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with Mikael Vejdemo-Johansson, Maria Joao Gouveia 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 this poster we shall look at the topos of sheaves over such algebra, discuss its construction and potential for a generalized simplicial homology over it. 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.
THE PERSISTENCE LATTICE
THE PERSISTENCE LATTICE
SCIENTIFIC PROJECTS
SCIENTIFIC PROJECTS
TOPOLOGICAL MEDICAL DATA ANALYSIS FOR EPIDEMIOLOGY
TOPOLOGICAL MEDICAL 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.
A TOPOS FOUNDATION FOR PERSISTENCE
A TOPOS FOUNDATION FOR PERSISTENCE
with Primož Škraba (Inštitut Jozef Štefan, Ljubljana)
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. 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.
EN / PT
THE DUALITY THEORY OF SKEW LATTICES
THE DUALITY THEORY OF SKEW LATTICES
with K. Cvetko-Vah (University of Ljubljana) and M. J. Gouveia (University of Lisbon)
Skew lattices are noncommutative generalizations of lattices and have been studied in the past 20 years. Topological representations of algebras revealed to be useful in many settings. One classical example of such a representation is Stone duality which was recently generalized for varieties of skew lattices, namely the variety of skew Boolean algebras with intersections. Priestley duality for distributive lattices motivated a recent topic of interest and involvement: to generalize Priestley duality and obtain a representation for skew distributive lattices. This project views the establishment of topological representations for another classes of skew lattices, preferably in the spirit of natural dualities and the use of these representations to obtain concrete constructions of canonical extensions of skew lattices. The study of skew lattices’ coset structure seems to be close related. We shall look at several open problems using this new machinery, and aim for a deeper knowledge on skew lattices.
Key Words: noncommutative lattices, coset structure, Stone's duality, Priestley's duality, skew Boolean algebra.
EN / PT
THE COSET STRUCTURE OF SKEW LATTICES
THE COSET STRUCTURE OF SKEW LATTICES
with K. Cvetko-Vah (University of Ljubljana)
During the last years, several authors were interested on the study of specific properties of the skew lattices that relate to fundamental properties of lattices. Unlike what happens in lattices in general, in skew lattices the properties of distributivity and cancellation are independent. In fact, cancellation is related with the property of symmetry that indicates the instances of commutativity in a skew lattice.
A skew lattice is a set S equipped with two associative, idempotent binary operations v and ^ that satisfy the absorption laws (b ^ a)v a = a = a v (a ^ b) and their duals. The class of skew lattices forms a variety largely studied in the past twenty years motivated by semigroup theory and lattice theory.
Key Words: noncommutative lattices, coset structure, bands of semigroups, index, skew Boolean algebra.
EN / PT
THE STAR OPERATION AND OTHER STORIES
THE STAR OPERATION AND OTHER STORIES
The notion of equivalence has been playing a fundamental role through the History of Mathematics. Equivalence relations are so ubiquitous in everyday life that we often forget about their proactive existence. By 1994, in a conference held Pontignano, Rota regarded the studdy of linear lattices, made of commutable equivalence relations. This dissertation is about the algebraic and order properties of equivalence relation latticas and, specially, the algebraic studdy of different operations definable in these lattices.
Key Words: partition lattices, commutable equivalences, independent partitions, classification of pairs of equivalences of finite type, star operation, n-trusting organizations.
with Luis Sequeira (University of Lisbon)
lecture
Some Operations on the Lattice of Equivalence Relations
as author at Solomon seminar,
573 views
GRIPENET
GRIPENET
The Project Gripenet was born in Portugal by the hands of the mathematician Gabriela Gomes leading a multidisciplinary group hosted by the Gulbenkian Institut of Science in Lisbon, Portugal. It intends to create a forecast of the epidemic of the virus influenza with basis on the updated mathematical models fed by online public subscriptions by volunteers over the internet. It is parallel to the National Health System and integrated in the larger project Influenzanet. This project was involved in workshops for primary school, middle school and highschool, together with training of teachers, in cooperation with the National Center for Education and the Mathematics Teachers Association, both in Lisbon.
Key Words: influenza, epidemic, dynamical systems, health, Epiwork.
JOÃO PITA COSTA 2016
JOÃO PITA COSTA 2016
with Gabriela Gomes and Vitor Faustino (Instituto Gulbenkian de Ciencia, Lisbon)
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