# Publications

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Author Title Type [ Year] "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)"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)"Lawvere theories and Jf-relative monads",

*arXiv 1601.02158*, pp. 1--21, 2016."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 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)"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.

"Martin-Lof identity types in the C-systems defined by a universe category",

*arXiv 1505.06446, submitted*, pp. 1–51, 2015."Products of families of types in the C-systems defined by a universe category",

*arXiv 1503.07072, submitted*, pp. 1–30, 2015."A univalent formalization of the p-adic numbers",

*Mathematical Structures in Computer Science*, vol. FirstView, pp. 1–25, 2, 2015.The arXiv version is here.

"B-systems",

*arXiv 1410.5389, submitted*, pp. 1–17, 2014."C-system of a module over a monad on sets",

*arXiv 1407.3394, submitted*, pp. 1–20, 2014."Subsystems and regular quotients of C-systems",

*arXiv 1406.5389, submitted*, pp. 1–11, 2014."On motivic cohomology with Z/l-coefficients",

*Ann. of Math. (2)*, vol. 174, pp. 401–438, 2011. (505.13 KB)"Homotopy theory of simplicial sheaves in completely decomposable topologies",

*J. Pure Appl. Algebra*, vol. 214, pp. 1384–1398, 2010. (762.49 KB)"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."Unstable motivic homotopy categories in Nisnevich and cdh-topologies",

*J. Pure Appl. Algebra*, vol. 214, pp. 1399–1406, 2010. (506.91 KB)"Voevodsky's Nordfjordeid lectures: motivic homotopy theory",

*Motivic homotopy theory*: Springer, Berlin, pp. 147–221, 2007. (1.12 MB)"Lecture notes on motivic cohomology",

*Clay Mathematics Monographs*, vol. 2: American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, pp. xiv+216, 2006. (1.34 MB)"A very short note on homotopy $\lambda$-calculus",

*Unpublished*, pp. 1–7, September, 2006. (113.85 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).

"Reduced power operations in motivic cohomology",

*Publ. Math. Inst. Hautes Études Sci.*, pp. 1–57, 2003. (542.94 KB)"Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic",

*Int. Math. Res. Not.*, pp. 351–355, 2002. (87.25 KB)"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)"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)"Bivariant cycle cohomology",

*Cycles, transfers, and motivic homology theories*, vol. 143: Princeton Univ. Press, Princeton, NJ, pp. 138–187, 2000. (348.23 KB)"Bloch-Kato conjecture and motivic cohomology with finite coefficients",

*The arithmetic and geometry of algebraic cycles (Banff, AB, 1998)*, vol. 548: Kluwer Acad. Publ., Dordrecht, pp. 117–189, 2000. (618.57 KB)"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)"Relative cycles and Chow sheaves",

*Cycles, transfers, and motivic homology theories*, vol. 143: Princeton Univ. Press, Princeton, NJ, pp. 10–86, 2000. (524.85 KB)"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)"A^1-homotopy theory of schemes",

*Inst. Hautes Études Sci. Publ. Math.*, pp. 45–143 (2001), 1999. (9.25 MB)"Categories and functors in mathematics",

*Unpublished*, pp. 1–18, 1999."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)"A^1-homotopy theory",

*Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998)*, 1998. (355.89 KB)"Singular homology of abstract algebraic varieties",

*Invent. Math.*, vol. 123, pp. 61–94, 1996. (1.56 MB)*Homology of schemes and covariant motives,*: ProQuest LLC, Ann Arbor, MI, pp. 64, 1992.

Thesis (Ph.D.)–Harvard University

(3.48 MB)