Time and place: Feb. 20, 2019 10:15 AM - 12:00 PM, B1119

The classical Cayley-Dickson construction produces a sequence of algebras, including the quaternion and octonion algebras, from which we get H-space structures on the three- and seven-spheres by taking unit spheres, and hence we get the quaternionic and octonionic Hopf fibrations. I will describe a version of the Cayley-Dickson construction that works directly with the unit spheres, using homotopy type theory. Homotopy type theory can (conjecturally) be seen as an internal language to reason about higher toposes, giving rise to a kind of synthetic homotopy theory. Indeed, this version of the Cayley-Dickson construction works in any higher topos.


Time and place: Feb. 6, 2019 10:15 AM - 12:00 PM, B1119

This is the first in a series of four talks which aims at an introduction to the theory of motives for rigid-analytic varieties as developed by Ayoub. In the first talk, I will mostly discuss the motivations for defining and studying rigid-analytic varieties and formulate some results (by Ayoub and Vezzani) that can be proved for the categories of motives of rigid-analytic varieties. In particular, I will formulate the recent rigidity theorem for rigid-analytic motives, proved by Bambozzi and Vezzani. While the first talk should mainly convey ideas and motivation, the remaining three talks will give more details to understand the proof of the rigidity theorem.

Time and place: Jan. 30, 2019 10:15 AM - 11:15 AM, B1119

Since Suslin and Voevodsky's introduction of finite correspondences, several alternate correspondence categories have been constructed in order to provide different linear approximations to the motivic stable homotopy category. In joint work with Andrei Druzhinin, we provide an axiomatic approach to a class of correspondence categories that are defined by an underlying cohomology theory. For such cohomological correspondence categories, one can prove strict homotopy invariance and cancellation properties, resulting in a well behaved associated derived category of motives.

Time and place: Jan. 22, 2019 2:15 PM - 4:00 PM, B1120

Grothendieck proved that the small etale site is invariant under universal homeomorphism of schemes and calls this the "remarkable equivalence." The statement is false for Nisnevich/etale sheaves on big sites. However, after the inverting the residual characteristics, it turns out that the stable motivic homotopy category is. We will try to give a complete proof of this theorem, state some applications and future directions. This is joint work with A. A. Khan.

Time: Nov. 21, 2018 10:15 AM - 12:00 PM

To extend A1-homotopy theory so that non A1-invariant cohomology theories like algebraic k-theory and algebraic de Rham cohomology are representable, the so-called box-invariance has been suggested. However, the usual Sing construction for box does not work well since box is not an interval object. In this talk, I will give a new Sing construction for box using calculus of fractions. This is a partial result of an ongoing project joint with Federico Binda and Paul Arne Østvær.

Time: Nov. 15, 2018

Paul Arne Østvær will speak at the conference New trends in K-theory and homotopy theory at Institut Henri Poincaré in November.

Time and place: Nov. 1, 2018 4:15 PM - 5:15 PM, B1119

We construct the homomorphism of presheaves $\mathrm{K}^\mathrm{MW}_{*}\to \pi^{*,*}_s$, where $\mathrm{K}^\mathrm{MW}_{*}$ is the naive Milnor-Witt K-theory presheaf, and $\pi^{*,*}_s$ are stable motivic homotopy groups over a base $S$. The Garkusha-Panin’s theory of framed motives and the Neshitov’s computation of $\pi^{*,*}_s(k)$ for $char k=0$, gives the alternative proof of the stable version of Morel’s theorem on zero motivic homotopy groups, namely the isomorphism $\mathrm{K}^\mathrm{MW}_{*}(k)\to \pi^{*,*}_s(k)$, for the case of fields $k$, $char k=0$. We extend this proof to the case of perfect fields of odd characteristic, and deduce that the above homomorphism induces isomorphism of unramified Milnor-Witt K-theory sheaf $\mathbf{K}^\mathrm{MW}_*$ and the associated (Nisnevich and Zariski) sheaf $\underline{\pi}^{*,*}_s$ over such fields. The talk is based on the joint work with Jonas Irgens Kylling.

Time and place: Oct. 25, 2018 4:15 PM - 5:15 PM, B1119

The weak factorization theorem for varieties roughly says that any proper birational map of smooth varieties factors as a sequence of blow-ups and blow-downs in smooth centres. I will show that a similar theorem holds for Deligne-Mumford stacks, provided that we enlarge the class of birational modifications used to include so called root stacks (there also are independent proofs for this by Harper and by Rydh). Furthermore, I will show how to use this to get a presentation of the Grothendieck group of Deligne-Mumford stacks with generators given by smooth and proper Deligne-Mumford stacks. Time permitting I will also mention some joint work with Gorchinskiy, Larsen and Lunts, where we use the results above to prove a conjecture by Galkin-Shinder on the categorical zeta function.

Time and place: Oct. 24, 2018 10:15 AM - 12:00 PM, B1120

In the talk I will discuss the cohomological interpretation of the existence of a nowhere vanishing section of a rank n vector bundle over a smooth algebraic variety of dimension n. I will briefly cover the classical statement for projective varieties involving the top Chern class and describe the approach to the affine case involving the techniques from the motivic homotopy theory and the motivic Euler class. Then I will discuss some special cases when the vanishing of the top Chern class yields the vanishing of the Euler class.

Time: Oct. 17, 2018 10:15 AM - 12:00 PM

The genuine analog of an $E_{\infty}$-ring spectrum in algebraic geometry is the notion of a normed motivic spectrum, which carries multiplicative transfers along finite etale morphisms. The homological shadows of an $E_{\infty}$-ring structure are the Dyer-Lashof operations which acts on the homology an $E_{\infty}$-ring spectrum. We will construct analogs of these operations in motivic homotopy theory, state their basic properties and discuss some consequences such as splitting results for normed motivic spectra. The construction mixes two ingredients: the theory of motivic colimits and equivariant motivic homotopy theory. This is joint work with Tom Bachmann and Jeremiah Heller.

Time: Oct. 11, 2018 4:15 PM - 5:00 PM

Let C be a generalised based category (to be defined) and R a commutative ring with identity.  In this talk, we construct a cohomology theory in the category B_R(C)  of contravariant functors from C  to the category of R-modules in an axiomatic way,  This cohomology theory generalises simultaneously Bredon cohomology involving finite, profinite, and discrete groups.  We also study higher K-theory of the categories of finitely generated projective objects and and finitely generated objects in B_R(C) and obtain some finiteness and other results.

Time: Oct. 10, 2018 10:15 AM - 12:00 PM

This is a partial report on a joint work with G. Garkusha. The triangulated category of framed bispectra SH^fr_nis(k) is introduced. This triangulated category only uses Nisnevich local equivalences and has nothing to do with any kind of motivic equivalences. It is proved that SH^fr_nis(k) recovers the classical Morel-Voevodsky triangulated categories of bispectra SH(k), provided the base field k is infinite and perfect.

Time: Oct. 4, 2018 4:15 PM - 5:15 PM

A modern approach to the motivic stable homotopy category allows one to express its mapping spaces in terms of geometric data called "framed correspondences". We will explain this approach and illustrate it by computing Gm-homotopy groups of the special linear algebraic cobordism spectrum MSL.

Time: Sep. 13, 2018 - Sep. 14, 2018

Paul Arne Østvær will give a talk in the parallel session at the National Mathematicians meeting in Bergen held on September 13--14, 2018.

Time: Sep. 10, 2018

Paul Arne Østvær will speak at the conference Motives and their applications, at the Euler International Mathematical Institute in St. Petersburg.

Time: Sep. 5, 2018

Håkon Kolderup will speak at the conference Motives in St. Petersburg at the Euler International Mathematical Institute in St. Petersburg, Russia, on "Cohomological correspondence categories".

Time: Sep. 5, 2018

Jonas Irgens Kylling will speak at the conference Motives in St. Petersburg at the Euler International Mathematical Institute in St. Petersburg, Russia, on "Slice spectral sequence calculations of hermitian K-theory and Milnor’s conjecture on quadratic forms for rings of integers".

Time: Aug. 29, 2018 10:15 AM - 12:00 PM

The Mahowald invariant is a method for constructing nontrivial classes in the stable homotopy groups of spheres from lower dimensional classes. I will introduce this construction and recall Mahowald and Ravenel's computation of the Mahowald invariant of 2^i for all i . I'll then introduce motivic and equivariant analogs of the Mahowald invariant, outline the computation of the generalized Mahowald invariants of 2^i and \eta^i for all i, and discuss the relationship between these generalized computations and exotic periodicity in the equivariant and motivic stable homotopy groups of spheres.

Time: Aug. 6, 2018 2:45 PM - 3:30 PM

Ivan Panin will speak at the International Congress of Mathematicians in Rio de Janeiro.

Time: July 23, 2018 10:15 AM - July 27, 2018 12:00 PM

Charanya Ravi will speak at K-Theory Workshop, a satellite event of the ICM 2018.

Time: July 10, 2018

Paul Arne Østvær will give a talk on A1-contractible varieties at the London Mathematical Society and Clay Mathematics Institute Research School on Homotopy Theory and Arithmetic Geometry: Motivic and Diophantine Aspects, Imperial College, 9-13 July 2018.

Time: July 4, 2018 2:30 PM - 3:30 PM

Paul Arne Østvær will give a lecture at the University of Oxford on A1-contractible varieties.

Time: June 1, 2018 - July 31, 2018

Paul Arne Østvær will hold a lecture series as a Nelder visiting fellow at the Imperial College London during June and July of 2018.

Time: May 31, 2018 9:30 AM - 10:20 AM

Sabrina Pauli will speak at the conference Algebraic Geometry - Mariusz Koras in memoriam at the Institute of Mathematics, Polish Academy of Sciences on "A1-contractibility of Koras-Russell like varieties."

Time and place: May 29, 2018 10:00 AM - May 30, 2018 1:00 PM, Georg Sverdrups hus Auditorium 2 (BL271511)

On the 29th and 30th of May, a topology meeting will convene at the University of Oslo to discuss a broad range of topics in topology, including topological Hochschild homology, motivic homotopy theory, symplectic geometry, and low-dimensional geometry.