# Publications

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[ Author] Title Type Year "Homotopy theory of simplicial sheaves in completely decomposable topologies",

, *J. Pure Appl. Algebra*, vol. 214, pp. 1384–1398, 2010. (762.49 KB)"The equivalence axiom and univalent models of type theory",

, *arXiv 1402.5556*, pp. 1–11, 2010."Cohomological theory of presheaves with transfers",

, *Cycles, transfers, and motivic homology theories*, vol. 143: Princeton Univ. Press, Princeton, NJ, pp. 87–137, 2000. (346.26 KB)"Equilateral triangulations of Riemann surfaces, and curves over algebraic number fields",

, *Dokl. Akad. Nauk SSSR*, vol. 304, pp. 265–268, 1989. (439.49 KB)"C-system of a module over a monad on sets",

, *arXiv 1407.3394, submitted*, pp. 1–20, 2014."A very short note on homotopy $\lambda$-calculus",

, *Unpublished*, pp. 1–7, September, 2006. (113.85 KB)"Reduced power operations in motivic cohomology",

, *Publ. Math. Inst. Hautes Études Sci.*, pp. 1–57, 2003. (542.94 KB)"The C-systems defined by universe categories: presheaves",

, *Theory and Applications of Categories*, vol. 32, issue 3, pp. 53-112, 01/2017. (618.12 KB)"Galois groups of function fields over fields of finite type over Q",

, *Uspekhi Mat. Nauk*, vol. 46, pp. 163–164, 1991. (148.48 KB)"On motivic cohomology with Z/l-coefficients",

, *Ann. of Math. (2)*, vol. 174, pp. 401–438, 2011. (505.13 KB)"Piece-Wise Euclidean Approximation of Jacobians of Algebraic Curves",

, *CSTARCI Math. Preprints*, 1990. (1.23 MB)"Unstable motivic homotopy categories in Nisnevich and cdh-topologies",

, *J. Pure Appl. Algebra*, vol. 214, pp. 1399–1406, 2010. (506.91 KB)"Products of families of types in the C-systems defined by a universe category",

, *arXiv 1503.07072, submitted*, pp. 1–30, 2015."Triangulated categories of motives over a field",

, *Cycles, transfers, and motivic homology theories*, vol. 143: Princeton Univ. Press, Princeton, NJ, pp. 188–238, 2000. (346.26 KB)"infty-groupoids as a model for a homotopy category",

, *Uspekhi Mat. Nauk*, vol. 45, pp. 183–184, 1990.Warning: the main theorem of this paper was shown by Carlos Simpson to be false.

(202.35 KB)"B-systems",

, *arXiv 1410.5389, submitted*, pp. 1–17, 2014."Four functors formalism",

, *Unpublished*, pp. 1–20, March, 1999.One of the most important ideas of the four functor formalism was the proof that a projective morphism is lower transversal (satisfies a generalized analog of the proper base change theorem). Unfortunately no complete proof remained but all of the main ideas are contained in these notes:
1998-11-13-notes.pdf
1998-11-21-notes.pdf
2002-01-08-notes.pdf

(77.92 KB)"Motivic cohomology with Z/2-coefficients",

, *Publ. Math. Inst. Hautes Études Sci.*, pp. 59–104, 2003.There were several papers that preceded this paper that explored other proofs of the Milnor Conjecture based on the same general direction of approach. Ultimately it was the present paper that was published in a journal since the proof that it contained was most direct and required the least amount of preliminary work. The earlier versions of the proof are available here:
Bloch-Kato conjecture for Z/2-coefficients and algebraic Morava K-theories (June 1995).
The Milnor Conjecture (December 1996)
On 2-torsion in motivic cohomology (July 2001)
Motivic cohomology with Z/2-coefficients (published version, 2003).

"Products of families of types and $(\Pi,\lambda)$-structures on C-systems",

, *Theory Appl. Categ.*, vol. 31, pp. No. 36, 1044-1094, 11/2016. (565.01 KB)"A nilpotence theorem for cycles algebraically equivalent to zero",

, *Internat. Math. Res. Notices*, pp. 187–198 (electronic), 1995. (98.87 KB)"Flags and Grothendieck Cartographical Group in Higher Dimensions",

, , *CSTARCI Math. Preprints*, 1990. (583.06 KB)"Martin-Lof identity types in the C-systems defined by a universe category",

, *arXiv 1505.06446, submitted*, pp. 1–51, 2015."Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic",

, *Int. Math. Res. Not.*, pp. 351–355, 2002. (87.25 KB)"Etale topologies of schemes over fields of finite type over Q",

, *Izv. Akad. Nauk SSSR Ser. Mat.*, vol. 54, pp. 1155–1167, 1990. (713.56 KB)"Subsystems and regular quotients of C-systems",

, *arXiv 1406.5389, submitted*, pp. 1–11, 2014."Categories and functors in mathematics",

, *Unpublished*, pp. 1–18, 1999."On the zero slice of the sphere spectrum",

, *Tr. Mat. Inst. Steklova*, vol. 246, pp. 106–115, 2004. (207.32 KB)"Lawvere theories and Jf-relative monads",

, , *arXiv 1601.02158*, pp. 1--21, 2016."Galois Group over Q and Teihmuller Modular Groups",

, , , *Proc. Conf. Constr. Methods and Alg. Number Theory*, Minsk, 1989."A possible new approach to the motivic spectral sequence for algebraic K-theory",

, , *Recent progress in homotopy theory Baltimore, MD, 2000)*, vol. 293: Amer. Math. Soc., Providence, RI, pp. 371–379, 2002. (810.81 KB)"The free n-category generated by a cube, oriented matroids and higher Bruhat orders",

, *Funktsional. Anal. i Prilozhen.*, vol. 25, pp. 62–65, 1991. (235.89 KB)"An experimental library of formalized Mathematics based on the univalent foundations",

, *Mathematical Structures in Computer Science*, vol. FirstView, pp. 1–17, 2, 2015.Pre-publication version is here.

"Triangulations of Oriented Manifolds and Ramified Coverings of Sphere (in Russian)",

, *Proc. Conf. of Young Scientists*: Moscow Univ. Press, 1990."Voevodsky's Nordfjordeid lectures: motivic homotopy theory",

, *Motivic homotopy theory*: Springer, Berlin, pp. 147–221, 2007. (1.12 MB)"A C-system defined by a universe category",

, *Theory Appl. Categ.*, vol. 30, pp. No. 37, 1181–1215, 2015.Prepublication versions in arXiv.

(449.89 KB)"A^1-homotopy theory",

, *Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998)*, 1998. (355.89 KB)*Homology of schemes and covariant motives*: ProQuest LLC, Ann Arbor, MI, pp. 64, 1992.

Thesis (Ph.D.)–Harvard University

(3.48 MB)"Motivic Eilenberg-Maclane spaces",

, *Publ. Math. Inst. Hautes Études Sci.*, pp. 1–99, 2010. (915.29 KB)"Univalent Foundations Project",

, *a modified version of an NSF grant application*, pp. 1–12, October, 2010."Open problems in the motivic stable homotopy theory. I",

, *Motives, polylogarithms and Hodge theory, Part I (Irvine, CA, 1998)*, vol. 3: Int. Press, Somerville, MA, pp. 3–34, 2002. (3.04 MB)"The $(\Pi,\lambda)$-structures on the C-systems defined by universe categories",

, *Theory and Applications of Categories*, vol. 32, issue 4, pp. 113-121, 01/2017. (296.95 KB)"Galois representations connected with hyperbolic curves",

, , *Izv. Akad. Nauk SSSR Ser. Mat.*, vol. 55, pp. 1331–1342, 1991. (667.54 KB)"Multidimensional Categories (in Russian)",

, *Proc. Conf. of Young Scientists*: Moscow Univ. Press, 1990.