Events

Abstract:  "The study of foliations falls largely into two parts: One can study the leaf geometry or one can study transversally elliptic operators. The leaf geometry consists of studying the individual submanifolds and how they lie within the manifold. On the other hand, the study of transversally elliptic operators was initiated in the seminal work of Atiyah. This talk will be arranged as follows: The first part will be an introduction to Riemannian foliations and transverse geometry, and the second part is a survey of the results on transversally elliptic operators, foliated gauge theory and some recent work."  

This lecture is the second of a mini-course consisting of seven lectures given by Professor Robert Bruner (Wayne State University, Detroit, USA), the author of the package.  Early lectures will focus on the use of the software.  Later lectures will describe the algorithms, data structures, and file structures.  

This lecture is the first of a mini-course consisting of seven lectures given by Professor Robert Bruner (Wayne State University, Detroit, USA), the author of the package.  Attendees are encouraged to bring their laptops and do the calculations in real time with the speaker.  

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.  

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.  

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.  

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.

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

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.  

In this talk, I will discuss how moduli spaces of Morse flow trees in Legendrian contact homology (LCH) can be oriented in a coherent and computable manner, obtaining a Morse-theoretic way to compute LCH with integer coefficients. This is built on the machinery of capping disks, and I will briefly explain how different systems of capping disks affect the orientations. This, in turn, uses the fact that an exact Lagrangian cobordism with cylindrical Legendrian ends induces a morphism between the LCH-complexes of the ends, which can be proven to hold also with integer coefficients.  

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.

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.  

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

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.

University of Oslo, Departement of Mathematics, 29-30 May 2018

Georg Sverdrups hus Auditorium 2

Further details see 

In this second talk I will prove the local slice theorem and give examples of applications, discuss compactness properties of instanton moduli spaces, and explain the definition and some properties of instanton homology.

In their book "Riemann-Roch Algebra", Fulton and Lang give an account of Chern classes in lambda-rings and a general version of Grothendieck's Riemann-Roch theorem. Their definition of Chern classes is based on the additive formal group law.  In work on connective K-theory, Greenlees and I have given an account of Chern classes in lambda-rings based on the multiplicative formal group law.  This account has an evident generalization to any formal group law.  The course will be an attempt to carry out Fulton and Lang's program in this more general setting.  Hoped for applications include generalizations of results relating rational lambda-modules to twisted Dirichlet characters. ---

Waldhausen's algebraic K-theory of spaces is an extension of algebraic K-theory from rings to spaces (or ring spectra) which also encodes important geometric information about manifolds. Bivariant A-theory is a bivariant extension of algebraic K-theory from spaces to fibrations of spaces. In this talk, I will first recall the definition and basic properties of bivariant A-theory and the A-theory Euler characteristic of Dwyer-Weiss-Williams. I will then introduce a bivariant version of the cobordism category and explain how this may be regarded as a universal space for the definition of additive characteristic classes of smooth bundles. Lastly, I will introduce a bivariant extension of the Dwyer-Weiss-Williams characteristic and discuss the Dwyer-Weiss-Williams smooth index theorem in this context. Time permitting, I will also discuss some ongoing related work on the cobordism category of h-cobordisms. This is joint work with W. Steimle.