Projects

Current Project:  Sheaves on moment graphs, Kazhdan-Lusztig theory and generalized Schubert calculus.

  • Sheaves on moment graphs and Riemann-Roch type formulas
  • Structure algebras, finite real reflection groups and equivariant motives
  • Kazhdan-Lusztig theory and generalized Schubert calculus

Slides of talks/Lecture Notes:

Papers:

  • Lanini, M., Zainoulline, K. A Riemann-Roch type theorem for twisted fibrations of moment graphs.
  • Zainoulline, K. Localized operations on T-equivariant oriented cohomology of projective homogeneous varieties
  • Lanini, M., Zainoulline, K. Twisted quadratic foldings of root systems.
  • Devyatov, R., Lanini, M., Zainoulline, K. Oriented cohomology sheaves on double moment graphs.
  • Lenart, C., Su, C., Zainoulline, K., Zhong, C. Geometric properties of the Kazhdan-Lusztig Schubert basis.
  • Lenart, C., Zainoulline, K., Zhong, C. Parabolic Kazhdan-Lusztig basis, Schubert classes and equivariant elliptic cohomology.
  • Lenart, C., Zainoulline, K. A Schubert basis in equivariant elliptic cohomology.
  • Lenart, C., Zainoulline, K. Towards generalized cohomology Schubert calculus via formal root polynomials.

Current Project:  Equivariant algebraic oriented cohomology and localization techniques

  • Equivariant oriented cohomology theories.
  • Representations of Hecke-type algebras and motives of twisted flag varieties. 

Slides of talks/Lecture notes:

Papers:

  • Calmes, B., Zainoulline, K., Zhong, C. Formal affine Demazure and Hecke algebras of Kac-Moody root systems.
  • Calmes, B.; Zainoulline, K.; Zhong, C. Equivariant oriented cohomology of flag varieties.
  • Calmes, B., Zainoulline, K., Zhong, C. Push-pull operators on the formal affine Demazure algebra and its dual.
  • Calmes, B., Zainoulline, K., Zhong, C. A coproduct structure on the formal affine Demazure algebra.
  • Hoffnung, A., Malagon-Lopez, J., Savage, A., Zainoulline, K. Formal Hecke algebras and algebraic oriented cohomology theories. 
  • Calmes, B.; Petrov, V.; Zainoulline, K. Invariants, torsion indices and cohomology of complete flags.  
  • Gille, S.; Zainoulline, K. Equivariant pretheories and invariants of torsors. 
  • Nenashev, A.; Zainoulline, K. Oriented cohomology and motivic decompositions of relative cellular spaces. 

Applications to motives and representations:

  • Calmes, B., Neshitov, A., Zainoulline, K. Relative equivariant motives versus modules.
  • Neshitov, A., Petrov, V., Semenov, N., Zainoulline, K. Motivic decompositions of twisted flag varieties and representations of Hecke-type algebras.

Project: Topological and the gamma-filtration on twisted flag varieties.

  • Topological and gamma-filtration of twisted flag varieties.
  • Applications to cohomological invariants (the Rost invariant)
  • Applications to the torsion problem in Chow groups.

Slides of talks:

Papers:

  • Baek, S., Devyatov, R., Zainoulline, K. The K-theory of versal flags and cohomological invariants of degree 3. Documenta Math. 22 (2017), 1117-1148.
  • Merkurjev, A., Neshitov, A., Zainoulline, K. Cohomological Invariants in degree 3 and torsion in the Chow group of a versal flag. Compositio Math. 151, Issue 8, (2015), 1416–1432.
  • Malagon-Lopez, J.; Zainoulline, K.; Zhong, C. Invariants, exponents and formal group laws. J. of Algebra 409, July 1, (2014), 244-264.
  • Baek, S., Zainoulline, K., Zhong, C. On the torsion of Chow groups of twisted Spin-flags. Math. Research Letters 20 (2013), no.4, 601-614.
  • Garibaldi, S.; Zainoulline, K. The gamma-filtration and the Rost invariant. J. Reine und Angew. Math. 696 (2014), 225–244.
  • Zainoulline, K. Twisted gamma-filtration of a linear algebraic group. Compositio Math. 148 (2012), no.5, 1645-1654. 
  • Queguiner-Mathieu A.; Semenov, N.; Zainoulline, K. The J-invariant, Tits algebras and Triality. J. of Pure and Applied Algebra 216 (2012), 2614-2628. 
  • Baek, S.; Neher, E.; Zainoulline, K. Basic polynomial invariants, fundamental representations and the Chern class map. Documenta Math. 17 (2012), 135–150. 

Project: Motivic decompositions and algebraic cycles on twisted flag varieties

  • Motivic decomposition of generically split twisted flag varieties.
  • The J-invariant of a linear algebraic group.
  • The canonical dimension of a linear algebraic group.

Maple package to work with algebraic cycles

Papers:

  • Zainoulline, K. Degree formula for connective K-theory. Invent. Math. 179 (2010), no.3, 507-522. 
  •  Zainoulline, K. Special correspondences and Chow traces of Landweber-Novikov operations. J. Reine und Angew. Math. 628 (2009), 195-204. 
  • Semenov, N.; Zainoulline, K. Essential dimension of Hermitian spaces. Math. Annalen 346 (2009), no.2, 499-503. 
  • Nikolenko, S.; Semenov, N.; Zainoulline, K. Motivic decompositions of anisotropic varieties of type F4 into generalized Rost motives. J. of K-theory 3 (2009), no.1, 85-102.
  • Petrov, V.; Semenov, N.; Zainoulline, K. J-invariant of linear algebraic groups. Ann. Sci. Ecole Norm. Sup. (4) 41 (2008), no.6, 1023-1053. 
  • Vishik, A.; Zainoulline, K. Motivic Splitting Lemma. Documenta Math. 13 (2008), 81-96. 
  • Petrov, V.; Semenov, N.; Zainoulline, K. Zero-cycles on a twisted Cayley plane. Canadian Math. Bull. 51 (2008), no.1, 114-124. 
  • Zainoulline, K. Canonical p-dimensions of algebraic groups and degrees of basic polynomial invariants. London Math. Bull. 39 (2007), 301-304. 
  • Calmes, B.; Petrov, V.; Semenov, N.; Zainoulline, K. Chow motives of twisted flag varieties. Compositio Math. 142 (2006), no.4, 1063-1080. 
  • Semenov, N.; Zainoulline, K. A classification of projective homogeneous varieties up to motivic isomorphism (English. Russian original). J. Math. Sciences 140 (2007), no.5, 692-698; transl. Zap. Nauchn. Semin. POMI 330 (2006), 158-172. 

Project: The Grothendieck-Serre conjecture, the Purity and the Gersten conjecture.

(an old project related to my PhD thesis)

The conjecture has been recently proven by Fedorov and Panin 

(see arXiv.org preprints http://arxiv.org/abs/1211.2678, http://arxiv.org/abs/1406.0241, http://arxiv.org/abs/1406.0247 and http://arxiv.org/abs/1406.1129)

Directions:

  • Gersten resolutions, Gersten conjecture and Purity problem for presheaves with transfers
  • Geometric presentation lemmas via general position arguments
  • Grothendieck-Serre’s conjecture on G-torsors.  (short overview on what was done before 2007)

Papers:

  • Panin, I.; Zainoulline, K. Gersten resolution with support. Manuscripta Math. 128 (2009), no. 4, 443-452. 
  • Ojanguren, M.; Panin, I.; Zainoulline, K. On the norm principle for quadratic forms. J. Ramanujan Math. Soc. 19 (2004), no.4, 1-12. 
  • Zainoulline, K. On Knebusch’s Norm Principle for quadratic forms over semi-local rings. Math. Zeitschrift 251 (2005), no.2, 415-425. 
  • Panin, I.; Zainoulline, K. Variations on the Bloch-Ogus Theorem. Documenta Math. 8 (2003), 51-67. 
  • Schmidt, A.; Zainoulline, K. Generic injectivity for etale cohomology and pretheories. J. of Algebra 263 (2003), no.2, 215-227. 
  • Zainoulline, K. The Purity Theorem for functors with transfers. K-Theory J. 22 (2001), no.4, 303-333. (came out of PhD thesis) 
  • Zainullin, K. On Grothendieck’s conjecture about principal homogeneous spaces for some classical algebraic groups. (English. Russian original) St. Petersburg Math. J. 12 (2001), no.1, 117-143; translation from Algebra i Analiz 12 (2000), no.1, 150-184. (came out of PhD thesis)