Related papers: An introduction to geometric complexity theory
Algebraic statistics is concerned with the study of probabilistic models and techniques for statistical inference using methods from algebra and geometry. This article presents a list of open mathematical problems in this emerging field,…
We discuss some basic problems and conjectures in a program to construct general orbifold conformal field theories using the representation theory of vertex operator algebras. We first review a program to construct conformal field theories.…
By combining well-known techniques from both noncommutative algebra and computational commutative algebra, we observe that an algorithmic approach can be applied to the study of irreducible representations of finitely presented algebras. In…
Computational complexity is a new quantum information concept that may play an important role in holography and in understanding the physics of the black hole interior. We consider quantum computational complexity for $n$ qubits using…
This paper argues that the requirement of applicableness of quantum linearity to any physical level from molecules and atoms to the level of macroscopic extensional world, which leads to a main foundational problem in quantum theory…
We give a short introduction to the methods of representing polynomial and trigonometric series that are often used in Celestial Mechanics. A few applications are also illustrated.
In the first section we recall some basic notions on Lie algebras. In a second time we study the algebraic variety of complex $n$-dimensional Lie algebras. We present different notions of deformations : Gerstenhaber deformations,…
Tropical geometry is a piecewise linear "shadow" of algebraic geometry. It allows for the computation of several cohomological invariants of an algebraic variety. In particular, its application to enumerative algebraic geometry led to…
The purpose of this paper is to explore the question "to what extent could we produce formal, machine-verifiable, proofs in real algebraic geometry?" The question has been asked before but as yet the leading algorithms for answering such…
In this note we discuss some arithmetic and geometric questions concerning self maps of projective algebraic varieties.
What makes a computational problem easy (e.g., in P, that is, solvable in polynomial time) or hard (e.g., NP-hard)? This fundamental question now has a satisfactory answer for a quite broad class of computational problems, so called…
A (global) determinantal representation of hypersurface in P^n is a matrix, whose entries are linear forms in homogeneous coordinates and whose determinant defines the hypersurface. We study the properties of such representations for…
Persistent homology is a method for computing the topological features present in a given data. Recently, there has been much interest in the integration of persistent homology as a computational step in neural networks or deep learning. In…
A survey on recent developments in (algebraic) integral geometry is given. The main focus lies on algebraic structures on the space of translation invariant valuations and applications in integral geometry.
This paper introduces arithmetic geometry for polynomial identity algebras using non-commutative (formal) deformation theory. Since formal deformation theory is inherently local the arithmetic and geometric results that follow give local…
This preprint is the introduction of my habilitation thesis for Paris7 university. It is a sumary of a collection of works on the 2 matrix model. In an introduction, 3 different and unequivalent definitions of matrix models are given…
In math.RT/0302174 we developed a framework to study representations of groups of the form $G((t))$, where $G$ is an algebraic group over a local field $K$. The main feature of this theory is that natural representations of groups of this…
We explain how recent developments in the fields of realisability models for linear logic -- or geometry of interaction -- and implicit computational complexity can lead to a new approach of implicit computational complexity. This…
The traditional study of plane and space algebraic curves by looking at their tangent vectors, curvatures and torsions provides geometric, but unfortunately not sufficient information about individual curves in order to be able to…
This is the first paper in a series of eight where in the first three we develop a systematic approach to the geometric algebras of multivectors and extensors, followed by five papers where those algebraic concepts are used in a novel…