Time and place: May 29, 2019 12:15 PM2:00 PM, B1120 NHA

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.  

Time and place: May 27, 2019 12:15 PM2:00 PM, B1120 NHA

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.  



Time: May 15, 2019 10:15 AM12:00 PM

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: Apr. 3, 2019 10:15 AM12:00 PM, B1119 NHA

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, B 723 NHA

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: Feb. 5, 2019 2:15 PM4:00 PM, NHA B1120

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.  

Time and place: Oct. 17, 2018 10:15 AM12:00 PM, B1120 NHA

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 and place: Oct. 11, 2018 4:15 PM5:00 PM, B 1119 NHA

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 and place: Oct. 10, 2018 10:15 AM12:00 PM, B1120 NHA

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 and place: Aug. 29, 2018 10:15 AM12:00 PM, 1120 N.H.A.

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 and place: May 2, 2018 2:15 PM4:00 PM, Gates of Eden, Sognsveien 77 B

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.

Time and place: May 2, 2018 11:20 AM12:00 PM, Desolation Row Sognsveien 77 B

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. ---

Time and place: May 2, 2018 10:15 AM11:05 AM, Desolation Row Sognsveien 77 B
Time and place: Apr. 11, 2018 2:15 PM4:00 PM, Desolation Row, Sognsveien 77 B

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.  

Time and place: Mar. 21, 2018 2:15 PM4:00 PM, Desolation Row, Sognsveien 77 B

I will review Witt vectors, KÀhler forms and logarithmic rings, and outline how they merge in the logarithmic de Rham-Witt complex. This structure gives an algebraic underpinning for the Hesselholt-Madsen (2003) calculation of logarithmic topological cyclic homology of many discrete valuation rings.   

Time and place: Mar. 20, 2018 10:15 AM12:00 PM, End of the Line, Sognsveien 77 B

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. 

Time and place: Mar. 8, 2018 10:15 AM2:00 PM, Desolation Row Sognsveien 77 B

I discuss how Bökstedt and Madsen (1994/1995) calculate mod p homotopy for THH(Z) and the fixed-point spectra THH(Z)^{C_{p^n}}, together with the R- and F-operators. This leads to a calculation for TC(Z; p) and K(Z_p), confirming the Lichtenbaum-Quillen conjecture in this case. 

Time and place: Mar. 7, 2018 2:15 PM4:00 PM, Desolation Row, Sognsveien 77 B

I will review Bökstedt, Hesselholt and Madsen's calculations of the topological cyclic homology of prime fields and the integers, again taking into account simplifications made in later papers. (If necessary, I will continue on Thursday.)   

Time and place: Mar. 7, 2018 10:15 AM12:00 PM, Desolation Row, Sognsveien 77 B
Time and place: Mar. 6, 2018 10:15 AM12:00 PM, End of the Line, Sognsveien 77B

Recently two different refinements of Voevodsky's theory of presheaves with transfers were introduced: the first one is the theory of framed presheaves based on the unpublished notes by Voevodsky and developed by Garkusha and Panin and the second one is the theory of Milnor-Witt presheaves due to Calmes and Fasel. I will review some relations between these theories and explain that the hearts of the homotopy t-structures on the corresponding categories of motives are naturally equivalent. The talk is based on a joint work with A. Neshitov. 

Time and place: Feb. 28, 2018 2:15 PM4:00 PM, Desolation Row Sognsveien 77B

In this third talk we will define Legendrian contact homology for Legendrian submanifolds in the 1-jet space of a smooth manifold M. Again, this will be the homology of a DGA generated by the double points of the Legendrian under the Lagrangian projection. The differential is defined by a count of punctured pseudo-holomorphic disks in the cotangent bundle of M, with boundary on the projected Legendrian. To prove that this indeed gives a differential we will use the theory of Fredholm operators from functional analysis. I will also say something about Floer theories in general. In particular, one of the main difficulties when defining Floer theories via pseudo-holomorphic curve techniques is to achieve transversality for the dbar-operator. There has been a development of several different machineries to solve these problems, for examle Polyfolds by Hofer et al., and Pardon's work on Virtual fundamental cycles. In our case, however, it is enough to perturb either the Legendrian submanifold or the almost complex structure.   

Time and place: Feb. 28, 2018 10:15 AM12:00 PM, Desolation Row Sognsveien 77B
Time and place: Feb. 21, 2018 2:15 PM4:00 PM, Desolation Row, Sognsveien 77B

I will review Marcel Bökstedt's calculation of the topological Hochschild homology of prime fields and the integers, taking into account simplifications made in papers by Angeltveit-R. (where BP<m-1> specializes to HFp for m=0 and to HZ(p) for m=1) and Ausoni (proof of Lemma 5.3).

Time and place: Feb. 21, 2018 10:15 AM12:00 PM, Desolation Row Sognsveien 77 B

Inspired by the Voevodsky machinery of standard triples a machinery of nice triples was invented in [PSV]. We develop further the latter machiny such that it works also in the finite field case [P]. This machinary is a tool to prove many interesting moving lemmas. It leads to a serios of applications. One of them is a proof of the Grothendieck--Serre conjecture in the finite field case. Another is a proof of Gersten type results for arbitrary cohomology theories on algebraic varieties. The Gersen type results allows to conclude the following: a presheaf of S1-spectra E on the category of k-smooth schemes is A1-local iff all its Nisnevich sheaves of stable A1-homotopy groups are strictly homotopy invariant. If the field k is infinite, then the latter result is due to Morel [M]. An example of moving lemma is this. Let X be a k-smooth quasi-projective irreducible k-variety, Z be its closed subset and x be a finite subset of closed points in X. Then there exists a Zariski open U containing x and a naive A1-homotopy between the motivic space morphism U--> X--> X/U and the morphism U--> X/U sending U to the distinguished point of X/U. Application: suppose E is a cohomology theory on k-smooth varieties and alpha is an E-cohomology class on X which vanishes on the complement of Z, then it vanishes on U from the lemma above.   

Time and place: Feb. 14, 2018 2:15 PM4:00 PM, Desolation Row Sognsveien 77B

In this second talk, I will define Chekanov's version of Legendrian contact homology (LCH) for Legendrian knots in R3. I will begin with an example, showing that LCH is more sensitive than the classical invariants. This will use a linearized version of the homology. In the second part of the talk I will focus on the proof that the differential indeed squares to zero, and also say something about invariance under Legendrian Reidemeister moves. This is intended to be a smooth introduction to the next talk, where we will consider Legendrian contact homology defined for Legendrians in arbitrary 1-jet spaces. This case is more delicate, and we have to understand the concept of Gromov compactness for pseudo-holomorphic curves to prove that we get a differential graded algebra associated to each Legendrian, whose homology will give a Legendrian invariant.