RESEARCH SUMMARY PETER BUBENIK

Transcription

1 RESEARCH SUMMARY PETER BUBENIK I am active in three areas of research: computational algebraic topology and data analysis, directed homotopy theory and concurrent computing, and homotopy theory, differential graded algebra and toric topology. Together with my collaborator Peter T. Kim, I am combining topological and statistical methods to aid practitioners in analyzing large, high-dimensional data sets [11, 7]. Independently and with various collaborators I am developing a directed version of homotopy theory for the purpose of modeling concurrent (parallel) computing [13, 6, 3, 9, 12]. My research background is in homotopy theory, in which I have made contributions to the classical question of how the attachment of a cell affects invariants such as the loop space homology [2] and the homotopy Lie algebra [4]. Currently I am working with Leah Gold to combine these topological techniques with algebraic techniques to the new field of toric topology [8]. In addition, I have published work with John Holbrook on constructing statistical samples [10], I have provided analytic support for George Bubenik s work on Moose hearing [1], and I have contributed to Zhiming Luo s work in homotopical algebra [29]. 1. Computational Algebraic Topology and Data Analysis 1.1. Motivation and overview. Mathematical scientists are being asked to apply their techniques to large, complex data sets, for which traditional methods are inadequate. Examples include data from computer images, gene chips, neuroscience, protein interactions, sensor networks, astronomy, biomechanics, medical imaging and music. These present challenges in representation, visualization, interpretation and analysis. Genome chips provide gene expression data for thousands of genes. From genome chip data, can it be determined which genes are involved in a given biological process or a certain medical condition? Drug discovery relies on sifting through a vast number of potential molecules for those which are likely to strongly interact with a chosen target. Can protein structure repositories be sorted to look for promising shapes? Sensor networks consist of a network of autonomous devices which monitor an environment. Can it be guaranteed that the sensor network cannot be evaded? In each of these settings, traditional techniques are inadequate for the analysis of the massive amount of data which are generated. In particular, standard statistical techniques are unable to capture certain nonlinear features which may be crucial for a proper understanding of the data. The mathematical field of topology has long been used to reveal broad, qualitative, geometric structures, such as the number of components or the presence of holes. Only recently, however, have the techniques of topology been adapted to study real-world data. One example of such a technique, called persistent topology, has been successfully applied 1

2 RESEARCH SUMMARY PETER BUBENIK 2 to the examples listed above. However these new techniques were developed without any statistical considerations. The aim of my research in this area is to significantly advance the new field of computational topology as follows. I am adapting current statistical and topological results and joining the two in order to provide stronger techniques than either approach could provide by itself. In particular, I am working to produce an optimal statistical procedure for computing topological information from sampled data [7] Background. Topology is a field of mathematics whose purpose is to identify broad, qualitative features of complex geometric objects, such as the number of components or the presence of holes. While the tools of topology have proved to be powerful theoretical techniques where the object under study is completely determined, it was not clear how to apply them in cases where the object is only partially known. For example, it was not known how to apply topology to a data sample. A breakthrough came with the concept of persistent topology. Its main idea is to consider the data under study at different scales of magnification and to apply existing mathematical techniques to characterize how the topology of the data changes as the scale changes. The pioneering articles are by Edelsbrunner, Letscher, Zomorodian [20] and Zomorodian and Carlsson [37]. Most articles on persistent topology do not incorporate a statistical perspective, though they do observe that heuristically, their techniques seem to be robust in the sense of confidence intervals. As it is becoming popularized, persistent topology is finding a growing list of applications. Examples include protein structure analysis [33], gene expression [19], and sensor networks [17]. One approach that combines topology and statistics is that of Niyogi, Smale and Weinberger [31]. Their article calculates how much data is needed to guarantee recovery of the underlying topology of the data. A drawback of their technique is that it supposes that the size of the smallest features of the data is known a priori. The stability result of Cohen-Steiner, Edelsbrunner and Harer [15] opens the way to another statistical approach. This result shows that if an estimate of the underlying function that is generating the data is sufficiently good in a precise technical sense, then the estimated persistent topology will be close to the actual persistent topology. Given this stability result, a good estimate of the underlying function implies a good estimate of the underlying topology. The statistical minimax method provides a technique for finding the optimal estimate which describes the data. Minimax techniques have been used to find optimal estimates in related situations [35, 27] Completed work. In my joint work with Peter T. Kim [11], we demonstrated the feasibility of combing statistical and topological techniques. We laid the groundwork to such an approach for samples in which the underlying distributions are parametrized by a finite-dimensional vector space [11]. We showed that the classical theory of spacings can be used to calculate the exact expectations of the persistent homology of samples from the uniform distribution on the circle, together with their

3 RESEARCH SUMMARY PETER BUBENIK 3 asymptotic behavior. We also made some nice connections between the persistent homology of some classical statistical distributions and well-known topological constructions. The filtrations of the n-dimensional sphere using sublevel sets of the von Mises-Fisher, Watson, and Bingham distributions correspond to different CW structures. Similarly, the matrix von Mises distribution corresponds to a relative CW structure on SO(3) using the Hopf fibration S 0 S 3 RP Current work. Currently I am working with Peter T. Kim (Guelph), his student Zhiming Luo, and Gunnar Carlsson (Stanford) to combine the current stability result of Cohen-Steiner, Edelsbrunner and Harer [15] with a new statistical estimation of functions in the relevant context. These theoretical advances are designed to provide powerful new tools for the analysis of large data sets in biology, medicine, and engineering. We will apply our techniques to analyze visual data [28, 14]. The analysis of this large (250MB) dataset will require use of the Ohio Supercomputer. Our advances will provide new algorithms for practitioners in science, engineering, medicine, security and commerce to obtain statistically quantifiable global information from large complex data sets. Mathematically, the sampled data are assumed to be generated by some unknown function on some underlying structure. The goal is to reconstruct the function from the sampled data. This paradigm is also a main part of a subject known as machine learning. One technique that does this is that of Niyogi, Smale and Weinberger [31]. Their article calculates how much data is needed to guarantee recovery of the underlying topology of the function generating the data. A drawback of their technique is that it supposes that the size of the smallest features of the data is known a priori. The primary objective of this project is to obtain a minimax estimate of functions in a way that can be combined with the result of [15] that implies that good estimates of a function will imply good estimates of the underlying topology. Furthermore, having such a method provides a general framework suitable for almost any data. Finally, a sharp adaptive minimax estimate will be optimal. For this objective, comparable minimax estimates exist in the literature [35, 27]. In order to obtain our minimax estimates (for functions on manifolds with respect to the supremum norm), we will generalize previous minimax estimates (for functions on the line, the circle and the sphere). While nontrivial, adapting these prior works to our situation is feasible. These theoretical results will then be used to study pertinent applications Preliminary results of this research are being published in an extended abstract [5] resulting from my invited stay at the Oberwolfach Mathematical Institute in Germany. More complete results are available in the preprint [7]. Together with G. Carlsson (Stanford) and P. Kim (Guelph), I am organizing a conference at the Banff International Research Station to stimulate research in this emerging area. 2. Directed homotopy theory and concurrent computing 2.1. Motivation and overview. Advances in computational hardware are, for the time being, no longer driven by an inexorable increase in chip speed. Instead, modern chips

4 RESEARCH SUMMARY PETER BUBENIK 4 contain an increasing number of computing cores. To fully exploit this parallel computing requires the concurrent use of shared resources. By using topological models, important theoretical and practical questions can be addressed [23]. Directed homotopy theory can provide a classification of the essential schedules, and can detect structural hazards in instruction pipelines, such as deadlocks and unsafe regions [22]. While non-concurrent computations can be modeled by one-dimensional directed graphs, an appropriate model for concurrent computations is a higher dimensional directed space, in which there is one dimension for each process. The schedules (executions) are given by paths from the initial state to the final state where the coordinates of the path track the progress of one of the processes. Execution paths which can be continuously deformed into one another correspond to essentially the same schedule. The understanding of continuous deformations is provided by the field of homotopy theory. The study of concurrency requires a directed version of homotopy theory, since the execution paths are not reversible. The directed spaces can by described by partiallyordered spaces (pospaces) or, if loops are allowed, by locally partially-ordered spaces (local pospaces). The combinatorial version of these models is given by higher dimensional automata, also known as cubical sets. Another application of a directed spaces is the space-time in general relativity, and this work may prove useful in that context. The goal of my research in this area is to develop mathematical models that will aid practitioners in analyzing concurrent systems Completed work. Together with Kris Worytkiewicz I used Quillen s model categories [32] to provide a framework for studying the homotopy theory of local pospaces [13]. The technical difficulty for developing such a theory is that the category of local pospaces is not known to have colimits, a necessary condition for a model category. Our approach in overcoming this shortcoming is inspired by the construction of Voevodsky s A 1 -homotopy theory [30] and Dugger s universal homotopy theories [18]. We passed to the simplicial presheaf category and applied Jardine s model structure [26]. We analyzed the weak equivalences in this model category, and finally we localized with respect to the class of (relative) directed homotopy equivalences. In other work, I showed relative directed homotopy is useful for a piece-by-piece simplification of pospaces [6] Current work. Since directed paths are not reversible in general, instead of the usual fundamental groupoid, one has the fundamental category. This captures the execution paths up to directed homotopy. The difficulty with the fundamental category is that it is too large for an algorithmic analysis. In my present work, I have showed that the analogue of the fundamental group for directed spaces is a directed graph, which for pospaces is bipartite. Furthermore, these directed topological invariants are faithfully inherited by certain full (co)reflective subcategories. As a result, I have constructed models of the fundamental category that are often finite, and still capture the essential executions. Furthermore, I proved that these construction lend themselves to a piece by piece analysis using van Kampen theorems [3].

5 RESEARCH SUMMARY PETER BUBENIK 5 There are many approaches to directed algebraic topology. Together with collaborators Eric Goubault, Eric Haucourt, and Sanjeevi Krishnan (Paris), I am working on connecting the component category approach of Goubault et. al. [21, 24] and the past and future retract approach of Grandis [25] that I used in my construction of models of the fundamental category [3]. We are also working on extending the component category approach to spaces which may have loops. In another project with David Spivak (Oregon), we are working on developing a model for concurrent systems in which not the states, but the execution paths are taking as the basic objects. A model is then constructed by considering the sheaves on the paths [12]. 3. Homotopy theory, differential graded algebra and toric topology 3.1. Motivation and overview. Understanding the homotopy groups of finite complexes remains one of the central challenges of homotopy theory. One fruitful approach to this problem has been to study the homology of the loop space. When the coefficients are taken to be the rational numbers, the loop space homology completely determines the rational homotopy groups of the space. In fact, it is a classical result that the rational homotopy groups have the structure of a graded Lie algebra and the loop space homology is a universal enveloping algebra on this Lie algebra. Studying the loop space homology with other coefficient rings has been sufficient in some cases to show that the loop space is homotopy equivalent to a product of simpler atomic spaces. In this case the homotopy groups of the given space are completely determined by the homotopy groups of the atomic spaces. A natural topological problem, perhaps first studied by J.H.C. Whitehead around 1940 [36] is to understand the effect of adding one or more cells to a given cell complex. In particular one can study the effect of attaching cells on the loop space homology. A new connection to this old problem has been provided by recent exciting work on (generalized) moment angle complexes. These structures, defined by Davis and Januszkiewicz [16], and generalized by Strickland [34] have important connections to combinatorics, algebraic geometry, symplectic geometry, and algebraic topology. For example, every smooth projective toric variety is the quotient of a moment angle complex by the free action of a real torus. The aim of my research in this area is to calculate the rational homotopy Lie algebra of generalized moment angle complexes using techniques that I have previously developed [2, 4] and noncommutative Grobner bases, an area in which my colleague and collaborator Leah Gold is an expert. This work has already uncovered surprising connections between distant areas of mathematics and promises further discoveries Completed Work. In my dissertation I generalized the previously studied inert cell attachments to free and semi-inert attachments. In these two cases, I constructed a differential graded Lie algebra model and determined the loop space homology as a module and as an algebra respectively. Under a further condition I determined the homotopy-type of the space. These results have been published in [2].

6 RESEARCH SUMMARY PETER BUBENIK 6 Furthermore, I carried out a more detailed analysis of the case with rational coefficients [4]. For a simply connected topological space X there is a differential graded Lie algebra, called the Quillen model, which determines the rational homotopy-type of X. Its homology is isomorphic to the rational homotopy groups of the loop space on X, called the homotopy Lie algebra. I showed that for spaces with finite (rational) LS category, these can be assumed to satisfy a condition I call separated, which is useful for calculations of the homotopy Lie algebra. This separated condition implies the free condition introduced in [2]. The separated Lie models give a nice characterization of the rational homotopy Lie algebra in the case where the top differential creates at least two new homology classes: the radical is contained in previous homology and the rational homotopy Lie algebra contains a free Lie algebra on two generators. So it satisfies a conjecture of Avramov and Félix: the homotopy Lie algebra is either finite or contains a free Lie subalgebra on two generators. Furthermore, I proved that any rational space constructed using a sequence of cell attachments of length N is equivalent to a space constructed using a sequence of free cell attachments of length N+1. This is shown by proving a similar result for differential graded Lie algebras (dgls). As a results, one obtains a method for calculating the homotopy Lie algebra, and the homology of certain dgls. These models hold the promise for allowing new calculations in homotopy theory Current work. Currently, I have applied the generalized moment angle complex construction to a collection of pointed spheres. I am working towards using Quillen models to calculate the homotopy Lie algebra of these spaces. A subsequent step would be apply this analysis to the Davis Januszkiewicz spaces which are central to the area of toric topology. It turns out that these spaces correspond to finite simple graphs in which each vertex has been labeled by a natural number. Together with Leah Gold, we have applied the theory of noncommutative Grobner bases to show that the Hilbert series of an algebra derived from these spaces is the inverse of the clique polynomial of the graph. In particular this algebra and the corresponding attaching map are inert if and only if the graph does not contain a triangle [8]. I have agreed to help write up work of Zhiming Luo on a model category structure for presheaves of simplicial groupoids so that it may be published [29]. 4. Other research 4.1. Completed work. Together with John Holbrook, I developed a method for constructing a particular type of statistical sample [10]. A random balanced sample (RBS) is a multivariate distribution with n components X k, each uniformly distributed on [ 1, 1], such that the sum of these components is precisely 0. The corresponding vectors X lie in an (n 1) dimensional polytope M(n). Applications of RBS distributions include sampling with antithetic coupling to reduce variance, isolation of nonlinearities, and goodness of fit testing.

7 RESEARCH SUMMARY PETER BUBENIK 7 We developed a new method for the construction of such RBS via densities over M(n) and these apply for arbitrary n. While simple densities had been known previously for small values of n (2, 3, 4), for larger n the known distributions with large support were fractal distributions (with fractal dimension asymptotic to n as n ). We also showed that the previously known densities (for n 4) are in fact the only solutions in a natural and very large class of potential RBS densities. In other work, I assisted George Bubenik in showing the palmated antlers of moose may serve as a parabolic reflector of sounds [1]. The external surface of the moose ear (the pinna) is sixty times larger than that of humans, and the pinnae can move independently and in almost any direction. We showed that when faced with a frontal sound source, the acoustic pressure in the ear increased by 19% if it was positioned towards the antlers rather than towards the sound source. References [1] George A. Bubenik and Peter G. Bubenik. Palmated antlers of moose may serve as a parabolic reflector of sounds. European Journal of Wildlife Research, 54(3): , [2] Peter Bubenik. Free and semi-inert cell attachments. Trans. Amer. Math. Soc., 357(11): , [3] Peter Bubenik. Models and van kampen theorems for directed algebraic topology. submitted, [4] Peter Bubenik. Separated Lie models and the homotopy Lie algebra. J. Pure Appl. Algebra, 212(2): , [5] Peter Bubenik. Statistical persistent homology. In Gunnar Carlsson and Dmitry Kozlov, editors, Computational algebraic topology, number 29 in Mathematisches Forschungsinstitut Oberwolfach Report, pages 9 11, [6] Peter Bubenik. Context for models of concurrency. Elect. Notes in Theor. Comp. Sci., (to appear). [7] Peter Bubenik, Gunnar Carlsson, Peter T. Kim, and Zhiming Luo. Optimal nonparametric regression on manifolds with application to topology. preprint, [8] Peter Bubenik and Leah Gold. Noncommutative graded algebras from graphs and their hilbert series. in preparation, [9] Peter Bubenik, Eric Goubault, Emmanuel Haucourt, and Sanjeevi Krishnan. Connections between approaches to directed algebraic topology. in preparation, [10] Peter Bubenik and John Holbrook. Densities for random balanced sampling. J. Multivariate Anal., 98(2): , [11] Peter Bubenik and Peter T. Kim. A statistical approach to persistent homology. Homology, Homotopy Appl., 9(2): , [12] Peter Bubenik and David Spivak. Path generated spaces. in preparation, [13] Peter Bubenik and Krzysztof Worytkiewicz. A model category for local po-spaces. Homology, Homotopy Appl., 8(1): , [14] Gunnar Carlsson, Tigran Ishkhanov, Vin de Silva, and Afra Zomorodian. On the local behavior of spaces of natural images. Int. J. Comput. Vision, 76(1):1 12, [15] David Cohen-Steiner, Herbert Edelsbrunner, and John Harer. Stability of persistence diagrams. In SCG 05: Proceedings of the twenty-first annual symposium on Computational geometry, pages , New York, NY, USA, ACM Press. [16] Michael W. Davis and Tadeusz Januszkiewicz. Convex polytopes, Coxeter orbifolds and torus actions. Duke Math. J., 62(2): , [17] Vin de Silva and Robert Ghrist. Homological sensor networks. Notic. Amer. Math. Soc., 54(1):10 17, 2007.

8 RESEARCH SUMMARY PETER BUBENIK 8 [18] Daniel Dugger. Universal homotopy theories. Adv. Math., 164(1): , [19] H. Edelsbrunner, M-L. Dequent, Y. Mileyko, and O. Pourquie. Assessing periodicity in gene expression as measured by microarray data. preprint, [20] Herbert Edelsbrunner, David Letscher, and Afra Zomorodian. Topological persistence and simplification. Discrete Comput. Geom., 28(4): , Discrete and computational geometry and graph drawing (Columbia, SC, 2001). [21] L. Fajstrup, M. Raussen, E. Goubault, and E. Haucourt. Components of the fundamental category. Appl. Categ. Structures, 12(1):81 108, Homotopy theory. [22] Lisbeth Fajstrup, Eric Goubault, and Martin Raußen. Detecting deadlocks in concurrent systems. In CONCUR 98: concurrency theory (Nice), volume 1466 of Lecture Notes in Comput. Sci., pages Springer, Berlin, [23] Lisbeth Fajstrup, Martin Raußen, and Eric Goubault. Algebraic topology and concurrency. Theoret. Comput. Sci., 357(1-3): , [24] E. Goubault and E. Haucourt. Components of the fundamental category. II. Appl. Categ. Structures, 15(4): , [25] Marco Grandis. The shape of a category up to directed homotopy. Theory Appl. Categ., 15:No. 4, (electronic), 2005/06. [26] J. F. Jardine. Simplicial presheaves. J. Pure Appl. Algebra, 47(1):35 87, [27] Jussi Klemelä. Asymptotic minimax risk for the white noise model on the sphere. Scand. J. Statist., 26(3): , [28] Ann B. Lee, Kim S. Pedersen, and David Mumford. The nonlinear statistics of high-contrast patches in natural images. Int. J. Comput. Vision, 54(1-3):83 103, [29] Zhiming Luo, Peter Bubenik, and Peter T. Kim. Closed model categories for presheaves of simplicial groupoids and presheaves of 2 groupoids. preprint, [30] Fabien Morel and Vladimir Voevodsky. A 1 -homotopy theory of schemes. Inst. Hautes Études Sci. Publ. Math., (90): (2001), [31] P. Niyogi, S. Smale, and S. Weinberger. Finding the homology of submanifolds with high confidence from random samples. Discrete and Computational Geometry, to appear. [32] Daniel G. Quillen. Homotopical algebra. Lecture Notes in Mathematics, No. 43. Springer-Verlag, Berlin, [33] Ahmet Sacan, Ozgur Ozturk, Hakan Ferhatosmanoglu, and Yusu Wang. Lfm-pro: A tool for detecting significant local structural sites in proteins. Bioinformatics, to appear. [34] N.P. Strickland. Toric spaces [35] A. B. Tsybakov. Pointwise and sup-norm sharp adaptive estimation of functions on the Sobolev classes. Ann. Statist., 26(6): , [36] J. H. C. Whitehead. On adding relations to homotopy groups. Ann. of Math. (2), 42: , [37] Afra Zomorodian and Gunnar Carlsson. Computing persistent homology. Discrete Comput. Geom., 33(2): , 2005.