Related papers: On the principal ideal theorem in arithmetic topol…
We present a new topological proof of the infinitude of prime numbers with a new topology. Furthermore, in this topology, we characterize the infinitude of any non-empty subset of prime numbers.
Following the analogies between knots and primes, 3-manifolds and number rings in arithmetic topology, we show a topological analogue of the Hasse norm principle for finite cyclic coverings of 3-manifolds, which was originally stated for…
We introduce a very natural topology on the set of total orderings of monomials of any algebra having a countable basis over a field. This topological space and some notable subspaces are compact. This topological framework allows us to…
We develop a new approach to the representation theory of quantum algebras supporting a torus action via methods from the theory of finite-state automata and algebraic combinatorics. We show that for a fixed number $m$, the torus-invariant…
The purpose of this note is to give a survey of the algebraic properties of multiplier ideals, and illustrate some of their applications to classical projective geometry.
The purpose of this paper is to give an elementary proof to the theorem due to Avramov on certain determinantal ideals of linear type.
Proofs of the fundamental theorem of algebra can be divided up into three groups according to the techniques involved: proofs that rely on real or complex analysis, algebraic proofs, and topological proofs. Algebraic proofs make use of the…
In this paper we explore the topological properties of self-replicating, 3-dimensional manifolds, which are modeled by idempotents in the (2+1)-cobordism category. We give a classification theorem for all such idempotents. Additionally, we…
These are the extended notes of a talk I gave at the Geometric Topology Seminar of the Max Planck Institute for Mathematics in Bonn on January 30th, 2012. My goal was to familiarize the topologists with the basics of arithmetic hyperbolic…
A set of meet-irreducible ideals is described for a class of maximal triangular almost finite algebras. This set forms a topological space under the hull-kernel closure, and there is a one-to-one correspondence between closed sets in this…
We study a topological generalization of ideal co-maximality in topological rings and present some of its properties, including a generalization of the Chinese remainder theorem. Using the hyperspace uniformity, we prove a stronger version…
This book is expository and is in Russian. It is shown how in the course of solution of interesting geometric problems (close to applications) naturally appear main notions of algebraic topology (homology groups, obstructions and…
We study the ideal of the algebraic relations among 3-point functions from a combinatorial and topological perspective. We place this problem in the broader setting of incidence toric ideals associated with incidence matrices of t-subsets…
A general method for constructing essential uniform algebras with prescribed properties is presented. Using the method, the following examples are constructed: an essential, natural, regular uniform algebra on the closed unit disc; an…
A complete determination of the prime ideals invariant under winding automorphisms in the generic 3 by 3 quantum matrix algebra is obtained. Explicit generating sets consisting of quantum minors are given for all of these primes, thus…
This is an expository article of our work on analogies between knot theory and algebraic number theory. We shall discuss foundational analogies between knots and primes, 3-manifolds and number rings mainly from the group-theoretic point of…
We present a simple short proof of the Fundamental Theorem of Algebra, without complex analysis and with a minimal use of topology. It can be taught in a first year calculus class.
The chapter provides an introduction to the basic concepts of Algebraic Topology with an emphasis on motivation from applications in the physical sciences. It finishes with a brief review of computational work in algebraic topology,…
The classical Neukirch-Uchida theorem states that the absolute Galois group determines a number field up to isomorphism. We prove an analogue of this theorem for 3-manifolds in the framework of arithmetic topology. We study infinite links…
In this paper we define a topology with sufficiently good properties on the set of prime ideals of a number field.