[Turkmath:7436] Cycles in İstanbul March 28 - April 2, 2011 TÜBİTAK - FEZA GÜRSEY INSTITUTE

[Kursat Aker]

 Cycles in İstanbul March 28 - April 2, 2011

TÜBİTAK - FEZA GÜRSEY INSTITUTE

*Speakers:*

   - Alexander Beilinson, University of Chicago
   - Spencer Bloch
   - Hélène Esnault, Universität Duisburg-Essen
   - Moritz Kerz, Universität Duisburg-Essen
   - Marc Levine, Universität Duisburg-Essen
   - Sinan Ünver, Koç Üniversitesi

*Lectures:*

   - *Infinitesimal Cycles* *by* Spencer Bloch, Hélène Esnault, Moritz Kerz

   *Kerz:* *Characteristic Zero*:
   Lifting Cycle Classes: The formal lifting, algebraization theorems (out
   of reach), base field of higher transcendence degree, relaton to the
   relative Hodge conjecture.

   *Esnault:* *Basic global facts*:
   Existence of crystalline cohomology with frobenius, iso. crystalline = DR
   of lifting, Hodge filtration, cycle class, canonical filtration, slopes.

   *Bloch:* DR-Witt cohomology, DR-W dilogarithm, results of
   Illusie-Raynaud.

   *Kerz:* Facts about Milnor K-theory, syntomic complexes.

   *Bloch and Esnault:* Discussion of proof.

   *References:*
   1. Illusie, L.: Complexe de de Rham-Witt et cohomologie cristalline,
      Annales ENS 12 (1979), 501-661.
      2. Bloch, S.: Semi-regularity and de Rham cohomology, Inventiones 17
      (1972), 51-66.
      3. Berthelot, P., Ogus, A., F-isocrystals and de Rham cohomology I,
      Inv. Math 72, 159-199 (1983).
      4. Mazur, B., Frobenius and the Hodge filtration, Ann. Math. 98, no.
      1, 58-95 (1973).
      5. Gros, M., Classes de Chern et classes de cycles en cohomologie de
      Hodge-Witt logarithmique, Mémoires de la S.M.F. 2e série, t. 21
      (1985), p. 1-87.

   - *Algebraic Cobordism* *by* Marc Levine

   Algebraic cobordism is an algebraic version of the theory of complex
   cobordism, just as the Chow ring is an algebraic version of singular
   cohomology. We will give a description of the construction of algebraic
   cobordism and its basic properties, following the original construction of
   Levine and Morel, as well as the more geometric presentation by Levine and
   Pandharipande. Besides outlining a number of computations, we will discuss a
   wide range of applications, including Riemann-Roch theorems, degree formulas
   and their applications to incompressibility, cobordism motives, as well as
   applications to Gromov-Witten theory and Donaldson-Thomas theory.

   *Main References:*
   1. Levine, M.; Pandharipande, R., Algebraic cobordism revisited. Invent.
      Math. 176 (2009), no. 1, 63130.
      2. Levine, M.; Morel, F., Algebraic cobordism. Springer Monographs in
      Mathematics. Springer, Berlin, 2007. xii+244 pp.

   *Supplementary References:*
   1. Baptiste Calmès, Victor Petrov, Kirill Zainoulline, Invariants,
      torsion indices and oriented cohomology of complete flags arXiv:0905.1341
      2. Brosnan, Patrick, Steenrod operations in Chow theory. Trans. Amer.
      Math. Soc. 355 (2003), no. 5, 18691903.
      3. Coates, Tom; Givental, Alexander, Quantum cobordisms and formal
      group laws. The unity of mathematics, 155171, Progr. Math., 244,
Birkhauser
      Boston, Boston, MA, 2006.
      4. Dai, Shouxin, Algebraic cobordism and Grothendieck groups over
      singular schemes. Homology, Homotopy Appl. 12 (2010), no. 1, 93110
      5. Jens Hornbostel, Valentina Kiritchenko, Schubert calculus for
      algebraic cobordism. arXiv:0903.3936
      6. Y.-P. Lee, R. Pandharipande, Algebraic cobordism of bundles on
      varieties. arXiv:1002.1500
      7. Levine, Marc, Comparison of cobordism theories. J. Algebra 322
      (2009), no. 9, 32913317
      8. Levine, Marc, Oriented cohomology, Borel-Moore homology, and
      algebraic cobordism. Special volume in honor of Melvin Hochster. Michigan
      Math. J. 57 (2008), 523572
      9. Levine, Marc, Steenrod operations, degree formulas and algebraic
      cobordism. Pure Appl. Math. Q. 3 (2007), no. 1, part 3, 283306.
      10. Merkurjev, Alexander, Steenrod operations and degree formulas. J.
      Reine Angew. Math. 565 (2003), 1326.
      11. Anatoly Preygel, Algebraic cobordism of varieties with G-bundles.
      arXiv:1007.0224
      12. Quillen, Daniel, Elementary proofs of some results of cobordism
      theory using Steenrod operations. Advances in Math. 7 1971 2956 (1971).
      13. Quillen, Daniel, On the formal group laws of unoriented and
      complex cobordism theory. Bull. Amer. Math. Soc. 75 1969 12931298.
      14. Semenov, N.; Zainoulline, K., Essential dimension of Hermitian
      spaces. Math. Ann. 346 (2010), no. 2, 499503.
      15. Yu-jong Tzeng, A Proof of the Göttsche-Yau-Zaslow Formula.
      arXiv:1009.5371
      16. Vishik, Alexander. Fields of u-invariant 2r+1. Algebra,
      arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II,
661685, Progr.
      Math., 270, Birkhauser Boston, Inc., Boston, MA, 2009.
      17. Vishik, A., Symmetric operations in algebraic cobordism. Adv.
      Math. 213 (2007), no. 2, 489552.
      18. Vishik, A.; Yagita, N., Algebraic cobordisms of a Pfister quadric.
      J. Lond. Math. Soc. (2) 76 (2007), no. 3, 586604.

   - *P-adic periods and derived de Rham cohomology* *by* Alexander
   Beilinson

   A construction of the p-adic period map which uses Illusie's derived de
   Rham cohomology will be illustrated.

   *References:*
   1. Faltings, G.: Almost etale extensions. Cohomologies p-adiques et
      applications arithmetiques, II. Asterisque No. 279 (2002), 185–270.
      2. Illusie, L.: Complexe cotangent et deformations. I. Lecture Notes
      in Mathematics, Vol. 239. Springer-Verlag, Berlin-New York,
1971. xv+355 pp.

      3. Illusie, L.: Complexe cotangent et deformations. II. Lecture Notes
      in Mathematics, Vol. 283. Springer-Verlag, Berlin-New York, 1972. vii+304
      pp.
      4. Illusie, L.: Cohomologie de de Rham et cohomologie etale p-adique,
      Seminaire Bourbaki 1989/90, Exp. 726, Asterisque 189-190, 1990,
p. 325–374.
      5. Niziol, W.: p-adic motivic cohomology in arithmetic. International
      Congress of Mathematicians. Vol. II, 459–472, Eur. Math. Soc.,
Zürich, 2006.

      6. Tsuji, T.: Semi-stable conjecture of Fontaine-Jannsen: a survey.
      Cohomologies p-adiques et applications arithmetiques, II.
Asterisque No. 279
      (2002), 323–370.

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

*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 25 people.*

*Deadline:* March 1, 2011

*To Apply:* http://www.gursey.gov.tr/apps/app-frm-gen.php?id=cycles1103

*Web site:* http://www.gursey.gov.tr/new/cycles1103/

*Organizers:*
Sinan Ünver (Koç Üniversitesi),
Kürşat Aker (TÜBİTAK - Feza Gürsey Enstitüsü).
*Contact:* sunver[]ku.edu.tr
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