Related papers: An introduction to geometric complexity theory
This paper presents a novel and straight formulation, and gives a complete insight towards the understanding of the complexity of the problems of the so called NP-Class. In particular, this paper focuses in the Searching of the Optimal…
We present and discuss applications of the category of probabilistic morphisms, initially developed in \cite{Le2023}, as well as some geometric methods to several classes of problems in statistical, machine and manifold learning which shall…
A computationally challenging classical elimination theory problem is to compute polynomials which vanish on the set of tensors of a given rank. By moving away from computing polynomials via elimination theory to computing pseudowitness…
The basic elements of the geometric approach to a consistent quantization formalism are summarized, with reference to the methods of the old quantum mechanics and the induced representations theory of Lie groups. A possible relationship…
This article is an interdisciplinary review and an on-going progress report over the last few years made by myself and collaborators in certain fundamental subjects on two major theoretic branches in mathematics and theoretical physics:…
We survey large value problems, including the large value problem for Dirichlet polynomials, the restriction problem, and problems from computer science. We describe known techniques and open problems, drawing on perspectives from all three…
Constraint satisfaction problems are computational problems that naturally appear in many areas of theoretical computer science. One of the central themes is their computational complexity, and in particular the border between…
We survey some algebraic geometric aspects of mirror symmetry and duality in string theory. Some applications of computer algebra to algebraic geometry and string theory are shortly reviewed.
This paper covers two topics: first an introduction to Algorithmic Complexity Theory: how it defines probability, some of its characteristic properties and past successful applications. Second, we apply it to problems in A.I. - where it…
We present a characterization of states in generalized probabilistic models by appealing to a non-commutative version of geometric probability theory based on algebraic geometry techniques. Our theoretical framework allows for incorporation…
An infinite dimensional algebra, which is useful for deriving exact solutions of the generalized pairing problem, is introduced. A formalism for diagonalizing the corresponding Hamiltonian is also proposed. The theory is illustrated with…
This book encompasses both traditional and modern methods treating partial differential equation (PDE) of first order and second order. There is a balance in making a selfcontained mathematical text and introducing new subjects. The Lie…
The topics of Convexity and Concavity and Envelopes are central in Complex Analysis and extensively investigated. The aim of this paper is to find a possible counterpart in Algebraic Geometry. The article presents preliminary results on…
In this paper we explore fundamental concepts in computational complexity theory and the boundaries of algorithmic decidability. We examine the relationship between complexity classes \textbf{P} and \textbf{NP}, where $L \in \textbf{P}$…
We study geometric variations of the discriminating code problem. In the \emph{discrete version} of the problem, a finite set of points $P$ and a finite set of objects $S$ are given in $\mathbb{R}^d$. The objective is to choose a subset…
The main purpose of this book is to propose an introduction to the modern tools of algebraic complexity. To remain as simple as possible while providing meaningful examples, we chose to focus on effective linear algebra; this is certainly…
This article explores some geometric and algebraic properties of the dynamical system which is represented by matrix differential equations arising from inertial navigation problems, such as the symplecticity and the orthogonality.…
We initiate a study of determinantal representations with symmetry. We show that Grenet's determinantal representation for the permanent is optimal among determinantal representations respecting left multiplication by permutation and…
We study the classical problem of computing geometric thickness, i.e., finding a straight-line drawing of an input graph and a partition of its edges into as few parts as possible so that each part is crossing-free. Since the problem is…
Every real hyperbolic form in three variables can be realized as the determinant of a linear net of Hermitian matrices containing a positive definite matrix. Such representations are an algebraic certificate for the hyperbolicity of the…