Colloquium at Math Department at WVU, Spring 2020

Date: Wednesday, March 25, 4-4:50pm. Preceded with refreshments, 3:30-4:00pm, in the math faculty lounge.
Location: 315 Armstrong Hall
Speaker: Professor Jerzy Wojciechowski, Department of Mathematics, West Virginia University
Title: Compactifications of infinite graphs
Abstract:

Any graph can be considered to be the topological space obtained by taking a copy of the closed interval [0,1] for each edge and using the quotient space construction to identify endpoints of those intervals that correspond to the same vertex. A compactification of a graph G is a compact Hausdorff space containing G as a dense subspace.

Locally finite graphs can be compactified by adding so called ends, introduced by Halin in 1964. The resulting compact space is equivalent to a special case of the Freudental compactification defined for any locally compact Hausdorff space. If the graph G is not locally finite, adding the ends is not sufficient to compactify G. The longstanding quest to decide what beside the ends has to be added to the graph to obtain a compactification has been first solved by Diestel in 2018, who proved that a graph can be compactified by adding its \aleph_0-tangles also called the infinite-order tangles.

The notion of a tangle was introduced by Robertson and Seymour in 1991. Infinite-order tangles can be equivalently defined as inverse limits of a certain inverse limit system obtained by considering the components of the graph obtained after deleting a finite number of vertices. If the graph is locally finite and connected, the infinite-order tangles correspond to its ends.

Recently, Kurkofka and Pitz found a relationship between the tangle compactification of a graph and its Stone-Cech compactification. They also showed that a graph can be compactified using the ends and the so called critical vertex sets and related thus obtained compactification to the compactification defined by Diestel.