Related papers: Introduction to tropical algebraic geometry
The purpose of this paper is fourfold. The first is to develop the theory of tropical differential algebraic geometry from scratch; the second is to present the tropical fundamental theorem for differential algebraic geometry, and show how…
The notion of geometric construction is introduced. This notion allows to compare incidence configurations in the algebraic and tropical plane. We provide an algorithm such that, given a tropical instance of a geometric construction, it…
These notes are an expanded version of an introductory lecture on contact geometry given at the 2001 Georgia Topology Conference. They are intended to present some of the "topological" aspects of three dimensional contact geometry.
This survey paper, written in spanish, is an extended version of lecture notes for a mini-course taught at the 2022 Summer School in Geometric Group Theory, which took place in the Centro de Ciencias Matem\'aticas in Morelia, Mexico in July…
An extended summary of the lecture course given at the V School on Geometry and Physics, Bia\l owe\.za 2016, in which an algebraic approach to differentiation and integration that is characteristic for non-commutative geometry is described.
Tropical mathematics redefines the rules of arithmetic by replacing addition with taking a maximum, and by replacing multiplication with addition. After briefly discussing a tropical version of linear algebra, we study polynomials build…
These are detailed notes for a lecture on "Non-associative Algebraic Structures: Classification and Structure" which I presented as a part of my Agrega\c{c}\~ao em Matem\'atica e Applica\c{c}\~oes (University of Beira Interior, Covilh\~a,…
This paper collects and extends the lectures I gave at the "XXIV International Fall Workshop on Geometry and Physics" held in Zaragoza (Spain) August 31 - September 4, 2015. Within these lectures I review the formulation of Quantum…
In the last few years there has been a growing interest towards methods for statistical inference and learning based on computational geometry and, notably, tropical geometry, that is, the study of algebraic varieties over the min-plus…
This is an expanded version of the lecture notes for a minicourse that I gave at a summer school called "Advanced Course on Geometry and Dynamics of Integrable Systems" at CRM Barcelona, 9--14/September/2013. In this text we study the…
This is the written version of a lecture given at the ``VI Mexican School of Gravitation and Mathematical Physics" (Nov 21-27, 2004, Playa del Carmen, Mexico), introducing the basics of Loop Quantum Geometry. The purpose of the written…
These are the notes for the Clay Mathematics Institute Senior Scholar Lecture which was delivered by Bernd Sturmfels in Park City, Utah, on July 22, 2004. The topic of this lecture is the ``tropical approach'' in mathematics, which has…
Complex algebraic varieties become easy piecewise-linear objects after passing to the so-called tropical limit. Geometry of these limiting objects is known as tropical geometry. In this short survey we take a look at motivation and…
In this pages I give an overview of the relationship between Model Theory, Arithmetic and Algebraic Geometry. The topics will be the basic ones in the area, so this is just an invitation, in the presentation of topics I mainly follow the…
This book was intended to serve as supporting material for a mini-course on web geometry delivered at the 27th Brazilian Mathematical Colloquium which took place at IMPA in the last week of July 2009.
These lecture notes are based on a set of six lectures that I gave in Edinburgh in 2008/2009 and they cover some topics in the interface between Geometry and Physics. They involve some unsolved problems and conjectures and I hope they may…
This is an expanded version of the author's lecture at the Conference in Commutative Algebra and Algebraic Geometry in Messina Italy in June 1999. The purpose of the talk was to give a brief introduction to the subject of tight closure,…
These notes are a record of lectures given in the Workshop on Connections Between Algebra and Geometry at the University of Regina, May 29--June 1, 2012. The lectures were meant as an introduction to current research problems related to fat…
This is an expanded version of my Shaw Prize Lecture delivered at the Chinese University of Hong Kong.
This is an expository article for Elsevier's Encyclopedia of Mathematical Physics on the subject in the title. Comments/corrections welcome.