Bio


Dr. Carlsson has been a professor of mathematics at Stanford University since 1991. In the last ten years, he has been involved in adapting topological techniques to data analysis, under NSF funding and as the lead PI on the DARPA “Topological Data Analysis” project from 2005 to 2010. He is the lead organizer of the ATMCS conferences, and serves as an editor of several Mathematics journals

Academic Appointments


2019-20 Courses


Stanford Advisees


All Publications


  • Bounded G-theory with fibred control JOURNAL OF PURE AND APPLIED ALGEBRA Carlsson, G., Goldfarb, B. 2019; 223 (12): 5360–95
  • Parametrized homology via zigzag persistence ALGEBRAIC AND GEOMETRIC TOPOLOGY Carlsson, G., de Silva, V., Kalisnik, S., Morozov, D. 2019; 19 (2): 657–700
  • Towards a new approach to reveal dynamical organization of the brain using topological data analysis NATURE COMMUNICATIONS Saggar, M., Sporns, O., Gonzalez-Castillo, J., Bandettini, P. A., Carlsson, G., Glover, G., Reiss, A. L. 2018; 9: 1399

    Abstract

    Little is known about how our brains dynamically adapt for efficient functioning. Most previous work has focused on analyzing changes in co-fluctuations between a set of brain regions over several temporal segments of the data. We argue that by collapsing data in space or time, we stand to lose useful information about the brain's dynamical organization. Here we use Topological Data Analysis to reveal the overall organization of whole-brain activity maps at a single-participant level-as an interactive representation-without arbitrarily collapsing data in space or time. Using existing multitask fMRI datasets, with the known ground truth about the timing of transitions from one task-block to next, our approach tracks both within- and between-task transitions at a much faster time scale (~4-9 s) than before. The individual differences in the revealed dynamical organization predict task performance. In summary, our approach distills complex brain dynamics into interactive and behaviorally relevant representations.

    View details for PubMedID 29643350

  • Hierarchical clustering of asymmetric networks ADVANCES IN DATA ANALYSIS AND CLASSIFICATION Carlsson, G., Memoli, F., Ribeiro, A., Segarra, S. 2018; 12 (1): 65–105
  • Admissible Hierarchical Clustering Methods and Algorithms for Asymmetric Networks IEEE TRANSACTIONS ON SIGNAL AND INFORMATION PROCESSING OVER NETWORKS Carlsson, G., Memoli, F., Ribeiro, A., Segarra, S. 2017; 3 (4): 711–27
  • Uncovering precision phenotype-biomarker associations in traumatic brain injury using topological data analysis. PloS one Nielson, J. L., Cooper, S. R., Yue, J. K., Sorani, M. D., Inoue, T., Yuh, E. L., Mukherjee, P., Petrossian, T. C., Paquette, J., Lum, P. Y., Carlsson, G. E., Vassar, M. J., Lingsma, H. F., Gordon, W. A., Valadka, A. B., Okonkwo, D. O., Manley, G. T., Ferguson, A. R. 2017; 12 (3)

    Abstract

    Traumatic brain injury (TBI) is a complex disorder that is traditionally stratified based on clinical signs and symptoms. Recent imaging and molecular biomarker innovations provide unprecedented opportunities for improved TBI precision medicine, incorporating patho-anatomical and molecular mechanisms. Complete integration of these diverse data for TBI diagnosis and patient stratification remains an unmet challenge.The Transforming Research and Clinical Knowledge in Traumatic Brain Injury (TRACK-TBI) Pilot multicenter study enrolled 586 acute TBI patients and collected diverse common data elements (TBI-CDEs) across the study population, including imaging, genetics, and clinical outcomes. We then applied topology-based data-driven discovery to identify natural subgroups of patients, based on the TBI-CDEs collected. Our hypothesis was two-fold: 1) A machine learning tool known as topological data analysis (TDA) would reveal data-driven patterns in patient outcomes to identify candidate biomarkers of recovery, and 2) TDA-identified biomarkers would significantly predict patient outcome recovery after TBI using more traditional methods of univariate statistical tests. TDA algorithms organized and mapped the data of TBI patients in multidimensional space, identifying a subset of mild TBI patients with a specific multivariate phenotype associated with unfavorable outcome at 3 and 6 months after injury. Further analyses revealed that this patient subset had high rates of post-traumatic stress disorder (PTSD), and enrichment in several distinct genetic polymorphisms associated with cellular responses to stress and DNA damage (PARP1), and in striatal dopamine processing (ANKK1, COMT, DRD2).TDA identified a unique diagnostic subgroup of patients with unfavorable outcome after mild TBI that were significantly predicted by the presence of specific genetic polymorphisms. Machine learning methods such as TDA may provide a robust method for patient stratification and treatment planning targeting identified biomarkers in future clinical trials in TBI patients.ClinicalTrials.gov Identifier NCT01565551.

    View details for DOI 10.1371/journal.pone.0169490

    View details for PubMedID 28257413

  • Representable Hierarchical Clustering Methods for Asymmetric Networks Carlsson, G., Memoli, F., Ribeiro, A., Segarra, S., Palumbo, F., Montanari, A., Vichi, M. SPRINGER INTERNATIONAL PUBLISHING AG. 2017: 83–95
  • Symmetric and r-symmetric tropical polynomials and rational functions JOURNAL OF PURE AND APPLIED ALGEBRA Carlsson, G., Verovsek, S. K. 2016; 220 (11): 3610-3627
  • On modules over infinite group rings INTERNATIONAL JOURNAL OF ALGEBRA AND COMPUTATION Carlsson, G., Goldfarb, B. 2016; 26 (3): 451-466
  • The ring of algebraic functions on persistence barcodes Homology, homotopy, and applications Adcock, A., Carlsson , E., Carlsson , G. 2016; 18 (1): 381-402
  • Topological data analysis for discovery in preclinical spinal cord injury and traumatic brain injury NATURE COMMUNICATIONS Nielson, J. L., Paquette, J., Liu, A. W., Guandique, C. F., Tovar, C. A., Inoue, T., Irvine, K., Gensel, J. C., Kloke, J., Petrossian, T. C., Lum, P. Y., Carlsson, G. E., Manley, G. T., Young, W., Beattie, M. S., Bresnahan, J. C., Ferguson, A. R. 2015; 6

    Abstract

    Data-driven discovery in complex neurological disorders has potential to extract meaningful syndromic knowledge from large, heterogeneous data sets to enhance potential for precision medicine. Here we describe the application of topological data analysis (TDA) for data-driven discovery in preclinical traumatic brain injury (TBI) and spinal cord injury (SCI) data sets mined from the Visualized Syndromic Information and Outcomes for Neurotrauma-SCI (VISION-SCI) repository. Through direct visualization of inter-related histopathological, functional and health outcomes, TDA detected novel patterns across the syndromic network, uncovering interactions between SCI and co-occurring TBI, as well as detrimental drug effects in unpublished multicentre preclinical drug trial data in SCI. TDA also revealed that perioperative hypertension predicted long-term recovery better than any tested drug after thoracic SCI in rats. TDA-based data-driven discovery has great potential application for decision-support for basic research and clinical problems such as outcome assessment, neurocritical care, treatment planning and rapid, precision-diagnosis.

    View details for DOI 10.1038/ncomms9581

    View details for Web of Science ID 000364932600028

    View details for PubMedID 26466022

    View details for PubMedCentralID PMC4634208

  • NUDGED ELASTIC BAND IN TOPOLOGICAL DATA ANALYSIS TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS Adams, H., Atanasov, A., Carlsson, G. 2015; 45 (1): 247-272
  • Evasion paths in mobile sensor networks INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH Adams, H., Carlsson, G. 2015; 34 (1): 90-104
  • Topological methods reveal high and low functioning neuro-phenotypes within fragile X syndrome. Human brain mapping Romano, D., Nicolau, M., Quintin, E., Mazaika, P. K., Lightbody, A. A., Cody Hazlett, H., Piven, J., Carlsson, G., Reiss, A. L. 2014; 35 (9): 4904-4915

    Abstract

    Fragile X syndrome (FXS), due to mutations of the FMR1 gene, is the most common known inherited cause of developmental disability as well as the most common single-gene risk factor for autism. Our goal was to examine variation in brain structure in FXS with topological data analysis (TDA), and to assess how such variation is associated with measures of IQ and autism-related behaviors. To this end, we analyzed imaging and behavioral data from young boys (n = 52; aged 1.57-4.15 years) diagnosed with FXS. Application of topological methods to structural MRI data revealed two large subgroups within the study population. Comparison of these subgroups showed significant between-subgroup neuroanatomical differences similar to those previously reported to distinguish children with FXS from typically developing controls (e.g., enlarged caudate). In addition to neuroanatomy, the groups showed significant differences in IQ and autism severity scores. These results suggest that despite arising from a single gene mutation, FXS may encompass two biologically, and clinically separable phenotypes. In addition, these findings underscore the potential of TDA as a powerful tool in the search for biological phenotypes of neuropsychiatric disorders. Hum Brain Mapp 35:4904-4915, 2014. © 2014 Wiley Periodicals, Inc.

    View details for DOI 10.1002/hbm.22521

    View details for PubMedID 24737721

    View details for PubMedCentralID PMC4113391

  • Classification of hepatic lesions using the matching metric COMPUTER VISION AND IMAGE UNDERSTANDING Adcock, A., Rubin, D., Carlsson, G. 2014; 121: 36-42
  • A Klein-Bottle-Based Dictionary for Texture Representation INTERNATIONAL JOURNAL OF COMPUTER VISION Perea, J. A., Carlsson, G. 2014; 107 (1): 75-97
  • Topological pattern recognition for point cloud data ACTA NUMERICA Carlsson, G. 2014; 23: 289-368
  • Topology of viral evolution PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA Chan, J. M., Carlsson, G., Rabadan, R. 2013; 110 (46): 18566-18571

    Abstract

    The tree structure is currently the accepted paradigm to represent evolutionary relationships between organisms, species or other taxa. However, horizontal, or reticulate, genomic exchanges are pervasive in nature and confound characterization of phylogenetic trees. Drawing from algebraic topology, we present a unique evolutionary framework that comprehensively captures both clonal and reticulate evolution. We show that whereas clonal evolution can be summarized as a tree, reticulate evolution exhibits nontrivial topology of dimension greater than zero. Our method effectively characterizes clonal evolution, reassortment, and recombination in RNA viruses. Beyond detecting reticulate evolution, we succinctly recapitulate the history of complex genetic exchanges involving more than two parental strains, such as the triple reassortment of H7N9 avian influenza and the formation of circulating HIV-1 recombinants. In addition, we identify recurrent, large-scale patterns of reticulate evolution, including frequent PB2-PB1-PA-NP cosegregation during avian influenza reassortment. Finally, we bound the rate of reticulate events (i.e., 20 reassortments per year in avian influenza). Our method provides an evolutionary perspective that not only captures reticulate events precluding phylogeny, but also indicates the evolutionary scales where phylogenetic inference could be accurate.

    View details for DOI 10.1073/pnas.1313480110

    View details for Web of Science ID 000326830900062

    View details for PubMedID 24170857

    View details for PubMedCentralID PMC3831954

  • BIG-DATA VISUALIZATION FOR TRANSLATIONAL NEUROTRAUMA Nielson, J., Inoue, T., Paquette, J., Lin, A., Sacramento, J., Liu, A. W., Guandique, C. F., Irvine, K., Gensel, J. C., Beattie, M. S., Bresnahan, J. C., Manley, G. T., Carlsson, G., Lum, P., Ferguson, A. R. MARY ANN LIEBERT, INC. 2013: A61–A62
  • Persistent Topology and Metastable State in Conformational Dynamics PLOS ONE Chang, H., Bacallado, S., Pande, V. S., Carlsson, G. E. 2013; 8 (4)

    Abstract

    The large amount of molecular dynamics simulation data produced by modern computational models brings big opportunities and challenges to researchers. Clustering algorithms play an important role in understanding biomolecular kinetics from the simulation data, especially under the Markov state model framework. However, the ruggedness of the free energy landscape in a biomolecular system makes common clustering algorithms very sensitive to perturbations of the data. Here, we introduce a data-exploratory tool which provides an overview of the clustering structure under different parameters. The proposed Multi-Persistent Clustering analysis combines insights from recent studies on the dynamics of systems with dominant metastable states with the concept of multi-dimensional persistence in computational topology. We propose to explore the clustering structure of the data based on its persistence on scale and density. The analysis provides a systematic way to discover clusters that are robust to perturbations of the data. The dominant states of the system can be chosen with confidence. For the clusters on the borderline, the user can choose to do more simulation or make a decision based on their structural characteristics. Furthermore, our multi-resolution analysis gives users information about the relative potential of the clusters and their hierarchical relationship. The effectiveness of the proposed method is illustrated in three biomolecules: alanine dipeptide, Villin headpiece, and the FiP35 WW domain.

    View details for DOI 10.1371/journal.pone.0058699

    View details for Web of Science ID 000317717300006

    View details for PubMedID 23565139

    View details for PubMedCentralID PMC3614941

  • Classifying Clustering Schemes FOUNDATIONS OF COMPUTATIONAL MATHEMATICS Carlsson, G., Memoli, F. 2013; 13 (2): 221-252
  • Extracting insights from the shape of complex data using topology SCIENTIFIC REPORTS Lum, P. Y., Singh, G., Lehman, A., Ishkanov, T., Vejdemo-Johansson, M., Alagappan, M., Carlsson, J., Carlsson, G. 2013; 3

    Abstract

    This paper applies topological methods to study complex high dimensional data sets by extracting shapes (patterns) and obtaining insights about them. Our method combines the best features of existing standard methodologies such as principal component and cluster analyses to provide a geometric representation of complex data sets. Through this hybrid method, we often find subgroups in data sets that traditional methodologies fail to find. Our method also permits the analysis of individual data sets as well as the analysis of relationships between related data sets. We illustrate the use of our method by applying it to three very different kinds of data, namely gene expression from breast tumors, voting data from the United States House of Representatives and player performance data from the NBA, in each case finding stratifications of the data which are more refined than those produced by standard methods.

    View details for DOI 10.1038/srep01236

    View details for Web of Science ID 000314710400001

    View details for PubMedID 23393618

  • Building Markov state models with solvent dynamics 11th Asia Pacific Bioinformatics Conference (APBC) Gu, C., Chang, H., Maibaum, L., Pande, V. S., Carlsson, G. E., Guibas, L. J. BIOMED CENTRAL LTD. 2013

    Abstract

    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.We present a solvent signature that compactly summarizes the solvent distribution in the high-dimensional data, and then define a distance metric between different configurations using this signature. We next incorporate the solvent information into the construction of Markov state models and present a fast geometric clustering algorithm which combines both the solute-based and solvent-based distances.We have tested our method on several different molecular dynamical systems, including alanine dipeptide, carbon nanotube, and benzene rings. With the new solvent-based signatures, we are able to identify different solvent distributions near the solute. Furthermore, when the solute has a concave shape, we can also capture the water number inside the solute structure. Finally we have compared the performances of different Markov state models. The experiment results show that our approach improves the existing methods both in the computational running time and the metastability.In this paper we have initiated an study to build Markov state models for molecular dynamical systems with solvent degrees of freedom. The methods we described should also be broadly applicable to a wide range of biomolecular simulation analyses.

    View details for PubMedID 23368418

  • Persistent topology and metastable state in conformational dynamics. PloS one Chang, H., Bacallado, S., Pande, V. S., Carlsson, G. E. 2013; 8 (4)

    Abstract

    The large amount of molecular dynamics simulation data produced by modern computational models brings big opportunities and challenges to researchers. Clustering algorithms play an important role in understanding biomolecular kinetics from the simulation data, especially under the Markov state model framework. However, the ruggedness of the free energy landscape in a biomolecular system makes common clustering algorithms very sensitive to perturbations of the data. Here, we introduce a data-exploratory tool which provides an overview of the clustering structure under different parameters. The proposed Multi-Persistent Clustering analysis combines insights from recent studies on the dynamics of systems with dominant metastable states with the concept of multi-dimensional persistence in computational topology. We propose to explore the clustering structure of the data based on its persistence on scale and density. The analysis provides a systematic way to discover clusters that are robust to perturbations of the data. The dominant states of the system can be chosen with confidence. For the clusters on the borderline, the user can choose to do more simulation or make a decision based on their structural characteristics. Furthermore, our multi-resolution analysis gives users information about the relative potential of the clusters and their hierarchical relationship. The effectiveness of the proposed method is illustrated in three biomolecules: alanine dipeptide, Villin headpiece, and the FiP35 WW domain.

    View details for DOI 10.1371/journal.pone.0058699

    View details for PubMedID 23565139

  • Alternative Axiomatic Constructions for Hierarchical Clustering of Asymmetric Networks IEEE Global Conference on Signal and Information Processing (GlobalSIP) Carlsson, G., Memoli, F., Ribeiro, A., Segarra, S. IEEE. 2013: 791–794
  • Hierarchical Clustering Methods and Algorithms for Asymmetric Networks 47th Asilomar Conference on Signals, Systems and Computers Carlsson, G., Memoli, F., Ribeiro, A., Segarra, S. IEEE. 2013: 1773–1777
  • AXIOMATIC CONSTRUCTION OF HIERARCHICAL CLUSTERING IN ASYMMETRIC NETWORKS IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP) Carlsson, G., Memoli, F., Ribeiro, A., Segarra, S. IEEE. 2013: 5219–5223
  • A Topological Paradigm for Hippocampal Spatial Map Formation Using Persistent Homology PLOS COMPUTATIONAL BIOLOGY Dabaghian, Y., Memoli, F., Frank, L., Carlsson, G. 2012; 8 (8)

    Abstract

    An animal's ability to navigate through space rests on its ability to create a mental map of its environment. The hippocampus is the brain region centrally responsible for such maps, and it has been assumed to encode geometric information (distances, angles). Given, however, that hippocampal output consists of patterns of spiking across many neurons, and downstream regions must be able to translate those patterns into accurate information about an animal's spatial environment, we hypothesized that 1) the temporal pattern of neuronal firing, particularly co-firing, is key to decoding spatial information, and 2) since co-firing implies spatial overlap of place fields, a map encoded by co-firing will be based on connectivity and adjacency, i.e., it will be a topological map. Here we test this topological hypothesis with a simple model of hippocampal activity, varying three parameters (firing rate, place field size, and number of neurons) in computer simulations of rat trajectories in three topologically and geometrically distinct test environments. Using a computational algorithm based on recently developed tools from Persistent Homology theory in the field of algebraic topology, we find that the patterns of neuronal co-firing can, in fact, convey topological information about the environment in a biologically realistic length of time. Furthermore, our simulations reveal a "learning region" that highlights the interplay between the parameters in combining to produce hippocampal states that are more or less adept at map formation. For example, within the learning region a lower number of neurons firing can be compensated by adjustments in firing rate or place field size, but beyond a certain point map formation begins to fail. We propose that this learning region provides a coherent theoretical lens through which to view conditions that impair spatial learning by altering place cell firing rates or spatial specificity.

    View details for DOI 10.1371/journal.pcbi.1002581

    View details for Web of Science ID 000308553500001

    View details for PubMedID 22912564

  • Computational topology for configuration spaces of hard disks PHYSICAL REVIEW E Carlsson, G., Gorham, J., Kahle, M., Mason, J. 2012; 85 (1)

    Abstract

    We explore the topology of configuration spaces of hard disks experimentally and show that several changes in the topology can already be observed with a small number of particles. The results illustrate a theorem of Baryshnikov, Bubenik, and Kahle that critical points correspond to configurations of disks with balanced mechanical stresses and suggest conjectures about the asymptotic topology as the number of disks tends to infinity.

    View details for DOI 10.1103/PhysRevE.85.011303

    View details for Web of Science ID 000299122400004

    View details for PubMedID 22400561

  • Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA Nicolau, M., Levine, A. J., Carlsson, G. 2011; 108 (17): 7265-7270

    Abstract

    High-throughput biological data, whether generated as sequencing, transcriptional microarrays, proteomic, or other means, continues to require analytic methods that address its high dimensional aspects. Because the computational part of data analysis ultimately identifies shape characteristics in the organization of data sets, the mathematics of shape recognition in high dimensions continues to be a crucial part of data analysis. This article introduces a method that extracts information from high-throughput microarray data and, by using topology, provides greater depth of information than current analytic techniques. The method, termed Progression Analysis of Disease (PAD), first identifies robust aspects of cluster analysis, then goes deeper to find a multitude of biologically meaningful shape characteristics in these data. Additionally, because PAD incorporates a visualization tool, it provides a simple picture or graph that can be used to further explore these data. Although PAD can be applied to a wide range of high-throughput data types, it is used here as an example to analyze breast cancer transcriptional data. This identified a unique subgroup of Estrogen Receptor-positive (ER(+)) breast cancers that express high levels of c-MYB and low levels of innate inflammatory genes. These patients exhibit 100% survival and no metastasis. No supervised step beyond distinction between tumor and healthy patients was used to identify this subtype. The group has a clear and distinct, statistically significant molecular signature, it highlights coherent biology but is invisible to cluster methods, and does not fit into the accepted classification of Luminal A/B, Normal-like subtypes of ER(+) breast cancers. We denote the group as c-MYB(+) breast cancer.

    View details for DOI 10.1073/pnas.1102826108

    View details for Web of Science ID 000289888500107

    View details for PubMedID 21482760

    View details for PubMedCentralID PMC3084136

  • Higher topological cyclic homology and the Segal conjecture for tori ADVANCES IN MATHEMATICS Carlsson, G., Douglas, C. L., Dundas, B. I. 2011; 226 (2): 1823-1874
  • CONTROLLED ALGEBRAIC G-THEORY, I JOURNAL OF HOMOTOPY AND RELATED STRUCTURES Carlsson, G., Goldfarb, B. 2011; 6 (1): 119-159
  • Derived representation theory and the algebraic K-theory of fields JOURNAL OF TOPOLOGY Carlsson, G. E. 2011; 4 (3): 543-572
  • Covering homology ADVANCES IN MATHEMATICS Brun, M., Carlsson, G., Dundas, B. I. 2010; 225 (6): 3166-3213
  • Zigzag Persistence FOUNDATIONS OF COMPUTATIONAL MATHEMATICS Carlsson, G., De Silva, V. 2010; 10 (4): 367-405
  • Characterization, Stability and Convergence of Hierarchical Clustering Methods JOURNAL OF MACHINE LEARNING RESEARCH Carlsson, G., Memoli, F. 2010; 11: 1425-1470
  • COMPUTING MULTIDIMENSIONAL PERSISTENCE JOURNAL OF COMPUTATIONAL GEOMETRY Carlsson, G., Singh, G., Zomorodian, A. 2010; 1 (1): 72–100
  • Constructing multi-resolution Markov State Models (MSMs) to elucidate RNA hairpin folding mechanisms. Pacific Symposium on Biocomputing. Pacific Symposium on Biocomputing Huang, X., Yao, Y., Bowman, G. R., Sun, J., Guibas, L. J., Carlsson, G., Pande, V. S. 2010: 228-239

    Abstract

    Simulating biologically relevant timescales at atomic resolution is a challenging task since typical atomistic simulations are at least two orders of magnitude shorter. Markov State Models (MSMs) provide one means of overcoming this gap without sacrificing atomic resolution by extracting long time dynamics from short simulations. MSMs coarse grain space by dividing conformational space into long-lived, or metastable, states. This is equivalent to coarse graining time by integrating out fast motions within metastable states. By varying the degree of coarse graining one can vary the resolution of an MSM; therefore, MSMs are inherently multi-resolution. Here we introduce a new algorithm Super-level-set Hierarchical Clustering (SHC), to our knowledge, the first algorithm focused on constructing MSMs at multiple resolutions. The key insight of this algorithm is to generate a set of super levels covering different density regions of phase space, then cluster each super level separately, and finally recombine this information into a single MSM. SHC is able to produce MSMs at different resolutions using different super density level sets. To demonstrate the power of this algorithm we apply it to a small RNA hairpin, generating MSMs at four different resolutions. We validate these MSMs by showing that they are able to reproduce the original simulation data. Furthermore, long time folding dynamics are extracted from these models. The results show that there are no metastable on-pathway intermediate states. Instead, the folded state serves as a hub directly connected to multiple unfolded/misfolded states which are separated from each other by large free energy barriers.

    View details for PubMedID 19908375

    View details for PubMedCentralID PMC4423759

  • Statistical Topology via Morse Theory Persistence and Nonparametric Estimation AMS Special Session on Algebraic Methods in Statistics and Probability Bubenik, P., Carlsson, G., Kim, P. T., Luo, Z. AMER MATHEMATICAL SOC. 2010: 75–92
  • Multiparameter Hierarchical Clustering Methods Carlsson, G., Memoli, F., LocarekJunge, H., Weihs, C. SPRINGER-VERLAG BERLIN. 2010: 63–70
  • The Theory of Multidimensional Persistence 23rd Annual Symposium on Computational Geometry Carlsson, G., Zomorodian, A. SPRINGER. 2009: 71–93
  • Topological methods for exploring low-density states in biomolecular folding pathways JOURNAL OF CHEMICAL PHYSICS Yao, Y., Sun, J., Huang, X., Bowman, G. R., Singh, G., Lesnick, M., Guibas, L. J., Pande, V. S., Carlsson, G. 2009; 130 (14)

    Abstract

    Characterization of transient intermediate or transition states is crucial for the description of biomolecular folding pathways, which is, however, difficult in both experiments and computer simulations. Such transient states are typically of low population in simulation samples. Even for simple systems such as RNA hairpins, recently there are mounting debates over the existence of multiple intermediate states. In this paper, we develop a computational approach to explore the relatively low populated transition or intermediate states in biomolecular folding pathways, based on a topological data analysis tool, MAPPER, with simulation data from large-scale distributed computing. The method is inspired by the classical Morse theory in mathematics which characterizes the topology of high-dimensional shapes via some functional level sets. In this paper we exploit a conditional density filter which enables us to focus on the structures on pathways, followed by clustering analysis on its level sets, which helps separate low populated intermediates from high populated folded/unfolded structures. A successful application of this method is given on a motivating example, a RNA hairpin with GCAA tetraloop, where we are able to provide structural evidence from computer simulations on the multiple intermediate states and exhibit different pictures about unfolding and refolding pathways. The method is effective in dealing with high degree of heterogeneity in distribution, capturing structural features in multiple pathways, and being less sensitive to the distance metric than nonlinear dimensionality reduction or geometric embedding methods. The methodology described in this paper admits various implementations or extensions to incorporate more information and adapt to different settings, which thus provides a systematic tool to explore the low-density intermediate states in complex biomolecular folding systems.

    View details for DOI 10.1063/1.3103496

    View details for Web of Science ID 000265617200017

    View details for PubMedID 19368437

    View details for PubMedCentralID PMC2719471

  • On the Nonlinear Statistics of Range Image Patches SIAM JOURNAL ON IMAGING SCIENCES Adams, H., Carlsson, G. 2009; 2 (1): 110-117

    View details for DOI 10.1137/070711669

    View details for Web of Science ID 000278101000006

  • Zigzag Persistent Homology and Real-valued Functions 25th Annual Symposium on Computational Geometry Carlsson, G., De Silva, V., Morozov, D. ASSOC COMPUTING MACHINERY. 2009: 247–256
  • TOPOLOGY AND DATA BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Carlsson, G. 2009; 46 (2): 255–308
  • Computing Multidimensional Persistence 20th International Symposium on Algorithms and Computations (ISAAC 2009) Carlsson, G., Singh, G., Zomorodian, A. SPRINGER-VERLAG BERLIN. 2009: 730–739
  • Localized homology COMPUTATIONAL GEOMETRY-THEORY AND APPLICATIONS Zomorodian, A., Carlsson, G. 2008; 41 (3): 126-148
  • Structural insight into RNA hairpin folding intermediates JOURNAL OF THE AMERICAN CHEMICAL SOCIETY Bowman, G. R., Huang, X., Yao, Y., Sun, J., Carlsson, G., Guibas, L. J., Pande, V. S. 2008; 130 (30): 9676-?

    Abstract

    Hairpins are a ubiquitous secondary structure motif in RNA molecules. Despite their simple structure, there is some debate over whether they fold in a two-state or multi-state manner. We have studied the folding of a small tetraloop hairpin using a serial version of replica exchange molecular dynamics on a distributed computing environment. On the basis of these simulations, we have identified a number of intermediates that are consistent with experimental results. We also find that folding is not simply the reverse of high-temperature unfolding and suggest that this may be a general feature of biomolecular folding.

    View details for DOI 10.1021/ja8032857

    View details for Web of Science ID 000257902500027

    View details for PubMedID 18593120

    View details for PubMedCentralID PMC2652247

  • Derived completions in stable homotopy theory JOURNAL OF PURE AND APPLIED ALGEBRA Carlsson, G. 2008; 212 (3): 550-577
  • A Near Optimal Coder For Image Geometry With Adaptive Partitioning 15th IEEE International Conference on Image Processing (ICIP 2008) Maleki, A., Shahram, M., Carlsson, G. IEEE. 2008: 1065–1068
  • Topological analysis of population activity in visual cortex JOURNAL OF VISION Singh, G., Memoli, F., Ishkhanov, T., Sapiro, G., Carlsson, G., Ringach, D. L. 2008; 8 (8)

    Abstract

    Information in the cortex is thought to be represented by the joint activity of neurons. Here we describe how fundamental questions about neural representation can be cast in terms of the topological structure of population activity. A new method, based on the concept of persistent homology, is introduced and applied to the study of population activity in primary visual cortex (V1). We found that the topological structure of activity patterns when the cortex is spontaneously active is similar to those evoked by natural image stimulation and consistent with the topology of a two sphere. We discuss how this structure could emerge from the functional organization of orientation and spatial frequency maps and their mutual relationship. Our findings extend prior results on the relationship between spontaneous and evoked activity in V1 and illustrates how computational topology can help tackle elementary questions about the representation of information in the nervous system.

    View details for DOI 10.1167/8.8.11

    View details for Web of Science ID 000258708900011

    View details for PubMedID 18831634

    View details for PubMedCentralID PMC2924880

  • On the local behavior of spaces of natural images INTERNATIONAL JOURNAL OF COMPUTER VISION Carlsson, G., Ishkhanov, T., De Silva, V., Zornorodian, A. 2008; 76 (1): 1-12
  • A Near Optimal Coder For Image Geometry With Adaptive Partitioning 15th IEEE International Conference on Image Processing (ICIP 2008) Maleki, A., Shahrarn, M., Carlsson, G. IEEE. 2008: 1061–1064
  • epsilon-entropy of piecewise polynomial functions and tree partitioning compression 33rd IEEE International Conference on Acoustics, Speech and Signal Processing Maleki, A., Carlsson, G. IEEE. 2008: 1181–1184
  • Localized homology 9th International Conference on Shape Modeling and Applications Zomorodian, A., Carlsson, G. IEEE COMPUTER SOC. 2007: 189-?
  • An algebraic topological method for feature identification INTERNATIONAL JOURNAL OF COMPUTATIONAL GEOMETRY & APPLICATIONS Carlsson, E., Carlsson, G., De Silva, V. 2006; 16 (4): 291-314
  • Computing persistent homology DISCRETE & COMPUTATIONAL GEOMETRY Zomorodian, A., Carlsson, G. 2005; 33 (2): 249-274
  • A barcode shape descriptor for curve point cloud data COMPUTERS & GRAPHICS-UK Collins, A., Zomorodian, A., Carlsson, G., Guibas, L. J. 2004; 28 (6): 881-894
  • The integral K-theoretic Novikov conjecture for groups with finite asymptotic dimension INVENTIONES MATHEMATICAE Carlsson, G., Goldfarb, B. 2004; 157 (2): 405-418
  • A geometric framework for sparse matrix problems ADVANCES IN APPLIED MATHEMATICS Carlsson, G., de Silva, V. 2004; 33 (1): 1-25
  • On homological coherence of discrete groups JOURNAL OF ALGEBRA Carlsson, G., Goldfarb, B. 2004; 276 (2): 502-514
  • LORA: Robust and simple routing algorithms for ad hoc mobile wireless networks IEEE Global Telecommunications Conference (GLOBECOM 01) Chiang, M., Carlsson, G. IEEE. 2001: 2793–2797
  • Admission control, power control and QoS analyses for ad hoc wireless networks IEEE International Conference on Communications Chiang, M., Carlsson, G. IEEE. 2001: 245–249
  • Cech homology and the novikov conjectures for K- and L-theory MATHEMATICA SCANDINAVICA Carlsson, G., Pedersen, E. K. 1998; 82 (1): 5-47
  • Continuously controlled algebraic K-theory of spaces and the Novikov conjecture MATHEMATISCHE ANNALEN Carlsson, G., Pedersen, E. K., Vogell, W. 1998; 310 (1): 169-180
  • On the algebraic K-theory of simply connected spaces DUKE MATHEMATICAL JOURNAL Bokstedt, M., Carlsson, G., Cohen, R., Goodwillie, T., HSIANG, W. C., Madsen, I. 1996; 84 (3): 541-563
  • On the algebraic K-theory of simply connected spaces Duke Mathematical Journal Bökstedt, M., Carlsson , G., Cohen, R., Goodwillie, T., Hsiang, W., Madsen, I. 1996; 84 (3)
  • CONTROLLED ALGEBRA AND THE NOVIKOV CONJECTURES FOR K-THEORY AND L-THEORY TOPOLOGY Carlsson, G., Pedersen, E. K. 1995; 34 (3): 731-758
  • Bounded K-theory and the assembly map in algebraic K-theory Novikov conjectures, index theory, and rigidity Carlsson , G. Cambridge University Press. 1995: 5–127
  • On the K-theory of infinite product categories K-theory Carlsson , G. 1995; 9 (4): 305-322
  • Proper homotopy and transfers for infinite groups Algebraic Topology and its Applications Carlsson , G. Springer Verlag. 1994: 1–14
  • A SURVEY OF EQUIVARIANT STABLE-HOMOTOPY THEORY TOPOLOGY Carlsson, G. 1992; 31 (1): 1-27
  • On the homotopy fixed point problem for free loop spaces and other function complexes K-theory Carlsson , G. 1991; 4 (4): 339-361
  • Equivariant stable theory and the finite descent problem for unstable K-theories American Journal of Mathematics Carlsson , G. 1991; 113 (6): 963-973

    View details for DOI 10.2307/2374897

  • Equivariant stable homotopy and Sullivan's conjecture Inventiones Mathematicae Carlsson , G. E. 1991; 103: 497-525

    View details for DOI 10.1007/BF01239524

  • THE CYCLIC GROUPS AND THE FREE LOOP SPACE COMMENTARII MATHEMATICI HELVETICI Carlsson, G. E., Cohen, R. L. 1987; 62 (3): 423-449
  • Segal's Burnside ring conjecture and the homotopy limit problem. Homotopy Theory (Durham 1985) Carlsson , G. Cambridge University Press. 1987: 6–34
  • Equivariant stable homotopy and Segal's Burnside conjecture Annals of Mathematics Carlsson , G. E. 1984; 120 (2): 189-224

    View details for DOI 10.2307/2006940

  • G.B. Segal's Burnside ring conjecture for (Z/2)^k Topology Carlsson , G. 1983; 22 (1): 83-103
  • On the homology of finite free (Z/2)^n-complexes Inventiones Mathematicae Carlsson , G. 1983; 74 (1): 139-147

    View details for DOI 10.1007/BF01388534

  • On the rank of abelian groups acting freely on (S^n)^k Inventiones Mathematicae Carlsson , G. 1982; 69 (3): 393-400

    View details for DOI 10.1007/BF01393939

  • A counterexample to a conjecture of Steenrod Inventiones Mathematicae Carlsson , G. 1981; 64 (1): 171-174

    View details for DOI 10.1007/BF01393939

  • Some restrictions on groups acting freely on (S^n)^k Transactions of the American Mathematical Society Carlsson , G. 1981; 264 (2): 449-457
  • SOME EXACT SEQUENCES IN THE THEORY OF HERMITIAN FORMS JOURNAL OF PURE AND APPLIED ALGEBRA Carlsson, G., Milgram, R. J. 1980; 18 (3): 233-252
  • Wu invariants of Hermitian forms Journal of Algebra Carlsson , G. 1980; 65 (1): 188-205
  • On the Witt group of a 2-adic group ring Quarterly Journal of Mathematics Carlsson , G. 1980; 31 (3): 283-313
  • On the stable splitting of bo^bo and torsion operations in connective K-theory Pacific Journal of Mathematics Carlsson , G. 1980; 87 (2): 283-297
  • On the non-existence of free actions of elementary abelian groups on products of spheres American Journal of Mathematics Carlsson , G. 1980; 102 (6): 1147-1157
  • An Adams type spectral sequence for change of rings Houston Journal of Mathematics Carlsson , G. 1978; 4: 541-550