Projects

Equivariant motives for oriented theories

  • Motives of versal flags and equivariant motives
  • Motives of equivariant sections

Papers:

  • De Clercq, C., Marth, E., Zainoulline, K. Rost nilpotence for twisted Milnor hypersurfaces, Preprint 2025, 3pp.
  • Xiong, R., Zainoulline, K., Motivic Lefschetz theorem for twisted Milnor hypersurfaces, Preprint 2024, 10pp.
  • Calmes, B., Neshitov, A., Zainoulline, K. Relative equivariant motives versus modules. Canadian Journal of Math. 73 (2021), Issue 1, 131-159.
  • Neshitov, A., Petrov, V., Semenov, N., Zainoulline, K. Motivic decompositions of twisted flag varieties and representations of Hecke-type algebras. Advances in Math. 340 (2018), 791-818.
  • Nenashev, A.; Zainoulline, K. Oriented cohomology and motivic decompositions of relative cellular spaces. J. of Pure and Applied Algebra 205 (2006), no.2, 323-340.

Equivariant Schubert calculus for oriented cohomology

  • Oriented cohomology sheaves on moment graphs
  • Kazhdan-Lusztig theory for oriented cohomology
  • Finite-real root systems and the associated virtual oriented theories

Papers:

  • Xiong, R., Zainoulline, K., Zhong, C. On the formal Peterson subalgebra and its dual, to appear in Canadian Journal of Math. (2025), 18pp.
  • Lanini, M., Xiong, R., Zainoulline, K. Structure algebras, Hopf-algebroids and oriented cohomology of a group, Math. Research Letters 32 (2025), no.3, 833-870.
  • Lenart, C., Su, C., Zainoulline, K., Zhong, C. Geometric properties of the Kazhdan-Lusztig Schubert basis. Algebra & Number Theory 17 (2023), no.2, 435-464.
  • Lanini, M., Zainoulline, K. A Riemann-Roch type theorem for twisted fibrations of moment graphs. Arkiv för Matematik 59 (2021), no.2, 359-384.
  • Lanini, M., Zainoulline, K. Twisted quadratic foldings of root systems. St.Petersburg Math. J. 33 (2022), no.1, 65-84.
  • Lenart, C., Zainoulline, K., Zhong, C. Parabolic Kazhdan-Lusztig basis, Schubert classes and equivariant oriented cohomology. J. Inst. Math. Jussieu 19 (2020), Issue 6, 1889-1929.
  • Devyatov, R., Lanini, M., Zainoulline, K. Oriented cohomology sheaves on double moment graphs. Documenta Math.  24 (2019), 563-608.
  • Lenart, C., Zainoulline, K. A Schubert basis in equivariant elliptic cohomology. New-York J. of Math. 23 (2017), 711-737.
  • Lenart, C., Zainoulline, K. Towards generalized cohomology Schubert calculus via formal root polynomials. Math. Research Letters 24, No. 3, (2017), 839-877.

Cohomological operations and algebraic cycles

  • Localized operations
  • Degree formula

Papers:

  • Zainoulline, K. Localized operations on T-equivariant oriented cohomology of projective homogeneous varieties, J. Pure Appl. Algebra 226 (2022), Issue 3, 17pp.
  • 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.

Equivariant oriented cohomology of G/P

  • Oriented cohomology of G/P
  • Hecke-type algebras for oriented theories
  • Oriented Kac-Moody flags

Papers:

  • Calmes, B., Zainoulline, K., Zhong, C. Formal affine Demazure and Hecke algebras of Kac-Moody root systems. Algebras and Representation Theory 23 (2020), 1031-1050. 
  • Calmes, B., Zainoulline, K., Zhong, C. Push-pull operators on the formal affine Demazure algebra and its dual. Manuscripta Math. 160 (2019), Issue 1-2, 9-50. 
  • Calmes, B., Zainoulline, K., Zhong, C. A coproduct structure on the formal affine Demazure algebra. Math. Zeitschrift 282 (2016), no.3, 1191-1218.
  • Calmes, B.; Zainoulline, K.; Zhong, C. Equivariant oriented cohomology of flag varieties. Documenta Math. Extra Volume: Alexander S. Merkurjev’s Sixtieth Birthday (2015), 113-144.
  • Hoffnung, A., Malagon-Lopez, J., Savage, A., Zainoulline, K. Formal Hecke algebras and algebraic oriented cohomology theories.  Selecta Math. 20 (2014), no.4, 1213-1245.
  • Calmes, B.; Petrov, V.; Zainoulline, K. Invariants, torsion indices and cohomology of complete flags. Ann. Sci. Ecole Norm. Sup. (4) 46 (2013), no.3, 405-448.

The canonical dimension of a linear algebraic group

  • The p-canonical dimension
  • Upper bounds

Papers:

  • Zainoulline K. The canonical dimension of a semisimple group and the unimodular degree of a root system. Intern. Math. Res. Notices (2023), no.4, 2834-2845.
  • Semenov, N.; Zainoulline, K. Essential dimension of Hermitian spaces. Math. Annalen 346 (2009), no.2, 499-503. 
  • Zainoulline, K. Canonical p-dimensions of algebraic groups and degrees of basic polynomial invariants. London Math. Bull. 39 (2007), 301-304. 

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.

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. 
  • Baek, S.; Neher, E.; Zainoulline, K. Basic polynomial invariants, fundamental representations and the Chern class map. Documenta Math. 17 (2012), 135–150. 

Chow motives of twisted flag varieties, the J-invariant and the Rost nilpotence

  • Motives of generically split twisted flag varieties.
  • The J-invariant of a linear algebraic group.
  • Rost nilpotence for generically split varieties

Maple package to work with algebraic cycles

Papers:

  • Queguiner-Mathieu A.; Semenov, N.; Zainoulline, K. The J-invariant, Tits algebras and Triality. J. of Pure and Applied Algebra 216 (2012), 2614-2628. 
  • 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. 
  • 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. 
  • 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. 

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

(an old project related to my PhD thesis)

The conjecture has been proven by Fedorov and Panin in the geometric case

(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)