Motivic Invariants: Nelder Fellow Lecture Series

11-22 June 2018
Imperial College London

Professor Paul Årne Østvær will visit Imperial College London during June 9-July 13 as a Nelder Visiting Professor, and during his visit he will give a series of lectures entitled Motivic Invariants. In this lecture series Professor Østvær will present a body of results on motivic invariants such as algebraic cobordism, motivic cohomology, and motivic spheres. Starting with the foundations of motivic spaces he will discuss Milnor's influential conjectures relating \(K\)-theory to Galois cohomology and quadratic forms, and calculations of universal motivic invariants. The lectures will be of interest to PhD students, postdocs, and members of staff with research interests in algebra, number theory, and topology. Below is a brief outline of the lectures accompanied with references. Professor Østvær will also give a lecture at the summer school Homotopy Theory and Arithmetic Geometry: Motivic and Diophantine Aspects, Imperial College, 9-13 July 2018.

First Lecture: June 11th, Monday, 13:00-15:00 at 140, Huxley building

Brief syllabus: The first lecture reviews with ample motivation the foundation of motivic homotopy theory from ground up. As main examples we discuss vector bundles and \(K\)-theory, Grassmannians, algebraic cobordism, and motivic spheres.


Dundas, Levine, Rondigs, Østvær, Voevodsky, Motivic homotopy theory, Lectures at a summer school in Nordfjordeid, Springer-Verlag, Universitext, 2007.

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

Voevodsky, \(\mathbb A^1\)-homotopy theory, Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), 1998.

Second Lecture: June 12th, Tuesday, 13:00-15:00 at 130, Huxley building

Brief syllabus: In the second lecture we explain the basics of motivic cohomology and its fundamental role in Voevodsky's proof of the Bloch-Kato and Milnor conjectures relating \(K\)-theory to Galois cohomology. A central technique is the use of motivic Steenrod operations acting on motivic cohomology groups.


Milnor, Algebraic \(K\)-theory and quadratic forms, Inventiones Math., 9 (1970), 318-344.

Voevodsky, Motivic cohomology with \(\mathbb Z/2\)-coefficients, Publ. Math. Inst. Hautes Etudes Sci., 98 (2003), 59-104.

Voevodsky, On motivic cohomology with \(\mathbb Z/l\)-coefficients, Ann. of Math., 174 (2011), 401-438.

Third Lecture: June 18th, Monday, 13:00-15:00 at 140, Huxley building

Brief syllabus: Continuing the theme of the second lecture we use the slice filtration and higher Witt-theory to prove Milnor's conjecture on quadratic forms. In the course of the proof we review the slices of \(K\)-theory, hermitian \(K\)-theory, and higher Witt-theory.


Orlov, Vishik, Voevodsky, An exact sequence for \(K^M_*/2\) with applications to quadratic forms, Ann. of Math., 165 (2007), 1-13.

Röndigs, Østvær, Slices of hermitian \(K\)-theory and Milnor's conjecture on quadratic forms, Geometry and Topology, 20 (2016), 1157-1212.

Voevodsky, Open problems in the motivic stable homotopy theory, Motives, polylogarithms and Hodge theory, Part I (Irvine, CA, 1998), vol. 3: Int. Press, Somerville, MA, pp. 3-34, 2002.

Fourth Lecture: June 19th, Tuesday, 13:00-15:00 at 140, Huxley building

Brief syllabus: The fourth lecture concerns calculations of universal motivic invariants. This is a topic under intense investigation with recent breakthroughs inspired by Morel's identification of the \(0\)th homotopy of the motivic sphere with the Grothendieck-Witt ring.


Morel, \(\mathbb A^1\)-algebraic topology over a field, Lecture Notes in Mathematics 2052, Springer-Verlag, 2012.

Röndigs, Spitzweck, Østvær, The first stable homotopy groups of motivic spheres, arXiv:1604.00365.