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TOPOSYS
TOPOSYS
The contributions of lattice theory for the study of topological data analysis.
The contributions of lattice theory for the study of topological data analysis.
The author JPC would like to acknowledge that his work was funded by the EU project TOPOSYS (FP7-ICT-318493-STREP)
#1. THE PERSISTENCE LATTICE
with Mikael Vejdemo-Johansson and Primož Škraba
Ever since the early developments of Universal Algebra at the service of the constructions of new logics and the development of Quantum Theory in the 50's, lattices took a fundamental role in the achievement of great new results, unveiling outstanding new perspectives. On the other hand, Topological Persistence has been in the center of the interest of Computational Topology for the past twenty years. With the aim of its generalization reaching deeper levels of understanding, the unveil of universal rules and the further study of the structures of these objects of study is of great interest. Here the study of lattice-like structures that can model the ideas that Persistent Homology deals with can give great contribution to this study. In [1] the authors describe the lattice operations over an input diagram of vector spaces and linear maps corresponding to the filtration of a given topological space. Particularly, the Krull-Schmidt-Remak decomposition, shown to be very useful in the decomposition of persistence modules, both in the standard and the zig-zag persistence case. It is well known that such a result, for modules satisfying certain finiteness conditions, follows from the Jordan-Holder theorem applied to the modular lattice of submodules of a given module (see [6]). Moreover, the fact that the underlying lattice structure constitutes a Heyting algebra provides us with an internal logic for persistence, analysed in [2]. Motivated by the role of topological exactness in the proof of the modularity of the underlying algebra, the authors describe in [5] the potential of a general lattice duality for modular lattices based on the exactness of the correspondent dual space. Furthermore, several algorithmic applications are derived from the good properties of this lattice structure in [3].
This research is integrated in the FP7 EU project Toposys, hosted by the Institut Jozef Štefan in Ljubljana.
Key Words: posets, lattices, decomposition, persistence, homology, barcodes.
Report 2014 arXiv 2013 arXiv 2014
#2. A TOPOS FOUNDATION FOR PERSISTENCE
#2. A TOPOS FOUNDATION FOR PERSISTENCE
with Mikael Vejdemo-Johansson and Primož Škraba
A topos theoretic approach to set theory permits ideas like time variable sets and provides tools for unification of techniques for mathematics having had a great importance in the recent developments of Quantum Theory. Persistent homology is a central tool in topological data analysis, which examines the structure of data through topological structure. The basic technique is extended in many different directions, permuting the encoding of topological features by barcodes and correspondent persistence diagrams. The set of points of all such diagrams determines a complete Heyting algebra that can explain aspects of the relations between correspondent persistence bars and provide a global perspective over this approach. We are fundamentally interested in the algebraic foundations of applied and computational algebraic topology, in particular in a unifying theory for the various flavors of persistent homology that have emerged so far. In [4] we shall look at the topos of sheaves over such algebra, discuss its construction and potential for a generalized simplicial homology over it. Furthermore, we describe in [8] the sheafification process through the etale space construction that permits us to compute persistent homology in the most concrete cases.
This research is integrated in the FP7 EU project Toposys, hosted by the Institut Jozef Štefan in Ljubljana.
Key Words: posets, lattices, Heyting algebras, persistence diagrams, sheaves, Grothendieck topos.
Given a real-valued function f ∶ X → R, extended persistence is the collection of pairs arising from a sequence of absolute and relative homology groups. The corespondent pairs in the extended persistence diagram keep track on the changes in the homology of the input function. Every class which is born at some point of the two-stage process will eventually die, being associated with a pair of critical values. The ordered pairs now fall into three classes: ordinary, relative and extended. Due to the need of their representation in the same persistence diagram, we assume the extended persistence diagram, from Edelsbrunner et al., and describe in [7] an extended underlying lattice structure that is compatible with the lattice presented as the internal logic for the topos of persistence in earlier works. We also describe the Alexandrov space of irreducibles, and the technique to recover the algebraic structure from it. Furthermore, in this work we explore Pultr's lattice duality between complete Heyting algebras and f-spaces, ie, Priestley spaces sufficing certain properties that permit this relation. We shall also look at Banaschewski duality for continuous distributive lattices and describe the dual space correspondent to the Heyting algebra of lifetimes. Moreover, we will use this duality to analyse the dual space in order to get a further insight on the algebra of lifetimes itself. This work started with my University visit to Maria Joao Gouveia in Lisbon, in November 2013, and was first presented at the Toposys project meeting in Vienna, in September 2014. A preview of its results can be found in [7].
Key Words: lattices, ideals and filters, extended persistence, barcodes.
#4. SKEW DISTRIBUTIVE LATTICES OVER THE ALGEBRA OF LIFETIMES
#4. SKEW DISTRIBUTIVE LATTICES OVER THE ALGEBRA OF LIFETIMES
with Karin Cvetko-Vah, Maria Joao Gouveia and Primož Škraba
#3. THE ORDER STRUCTURE OF EXTENDED PERSISTENCE
with Maria Joao Gouveia and Primož Škraba
Skew lattices have been the most successful variation of noncommutative lattices. A special attention has been always devoted to skew lattices in rings, that constitute a large class of examples, where Karin Cvetko-Vah and JPC answered several open questions. Today the classical dualities as Stone’s and Priestley’s are a focus of research in this context. Of great interest is the equivalence between skew distributive lattices and sheaves of sets over local Priestley spaces. This result was achieved in 2013 with the study of a non-commutative version of Priestley duality. A Priestley-like duality between complete Heyting algebras and Priestley spaces with certain extra conditions is known since the end of the last century. With this study we intend to clarify the connection between the latter dualities, identify which skew distributive lattices are associated to complete Heyting algebras, and understand how can we take profit of that knowledge in order to deal with time-variable sets as noncommutative lattices. JPC has been working with skew distributive lattices for several years, having submitted a paper on the subject together with J. Leech and M. Kynion. This research topic was first presented in October 2014 at the algebra seminar in FMF, Ljubljana.
Key Words: lattices, ideals and filters, etale space, Priestley duality, skew distributive lattice.
#5. TOPOLOGICAL DATA ANALYSIS FOR EPIDEMIOLOGY
#5. TOPOLOGICAL DATA ANALYSIS FOR EPIDEMIOLOGY
with Primož Škraba
Influenzanet is a system to monitor the activity of influenza-like-illness (ILI) with the aid of volunteers via the internet. It has been operational for more than 10 years at the EU level since 2008. In contrast with the traditional system of sentinel networks of mainly primary care physicians, Influenzanet obtains its data directly from the population. This creates a fast and flexible monitoring system whose uniformity allows for direct comparison of ILI rates between countries. Persistent homology is a central tool in topological data analysis, which examines the structure of data through topological structure. In the past years it has taken an important role in the development of medicine. It is an area of mathematics interested in identifying a global structure by inferring high-dimensional structure from low-dimensional representations and studying properties of a often continuous space by the analysis of a discrete sample of it, assembling discrete points into global structure. The basic technique can be extended in many different directions, permuting the encoding of topological features by barcodes and correspondent persistence diagrams. Using persistence we are able to analyze the Influenzanet data identifying several topological features relevant to the epidemiological study. In particular, we can identify data noise, distinguish higher dimension features and look at join spaces between countries. This is done both in terms of the overall structure of a disease as well as its evolution. Finally, it provides a way to test agreement at a global scale arising from standard local models.
Key Words: Topological Data Analysis, Epidemiology, persistence diagrams, algorithms, Influenza.
Research Papers
Research Papers
J. Pita Costa. Topological data analysis and applications. In: Proceedings of the 40th Jubilee International Convention MIPRO 2017 - Modeling System Behaviour Workshop (2017).
J. Pita Costa and Tihana Galinac Grbac. The Topological Data Analysis of Time Series Failure Data in Software Evolution. In: Proceedings of the 8th ACM/SPEC on International Conference on Performance Engineering Companion. ACM, 2017. p. 25-30. doi>10.1145/3053600.3053604
J. Pita Costa, P. Škraba and M. Vejdemo Johansson. Variable sets over an algebra of lifetimes. arXiv:1409.8613. To be submitted to Foundations of Computational Mathematics (2016)
J. Pita Costa, P. Škraba and M. Vejdemo Johansson. The power of pullbacks: meets and joins in persistence. To be submitted to Discrete and Computational Geometry (2016).
L. Fajstrup and J. Pita Costa. On the hierarchy of d-structures . To appear in Order (2016)
J. Pita Costa and P. Škraba. A topological data analysis approach to the epidemiology of Influenza. SIKDD, Ljubljana, Slovenia (2015).
Joao Pita Costa, Daniela Paolotti, Flavio Fuart, Primož Škraba, Evgenia Belayeva, Inna Novalija. Supporting Epidemic Intelligence, Personalised and Public Health with advanced computational methods. EM-Health Workshop, Ljubljana, Slovenia (2015).
J. Pita Costa and P. Škraba. Topological epidemiological data analysis, Poster at the ACM Digital Health Conference 2015, Firenze, Italy (2015).
J. Pita Costa and P. Škraba. Topological analysis of Influenza data, Poster at the ECCS14, Lucca, Italy (2014).
M. Kinyon, J. Leech and J. Pita Costa. Distributivity in skew lattices. Semigroup Forum 91 (2015):378-400.
J. Pita Costa and K. Cvetko-Vah. Flat coset decompositions of skew lattices. Semigroup Forum 92 (2016): 361-376.
J. Pita Costa, P. Škraba and M. Vejdemo Johansson. Aspects of an internal logic of persistence. arXiv:1409.3762 (2014)
J. Pita Costa and P. Škraba. A lattice for persistence. arXiv:1307.4192.
Conference Talks & Posters
Conference Talks & Posters
2015.10.12 Supporting epidemic intelligence, personalised and public health with advanced computational methods. Delavnica EM-Zdravje, Ljubljana, Slovenia.
2015.10.05 A topological data analysis approach to the epidemiology of Influenza. SIKDD, Ljubljana, Slovenia.
2015.07.20 The topos foundation of persistence, Applications of Algebra, Kalamata, Greece.
2015.06.22 Persistence on Sheaves over Lifetimes, ACAT Summer School, Ljubljana, Slovenia.
2015.06.16 The topos foundation of persistence, Conference on Dynamics, Topology and Applications, Bedlewo, Poland.
2015.05.18 Topological epidemiological data analysis, ACM Digital Health Conference 2015, Firenze, Italy.
2015.03.01 Skew distributive lattices and computational topology, AAA89 General Algebra Conference, Dresden, Germany.
2015.02.04 On the topos foundation of persistence , Computational Topology Meeting, Lisbon, Portugal.
2014.09.23 Topological analysis of Influenza data, ECCS14, Lucca, Italy.
2014.09.08 The order structure of extended persistence, Toposys Meeting, IST Vienna.
2014.06.20 Recent contributions of lattice theory to topological data analysis, AAA88, Warsaw, Poland.
2014.05.27 Towards a Topos for Persistence, ATMCS 6, Vancouver, Canada
2013.09.18 Order structures for Topological Data Analysis. ECCS 2013, Barcelona, Spain.
2013.07.22 The Persistence Lattice, Applied Topology Conference, Bedlewo, Poland.
2013.07.04 Heyting algebra over the persistence diagram. Summer School on Computational Topology and Topological Data Analysis, Ljubljana, Slovenia.
2013.06.11 The Persistence Lattice, CSASC Joint Mathematical Conference, Koper, Slovenia.
2013.06.08 The Persistence Lattice, Novi Sad Algebra Conference, Novi Sad, Serbia.
2012.06.19 Coset Laws for Skew Lattices. 5th Croatian Mathematical Congress. Rijeka, HR.
Seminar Talks & Workshops
Seminar Talks & Workshops
2015.09.02 On the stability of the persistence topos, Toposys Final Meeting, Krakow University, Poland.
2015.06.10 Topological analysis of Influenza data, BioTech Department, University of Rijeka, Croatia.
2015.05.27 A topos foundation of persistence. Seminar za Algebro, Fakulteta za Matematika, Ljubljana, SI.
2015.05.27 Skew distributive lattices in the topos foundation of persistence, Seminar za Algebro, Fakulteta za Matematika, Ljubljana, SI.
2014.11.19 Topological analysis of Influenza data, AI Lab, Inštitut Jozef Štefan, Ljubljana, Slovenija.
2014.10.08 Variable sets over an algebra of lifetimes: a contribution of lattice theory to the study of computational topology. arxiv:1409.8613. Seminar za Algebro, Fakulteta za Matematika, Ljubljana, SI.
2014.06.04 Towards a Topos for Persistence, AI Lab, Inštitut Jozef Štefan, Ljubljana, Slovenija.
2014.04.25 Towards a topos foundation for persistence with applications. Seminar of the Society of Mathematicians and Physicists of Rijeka, Rijeka, Croatia.
2014.03.18 Towards a topos foundation for persistence part II. Seminar of Theoretical Computer Science, University of Ljubljana, Slovenija.
2014.02.19 A topos for topological data analysis with possible applications for Influenzanet. AI Lab, Inštitut Jozef Štefan, Ljubljana, Slovenija.
2014.02.18 Towards a topos foundation for persistence part I. Seminar of Theoretical Computer Science, University of Ljubljana, Slovenija.
2013.11.06 On the contributions of lattice theory to the study of persistent homology. Topology and Category Theory Seminar, University of Coimbra, Portugal.
2013.11.01 Locales for topological data analysis. CAUL Algebra Seminar, University of Lisbon, Portugal.
2013.10.24 Order structures for topological data analysis. Seminar of the Society of Mathematicians and Physicists of Rijeka, Rijeka, Croatia.
2013.10.23 Heyting algebra over the persistence diagram. Seminar za Algebro, Fakulteta za Matematika, Ljubljana, SI.
2013.08.15 Heyting algebra over the persistence diagram. FP7 Toposys EU Project Meeting, Israel.
2013.05.28 The Persistence Lattice, Algebra Seminar, Faculty of Mathematics, University of Ljubljana, Slovenia.
2013.05.16 The Persistence Lattice, Artificial Inteligence Seminar, Inštitut Jozef Štefan, Ljubljana, Slovenia.
2012.10.23 Contributions to skew lattice theory. Solomonov Seminar, Inštitut Jozef Štefan, Ljubljana, SI.
2012.10.19 Flat coset decomposition of skew lattices. Seminar za Algebro, Fakulteta za Matematika, Ljubljana, SI.
2012.10.04-12 Working seminars on noncommutative lattices. CAUL, University of Lisboa, PT.
References
References
Primož Škraba, Institut Jozef Stefan, Ljubljana.
Mikael Vejdemo Johansson, KTH, Sweden.
Karin Cvetko-Vah, University of Ljubljana, Slovenia.
Andrej Bauer, University of Ljubljana, Slovenia.
Maria Joao Gouveia, University of Lisbon, Portugal.
Graham Ellis, University of Galway, Ireland.
Herbert Edelsbrunner, Institute of Science and Technology, Austria
JOÃO PITA COSTA 2016
JOÃO PITA COSTA 2016
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