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The computer algebra system CHEVIE is designed to facilitate computations with various combinatorial structures arising in Lie theory, like finite Coxeter groups and Hecke algebras. We discuss some recent examples where CHEVIE has been…
Modern statistical learning theory and deep learning characterize generalization primarily in terms of continuous capacity control (e.g., norm-based regularization, margin maximization, low-rank bias). While highly successful in continuous…
Algebras of Logic deal with some algebraic structures, often bounded lattices, considered as models of certain logics, including logic as a domain of order theory. There are well known their importance and applications in social life to…
We classify, up to isomorphism and up to equivalence, division gradings (by abelian groups) on finite-dimensional simple real algebras. Gradings on finite-dimensional simple algebras are determined by division gradings, so our results give…
Cluster algebras are a recent topic of study and have been shown to be a useful tool to characterize structures in several knowledge fields. An important problem is to establish whether or not a given cluster algebra is of finite type.…
For any finitely generated abelian group $Q$, we reduce the problem of classification of $Q$-graded simple Lie algebras over an algebraically closed field of "good" characteristic to the problem of classification of gradings on simple Lie…
We survey some recent applications of machine learning to problems in geometry and theoretical physics. Pure mathematical data has been compiled over the last few decades by the community and experiments in supervised, semi-supervised and…
Deep learning was recently successfully used in deriving symmetry transformations that preserve important physics quantities. Being completely agnostic, these techniques postpone the identification of the discovered symmetries to a later…
This thesis studies arithmetic of linear algebraic groups. It involves studying the properties of linear algebraic groups defined over global fields, local fields and finite fields, or more generally the study of the linear algebraic groups…
A finite-dimensional unital and associative algebra over $\mathbb{R}$, or what we shall call simply "an algebra" in this paper for short, generalities the construction by which we derive the complex numbers by "adjoining an element $i$" to…
Automata learning is a popular technique used to automatically construct an automaton model from queries. Much research went into devising ad hoc adaptations of algorithms for different types of automata. The CALF project seeks to unify…
Abstraction is key to human and artificial intelligence as it allows one to see common structure in otherwise distinct objects or situations and as such it is a key element for generality in AI. Anti-unification (or generalization) is…
In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has…
We aim at studying collections of algebraic structures defined over a commutative ring and investigating the complexity of significant constructions carried out on these objects. The assignment of measures of size, via a multiplicity…
Statistics and Optimization are foundational to modern Machine Learning. Here, we propose an alternative foundation based on Abstract Algebra, with mathematics that facilitates the analysis of learning. In this approach, the goal of the…
Machine learning algorithms use error function minimization to fit a large set of parameters in a preexisting model. However, error minimization eventually leads to a memorization of the training dataset, losing the ability to generalize to…
We consider the problem of classifying gradings by groups on a finite-dimensional algebra $A$ (with any number of multilinear operations) over an algebraically closed field. We introduce a class of gradings, which we call almost fine, such…
We use machine learning to classify examples of braids (or flat braids) as trivial or non-trivial. Our ML takes form of supervised learning using neural networks (multilayer perceptrons). When they achieve good results in classification, we…
This is an introduction to linear algebra and group theory. We first review the linear algebra basics, namely the determinant, the diagonalization procedure and more, and with the determinant being constructed as it should, as a signed…
A method for machine learning and serving of discrete field theories in physics is developed. The learning algorithm trains a discrete field theory from a set of observational data on a spacetime lattice, and the serving algorithm uses the…