Order-compactifications of totally ordered spaces were described by Blatter (J Approx Theory 13:56–65, 1975) and by Kent and Richmond (J Math Math Sci 11(4):683–694, 1988). Their results generalize a similar characterization of order-compactifications of linearly ordered spaces, obtained independently by Fedorčuk (Soviet Math Dokl 7:1011–1014, 1966; Sib Math J 10:124–132, 1969) and Kaufman (Col...

In this paper we explore the challenges surrounding searching effectively in problems with preferences. These problems are characterized by a relative abundance of goal states: at one extreme, if every goal is soft, every state is a goal state. We present techniques for planning in such search spaces, managing the sometimes-conflicting aims of intensifying search around states on the open list ...

Hodge theory is a beautiful synthesis of geometry, topology, and analysis, which has been developed in the setting of Riemannian manifolds. On the other hand, spaces of images, which are important in the mathematical foundations of vision and pattern recognition, do not fit this framework. This motivates us to develop a version of Hodge theory on metric spaces with a probability measure. We bel...

We describe a compositional approach to Craig interpolation based on the heuristic that simpler proofs of special cases are more likely to generalize. The method produces simple interpolants because it is able to summarize a large set of cases using one relatively simple fact. In particular, we present a method for finding such simple facts in the theory of linear rational arithmetic. This make...

Large scale phenomena in homotopy theory (Jordan Hall 124) Paul Goerss (Northwestern University) In the late 1940s it wasn’t clear which was the harder problem: the cohomology of Eilenberg-MacLane spaces or the stable homotopy groups of spheres. After the Cartan-Serre work on Eilenberg-MacLane spaces in the early ’50s, people then optimistically set about computing the homotopy groups of sphere...