Dispersing Obnoxious Facilities on a Graph

We study a continuous facility location problem on a graph where all edges have unit length and where the facilities may also be positioned in the interior of the edges. The goal is to position as many facilities as possible subject to the condition that any two facilities have at least distance $$\delta$$ δ from each other. We investigate the complexity of this problem in terms of the rational parameter $$\delta$$ δ . The problem is polynomially solvable, if the numerator of $$\delta$$ δ is 1 or 2, while all other cases turn out to be NP-hard.

Particles and Spacetime Diagrams

This chapter begins the process of making a 3D interpretation of the plaid model. The idea is to group together certain of the light points and think of them as instances of 1-dimensional worldlines rather than as a succession of points. It fixes an even rational parameter p/q and uses the notation from Section 1.2, i.e., ω‎ = p + q. Section 4.2 explains a different notion of adjacency for the ω‎ × ω‎ blocks, dividing up the plaid model. Section 4.3 says what it means for two horizontal light points in remotely adjacent blocks to be different instances of the same particle. Section 4.4 does the same for the vertical particles. Section 4.5 shows a few pictures of spacetime diagrams and discuss their symmetries. Section 4.6 proves a technical lemma, the Bad Tile Lemma, which is very similar in spirit to Theorem 1.4.

Connection to the Truchet Tile System

This chapter fixes some even rational parameter p/q as usual. It shows that the pixelated spacetime slices of capacity 2p are combinatorially equivalent to certain of the tilings from P. Hooper's Truchet tile system [H]. Section 7.2 describes the Truchet tile system. Section 7.3 states the main result, the Truchet Comparison Theorem. One can view the Truchet Comparison Theorem as a computational tool for understanding some of the pixelated spacetime diagrams. Section 7.4 uses the Truchet Comparison Theorem to get more information about the surface Σ‎(p/q) from Corollary 6.6. Section 7.5 proves a curious result from elementary number theory which underlies the Truchet Comparison Theorem. Section 7.6 puts together the ingredients and proves the Truchet Comparison Theorem.

Wall-crossing in genus zero Landau–Ginzburg theory

We study a family of moduli spaces and corresponding quantum invariants introduced recently by Fan–Jarvis–Ruan. The family has a wall-and-chamber structure relative to a positive rational parameter ϵ. For a Fermat quasi-homogeneous polynomial