebraic geometry. Among main objects of global singularity theory are
the Thom polynomials of singularity classes. We shall consider Thom
polynomials of singularities of mappings and Lagrangian and Legendrian Thom
polynomials. We shall show that in some bases coming from representation
theory, they admit positive expansions.
*References:*
1. Pragacz, Weber; Thom polynomials of invariant cones, Schur functions,
and positivity, in: "Algebraic cycles, sheaves, shtukas, and moduli",
"Trends in Mathematics", Birkhauser, 2007, 117-129.
2. Pragacz, Mikosz, Weber; Positivity of Thom polynomials II: the
Lagrange singularities , Fundamenta Math. 202 (2009), 65-79.
3. Pragacz, Mikosz, Weber; Positivity of Legendrian Thom polynomials,
arXiv:1005.1283v1 [math.AG].
- *Chern Classes of Singular Varieties* *by* Piotr Pragacz
For a singular variety, the family of the tangent spaces to its points
does not give rise to a vector bundle. Nevertheless, it is possible to have
well defined characteristic classes. They will be the subject of the talk.
*References:*
1. Pragacz, Parusinski; Characteristic classes of hypersurfaces and
characteristic cycles, Journal of Algebraic Geometry 10 (2001), 63-79.
2. Pragacz, Parusinski; Chern-Schwartz-MacPherson classes and the
Euler characteristic of degeneracy loci and special divisors,
Journal of the
A.M.S. 8(1995), 793-817.
------------------------------
*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.
All participants, including those from Istanbul, are *strongly* encouraged
to fill in the following *application form*. Filling in the application form
is *critical*: It will help us at TÜBİTAK - Feza Gürsey Institute to keep
better records and provide the best service for all participants, including
meal arrangements and alike.
*Number of participants is limited to 30 people.*
*Deadline:* May 27, 2011
*To Apply:* http://www.gursey.gov.tr/apps/app-frm-gen.php?id=co1106
*Web site:* http://www.gursey.gov.tr/new/co1106/
*Organizers:*
Özer Öztürk (MSGSÜ),
Kürşat Aker (TÜBİTAK - Feza Gürsey Enstitüsü).
*Contact:* aker[/]gursey.gov.tr
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