My research career has been close to topics as univeral algebra and semigroup theory, having encountered computational topology in the past year. Indeed, my recent results show how skew lattices can provide great contribution to the study of topological data analysis. In the following sections I present my previous and ongoing research. The text intends to be self contained and present clearly the open problems that I am currently dealing with.
1. Topological data analysis.
1.1 Introduction. Topological Data Analysis is a vibrant area of research due to the developments in applied and computational algebraic topology. It applies the qualitative methods of topology to problems of machine learning, data mining and computer vision. In particular, persistent homology 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 continuous space by the analysis of a discrete sample of it, assembling points into global structure. When considering a notion of distance on the space, one gets a perspective of this space under different scales, where small features will eventually disappear. Persistence allows us to compute the homology at all scales, thereby giving us the ability to find ranges of scales where the structure of the space is stable. Techniques of persistence can be used to infer topological structure in data sets while certain variations on the method can be applied to study aspects of the shape of point clouds. This research has had a large impact on: social media (applying topological techniques to the analysis of text data); robotic grasping and machine learning (using homology generators, winding numbers and Gauss linking integrals); cancer research (combining geometric data transformations and topological methods to classify new subtypes); natural image statistics (analyzing of the local behavior of the space of natural images); and archeology (using an object descriptor based on the Euler characteristic at multiple levels for supervised classification).
1.2 Problem. The topological information of a filtration can be encoded in a set of closed intervals, named a barcode, where the birth and death time of every connected component of the sample of a given topological space is recorded. Each barcode can also be represented by vertices in a plane, named a persistence diagram, where the first coordinate corresponds to the birth time and the second coordinate corresponds to the death time. Barcodes, seen as multi-scale signatures, are important to the development of new techniques in machine learning, nonlinear systems and big data. Many of the applications require the manipulation and comparison of persistence diagrams. Sheaf theory is the standard tool in topology for tracking locally defined information attached to the open sets of a topological space and transferring it to a global scenario. I use these tools in [14] to organize the mathematical information enabling an easy shift in perspective between general, particular and even singular approaches to the same geometric objects. Persistence diagrams can also be dealt with as objects of a topos permitting new approaches to the study and leading to a deeper understanding of the relationship between such structures. Topoi behave as categories of sheaves of sets on a topological space. Topos theory was proposed as an alternative to set theory for the foundation of Mathematics, providing an effective means for transferring results and techniques between distinct fields. Hence, it has a fundamental role in the unification of mathematics, illuminating the nature of concepts, establishing new theorems, suggesting more general examples and promoting new lines of investigation. Lattices are posets that satisfy minimal conditions to acquire an algebraic structure compatible with their underlying order structure. Heyting algebras are bounded distributive lattices, capturing essential properties of both set operations and logic operations. The lattice of open sets of a topological space forms a Heyting algebra under the operations of union and intersection. Duality theory establishes an equivalence between Heyting algebras and Esakia spaces, permitting us to solve problems defined in one framework in a different framework. The category of sheaves on a Heyting algebra is a topos. Distributive skew lattices are sheaves over distributive lattices. I studied their coset structure in [5]. Based on this, I present the idea of a topos-based approach in [14], where sheaves of sets are described as sets with a temporally varying shape. I aim to develop the theory of persistent homology as the internal homology of simplicial complexes over a set theory in which elements have lifetimes. In such a setting, a filtered topological space corresponds to a topological space where parts of the space come in at later times; the construction of the homology functor immediately provides homology groups where elements come in and go away as time flows. This leads to the algebraic foundations of computational algebraic topology, in particular to a formulation in the topos setting for the various flavors of persistence that have emerged: standard, multidimensional and zigzag persistence. For each of these cases, the recognition of an underlying algebraic structure in [13] has contributed both to the identification of new problems and to the development of new algorithms. The Heyting algebra presented in this paper permits the study of an internal logic in [7] where I discuss several consequences of the nice properties of this order structure.
1.3 Future work. A topos theoretic strategy consists of a framework of recognized interest with the following outcomes:
- Develop new approaches and better understanding of algorithmic constructions for persistence, applying the knowledge mainly to: model systems (studying locally defined structures and transferring their information to globally valid inferences by way of consistency relations) and machine learning problems (using sheaves to take compatible local descriptions and glue them together to a global cohesive picture, aiming an algebraic method to identify which global pictures are possible to combine from local pieces).
- Establish an internal logic for persistence: study the elementary topos based on the Heyting algebra of vertices in a persistence diagram where the order structure, represented by the lattice operations, corresponds to the set inclusion of the correspondent bars in the barcode.
- Extract topological information by the usage of duality theory over the Heyting algebra of a persistence diagram. Indeed, an original approach to the classical study of the topological features by barcodes. The study of its spectrum clarifies the ring underlying the correspondent spectral spaces.
- Explore consequences for lattice theory, with the development of clear applications that permit the creation of new problems within the theory and motivate new strategies and ideas, and the implementation of new techniques inspired by persistence in lattice theoretic proofs.
Sheaves over distributive lattices consist of skew lattices, a non-commutative version of lattices studied in [5]. Classical persistence will therefore emerge as a subcategory of the resulting topos of these sheaves, ensuring the feasibility of the project within its duration. The feasibility of the project is aided by the researcher’s joint expertise in lattice structures and computational topology. This makes the project ideally positioned as the intersection of these foundational areas has not been explored.
2. Noncommutative lattice theory
2.1 Introduction. Lattices are of great importance in the study of Universal Algebra permitting the construction of clear examples and revealing its presence in many circumstances. These algebraic structures are determined by sets with two associative, idempotent and commutative binary operations that satisfy the property of absorption that establishes a relation between them. Developments in metamathematics and category theory in the 1940s and 1950s furthered the field, particularly the work of A. Robinson, A. Tarski, A. Mostowski, and their students (Brainerd 1967). P. Jordan was the first to study noncommutative lattices in 1949. Skew lattices have been the most successful variation of noncommutative lattices. Jonathan Leech studied these algebras and was interested on skew Boolean algebras, that generalize Boolean algebras. The variety of skew Boolean algebras includes the class of algebras earlier studied by W. D. Cornish. Of particular interest are the skew Boolean intersection algebras where every pair of elements has a natural meet with respect to the natural partial order. Following ideas of Keimal and Werner, R.J. Bignall observed that the variety of skew Boolean intersection algebras constitutes a discriminator variety and thus is both congruence permutable and congruence distributive. J. Leech collaborated with R. Veroff, R. .J. Bignall and M. Spinks to study general properties of these algebras and use them in the study of multiple valued logic. A special attention has been devoted to skew lattices in rings that constitute a large class of examples; in particular K. Cvetko-Vah answered several open questions related with these algebras, often in collaboration with J. Leech.
2.2 Problem. Unlike what happens in lattices, the properties of cancellation and distributivity are independent for skew lattices. K. Cvetko-Vah, J. Leech, M. Spinks and M. Kinyon studied both of these properties and unveiled several characterizations and subvarieties that enriched the theory at a great level and permitted the topic of research of my Ph.D. dissertation in [9] to contribute to an important and actual discussion. The pilars for this particular research were set in [10] where results of categorical nature are established to open new perspectives on the study of the coset structure of skew lattices. This approach has already been fruitful in the Ph.D. Thesis of my mentor K. Cvetko-Vah, and in our common papers [2], where we identify laws that permit the characterization of cancellative skew lattices, and [1], where we establish these laws in a wider panorama, exhibiting several combinatorial consequences of interest. Following this line of work I presented in [8] the coset structure of categorical skew lattices, a class that includes distributive skew lattices characterized by its potencial to consitute a category where the objects are cosets and the morphisms are sets of coset bijections between them. This category is further studied in [11]. The established coset laws influence the study of categorical skew lattices by J. Leech and M. Kinyon with whom I continue my research in distributive skew lattices in [5]. Moreover, this work follows in [6] with the study of modular skew lattices and its relation with cancellative skew lattices that as started when I began my Ph.D. Of great interest is the flat coset structure in [3] that provides a deeper understanding of right and left cosets and their role in the wider coset structure of skew lattice. The study of ideals and filters in [12] clarifies the study of a generalization of Stone duality and Priestley duality for skew lattices, established in 2012 with authors as A. Bauer, K. Cvetko-Vah, G. Kudryavtseva, M. Gherke and S. Van Gool. I encounter this research in [4] with a coset approach clarifying some aspects of the sheaves described there. This ongoing work will lead to a further knowledge on the transfer of local information in distributive lattices to a global scenario of skew distributive lattices, a matter of interest for the study of locales in topological data analysis.
2.3 Future work. While Algebra is a wide field of Mathematics that has proven to bring great contributions to other natural sciences like chemistry or biology (through group theory for instance), it is nowadays an important base of knowledge for the recent developments in Computer Science and other fields of knowledge. With this research I shall accomplish the following listed objectives:
Systematic study of the coset structure of distributive skew lattices and its impact on the generalization of dualities in this context;
Exploration of some open problems on the recent generalization of Priestley’s duality for skew lattices using the new available approaches;
Construction of concrete canonical extensions of certain classes of skew lattices using topological representations;
Study of the dualizable varieties of skew lattices in the sense of the theory of natural dualities;
Contribution to the study of finitely generated varieties of skew lattices.
The relevance of such objectives is due to the contribution to the study of noncommutative lattices, within Lattice Theory and Universal Algebra, as well as to the general study of semigroups and its application to advanced Linear Algebra and Operator Theory as well as to Logic and Computer Science. The interdisciplinary status of such research will provide new approaches in these fields as it has proven to succeed to do in the recent past, through the works of J. Leech, R. Bignall or M. Spinks. The new machinery brought by the recently accomplished generalizations of Stone and Priestley dualities is of great importance and will ensure a new panoramic over open problems and research strategies.
3. References.
[1] K. Cvetko-Vah and J. Pita Costa. On the coset laws for skew lattices. Semigroup Forum 8 (2010), pp 395-411.
[2] K. Cvetko-Vah and J. Pita Costa. On the coset laws for skew lattices in rings. Novi Sad Journal of Mathematics Vol. 40 No. 3 (2010).
[3] K. Cvetko-Vah and J. Pita Costa. Flat coset structure. Algebra Universalis. Submitted (2013).
[4] K. Cvetko-Vah, M. J. Gouveia and J. Pita Costa. Coset structure approach to the study of non-commutative Priestley duality. In preparation (2014).
[5] J. Leech, M. Kinyon and J. Pita Costa. Distributive skew lattices. Semigroup Forum. Submitted (2013), arXiv: 1306.5598.
[6] J. Leech, M. Kinyon, K. Cvetko-Vah and J. Pita Costa. Modularity for skew lattices. In preparation (2013).
[7] J. Pita Costa and P. Škraba. An internal logic for persistence. Journal of Topological Methods in Nonlinear Analysis. In preparation (2013).
[8] J. Pita Costa. Coset laws for categorical skew lattices. Algebra Universalis (2012), DOI: 10.1007/s00012-012-0194-z.
[9] J. Pita Costa. On the coset structure of skew lattices. Ph. D. dissertation, University of Ljubljana (2012).
[10] J. Pita Costa. On the coset structure of a skew lattice. Demonstratio Mathematica Vol. XLIV No. 4. (2011), pp 673-692.
[11] J. Pita Costa. On the coset category of a skew lattice. Demonstratio Mathematica. Submmited (2013).
[12] J. Pita Costa. On ideals of a skew lattice. Discussiones Mathematicae 32 (2012), 5--21. DOI: doi:10.7151/dmgaa.1187.
[13] P. Škraba and J. Pita Costa. A lattice for persistence. Foundations of Computational Mathematics. Submitted (2013), arXiv: 1307.4192.
[14] M. Vejdemo-Johansson, P. Škraba and J. Pita Costa. A Topos Foundation for Persistence. In preparation (2013).