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This note is supposed to be an introduction to those concepts of toric geometry that are necessary to understand applications in the context of string and F-theory dualities. The presentation is based on the definition of a toric variety in…
In this survey, we describe the fundamental differential-geometric structures of information manifolds, state the fundamental theorem of information geometry, and illustrate some use cases of these information manifolds in information…
In this article we present a generalization of a Leibniz's geometrical theorem and an application of it.
These notes outline some basic notions of Tropical Geometry and survey some of its applications for problems in classical (real and complex) geometry. To appear in the Proceedings of the Madrid ICM.
In this Master of Science Thesis I introduce geometric algebra both from the traditional geometric setting of vector spaces, and also from a more combinatorial view which simplifies common relations and operations. This view enables us to…
These are lecture notes for a course on the theory of Clifford algebras, with special emphasis on their wide range of applications in mathematics and physics. Clifford algebra is introduced both through a conventional tensor algebra…
These lecture notes are a personal introduction to signed graphs, concentrating on the aspects that have been most persistently interesting to me. They are just a few corners of signed graph theory; I am leaving out a great deal. The…
In this paper we introduce elements of algebraic geometry over an arbitrary algebraic structure. We prove Unification Theorems which gather the description of coordinate algebras by several ways.
In this paper we make an overview of results relating the recent "discoveries" in differential geometry, such as higher structures and differential graded manifolds with some natural problems coming from mechanics. We explain that a lot of…
This is the first part of the lecture notes that grew out of the special course given during the 2021-2022 academic year. In these lecture notes we present an approach to the fundamental structures of differential geometry that uses the…
Over the past twenty years, lecture hall partitions have emerged as fundamental combinatorial structures, leading to new generalizations and interpretations of classical theorems and new results. In recent years, geometric approaches to…
In this text, we wish to provide the reader with a short guide to recent works on the theory of dilatations in Commutative Algebra and Algebraic Geometry. These works fall naturally into two categories: one emphasises foundational and…
This paper is a survey of computational issues in algebraic geometry, with particular attention to the theory of Grobner bases and the regularity of an algebraic variety. 1. A geometric introduction to Grobner bases. 2. An algebraic…
These notes, based on a graduate course I gave at Hamburg University in 2003, are intended to students having basic knowledges of differential geometry. Their main purpose is to provide a quick and accessible introduction to different…
These are general notes on tensor calculus which can be used as a reference for an introductory course on tensor algebra and calculus. A basic knowledge of calculus and linear algebra with some commonly used mathematical terminology is…
A new type of sectional curvature is introduced. The notion is purely algebraic and can be located in linear algebra as well as in differential geometry.
We introduce a notion of oriented dialgebra and develop a cohomology theory for oriented dialgebras based on the possibility to mix the standard chain complexes computing group cohomology and associative dialgebra cohomology. We also…
These are lecture notes for the course "MATS4120 Geometry of geodesics" given at the University of Jyv\"askyl\"a in Spring 2020. Basic differential geometry or Riemannian geometry is useful background but is not strictly necessary. Exercise…
The purpose of this paper is to introduce an algebraic cohomology and formal deformation theory of left alternative algebras. Connections to some other algebraic structures are given also.
These lecture notes give an overview of recent results in geometric Langlands correspondence which may yield applications to quantum field theory. We start with a motivated introduction to the Langlands Program, including its geometric…