An outline of polar spaces: basics and advances

TL;DR

This paper provides a comprehensive overview of polar spaces, covering fundamental concepts, classical results, and recent advances including infinite rank cases and embeddings in groups and projective spaces.

Contribution

It offers a detailed survey of polar spaces, highlighting recent research developments and open problems in the field.

Findings

Survey of fundamental results on polar spaces

Introduction to polar spaces of infinite rank

Discussion of embeddings in groups and projective spaces

Abstract

This paper is an extended version of a series of lectures on polar spaces given during the workshop and conference 'Groups and Geometries', held at the Indian Statistical Institute in Bangalore in December 2012. The aim of this paper is to give an overview of the theory of polar spaces focusing on some research topics related to polar spaces. We survey the fundamental results about polar spaces starting from classical polar spaces. Then we introduce and report on the state of the art on the following research topics: polar spaces of infinite rank, embedding polar spaces in groups and projective embeddings of dual polar spaces.