scholarly journals Surface Bundles in Topology, Algebraic Geometry, and Group Theory

2020 ◽  
Vol 67 (02) ◽  
pp. 1
Author(s):  
Nick Salter ◽  
Bena Tshishiku
Keyword(s):  
Algebraic Geometry ◽  
Group Theory ◽  
Surface Bundles

Thirteen Papers on Group Theory, Algebraic Geometry and Algebraic Topology

1968 ◽  
Author(s):  
A. Andrianov ◽  
V. Dem′janenko ◽  
S. Demuškin ◽  
N. Efimov ◽  
N. Fel′dman ◽  
...  
Keyword(s):  
Algebraic Geometry ◽  
Group Theory ◽  
Algebraic Topology

scholarly journals THE MONOID OF FAMILIES OF QUIVER REPRESENTATIONS

2002 ◽  
Vol 84 (3) ◽  
pp. 663-685 ◽  
Author(s):  
MARCUS REINEKE
Keyword(s):  
Algebraic Geometry ◽  
Group Theory ◽  
Representation Theory ◽  
Quantum Group ◽  
Subject Classification ◽  
Quiver Representations ◽  
Enveloping Algebras ◽  
Geometric Methods ◽  
Quantized Enveloping Algebras ◽  
Monoid Structure

A monoid structure on families of representations of a quiver is introduced by taking extensions of representations in families, that is, subvarieties of the varieties of representations. The study of this monoid leads to interesting interactions between representation theory, algebraic geometry and quantum group theory. For example, it produces a wealth of interesting examples of families of quiver representations, which can be analysed by representation-theoretic and geometric methods. Conversely, results from representation theory, in particular A. Schofield's work on general properties of quiver representations, allow us to relate the monoid to certain degenerate forms of quantized enveloping algebras.2000 Mathematical Subject Classification: 16G20, 14L30, 17B37.

scholarly journals Residually finite rationally p groups

2019 ◽  
Vol 22 (03) ◽  
pp. 1950016
Author(s):  
Thomas Koberda ◽  
Alexander I. Suciu
Keyword(s):  
Algebraic Geometry ◽  
Group Theory ◽  
Large Class ◽  
Algebraic Curves ◽  
Fundamental Groups ◽  
Finitely Generated ◽  
Residual Property ◽  
Residually Finite ◽  
Torsion Freeness ◽  
Do So

In this paper, we develop the theory of residually finite rationally [Formula: see text] (RFR[Formula: see text]) groups, where [Formula: see text] is a prime. We first prove a series of results about the structure of finitely generated RFR[Formula: see text] groups (either for a single prime [Formula: see text], or for infinitely many primes), including torsion-freeness, a Tits alternative, and a restriction on the BNS invariant. Furthermore, we show that many groups which occur naturally in group theory, algebraic geometry, and in 3-manifold topology enjoy this residual property. We then prove a combination theorem for RFR[Formula: see text] groups, which we use to study the boundary manifolds of algebraic curves [Formula: see text] and in [Formula: see text]. We show that boundary manifolds of a large class of curves in [Formula: see text] (which includes all line arrangements) have RFR[Formula: see text] fundamental groups, whereas boundary manifolds of curves in [Formula: see text] may fail to do so.

The Applications of Group Theory

2012 ◽  
Vol 430-432 ◽  
pp. 1265-1268
Author(s):  
Xiao Qiang Guo ◽  
Zheng Jun He
Keyword(s):  
Number Theory ◽  
Algebraic Geometry ◽  
Group Theory ◽  
Algebraic Number ◽  
Algebraic Topology ◽  
Polynomial Equation ◽  
Algebraic Number Theory ◽  
Simple Groups ◽  
Finite Simple Groups

Since the classification of finite simple groups completed last century, the applications of group theory are more and more widely. We first introduce the connection of groups and symmetry. And then we respectively introduce the applications of group theory in polynomial equation, algebraic topology, algebraic geometry , cryptography, algebraic number theory, physics and chemistry.

Presentations and Low-dimensional Homology

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Group Theory ◽  
Simple Proof ◽  
Mapping Class Group ◽  
Class Group ◽  
Euler Class ◽  
Mapping Class ◽  
Basic Invariant ◽  
Homology Groups ◽  
Surface Bundles ◽  
Low Dimensional

This chapter presents explicit computations of the first and second homology groups of the mapping class group. It begins with a simple proof, due to Harer, of the theorem of Mumford, Birman, and Powell; the proof includes the lantern relation, a relation in Mod(S) between seven Dehn twists. It then applies a method from geometric group theory to prove the theorem that Mod(Sɡ) is finitely presentable. It also provides explicit presentations of Mod(Sɡ), including the Wajnryb presentation and the Gervais presentation, and gives a detailed construction of the Euler class, the most basic invariant for surface bundles, as a 2-cocycle for the mapping class group of a punctured surface. The chapter concludes by explaining the Meyer signature cocycle and the important connection of this circle of ideas with the theory of Sɡ-bundles.

Crystal Properties via Group Theory

1995 ◽  
Author(s):  
Arthur S. Nowick
Keyword(s):  
Group Theory ◽  
Crystal Properties
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