Gunnar Carlsson - Academia.edu (original) (raw)

Papers by Gunnar Carlsson

BMC Bioinformatics, Jan 21, 2013

Background Markov state models have been widely used to study conformational changes of biologica... more Background Markov state models have been widely used to study conformational changes of biological macromolecules. These models are built from short timescale simulations and then propagated to extract long timescale dynamics. However, the solvent information in molecular simulations are often ignored in current methods, because of the large number of solvent molecules in a system and the indistinguishability of solvent molecules upon their exchange. Methods We present a solvent signature that compactly summarizes the ...

This report is published in the interest of scientific and technical information exchange, and it... more This report is published in the interest of scientific and technical information exchange, and its publication does not constitute the Government's approval or disapproval of its ideas or findings.

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The International Journal of Robotics Research

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...

Journal of Applied and Computational Topology

Topological data analysis is the study of data using techniques from algebraic topology. Often, o... more Topological data analysis is the study of data using techniques from algebraic topology. Often, one begins with a finite set of points representing data and a ''filter'' function which assigns a real number to each datum. Using both the data and the filter function, one can construct a filtered complex for further analysis. For example, applying the homology functor to the filtered complex produces an algebraic object known as a ''one-dimensional persistence module'', which can often be interpreted as a finite set of intervals representing various geometric features in the data. If one runs the above process incorporating multiple filter functions simultaneously, one instead obtains a multidimensional persistence module. Unfortunately, these are much more difficult to interpret. In this article, we analyze the space of multidimensional persistence modules from the perspective of algebraic geometry. We first build a moduli space of a certain subclass of easily analyzed multidimensional persistence modules, which we construct specifically to capture much of the information which can be gained by using multidimensional persistence instead of one-dimensional persistence. We argue that the global sections of this space provide interesting numeric invariants when evaluated against our subclass of multidimensional persistence modules. Finally, we extend these global sections to the space of all multidimensional persistence modules and discuss how the resulting numeric invariants might be used to study data. This paper extends the results of Adcock et al. (Homol Homotopy Appl 18(1), 381-402, 2016) by constructing numeric invariants from the computation of a multidimensional persistence module as given by Carlsson et al.

MATHEMATICA SCANDINAVICA, 1998

This paper introduces hierarchical quasiclustering methods, a generalization of hierarchical clus... more This paper introduces hierarchical quasiclustering methods, a generalization of hierarchical clustering for asymmetric networks where the output structure preserves the asymmetry of the input data. We show that this output structure is equivalent to a finite quasi-ultrametric space and study admissibility with respect to two desirable properties. We prove that a modified version of single linkage is the only admissible quasi-clustering method. Moreover, we show stability of the proposed method and we establish invariance properties fulfilled by it. Algorithms are further developed and the value of quasi-clustering analysis is illustrated with a study of internal migration within United States.

The persistence barcode is a well-established complete discrete invariant for finitely generated ... more The persistence barcode is a well-established complete discrete invariant for finitely generated persistence modules [5] [1]. Its definition, however, does not extend to multi- dimensional persistence modules. In this paper, we introduce a new discrete invariant: the exterior critical series. This invariant is complete in the one-dimensional case and can be defined for multi-dimensional persistence modules, like the rank invariant [2]. However, the exterior critical series can detect some features that are not captured by the rank invariant.

In this paper we present an intelligent tutoring system with affective learning, which is integra... more In this paper we present an intelligent tutoring system with affective learning, which is integrated into a social network for learning mathematics. The system is designed to help students of the second grade of primary education to improve the teaching-learning process. The system evaluates cognitive and affective aspects of the student by using a neural network and a fuzzy expert system to decide the following exercise to be resolved by the student, enabling personalized learning. We evaluate and compare our tutoring system against other well-known tutoring systems.

Homology theory has been a very effective tool in the study of homotopy invariants for topologica... more Homology theory has been a very effective tool in the study of homotopy invariants for topological spaces. An important reason for this is the fact that it is often easy to compute homology groups. For instance, if one is given a finite simplicial complex, computing its ...

Journal of Pure and Applied Algebra, 2016

A tropical polynomial in nr variables, divided into blocks of r variables each, is r-symmetric if... more A tropical polynomial in nr variables, divided into blocks of r variables each, is r-symmetric if it is invariant under the action of S n that permutes the blocks. For r = 1 we call these symmetric tropical polynomials. We can define r-symmetric and symmetric tropical rational functions in a similar manner. In this paper we identify generators for the sets of symmetric tropical polynomials and rational functions. While r-symmetric tropical polynomials are not finitely generated for r ≥ 2, we show that r-symmetric tropical rational functions are and provide a list of generators.

F1000posters, Jan 7, 2013

Journal of Computational Geometry, Nov 18, 2010

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. We recast this computation as a problem within computational commutative algebra, and utilize algorithms from this area to solve it. While the resulting problem is Expspace-complete and the standard algorithms take doubly-exponential time, we exploit the structure inherent within multifiltrations to yield practical algorithms. We implement all algorithms in the paper and provide statistical experiments to demonstrate their feasibility.

Homology, Homotopy and Applications, 2016

Topological methods, including persistent homology, are powerful tools for analysis of high-dimen... more Topological methods, including persistent homology, are powerful tools for analysis of high-dimensional data sets but these methods rely almost exclusively on thresholding techniques. In noisy data sets, thresholding does not always allow for the recovery of topological information. We present an easy to implement, computationally efficient pre-processing algorithm to prepare noisy point cloud data sets for topological data analysis. The topological de-noising algorithm allows for the recovery of topological information that is inaccessible by thresholding methods. We apply the algorithm to synthetically-generated noisy data sets and show the recovery of topological information which is impossible to obtain by thresholding. We also apply the algorithm to natural image data in R 8 and show a very clean recovery of topological information previously only available with large amounts of thresholding. Finally, we discuss future directions for improving this algorithm using zigzag persistence methods.

Homology Homotopy and Applications, 2001