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I will present a progress report on some work joint with John Oprea and Nick Scoville. ÌýAn n-dimensional digital image is a finite subset of the integer lattice in R^n, together with an adjacency relation. ÌýFor instance, a 2-dimensional digital image is an abstraction of an actual digital image consisting of pixels. ÌýOur work consists of developing notions and techniques from homotopy theory in the setting of digital images. Ìý

In an extensive literature, Ìýa number of authors have introduced concepts from topology into the study of digital images. Ìý But some of these notions, as they appear in the literature, do not seem satisfactory from a homotopy point of view. ÌýIndeed, some of the constructs most useful in homotopy theory, such as cofibrations and path spaces, are absent from the literature. ÌýWorking 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. ÌýI will indicate how our approach may be used, for example, to study Lusternik-Schnirelmann category in a digital setting. ÌýOne future goal is to develop a characterization of a "homotopy circle" (in the digital setting) using the notion of topological complexity. ÌýThis is with a view towards recognizing circles, and perhaps other features, using these ideas. ÌýThis talk will introduce someÌýof the basics of topological notions in the setting of digital images, and will not require any specialized background.

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