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

**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.

**References**:

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.

**References**:

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.

**References**:

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.

**References**:

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.