I realize it has been a while since I have posted anything. I’m afraid teaching and research took up enough of my late summer schedule to keep me from finishing up a number of posts I had started. This is alright; I’ve been working hard trying to make my first experience with teaching topology at Georgia State a success.

This post is a reminder about the upcoming PTM-DMV Conference in Poznan, Poland: the joint meeting of the German Mathematical Society (Deutsche Mathematiker-Vereinigung) and Polish Mathematical Societies (Polskie Towarzystwo Matematyczne).

I will be giving a talk in the Session on Wild Algebraic & Geometric Topology.

Abstracts can be found on the session link above. Here is the schedule:

**9/17/2014 (Wednesday)**

- 2:30 – 2:55 PM
**Wolfgang Herfort**– Cotorsion and Homology - 3:00 – 3:25 PM
**Wolfram Hojka**– Mapping the harmonic archipelago - 3:30 – 3:55 PM
**Isacc Goldbring**– The fundamental group of a locally finite graph with ends: a hyperfinite approach - 4:00 – 4:25 PM
**Benoit Loridant**– Fundamental group of Rauzy fractals - 4:30 – 5:00 PM Coffee Break
- 5:00 – 5:25 PM
**Jean-Francois LaFont**– One-dimensional geodesic spaces, Part I: Structure Theory - 5:30 – 5:55 PM
**David Constantine**– One-dimensional geodesic spaces, Part II: Marked length rigidity

**9/18/2014 (Thursday)**

- 2:30 – 2:55 PM
**Hanspeter Fischer**– Word calculus in the fundamental group of the Menger Curve - 3:00 – 3:25 PM
**Katsuya Eda**– Singular homology groups of one-dimensional Peano continua - 3:30 – 3:55 PM
**Janusz Przewocki**– Milnor-Thurston homology of some wild topological spaces - 4:00 – 4:25 PM
**Thilo Kuessner**– Measure homology and singular homology - 4:30 – 5:00 PM Coffee Break
- 5:00 – 5:25 PM
**Jeremy Brazas**– A characterization of the unique path lifting property for the whisker topology - 5:30 – 5:55 PM
**Ali Pakdaman**– One point unions preserve having the categorical universal covering - 6:00 – 6:25 PM
**Alvaro Sanchez-Gonzalez**– A shape topology for the universal path space

**9/19/2014 (Friday)**

- 2:30 – 2:55 PM
**Matija Cencelj**– Gropes and their fundamental groups - 3:00 – 3:25 PM
**Brendon LaBuz**– Big free groups acting on -trees - 3:30 – 3:55 PM
**Oleg Bogopolski**– Generalized presentations of groups, in particular of - 4:00 – 4:25 PM
**Andreas Zastrow**– The obstruction to contractibility of snake cones and Alternating cones - 4:30 – 5:00 PM Coffee Break

There should be many great talks in this session on some pretty wild stuff!

My own talk will address the following theoretical barrier: to what extent can the structure of the fundamental group of a path-connected metric space be understood using generalized covering maps based on unique lifting of paths and homotopies of paths? Various generalizations have proven to be particularly useful for studying one-dimensional spaces like the Menger curve (and will be used in Hanspeter Fischer’s talk) and studying free topological groups.

I’ll use some categorical ideas to show that if a subgroup corresponds to any such covering-like map (with the most primitive unique lifting properties), then it can be understood using generalized coverings constructed with the so-called “whisker topology.” The whisker construction is not specialized, it appears in most textbooks that include the classification of covering spaces, however, it doesn’t not seem to be as well-known that it has much more general applications. The obstruction to whether or not these generalized coverings exists is precisely the unique path lifting property of the endpoint projection . I’ll then characterize, for any arbitrary subgroup , the existence of unique path lifting for this map in terms of a more practical sequential closure-like condition using test maps from the following one-dimensional planar Peano continuum .

Here’s a sneak peak of my slides. Hope to see you there!