Seminars - Page 2

Time and place: May 15, 2019 10:15 AM12:00 PM, B1119

Subtle Stiefel-Whitney classes have been introduced by Smirnov and Vishik as a tool for classifying quadratic forms. Following this path, in this talk, I will introduce subtle characteristic classes for Hermitian forms, coming from the motivic cohomology ring of the Nisnevich classifying space of the unitary group associated to the standard split Hermitian form of a quadratic extension. Moreover, I will discuss the connection between these new classes and the subtle Stiefel-Whitney ones, deduce information on the kernel invariant for quadratic forms divisible by a 1-fold Pfister form, show that these classes see the triviality of Hermitian forms and express the motive of the torsor associated to a Hermitian form in terms of its subtle characteristic classes.

Time and place: May 13, 2019 10:15 AM12:00 PM, 723
Time: Apr. 30, 2019 2:15 PM4:00 PM

Unlike most cohomology theories in algebraic geometry, algebraic K-theory does not satisfy descent with respect to arbitrary blow-up squares. We explain why the only obstruction is the failure of the Mayer-Vietoris property for unions of closed subschemes. Since this obstruction vanishes after forcing A1-homotopy invariance, this gives a direct new proof of Cisinski's theorem that homotopy K-theory does satisfy cdh descent.

Time and place: Apr. 15, 2019 10:15 AM12:00 PM, B723

I will talk about how to prove an arithmetic refinement of the Yau-Zaslow formula by replacing the classical Euler characteristic in Beauville's argument by a variant of Levine's motivic Euler characteristic.  We derive several similar formulas for other related invariants, including Saito's determinant of cohomology, and a generalisation of a formula of Kharlamov and Rasdeaconu on counting real rational curves on real K3 surfaces. Joint work with Frank Neumann.

Time and place: Apr. 3, 2019 10:15 AM12:00 PM, B1119

There are several cohomology theories over a field like Hodge cohomology theory that are not A1-invariant but still having other fundamental properties like the Projective bundle formula. These are not representable in DM. I will explain how to extend DM to include them using log geometry and cube-invariance. Some fundamental properties like Gysin triangles and blow-up triangles will be also discussed. This is joint with Federico Binda and Paul Arne Østvær.

Time and place: Apr. 1, 2019 10:15 AM12:00 PM, B723

In preparation for the MHE seminar "log motives over a field", we give an introduction to ongoing work on motives for log schemes over fields. This is joint with Doosung Park and Paul Arne Østvær.

Time and place: Mar. 28, 2019 10:15 AM12:00 PM, B723
Time and place: Mar. 27, 2019 10:15 AM12:00 PM, B1119
Time and place: Mar. 25, 2019 10:15 AM12:00 PM, B723
Time and place: Mar. 21, 2019 10:15 AM12:00 PM, B723
Time: Mar. 20, 2019 2:00 PM2:40 PM

Sabrina Pauli is giving a conference talk at the women in homotopy theory and algebraic geometry workshop in Berlin.

Time and place: Mar. 6, 2019 10:15 AM12:00 PM, B1119

I will discuss the “isotropic motivic category”. This “local” version of Voevodsky motivic category (with finite coefficients), obtained from the “global” one by, roughly speaking, annihilating the motives of anisotropic varieties, has many remarkable properties. Considering such “local” versions for all finitely generated extensions of a ground field, permits to read global information in a rather simple form. For appropriate (so-called, “flexible”) fields, “isotropic motives” are more reminiscent of their topological counterparts. In particular, “isotropic Chow groups” hypothetically coincide with Chow groups modulo numerical equivalence (with finite coefficients) and so should be finite-dimensional (checked in various cases). On the other hand, the “isotropic motivic cohomology” ring of a point doesn’t depend on a field and encodes Milnor’s operations.

Time and place: Feb. 26, 2019 2:15 PM4:00 PM, B1120

This talk discusses a few properties of cones with respect to a single endomorphism of the unit in the motivic stable homotopy category.

Time: Feb. 21, 2019 4:00 PM5:30 PM

Håkon Kolderup is giving a talk at the seminar in arithmetic geometry at the Nagoya University.

Time and place: Feb. 20, 2019 10:15 AM12:00 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: Feb. 14, 2019 3:30 PM4:30 PM

Håkon Kolderup will speak at the International Workshop on motives in Tokyo.

Time and place: Feb. 6, 2019 10:15 AM12:00 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 AM11:15 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 PM4:00 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 AM12:00 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 PM5:15 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 PM5:15 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 AM12:00 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 AM12:00 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.