Events - Page 3

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

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.

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.

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.

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.

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.

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.

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.

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

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

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".

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".

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.

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

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

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

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

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."

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.

Paul Arne Østvær will give a talk at the KTH topology seminar on "A motivic Segal conjecture for the group of order two."

Paul Arne Østvær will speak at the conference Motivic homotopy theory and refined enumerative geometry at the Universität Duisburg-Essen, Essen, Germany on "A motivic Segal conjecture for the group of order two."

Paul Arne Østvær will speak at the conference Motives in Tokyo, which will be held in honor of Shuji Saito's 60th birthday.

This talk is supposed to be an Introductionary talk to the preprint arXiv:1409.4372v4 (joint work with G.Garkusha). More specifically, using the theory of framed correspondences developed by Voevodsky, the authors introduce and study framed motives of algebraic varieties. This study gives rise to a construction of the big frame motive functor. It is shown that this functor converts the classical Morel--Voevodsky motivic stable homotopy theory into an equivalent local theory of framed bispectra, and thus producing a new approach to stable motivic homotopy theory. As a topological application, it is proved that for the simplicial set $Fr(Delta^\bullet_C, S^1)$ has the homotopy type of the space $\Omega^{\infty} Sigma^{\infty} (S^1)$. Here C is the field complex numbers.