Papers by Gunnar Carlsson
arXiv (Cornell University), Apr 6, 2014
In this paper we study a model structure on a category of schemes with a group action and the res... more In this paper we study a model structure on a category of schemes with a group action and the resulting unstable and stable equivariant motivic homotopy theories. The new model structure introduced here samples a comparison to the one by Voevodsky and Hu-Kriz-Ormsby. We show that it allows to detect equivariant motivic weak equivalences on fixed points and how this property leads to a topologically convenient behavior of stable equivalences. We also prove a negative result concerning descent for equivariant algebraic K-theory. I want to thank my advisor Oliver Röndigs for the great support in that time.
arXiv (Cornell University), Jan 3, 2011
This paper extends the notion of geometric control in algebraic K-theory from additive categories... more This paper extends the notion of geometric control in algebraic K-theory from additive categories with split exact sequences to other exact structures. In particular, we construct exact categories of modules over a Noetherian ring filtered by subsets of a metric space and sensitive to the large scale properties of the space. The algebraic K-theory of these categories is related to the bounded K-theory of geometric modules of Pedersen and Weibel the way G-theory is classically related to K-theory. We recover familiar results in the new setting, including the nonconnective bounded excision and equivariant properties. We apply the results to the G-theoretic Novikov conjecture which is shown to be stronger than the usual K-theoretic conjecture.
arXiv (Cornell University), Apr 22, 2014
Controlled algebra plays a central role in many recent advances in geometric topology. This paper... more Controlled algebra plays a central role in many recent advances in geometric topology. This paper studies the iteration construction that was present from the very origins of the theory but started being exploited only recently. We develop the general framework for fibred control, prove localization theorems required for fibrewise excision, and then prove several versions of fibrewise excision theorems. However, we also demonstrate how the standard tools break down in the presence of new equivariant phenomena which require advanced localization methods. Contents 1. Introduction 1 2. Pedersen-Weibel categories, Karoubi filtrations 2.1. Pedersen-Weibel theory 2.2. Bounded excision theorem 2.3. Coarsely saturated coverings 2.4. Some coarse geometry and functoriality 3. K-theory with fibred control 3.1. Definition of fibred control 3.2. Fibrewise coarsely saturated coverings 11 3.3. Fibrewise excision theorems 4. Equivariant theory 4.1. Basic equivariant theory 4.2. Equivariant fibred theories 4.3. Fibrewise equivariant localization and excision 4.4. Constructions of bounded actions 5.
arXiv (Cornell University), Jul 5, 2020
In this paper, we develop a theory of Becker-Gottlieb transfer based on Spanier-Whitehead duality... more In this paper, we develop a theory of Becker-Gottlieb transfer based on Spanier-Whitehead duality that holds in both the motivic andétale settings for smooth quasi-projective varieties in as broad a context as possible: for example, for varieties over non-separably closed fields in all characteristics, and also for both theétale and motivic settings. In view of the fact that the most promising applications of the traditional Becker-Gottlieb transfer has been to torsors and Borel-style equivariant cohomology theories, we focus our applications to motivic cohomology theories for torsors as well as Borel-style equivariant motivic cohomology theories, both defined with respect to motivic spectra. We obtain several results in this direction, including a stable splitting in generalized motivic cohomology theories. Various further applications will be discussed in forthcoming papers. 2. Spanier-Whitehead duality and the construction of transfer in a general framework We begin by recalling the basics of Spanier-Whitehead duality and at the same time clarifying certain key concepts that appear in this framework. 2.1. Weak Dualizability, Reflexivity and Dualizability in a closed symmetric monoidal stable model category. This section is worked out in as broad a generality as possible, so that it becomes easy to define variants Proposition 2.5. Let X → Y → Z → X [1] denote a distinguished triangle in Spt. If two of the three objects X , Y, and Z are dualizable, so is the third. Proof. Clearly it suffices to prove that Z is dualizable if X and Y are. Since X and Y are assumed to be dualizable, one observes that the natural maps: Z ∧ DY → Hom(Y, Z) and Z ∧ DX → Hom(X , Z) are weak-equivalences. Now one has the commutative diagram:
The International Journal of Robotics Research, 2014
Suppose that ball-shaped sensors wander in a bounded domain. A sensor does not know its location ... more Suppose that ball-shaped sensors wander in a bounded domain. A sensor does not know its location but does know when it overlaps a nearby sensor. We say that an evasion path exists in this sensor network if a moving intruder can avoid detection. In ‘ Coordinate-free coverage in sensor networks with controlled boundaries via homology', Vin de Silva and Robert Ghrist give a necessary condition, depending only on the time-varying connectivity data of the sensors, for an evasion path to exist. Using zigzag persistent homology, we provide an equivalent condition that moreover can be computed in a streaming fashion. However, no method with time-varying connectivity data as input can give necessary and sufficient conditions for the existence of an evasion path. Indeed, we show that the existence of an evasion path depends not only on the fibrewise homotopy type of the region covered by sensors but also on its embedding in spacetime. For planar sensors that also measure weak rotation and...
International Journal of Algebra and Computation, 2016
Let [Formula: see text] be a commutative ring and [Formula: see text] be an infinite discrete gro... more Let [Formula: see text] be a commutative ring and [Formula: see text] be an infinite discrete group. The algebraic [Formula: see text]-theory of the group ring [Formula: see text] is an important object of computation in geometric topology and number theory. When the group ring is Noetherian, there is a companion [Formula: see text]-theory of [Formula: see text] which is often easier to compute. However, it is exceptionally rare that the group ring is Noetherian for an infinite group. In this paper, we define a version of [Formula: see text]-theory for any finitely generated discrete group. This construction is based on the coarse geometry of the group. It has some expected properties such as independence from the choice of a word metric. We prove that, whenever [Formula: see text] is a regular Noetherian ring of finite global homological dimension and [Formula: see text] has finite asymptotic dimension and a finite model for the classifying space [Formula: see text], the natural Ca...
This paper tackles the problem of computing topological invariants of geometric objects in a robu... more This paper tackles the problem of computing topological invariants of geometric objects in a robust manner, using only point cloud data sampled from the object. It is now widely recognised that this kind of topological analysis can give qualitative information about data sets which is not readily available by other means. In particular, it can be an aid to visualisation of high dimensional data. Standard simplicial complexes for approximating the topological type of the underlying space (such asČech, Rips, or α-shape) produce simplicial complexes whose vertex set has the same size as the underlying set of point cloud data. Such constructions are sometimes still tractable, but are wasteful (of computing resources) since the homotopy types of the underlying objects are generally realisable on much smaller vertex sets. We obtain smaller complexes by choosing a set of 'landmark' points from our data set, and then constructing a "witness complex" on this set using ideas motivated by the usual Delaunay complex in Euclidean space. The key idea is that the remaining (non-landmark) data points are used as witnesses to the existence of edges or simplices spanned by combinations of landmark points. Our construction generalises the topology-preserving graphs of Martinetz and Schulten [MS94] in two directions. First, it produces a simplicial complex rather than a graph. Secondly it actually produces a nested family of simplicial complexes, which represent the data at different feature scales, suitable for calculating persistent homology [ELZ00, ZC04]. We find that in addition to the complexes being smaller, they also provide (in a precise sense) a better picture of the homology, with less noise, than the full scale constructions using all the data points. We illustrate the use of these complexes in qualitatively analyzing a data set of 3 × 3 pixel patches studied by David Mumford et al [LPM03].
Proceedings of the twenty-fifth annual symposium on Computational geometry, 2009
We study the problem of computing zigzag persistence of a sequence of homology groups and study a... more We study the problem of computing zigzag persistence of a sequence of homology groups and study a particular sequence derived from the levelsets of a real-valued function on a topological space. The result is a local, symmetric interval descriptor of the function. Our structural results establish a connection between the zigzag pairs in this sequence and extended persistence, and in the process resolve an open question associated with the latter. Our algorithmic results not only provide a way to compute zigzag persistence for any sequence of homology groups, but combined with our structural results give a novel algorithm for computing extended persistence. This algorithm is easily parallelizable and uses (asymptotically) less memory.
Handbook of K-Theory, 2005
Equivariant homotopy methods developed over the last 20 years lead to recent breakthroughs in the... more Equivariant homotopy methods developed over the last 20 years lead to recent breakthroughs in the Borel isomorphism conjectures for Loday assembly maps in K-and L-theories. An important consequence of these algebraic conjectures is the topological rigidity of compact aspherical manifolds. Our goal is to strip the basic idea to the core and follow the evolution over time in order to explain the advantages of the flexible state that exists today. We end with an outline of the proof of the Borel conjecture in algebraic K-theory for groups of finite asymptotic dimension. Contents 1. Introduction 1 2. Bounded K-theory and the Loday assembly map 2.1. Homotopy fixed point method 2.2. The Loday assembly map 2.3. Geometric applications of the assembly map 2.4. Topological rigidity of aspherical manifolds 2.5. Bounded K-theory and its application 2.6. Example: free abelian groups 3. Applications of continuous control at infinity 3.1. Continuous control in good compactifications 3.2. Continuous control with large actions at infinity 4. Surjectivity of the K-theoretic assembly map 4.1. Preparation for the argument 4.2. The beginning of the argument 21 4.3. Fibrewise bounded G-theory 4.4. The end of the argument 4.5. Coarse coherence of fundamental groups 5. Comparison with the Farrell-Jones conjecture 31 References
In this paper we consider the problem of Galois descent for suitably completed algebraic K-theory... more In this paper we consider the problem of Galois descent for suitably completed algebraic K-theory of fields. One of the main results is a suitable form of rigidity for Borel-style generalized equivariant cohomology with respect to certain spectra. In order to apply this to the problem at hand, we need to invoke a derived Atiyah-Segal completion theorem for pro-groups. In the present paper, the authors apply such a derived completion theorem proven by the first author elsewhere. These two results provide a proof of the Galois descent problem for equivariant algebraic K-theory as formulated by the first author, at least when restricted to the case where the absolute Galois groups are pro-l groups for some prime l different from the characteristic of the base field and the K-theory spectrum is completed at the same prime l. Work in progress hopes to remove these restrictions.
Lecture Notes in Computer Science, 2009
The theory of multidimensional persistence captures the topology of a multifiltration-a multipara... more The theory of multidimensional persistence captures the topology of a multifiltration-a multiparameter family of increasing spaces. Multifiltrations arise naturally in the topological analysis of scientific data. In this paper, we give a polynomial time algorithm for computing multidimensional persistence.
PLoS Computational Biology, 2012
Oberwolfach Reports, 2008
Exercise: Show that any two geometric realizations are homeomorphic. (**) the link of every i-sim... more Exercise: Show that any two geometric realizations are homeomorphic. (**) the link of every i-simplex triangulates a sphere of dimension d − i − 1. Caveat: Not every triangulation of a manifold satisfies condition (**). Exercise: If K satisfies condition (**) then so does Sd(K).
Journal of Pure and Applied Algebra, 2008
Journal of Algebra, 2004
We explore a weakening of the coherence property of discrete groups studied by F. Waldhausen. The... more We explore a weakening of the coherence property of discrete groups studied by F. Waldhausen. The new notion is defined in terms of the coarse geometry of groups and should be as useful for computing their K-theory. We prove that a group Γ of finite asymptotic dimension is weakly coherent. In particular, there is a large collection of R[Γ ]-modules of finite homological dimension when R is a finite-dimensional regular ring. This class contains word-hyperbolic groups, Coxeter groups and, as we show, the cocompact discrete subgroups of connected Lie groups.
Computational Geometry, 2008
In this paper, we provide the theoretical foundation and an effective algorithm for localizing to... more In this paper, we provide the theoretical foundation and an effective algorithm for localizing topological attributes such as tunnels and voids. Unlike previous work that focused on 2-manifolds with restricted geometry, our theory is general and localizes arbitrary-dimensional attributes in arbitrary spaces. We implement our algorithm to validate our approach in practice.
In this paper, we develop a theory of Spanier-Whitehead duality in the context of motivic homotop... more In this paper, we develop a theory of Spanier-Whitehead duality in the context of motivic homotopy theory. Notable among the applications of this theory is a variant of the classical Becker-Gottlieb transfer in the framework of motivic homotopy theory, with several potential applications, and which will be dealt with in detail in a sequel. A variant of this theory in the context ofétale homotopy theory was already developed by the second author several years ago: we will explore the connections between these two theories as well.
This paper extends the notion of geometric control in algebraic K-theory from additive categories... more This paper extends the notion of geometric control in algebraic K-theory from additive categories with split exact sequences to other exact structures. We construct boundedly controlled exact categories over a general proper metric space and recover facts familiar from bounded K-theory of free geometric modules, including controlled excision. The framework and results are used in our forthcoming computation of the algebraic K-theory of geometric groups. Contents 1. Introduction 1 2. Controlled categories of filtered objects 7 3. Exact quotients of controlled categories 11 4. Localization in controlled K-theory 16 5. Nonconnective bounded excision 23 References 27
We construct a framework for studying clustering algorithms, which includes two key ideas: persis... more We construct a framework for studying clustering algorithms, which includes two key ideas: persistence and functoriality. The first encodes the idea that the output of a clustering scheme should carry a multiresolution structure, the second the idea that one should be able to compare the results of clustering algorithms as one varies the data set, for example by adding points or by applying functions to it. We show that within this framework, one can prove a theorem analogous to one of J. Kleinberg [Kle02], in which one obtains an existence and uniqueness theorem instead of a non-existence result. We explore further properties of this unique scheme, stability and convergence are established.
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Papers by Gunnar Carlsson