Titles and Abstracts of Talks

Pablo Pelaez - UNAM, Mexico
Talk 1 Title: The slice filtration in motivic homotopy theory.
We will discuss several aspects of the slice filtration in motivic homotopy theory.
Pablo Pelaez- UNAM, Mexico
Talk 2 Title: Towards a motivic spectral sequence for Quillen K-theory over non-reduced and singular schemes.
We will describe the construction of a spectral sequence converging to the Quillen K-theory of any scheme of finite type over a field.
Marco Schlichting - University of Warwick
Title: Homology stability for SL_n and Euler classes of projective modules
In this talk I will show that for a local ring R with infinite residue field, the relative homology groups H_i(SLnR,SL_{n-1}R) vanish for i less than n and equal to Milnor-Witt K-theory of R for i=n. If time permits, I will indicate how this leads to a generalization of a theorem of Morel on Euler classes of projective modules.
Wilberd van der Kallen - University of Utrecht
Title: Extrapolating an Euler class.
Let $R$ be a noetherian ring of dimension $d$ and let $n$ be an integer so that $n\leq d\leq 2n-3$. Let $(a_1,\dots,a_{n+1})$ be a unimodular row so that the ideal $J=(a_1,\dots,a_n)$ has height $n$. Jean Fasel has associated to this row an element $[(J,\omega_J)]$ in the Euler class group $E^n(R)$, with $\omega_J:(R/J)^n\to J/J^2$ given by $(\bar a_1,\dots,\bar a_{n-1},\bar a_n\bar a_{n+1})$. If $R$ contains an infinite field $F$ then we show that the rule of Fasel defines a homomorphism from $\WMS_{n+1}(R)=\Um_{n+1}(R)/E_{n+1}(R)$ to $E^n(R)$. The main problem is to get a well defined map on all of $\Um_{n+1}(R)$. Similar results have been obtained by Das and Zinna \cite{DasZinna}, with a different proof. Our proof uses that every Zariski open subset of $\SL_{n+1}(F)$ is path connected for walks made up of elementary matrices.
Kirsten Wickelgren - Georgia Tech
Title: Motivic desuspension
This talk will present an EHP sequence in A^1 algebraic topology at p=2, encoding the obstruction to simplicial desuspension. We will discuss a computation and equivariant analogues. This is joint work with Ben Williams, and includes joint work with Aravind Asok and Jean Fasel.

Serge Yagunov - Steklov Institute
Title: Motivic Cohomology Spectral Sequence and Steenrod Algebra
We will see that differentials d_n in the motivic cohomology spectral sequence with p-local coefficients vanish unless p-1 divides n-1. We also obtain an explicit formula for the first non-trivial differential d_p, expressing it in terms of motivic Steenrod p-power operations and Bockstein homomorphisms.
Jean Fasel - Institut Fourier
Title: Cohomological detection of complete intersections
Let k be a field and X a smooth affine variety over k. If Z\subset X is a closed subvariety equipped with a trivialization of its conormal bundle, then we will give cohomological criteria for Z to be a complete intersection in X. Along the way, we will compare Euler class groups as defined by Nori, Bhatwadekar and Sridharan with the Chow-Witt groups introduced by Barge and Morel and give a conditional answer to a conjecture of Murthy. Our method relies on the geometry of smooth split quadrics.
Oliver Roendigs - University of Osnabruck
Title: The slice filtration for hermitian K-theory and motivic stable homotopy
The talk will report on joint work with Paul Arne Oestvaer and Markus Spitzweck. Marc Levine showed that, at least over a field of characteristic zero, the slices of the motivic sphere spectrum can be read off from the second page of the topological Adams-Novikov spectral sequence. Information on the slices of hermitian K-theory and their multiplicative structure allows to determine the first slice differential for the motivic sphere spectrum in a certain range. This implies results for the first motivic stable stem.
Roy Joshua- Ohio State University
Title: Equivariant Algebraic K-theory and Derived completion
Abstract :
Derived completion is a technique that originated slightly over 20 years ago. However, Gunnar Carlsson may have been the first to realize the true potential of this technique, especially in the context of Algebraic K-theory. The goal of this talk is to discuss some applications of this technique, especially in the context of our joint work with Gunnar, to equivariant algebraic K-theory.
Teena Gerhardt - Michigan State University
Title: An approach to the algebraic K-theory of Z[C_2]
Methods in equivariant stable homotopy theory make some algebraic K-theory computations more accessible. Using these methods to compute the algebraic K-theory of pointed monoid algebras is particularly interesting, as the full power of equivariant homotopy groups is used. I will recall some successes of these methods and then describe how equivariant techniques contribute to a new strategy for answering a classical computational question. In particular, I will discuss a new approach to computing the algebraic K-theory of the group ring Z[C_2]. This is joint work in progress with Vigleik Angeltveit.

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Aravind Asok - University of Southern California, Los Angeles
Title: Algebraizing topological vector bundles
I will discuss the problem of when a topological vector bundle on a smooth complex affine variety admits an algebraic structure. A necessary condition that a topological vector bundle is algebraizable is that its integral Chern classes are algebraic, i.e., lie in the image of the cycle class map from Chow groups to integral cohomology. It is a folk conjecture (arguably one that can be attributed to P. Griffiths) that any topological vector bundle on a smooth complex affine variety with algebraic Chern classes is algebraizable. I will explain that this conjecture is true for smooth affine varieties of dimension <= 3, and show that it is false for smooth affine varieties of dimension >= 4. In particular, I will describe necessary and sufficient condition for algebraizability of rank 2 vector bundles on smooth complex affine 4-folds and an example that exhibits non-triviality of an obstruction beyond algebraicity of Chern classes. This is based on joint work with Jean Fasel and Mike Hopkins.
Charles Weibel - Rutgers University
Title: The Witt group of real algebraic varieties
It is a classical problem to compute the Witt group W(V) of an algebraic variety V over the real numbers. We introduce a computeable topological version WR(V) and show that the kernel and cokernel of W(V)-->WR(V) are 2-primary groups of bounded exponent, depending on dim(V). When V is a curve, W(V) and WR(V) are isomorphic. This is joint work with Max Karoubi.

Amalendu Krishna- Tata Institute
Title: Zero-cycles on singular schemes and class field theory
In this talk, we shall discuss some relation between the Chow group of 0-cycles on a singular projective scheme $X$ over a finite field and the class field theory of its function field. As applications, we shall show how to obtain the Bloch-Quillen formula for the Chow group of 0-cycles and prove the Bloch-Srinivas conjecture for such schemes. As another application, we deduce simple proofs of results of Kerz-Saito for a class of surfaces without any assumption on the charactersitic.

Jeremiah Heller - UIUC
Title: Endomorphisms of the equivariant motivic sphere
For a finite group G, I will discuss a tom Dieck style splitting theorem for certain stable equivariant motivic homotopy groups. As a corollary, we obtain a computation of the group of endomorphisms of the equivariant motivic sphere spectrum. This is joint work with D. Gepner.
David Gepner - Purdue University
Title: Localization sequences in the algebraic K-theory of ring spectra
Localization sequences in algebraic K-theory are important both theoretically and computationally. The idea is to understand the K-theory of a localization $R\to R[S^{-1}]$ of rings (more generally, schemes, ring spectra, derived schemes, etc.). It is well-known that the fiber of the map $K(R)\to K(R[S^{-1}])$ is the K-theory of the S-nilpotent perfect R-modules, and it turns out that, under mild conditions on S, this category admits a single compactly generator and so is again the K-theory of a ring spectrum A. In case S is generated by a single homogeneous element $f\in\pi_* R$ and the quotient $R/f$ is a ring spectrum, it is natural to ask whether or not the map $K(R/f)\to K(A)$ is an equivalence, say as a result of a very strong version of devissage. We show that this is not the case in general, and in particular answer a question of Rognes concerning higher chromatic analogues of a localization sequence involving Z due to Quillen and connective complex K-theory due to Blumberg-Mandell. This is joint work with Ben Antieau and Tobias Barthel.

Clark Barwick - MIT
Title: Equivariant algebraic K-theory
Abstract: I will give an overview of the general set-up of equivariant algebraic K-theory, and I'll describe how this framework is a natural one in which to contemplate etale cohomology, crystalline cohomology, topological Hochschild homology, and Goodwillie calculus.
Tyler Lawson - University of Minnesota
Title: Continuous homology and topological cyclic homology.
Abstract: Recent work in algebraic K-theory has aimed at understanding its connection to the chromatic filtration. In this talk I'll discuss an approach to studying the topological cyclic homology (TC) of spectra by use of a "continuous" homology theory. This results in computational data which is more tractable than the homotopy groups, but still provides valid input for an Adams-Novikov spectral sequence computing TC.
Gunnar Carlsson - Stanford University
Talk 1 Title : Representation theoretic models for the algebraic K-theory of fields:I
Abstract: It is often useful to identify algebraic K-theory spectra of complicated rings and schemes as the result of a homotopy colimit construction with input the K-theory spectrum of a simpler ring or scheme. The prototype of such constructions is the so-called assembly map, which often describes the K-theory spectrum of a group ring as the smash product of the classifying space of the group with the K-theory spectrum of the ring of coefficients of the group ring. One could ask whether the K-theory spectrum of a geometric field F (one containing an algebraically closed subfield k) could also be described in terms of a construction on the K-theory spectrum of k, from which one could obtain the structure of KF using Suslin's theorem. These talks will describe such a construction. It turns out that one must construct the K-theory spectrum of the category of continuous k-linear representations of the absolute Galois group of F. It also turns out that one must perform a homotopy invariant completion construction in order to achieve the equivalence, which is of interest in its own right. We will discuss the construction, and methods for proving it.

Gunnar Carlsson - Stanford University
Talk 2 Title: Representation theoretic models for the algebraic K-theory of fields:II
Abstract: See above.
Daniel Ramras - IUPUI
Title : Co-assembly for Bieberbach groups: beyond the Novikov conjecture.
Abstract: For a discrete group G, the (strong) Novikov conjecture states that the assembly map from the K-homology of BG to the K-theory of the group C*-algebra of G is rationally injective. The deformation K-theory spectrum of G, built from the category of finite-dimensional unitary representations of G, admits a co-assembly map to the K-theory of BG. This map is dual to assembly in the sense that rational surjectivity of co-assembly in high dimensions implies rational injectivity of assembly. These ideas were used in joint work with Willet and Yu to give a proof of the Novikov conjecture for Bieberbach groups (fundamental groups of flat manifolds). In this talk, I'll explain work in progress aimed at proving the co-assembly map is actually a rational isomorphism (in high dimensions) for such groups.

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