Related papers: Unsolved Problems in Group Theory. The Kourovka No…

We collect some open problems about minimal presentations of numerical semigroups and, more generally, about defining ideals and free resolutions of their semigroup rings and associated graded rings. We emphasize both long-standing problems…

Enumeration of tilings is the mathematical study concerning the total number of coverings of regions by similar pieces without gaps or overlaps. Enumeration of tilings has become a vibrant subfield of combinatorics with connections and…

We consider several algorithmic problems concerning geodesics in finitely generated groups. We show that the three geodesic problems considered by Miasnikov et al [arXiv:0807.1032] are polynomial-time reducible to each other. We study two…

We generalize the classical knapsack and subset sum problems to arbitrary groups and study the computational complexity of these new problems. We show that these problems, as well as the bounded submonoid membership problem, are P-time…

Mathematicians have long been fascinated by the resolution of algebraic and Diophantine equations in search of integer or rational solutions. This article presents a list of thirty-three open problems in number theory, posed in the 13th…

This book covers many of the recent results on group actions on the circle, with an emphasis in the differentiable case.

It is shown that the knapsack problem (introduced by Myasnikov, Nikolaev, and Ushakov) is undecidable in a direct product of sufficiently many copies of the discrete Heisenberg group (which is nilpotent of class 2). Moreover, for the…

This article is based on an invited talk given at the Workshop on Mathematical Physics Towards XXIst Century, held at Beer-Sheva, Israel in 1993. It contains an introduction to quantum gravity for mathematical physicists with an emphasis on…

This survey paper is an expanded version of lectures given at the Clay Mathematics Academy ; see http://www.claymath.org/programs/outreach/academy/colloquium2005.php These lectures were intended to very young (and motivated) college…

This is a replacement paper. There are 6 chapters. The first two chapters are introductory. The third chapter is on extremal graph theory. The fourth chapter is about algebra in graph theory. The fifth chapter is focused on algorithms. The…

Group theory is used in many textbooks of contemporary physics. However, electromagnetic community often considers group theory as an "exotic" tool. Graduate and postgraduate textbooks on electromagnetics and electrodynamics usually do not…

These are lecture notes of a C.I.M.E. course I gave at Cetraro, June 6-11 2005. The theory described is the version of Chen-Ruan's Gromov-Witten theory of orbifolds developed by Graber, Vistoli and me in the algebraic setting, but with…

This document is a collection of comments that I wrote down while reading the first four chapters of the book "Discrete Groups, Expanding Graphs and Invariant Measures" by Alexander Lubotzky. Most of them are more detailed versions of…

In this paper a special group of bijective maps of a normed plane, called the group of general rotations, is introduced; it contains the isometry group as a subgroup. The concept of general rotations leads to the notion of flexible motions…

We collect here various conjectures on congruences made by the author in a series of papers, some of which involve binary quadratic forms and other advanced theories. Part A consists of 100 unsolved conjectures of the author while…

This paper presents fifteen problems about mapping class groups. It is an expanded and updated version of the author's preprint "Ten problems on the mapping class groups". The paper will appear in the book "Problems on Mapping Class Groups…

This text contains over three hundred specific open questions on various topics in additive combinatorics, each placed in context by reviewing all relevant results. While the primary purpose is to provide an ample supply of problems for…

Denote the integers by Z and the positive integers by N. The groups Z^k (k a natural number) are discrete, and the classification up to isomorphism of their (topological) subgroups is trivial. But already for the countably infinite power…

These lecture notes cover classical undecidability results in number theory, Hilbert's 10th problem and recent developments around it, also for rings other than the integers. It also contains a sketch of the authors result that the integers…

The survey contains a brief description of the ideas, constructions, results, and prospects of the theory of hypergroups and generalized translation operators. Representations of hypergroups are considered, being treated as continuous…