Two lectures on the arithmetic of K3 surfaces

In these lecture notes we review different aspects of the arithmetic of K3 surfaces. Topics include rational points, Picard number and Tate conjecture, zeta functions and modularity.Comment: 26 pages; v4: typos corrected, references update

LearnFCA: A Fuzzy FCA and Probability Based Approach for Learning and Classification

Formal concept analysis(FCA) is a mathematical theory based on lattice and order theory used for data analysis and knowledge representation. Over the past several years, many of its extensions have been proposed and applied in several domains including data mining, machine learning, knowledge management, semantic web, software development, chemistry ,biology, medicine, data analytics, biology and ontology engineering. This thesis reviews the state-of-the-art of theory of Formal Concept Analysis(FCA) and its various extensions that have been developed and well-studied in the past several years. We discuss their historical roots, reproduce the original definitions and derivations with illustrative examples. Further, we provide a literature review of it’s applications and various approaches adopted by researchers in the areas of dataanalysis, knowledge management with emphasis to data-learning and classification problems. We propose LearnFCA, a novel approach based on FuzzyFCA and probability theory for learning and classification problems. LearnFCA uses an enhanced version of FuzzyLattice which has been developed to store class labels and probability vectors and has the capability to be used for classifying instances with encoded and unlabelled features. We evaluate LearnFCA on encodings from three datasets - mnist, omniglot and cancer images with interesting results and varying degrees of success. Adviser: Dr Jitender Deogu

LEARNFCA: A FUZZY FCA AND PROBABILITY BASED APPROACH FOR LEARNING AND CLASSIFICATION

Formal concept analysis(FCA) is a mathematical theory based on lattice and order theory used for data analysis and knowledge representation. Over the past several years, many of its extensions have been proposed and applied in several domains including data mining, machine learning, knowledge management, semantic web, software development, chemistry ,biology, medicine, data analytics, biology and ontology engineering. This thesis reviews the state-of-the-art of theory of Formal Concept Analysis(FCA) and its various extensions that have been developed and well-studied in the past several years. We discuss their historical roots, reproduce the original definitions and derivations with illustrative examples. Further, we provide a literature review of it’s applications and various approaches adopted by researchers in the areas of dataanalysis, knowledge management with emphasis to data-learning and classification problems. We propose LearnFCA, a novel approach based on FuzzyFCA and probability theory for learning and classification problems. LearnFCA uses an enhanced version of FuzzyLattice which has been developed to store class labels and probability vectors and has the capability to be used for classifying instances with encoded and unlabelled features. We evaluate LearnFCA on encodings from three datasets - mnist, omniglot and cancer images with interesting results and varying degrees of success. Adviser: Jitender Deogu

On mining complex sequential data by means of FCA and pattern structures

Nowadays data sets are available in very complex and heterogeneous ways. Mining of such data collections is essential to support many real-world applications ranging from healthcare to marketing. In this work, we focus on the analysis of "complex" sequential data by means of interesting sequential patterns. We approach the problem using the elegant mathematical framework of Formal Concept Analysis (FCA) and its extension based on "pattern structures". Pattern structures are used for mining complex data (such as sequences or graphs) and are based on a subsumption operation, which in our case is defined with respect to the partial order on sequences. We show how pattern structures along with projections (i.e., a data reduction of sequential structures), are able to enumerate more meaningful patterns and increase the computing efficiency of the approach. Finally, we show the applicability of the presented method for discovering and analyzing interesting patient patterns from a French healthcare data set on cancer. The quantitative and qualitative results (with annotations and analysis from a physician) are reported in this use case which is the main motivation for this work. Keywords: data mining; formal concept analysis; pattern structures; projections; sequences; sequential data.Comment: An accepted publication in International Journal of General Systems. The paper is created in the wake of the conference on Concept Lattice and their Applications (CLA'2013). 27 pages, 9 figures, 3 table

GENERATING KNOWLEDGE STRUCTURES FROM OPEN DATASETS' TAGS - AN APPROACH BASED ON FORMAL CONCEPT ANALYSIS

Under influence of data transparency initiatives, a variety of institutions have published a significant number of datasets. In most cases, data publishers take advantage of open data portals (ODPs) for making their datasets publicly available. To improve the datasets' discoverability, open data portals (ODPs) group open datasets into categories using various criteria like publishers, institutions, formats, and descriptions. For these purposes, portals take advantage of metadata accompanying datasets. However, a part of metadata may be missing, or may be incomplete or redundant. Each of these situations makes it difficult for users to find appropriate datasets and obtain the desired information. As the number of available datasets grows, this problem becomes easy to notice. This paper is focused on the first step towards decreasing this problem by implementing knowledge structures to be used in situations where a part of datasets' metadata is missing. In particular, we focus on developing knowledge structures capable of suggesting the best match for the category where an uncategorized dataset should belong to. Our approach relies on dataset descriptions provided by users within dataset tags. We take advantage of a formal concept analysis to reveal the shared conceptualization originating from the tags' usage by developing a concept lattice per each category of open datasets. Since tags represent free text metadata entered by users, in this paper we will present a method of optimizing their usage through means of semantic similarity measures based on natural language processing mechanisms. Finally, we will demonstrate the advantage of our proposal by comparing concept lattices generated using formal the concept analysis before and after the optimization process. The main experimental research results will show that our approach is capable of reducing the number of nodes within a lattice more than 40%