相关论文: Macaulay 2 and the geometry of schemes

This article is intended to an introductory lecture in material physics, in which the modern computational group theory and the electronic structure calculation are in collaboration. The effort of mathematicians in field of the group…

Algebraic geometry for groups and Lie algebraic has been recently defined and studied by many authors on the purpose to study set defined by algebraic equations on abstract groups and Lie algebras. The purpose of this paper is to present a…

Despite the advances in the development of numerical methods analytical approaches still play the key role on the way towards a deeper understanding of many-particle systems. In this regards, diagonalization schemes for Hamiltonians…

We give illustrative examples of how the computer algebra system OSCAR can support research in commutative algebra and algebraic geometry. We start with a thorough introduction to Groebner basis techniques, with particular emphasis on the…

This note is based on the plenary talk given by the second author at MACIS 2015, the Sixth International Conference on Mathematical Aspects of Computer and Information Sciences. Motivated by some of the work done within the Priority…

Real world data often lie on low-dimensional Riemannian manifolds embedded in high-dimensional spaces. This motivates learning degenerate normalizing flows that map between the ambient space and a low-dimensional latent space. However, if…

We discuss an algorithm computing the push-forward to projective space of several classes associated to a (possibly singular, reducible, nonreduced) projective scheme. For example, the algorithm yields the topological Euler characteristic…

Macaulay Duality, between quotients of a polynomial ring over a field, annihilated by powers of the variables, and finitely generated submodules of the ring's graded dual, is generalized over any Noetherian ring, and used to provide…

I will give a new way to understand the Geometric Langlands Program.

This text is a slightly edited version of lecture notes for a course I gave at ETH, during the Winter term 2000-2001, to undergraduate Mathematics and Physics students. Contents: Chapter 1 - Examples of Dynamical Systems Chapter 2 -…

Given a homotopy connecting two polynomial systems we provide a rigorous algorithm for tracking a regular homotopy path connecting an approximate zero of the start system to an approximate zero of the target system. Our method uses recent…

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 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…

This paper presents a mathematical framework for analyzing machine learning models through the geometry of their induced partitions. By representing partitions as Riemannian simplicial complexes, we capture not only adjacency relationships…

We describe a new software package for computing multiplier ideals in certain cases, including monomial ideals, monomial curves, generic determinantal ideals, and hyperplane arrangements. In these cases we take advantage of combinatorial…

Shape analysis and compuational anatomy both make use of sophisticated tools from infinite-dimensional differential manifolds and Riemannian geometry on spaces of functions. While comprehensive references for the mathematical foundations…

How to understand deep learning systems remains an open problem. In this paper we propose that the answer may lie in the geometrization of deep networks. Geometrization is a bridge to connect physics, geometry, deep network and quantum…

When two or more subsystems of a quantum system interact with each other they can become entangled. In this case the individual subsystems can no longer be described as pure quantum states. For systems with only 2 subsystems this…

We survey the application of computer algebra in the context of gravitational theories. After some general remarks, we show of how to check the second Bianchi-identity by means of the Reduce package Excalc. Subsequently we list some…

We present twelve numerical methods for evaluation of objects and concepts from Poisson geometry. We describe how each method works with examples, and explain how it is executed in code. These include methods that evaluate Hamiltonian and…