If is a connected topological manifold, and is a point in , the (topological) fundamental group of at is traditionally defined as the space of equivalence classes of loops starting and ending at , with two loops considered equivalent if they are homotopic to each other. 1,232 more words

## Tags » Math.AT

#### Cayley graphs and the algebra of groups

This is a sequel to my previous blog post “Cayley graphs and the geometry of groups“. In that post, the concept of a… 3,500 more words

#### The Hilbert-Smith conjecture

The classical formulation of Hilbert’s fifth problem asks whether topological groups that have the *topological* structure of a manifold, are necessarily Lie groups. This is indeed, the case, thanks to following theorem of… 2,861 more words

#### The Guth-Katz bound on the ErdÅ‘s distance problem

Combinatorial incidence geometry is the study of the possible combinatorial configurations between geometric objects such as lines and circles. One of the basic open problems in the subject has been the… 5,556 more words

#### Mazur's swindle

Let be a natural number. A basic operation in the topology of oriented, connected, compact, -dimensional manifolds (hereby referred to simply as *manifolds* for short) is that of… 858 more words

#### At the AustMS conference

This week I was in my home town of Adelaide, Australia, for the 2009 annual meeting of the Australian Mathematical Society. This was a fairly large meeting (almost 500 participants). 2,037 more words

#### Recent progress on the Kakeya conjecture

Below the fold is a version of my talk “Recent progress on the Kakeya conjecture” that I gave at the Fefferman conference. 4,082 more words