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.