In Fall 2018 the Topology, Geometry, and Algebra Seminar is held 1:00 PM - 2:00 pm in Rhodes Tower 1516 on Fridays. To download notes from a given day's lecture, please click on the title.
|October 12||Greg Lupton (CSU)||Digital Homotopy Theory||
I will describe work in progress, joint with John Oprea and Nick Scoville. An n-dimensional digital image is a finite subset of the integer lattice in R^n, together with an adjacency relation. For instance, a 2-dimensional digital image is an abstraction of an actual digital image consisting of pixels. Our work consists of developing notions and techniques from homotopy theory in the setting of digital images.
In an extensive literature, a number of authors have introduced concepts from topology into the study of digital images. But some of these notions, as they appear in the literature, do not seem satisfactory from a homotopy point of view. Indeed, some of the constructs most useful in homotopy theory, such as cofibrations and path spaces, are absent from the literature. Working in the digital setting, we develop some basic ideas of homotopy theory, including cofibrations and path fibrations, in a way that seems more suited to homotopy theory. We illustrate how our approach may be used, for example, to study Lusternik-Schnirelmann category and topological complexity in a digital setting. One future goal is to develop a characterization of a "homotopy circle" (in the digital setting) using the notion of topological complexity. This is with a view towards recognizing circles, and perhaps other features, using these ideas. The talk(s) will include a survey of the basics on topological notions in the setting of digital images.
|October 19||Greg Lupton (CSU)||Digital Homotopy Theory||Continued|
|October 26||Greg Lupton (CSU)||Digital Homotopy Theory||Continued|
|November 9||Viji Thomas (CSU)||A Group theoretical construction||We will introduce a group theoretical construction and show its relation to the Schur Multiplier (Second Homology group with coefficients in the integers), second stable homotopy group of the Eilenberg Maclane space and the Bogomolov multiplier. If time permits we will show its relation to a conjecture of Schur and also its relation to Noethers rationality problem.|