Related papers: A Singular Mathematical Promenade
An introductory guide to mathematical cosmology is given focusing on the issue of the genericity of various important results which have been obtained during the last thirty or so years. Some of the unsolved problems along with certain new…
Mathematics is often perceived as difficult or inaccessible, yet meaningful engagement can arise in unexpected places. In this article we describe a multi-year exploration of mathematical outreach through games, puzzles, exhibitions, and…
This paper presents mathematics as a general science of computation in a way different from the tradition. It is based on the radical philosophical standpoint according to which the content, meaning and justification of experience lies in…
The purpose of this note is to give an exposition of some interesting combinatorics and convex geometry concepts that appear in algebraic geometry in relation to counting the number of solutions of a system of polynomial equations in…
We build and investigate a pure gauge theory on arbitrary discrete groups. A systematic approach to the construction of the differential calculus is presented. We discuss the metric properties of the models and introduce the action…
The world of mathematics is often considered abstract, with its symbols, concepts, and topics appearing unrelated to physical objects. However, it is important to recognize that the development of mathematics is fundamentally influenced by…
We describe and explain the desire, common among mathematicians, both for unity and independence in its major themes. In the dialogue that follows, we express our spontaneous and considered judgment and reservations by contrasting the…
This article summarises a Web-book on "Complexity" that was developed to introduce undergraduate students to interesting complex systems in the biological, physical and social sciences, and the common tools, principles and concepts used for…
The purpose of this thesis is to study classical combinatorial objects, such as polytopes, polytopal complexes, and subspace arrangements, using tools that have been developed in combinatorial topology, especially those tools developed in…
This work in progress aims to provide a unified introduction to statistical learning, building up slowly from classical models like the GMM and HMM to modern neural networks like the VAE and diffusion models. There are today many internet…
This is a survey of known algorithms in algebraic topology with a focus on finite simplicial complexes and, in particular, simplicial manifolds. Wherever possible an elementary approach is chosen. This way the text may also serve as a…
These Notes deal with various areas of mathematics, and seek reciprocal combinations, explore mutual relations, ranging from abstract objects to problems in physics.
This book introduces to the theory of probabilities from the beginning. Assuming that the reader possesses the normal mathematical level acquired at the end of the secondary school, we aim to equip him with a solid basis in probability…
"Mathematicians, like physicists, are pushed by a strong fascination. Research in mathematics is hard, it is intellectually painful even if it is rewarding, and you would not do it without some strong urge." [D. Ruelle]. We shall give some…
In this paper, we give the explicit bounds for the data of objects involved in some basic theorems of Singularity theory: the Inverse, Implicit and Rank Theorems for Lipschitz mappings, Splitting Lemma and Morse Lemma, the density and…
This article is a survey of recent developments in, and a tutorial on, the approach to P v. NP and related questions called Geometric Complexity Theory (GCT). It is written to be accessible to graduate students. Numerous open questions in…
Many mathematicians find mathematics aesthetically beautiful and even comparable to art forms such as music or painting. On the other hand, every year a great number of school students leave mathematics with total disillusionment and…
These are expanded notes for a short series of lectures, presented at the University of Luxembourg in 2017, giving an introduction to some of the ideas of supersymmetry and supergeometry. In particular, we start from some motivating facts…
The purpose of this sophomore-level textbook is twofold: to introduce the student to classical electrodynamics and, at the same time, explain in simple terms the quantum theory of conducting substances (in particular, the solid ones). The…
This paper aims to provide a careful and self-contained introduction to the theory of topological degree in Euclidean spaces. It is intended for people mostly interested in analysis and, in general, a heavy background in algebraic or…