Lecture Notes and Slides

Lectures on motivic cohomology.
Conference on "Motives and Automorphic Forms" in Honour of Günter Harder's 85th Birthday, March 6 - 10, 2023.

Three lectures on quadratic enumerative geometry. "Moduli, Motives and Bundles – New Trends in Algebraic Geometry'' held at the BIRS Conference centre Oaxaca, Mexico, September 18-23, 2022.

Motivic cohomology. Lecture for the 2022 ICM. Proc. Int. Cong. Math. 2022, Vol. 3, pp. 2048–2106. EMS Press, Berlin, 2023.

Welschinger invariants and quadratic degrees. Slides for a talk at the conference "Periods and Motives", Berlin, July 10-12, 2019.

Lectures on quadratic enumerative geometry. Four lectures, given at the conference “Motivic homotopy theory and refined enumerative geometry”, May 14–18, 2018, at the Universität Duisburg-Essen.

A calculus of characteristic classes in Witt cohomology. Slides for a talk given at the Conference on Algebraic Geometry and Number Theory on the occasion of Jean-Louis Colliot-Thélène’s 70th birthday, December 4-6, 2017, Villa Finaly, Florence, Italy.

The slice tower and the Adams-Novikov spectral sequence. Slides for a talk, Sept. 2014.

Lausanne 2012 : Program for: The motivic fundamental group, October 1-5, 2012.

JAMI 2009: Tate motives and fundamental groups, March 25, 2009.

UCLA Colloquium: a survey of algebraic cobordism, Jan. 22, 2009.

BU Conference on motives QFT and pseudo-differential operators, June 2008
Lecture 1 Cycle complexes
Lecture 2 Motivic categories
Lecture 3 Tate motives.

ICTP school on algebraic K-theory and its applications, May 2007.
Six lectures on motives

French-Asian summer school in algebraic geometry and number theory, I.H.E.S. July 2006.
Six lectures on motives

Conference on Motives and Algebraic Cycles, in honor of Spencer Bloch,
Fields Institute-March 19-23, 2007
Motivic Postnikov towers
A lecture on Voevodsky's slice filtration, its relation to the homotopy coniveau tower,
with some applications to Severi-Brauer varieties and central simple algebras.

2nd German-Chinese conference on complex geometry, Shanghai-Sept. 2006
Algebraic cobordism
A lecture on algebraic cobordism, including a discussion of ``double-point" cobordism
and an application to Donaldson-Thomas invariants.

Three lectures on algebraic cobordism
Motives and Periods, June 5-12, 2006
UBC, Vancouver
Lecture 1, Lecture 2, Lecture 3

Introduction to algebraic cobordism
The first of 3 lectures on algebraic cobordism in the Distinguished Lecture Series
of the University of Western Ontario Mathematics Department,
Jan. 25-27, 2005.

A summer course in motivic homotopy theory
Algebraic geometry background, Nordfjordeid, 2003

A short course in K-theory
Morelia, 2002


(with H. Esnault) Tate motives and the fundamental group
Preprint (2007)

K-theory and motivic cohomology of schemes, I
Preprint (2004)


The intrinsic stable normal cone
Algebr. Geom. 8 (2021), no. 5, 518–561.

(with Yaping Yang, Gufang Zhao) Algebraic elliptic cohomology and flops II: SL-cobordism
Adv. Math. 384 (2021), Paper No. 107726, 66 pp.

Aspects of enumerative geometry with quadratic forms
Doc. Math. 25 (2020), 2179–2239.

(with A. Raksit) Motivic Gauss-Bonnet formulas
Algebra Number Theory 14 (2020), no. 7, 1801–1851.

Motivic Euler characteristics and Witt-valued characteristic classes
Nagoya Math. J. 236 (2019), 251–310.

(with Yaping Yang, Gufang Zhao, Joël Riou) Algebraic elliptic cohomology theory and flops I
Math. Ann. 375 (2019), no. 3-4, 1823–1855.

(with A. Chatzistamatiou) Torsion orders of complete intersections
Algebra Number Theory 11 (2017), no. 8, 1779–1835.

(with A. Ananyevskiy and I. Panin) Witt sheaves and the η -inverted sphere spectrum
J. Topol. 10 (2017), no. 2, 370–385.

Intersection theory in algebraic cobordism
J. Pure Appl. Algebra 221 (2017), no. 7, 1645–1690.

The Adams-Novikov spectral sequence and Voevodsky's slice tower
Geom. Topol. 19 (2015), no. 5, 2691–2740.

(with G.S. Tripathi) Quotients of MGL, their slices and their geometric parts
Doc. Math.(2015-Extra Vol.-Alexander S. Merkurjev's Sixtieth Birthday), 407–442.

(with H. Esnault and O. Wittenberg) Index of varieties over Henselian fields and Euler characteristic of coherent sheaves
J. Algebraic Geom. 24 (2015), no. 4, 693–718.

Motivic Landweber exact theories and their effective covers
Homology Homotopy Appl. 17 (2015), no. 1, 377–400.

A comparison of motivic and classical stable homotopy theories
J. Topol. 7 (2014), no. 2, 327–362.

(with S. Dai) Connective algebraic K-theory
arXiv:1212.0228, J. K-Theory 13 (2014), no. 1, 9–56.

Tate motives and the fundamental group
Cycles, Motives and Shimura Varieties, ed. V. Srinivas, Tata Institute of Fundamental Research, Mumbai, India, August 2010, 265--392.

Slices and transfers
Documenta Mathematica, Extra Volume: Andrei A. Suslin's Sixtieth Birthday (2010), 393--443.

(with J.-L. Colliot-Thélène) Une version du théorème d'Amer et Brumer pour les zéro-cycles
Quadratic forms, linear algebraic groups, and cohomology, 215–223, Dev. Math., 18, Springer, New York, 2010.

(with Bruno Kahn) Motives of Azumaya algebras
Journal of the Inst. of Math. Jussieu 9(3), 2010, 481–599.

Smooth Motives
Motives and algebraic cycles, 175--231, Fields Inst. Commun.,56, Amer. Math. Soc., Providence, RI, 2009.

Comparison of cobordism theories
J. Algebra 322 (2009), no. 9, 3291--3317.

(with Rahul Pandharipande) Algebraic Cobordism Revisited
Invent. Math. 176 (2009), no. 1, 63--130.

Six lectures on motives
Some recent developments in algebraic K-theory, 131--227, ICTP Lect. Notes, 23, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2008.

Motivic homotopy theory
Milan J. Math. 76 (2008), 165--199.

Oriented cohomology, Borel-Moore homology and algebraic cobordism
Michigan Math. J., Special Issue in honor of Melvin Hochster, 57, August 2008, 523-572

(with Amalendu Krishna) Additive higher Chow groups of schemes
J. Reine Angew. Math. 619 (2008), 75--140.

(with C. Serpé) On a spectral sequence for equivariant K-theory
K-Theory 38(2008), no. 2, 177--222.

The homotopy coniveau tower
J Topology 1 (2008) 217--267.

Steenrod operations, degree formulas and algebraic cobordism
Pure and Applied Mathematics Quarterly Journal, volume in honor of the 60th birthday of Robert MacPherson 3, No. 1 (2007) 283-306.

Motivic Tubular Neighborhoods 
Documenta Math. 12 (2007) 71--146.

(with Fabien Morel) Algebraic Cobordism
Monographs in Mathematics, 246 pp., Springer, Berlin 2007.

(with B. Dundas, P.A. Østvaer, O. Röndigs, V. Voevodsky) Background from algebraic geometry Link
Motivic homotopy theory Lectures at a summer school in Nordfjordeid, Norway, August 2002. Universitext 220 pp. Springer, Berlin 2007, 70--145.

Chow's moving lemma and the homotopy coniveau tower
K-theory 37 (1-2) 129-209 (2006).

Mixed Motives
An expository article in the Handbook of K-theory, vol 1,
E.M. Friedlander, D.R. Grayson, eds.,
429-522. Springer Verlag, 2005.

(with S. Bloch and H. Esnault)   Divisibility of the eigenvalues of
Frobenius acting on étale cohomology of singular projective varieties

Amer. J. Math. 127 (2005), no. 1, 193--207.

A survey of algebraic cobordism
Proceedings of the International Conference on Algebra.
Algebra Colloq. 11 (2004), no. 1, 79--90.

Fundamental classes in algebraic cobordism
K-Theory 30 (2) (2003), 129-135.

Algebraic cobordism
Proceedings of the 2002 ICM, August, 2002.

(with F. Morel)  Cobordisme algébrique, I
C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), no. 8, 723--728.

(with F. Morel)  Cobordisme algébrique, II
C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), no. 9, 815--820.

Techniques of localization in the theory of algebraic cycles
J. Alg. Geom. 10 (2001) 299-363.

Blowing up monomial ideals
JPAA  160 (2001) 67-103.

(with T. Geisser)  The Bloch-Kato conjecture and a theorem of

Jour. f.d. Reine u. Ang. Math 530 (2001), 55-103.

Inverting the motivic Bott element
K-Theory  19 (2000), no. 1, 1--28.

(with H. Esnault)  The Steinberg curve
Amer. J. Math. 122 (2000), no. 4, 783--804

(with T. Geisser)  The K-theory of fields in characteristic p
Invent. Math. 139 (2000), no. 3, 459--493.

Mixed Motives    On-line version
Math. Surveys and Monographs 57, x+515 pp., AMS, Prov. 1998.

(with H. Esnault, B. Kahn, E. Viehweg)
The Arason invariant and mod 2 algebraic cycles
J. of the AMS, 11 no. 1, (1998), 73-118.

Homology of algebraic varieties: an introduction
to the works of Suslin and Voevodsky

Bull. Amer. Math. Soc. (N.S.)  34 (1997), no. 3, 293-312.

Lambda operations, K-theory and motivic cohomology
Algebraic K-Theory, (volume in memory of R. Thomason)
ed. V. Snaith, Fields Inst. Communications 16 (1997) 131-184.

(with H. Esnault, E. Viehweg)
Chow groups of projective varieties of very small degree
Duke Math. J. 87 (1997), no. 1, 29-58.

The weight two K-theory of fields
K-theory 9 No. 5, (1995) 443-501.

Bloch's higher Chow groups revisited
K-theory Strasbourg 1992 Astérisque vol. 226,
ed. C. Kassel, J.L. Loday, N. Schappacher,
Soc. Math. de France(1994) 235-320.

(with H. Esnault)  Surjectivity of cycle maps
Journées de Geométrie Algébrique D'Orsay, Juillet 1992,
Astérisque vol.  218(1993) 203-226.

Tate motives and the vanishing conjectures for algebraic K-theory
Algebraic K-Theory and Algebraic Topology,
ed. P.G. Goerss and J.F. Jardine,
NATO ASI Series, Series C, Vol. 407 (1993) 167-188.

The algebraic K-theory of the classical groups and some twisted forms
Duke Math. Journ. 70 no. 2(1993) 405-443.

Deligne-Beilinson cohomology for singular varieties
Proceedings of the S.Margherita K-theory Conference, Cont. Math. 126 (1992) 113-146.

Relative Milnor K-theory
K-theory 6 (1992) 113-175.

(with V. Srinivas and J. Weyman), K-theory of twisted Grassmannians
K-theory 3 No. 2(1989) 99-122.

The indecomposable K3 of fields
Bulletin of the AMS, Nov. 1987.

The indecomposable K3 of fields
Ann. Sci. Ecole Norm. Sup.(4)22 No. 2(1989) 255-344.

Localization on singular varieties
Inv. Math. 91(1988) 423-464.

A remark on extremal Kähler metrics
J. Diff. Geom. 21(1985) 73-77.

A K-theoretic approach to multiplicities
Math. Ann. 271(1985) 451-458.

(with H. Gillet) The relative form of Gersten's conjecture over a discrete valuation ring: the smooth case
J. Pure and Appl. Alg. 46 (1987) 59-71.

Pluri-canonical divisors on Käahler manifolds II
Duke J. Math., 52 No. 1(1985) 61-65.

K-groups of surface singularities and modules of finite length
Compositio Math. 59 No. 1(1986) 21-40.

Bloch's formula for a singular surface
Topology 124 No. 2(1985) 165-174.

Zero-cycles and K-theory on singular varieties
Algebraic Geometry, Bowdoin 1985 S. Bloch et. al. ed. AMS Proc. Symp. Pure Math. Vol. 46 pt. 2(1987) 451-462.

(with C. Weibel) Zero-cycles and complete intersections on affine surfaces
J. Reine u. Ang. Math. 359(1985) 106-120.

Pluri-canonical divisors on Kähler manifolds
Inventiones Math. 74(1983) 293-303.

Torsion zero-cycles on singular varieties
Am. J. Math. 107(1985) 737-757.

(with V. Srinivas) Zero-cycles on certain singular elliptic surfaces
Compositio Math. 52 fasc. 2(1984) 179-196.

Deformations of irregular threefolds
Algebraic Threefolds Lecture Notes in Math. 947 269-286, Springer 1982.

(with P. Blass) Families of Zariski surfaces
Duke J. Math. 48(1981) 129-136.

Deformations of uni-ruled varieties
Duke J. Math. 48(1981) 467-473.

Some examples from the deformation theory of ruled varieties
Am. J. Math. 103(1981) 997-1020.

Stability of irrational ruled threefolds
Math. Ann. 262 (1983) 391-406.

The stability of certain classes of uni-ruled varieties(thesis)
Compositio Math. 52 fasc. 1(1984) 3-30.