# Oliver RöndigsUniversität Osnabrück | UOS · Institut für Mathematik

Oliver Röndigs

Prof. Dr. math.

## About

47

Publications

3,722

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772

Citations

Citations since 2016

Introduction

Oliver Röndigs currently works at the Institut für Mathematik, Universität Osnabrück.

Additional affiliations

January 2017 - December 2020

October 2006 - present

July 2004 - September 2006

## Publications

Publications (47)

We compute the 1-line of stable homotopy groups of motivic spheres over fields of characteristic not two in terms of hermitian and Milnor K-groups. This is achieved by solving questions about convergence and differentials in the slice spectral sequence.

We advance the understanding of K-theory of quadratic forms by computing the
slices of the motivic spectra representing hermitian K-groups and Witt-groups.
By an explicit computation of the slice spectral sequence for higher
Witt-theory, we prove Milnor's conjecture relating Galois cohomology to
quadratic forms via the filtration of the Witt ring b...

Let k be a field and X be a smooth projective curve over k with a rational point. Then X admits a theta characteristic if and only if the motivic stable homotopy type of X splits off the top cell. The constructed splitting lifts the splitting of the motive of X. © 2009. Published by Oxford University Press. All rights reserved.

We employ the Goodwillie spectral sequence for the iterated loop space functor in order to provide realizability conditions on certain unstable modules over the Steenrod algebra at an odd prime.

Coloured operads were introduced in the 1970s for the purpose of studying homotopy invariant algebraic structures on topological
spaces. In this paper, we introduce coloured operads in motivic stable homotopy theory. Our main motivation is to uncover
hitherto unknown highly structured properties of the slice filtration. The latter decomposes every...

Motivated by calculations of motivic homotopy groups, we give widely attained conditions under which operadic algebras and modules thereof are preserved under (co)localization functors.

We compute the 2-line of stable homotopy groups of motivic spheres over fields of characteristic not two in terms of motivic cohomology and hermitian K-groups.

The endomorphism ring of the projective plane over a field F of characteristic neither two nor three is slightly more complicated in the Morel-Voevodsky motivic stable homotopy category than in Voevodsky's derived category of motives. In particular, it is not commutative precisely if there exists a square in F which does not admit a sixth root.

We argue that the very effective cover of hermitian $K$-theory in the sense of motivic homotopy theory is a convenient algebro-geometric generalization of the connective real topological $K$-theory spectrum. This means the very effective cover acquires the correct Betti realization, its motivic cohomology has the desired structure as a module over...

We employ the slice spectral sequence, the motivic Steenrod algebra, and Voevodsky's solutions of the Milnor and Bloch-Kato conjectures to calculate the hermitian K-groups of rings of integers in number fields. Moreover, we relate the orders of these groups to special values of Dedekind ζ-functions for totally real abelian number fields. Our method...

The term "motivic Moore spectrum" refers to a cone of an element in the motivic stable homotopy groups of spheres. This article discusses some properties of motivic Moore spectra, among them the question whether the ring structure on the motivic sphere spectrum descends to a ring structure on a motivic Moore spectrum. This discussion requires an un...

We compute the homotopy groups of the {\eta}-periodic motivic sphere spectrum over a finite-dimensional field k such that (a) k has odd characteristic, (b) k has 2-cohomological dimension at most 2, or (c) k contains a square root of -1. We also study the general characteristic 0 case and show that the {\eta}-periodic slice spectral sequence over Q...

Motivated by calculations of motivic homotopy groups, we give widely attained conditions under which operadic algebras and modules thereof are preserved under (co)localization functors

We employ the slice spectral sequence, the motivic Steenrod algebra, and Voevodsky's solutions of the Milnor and Bloch-Kato conjectures to calculate the hermitian $K$-groups of rings of integers in number fields. Moreover, we relate the orders of these groups to special values of Dedekind $\zeta$-functions for totally real abelian number fields. Ou...

We determine systematic regions in which the bigraded homotopy sheaves of the motivic sphere spectrum vanish.

We provide a random simplicial complex by applying standard constructions to a Poisson point process in Euclidean space. It is gigantic in the sense that - up to homotopy equivalence - it almost surely contains infinitely many copies of every compact topological manifold, both in isolation and in percolation.

We solve affirmatively the homotopy limit problem for $K$-theory over fields of finite virtual cohomological dimension. Our solution employs the motivic slice filtration and the first motivic Hopf map.

Waldhausen's algebraic K-theory machinery is applied to motivic homotopy theory, producing an interesting motivic homotopy type. Over a field F of characteristic zero, its path components receive a surjective ring homomorphism from the Grothendieck ring of varieties over F.

We give concise formulas in terms of generators and relations for the multiplicative structures on the graded slices of hermitian K-theory and Witt-theory.

We advance the understanding of K–theory of quadratic forms by computing the slices of the motivic spectra representing hermitian K–groups andWitt groups. By an explicit computation of the slice spectral sequence for higher Witt theory, we prove Milnor’s conjecture relating Galois cohomology to quadratic forms via the filtration of the Witt ring by...

Hermitian K-theory and Witt-theory are cellular in the sense of stable motivic homotopy theory over any base scheme without points of characteristic two.

It is shown that the first and second homotopy groups of the $\eta$-inverted sphere spectrum over a field of characteristic not two are zero. A cell presentation of higher Witt theory is given as well, at least over the complex numbers.

We compute the 1-line of stable homotopy groups of motivic spheres over fields of
characteristic not two in terms of hermitian and Milnor K-groups. This is achieved by
solving questions about convergence and differentials in the slice spectral sequence.

This is a continuation, completion, and generalization of our previous joint
work with B. Chorny. We supply model structures and Quillen equivalences
underlying Goodwillie's constructions on the homotopy level for functors
between certain simplicial model categories.

We employ the Goodwillie spectral sequence for the iterated loop space
functor in order to provide realizability conditions on certain unstable
modules over the Steenrod algebra at an odd prime.

Under a certain normalization assumption we prove that the P1-spectrum BGL of Voevodsky which represents algebraic K-theory is unique over Spec.(Z). Following an idea of Voevodsky, we equip the P1-spectrum BGL with the structure of a commutative P1-ring spectrum in the motivic stable homotopy category. Furthermore, we prove that under a certain nor...

It is shown that the K-theory of every noetherian base scheme of finite Krull dimension is represented by a strict ring object in the setting of motivic stable homotopy theory. The adjective `strict' is used to distinguish between the type of ring structure we construct and one which is valid only up to homotopy. Both the categories of motivic func...

Quillen’s algebraic K-theory is reconstructed via Voevodsky’s algebraic cobordism. More precisely, for a ground field k the algebraic cobordism P1-spectrum MGL of Voevodsky is considered as a commutative P1-ring spectrum. Setting \(\mathrm{MGL}^i = \bigoplus_{p-2q =i}\mathrm{MGL}^{p,q}\) we regard the bigraded theory MGLp,q
as just a graded theory....

For fields of characteristic zero, we show that the homotopy category of modules over the motivic ring spectrum representing motivic cohomology is equivalent to Voevodsky's big category of motives. The proof makes use of some highly structured models for motivic stable homotopy theory, motivic Spanier–Whitehead duality, the homotopy theories of mot...

We show that extensions of algebraically closed fields induce full and faithful functors between the respective motivic stable
homotopy categories with finite coefficients.
Mathematics Subject Classification (2000)14F42–55P42–55U35

Twisted diagrams are "diagrams" with components in different categories. Structure maps are defined using auxiliary data which consists of functors relating the various categories to each other. Prime examples of the construction are spectra (in the sense of homotopy theory) and quasi-coherent sheaves on schemes. We develop the basic theory of twis...

An algebraic version of a theorem due to Quillen is proved. More precisely, for a ground field k we consider the motivic stable homotopy category SH(k) of P^1-spectra equipped with the symmetric monoidal structure described in arXiv:0709.3905v1 [math.AG]. The algebraic cobordism P^1-spectrum MGL is considered as a commutative monoid equipped with a...

Under a certain normalization assumption we prove that the $\Pro^1$-spectrum $\mathrm{BGL}$ of Voevodsky which represents algebraic $K$-theory is unique over $\Spec(\mathbb{Z})$. Following an idea of Voevodsky, we equip the $\Pro^1$-spectrum $\mathrm{BGL}$ with the structure of a commutative $\Pro^1$-ring spectrum in the motivic stable homotopy cat...

Motivic homotopy theory is a new and in vogue blend of algebra and topology. Its primary object is to study algebraic varieties
from a homotopy theoretic viewpoint. Many of the basic ideas and techniques in this subject originate in algebraic topology.

In this Note we summarize the main results and techniques in our homotopical algebraic approach to motives. A major part of this work relies on highly structured models for motivic stable homotopy theory. For any noetherian and separated base scheme of finite Krull dimension these frameworks give rise to a homotopy theoretic meaningful study of mod...

The category of small covariant functors from simplicial sets to simplicial sets supports the projective model structure [B. Chorny, W.G. Dwyer, Homotopy theory of small diagrams over large categories, preprint, 2005]. In this paper we construct various localizations of the projective model structure and also give a variant for functors from simpli...

In this paper we employ enriched category theory to construct a convenient model for several stable homotopy categories. This is achieved in a three-step process by introducing the pointwise, homotopy functor and stable model category structures for enriched functors. The general setup is shown to describe equivariant stable homotopy theory, and we...

The notion of motivic functors refers to a motivic homotopy theoretic analog of continuous functors. In this paper we lay the foundations for a homotopical study of these functors. Of particular interest is a model structure suitable for studying motivic functors which preserve motivic weak equivalences and a model structure suitable for motivic st...

The notion of motivic functors refers to a motivic homo- topy theoretic analog of continuous functors. In this paper we lay the foundations for a homotopical study of these functors. Of particular interest is a model structure suitable for studying motivic functors which preserve motivic weak equivalences and a model structure suit- able for motivi...

## Projects

Project (1)

Our research aims at formulating and solving ground-breaking problems in motivic homotopy theory. As a relatively new field of research this subject has quickly turned into a well-established area of mathematics drawing inspiration from both algebra and topology. By leverage of modern topological methods, this offers powerful ways of studying algebro-geometric objects defined by polynomial equations. Funded by the RCN Frontier Research Group Project. Total budget approximately EUR 2.3 million.