Georgia Institute of Technology College of Sciences

During the 17th Century the French priest and physicist Edme Mariotte observed that objects floating on a liquid surface can attract or repel each other, and he attempted (without success!) to develop physical laws describing the phenomenon. Initial steps toward a consistent theory came later with Laplace, who in 1806 examined the configuration of two infinite vertical parallel plates of possibly differing materials, partially immersed in an infinite liquid bath and rigidly constrained. This can be viewed as an instantaneous snapshot of an idealized special case of the Mariotte observations. Using the then novel concept of surface tension, Laplace identified particular choices of materials and of plate separation, for which the plates would either attract or repel each other. The present work returns to that two‐plate configuration from a more geometrical point of view, leading to characterization of all modes of behavior that can occur. The results lead to algorithms for evaluating the forces with arbitrary precision subject to the physical hypotheses, and embrace also some surprises, notably the remarkable variety of occurring behavior patterns despite the relatively few available parameters. A striking limiting discontinuity appears as the plates approach each other. A message is conveyed, that small configurational changes can have large observational consequences, and thus easy answers in less restrictive circumstances should not be expected.

We study the Maker-Breaker component game, played on the edge set of a regular graph. Given a graph G, the s-component (1:b) game is defined as follows: in every round Maker claims one free edge of G and Breaker claims b free edges. Maker wins this game if her graph contains a connected component of size at least s; otherwise, Breaker wins the game. For all values of Breaker's bias b, we determine whether Breaker wins (on any d-regular graph) or Maker wins (on almost every d-regular graph) and provide explicit winning strategies for both players. To this end, we prove an extension of a theorem by Gallai-Hasse-Roy-Vitaver about graph orientations without long directed simple paths. Joint work with Alon Naor.

I discuss "simple" dynamical systems on networks and examine how network structure affects dynamics of processes running on top of networks. I consider results based on "locally tree-like" and/or mean-field and pair approximations and examine when they seem to work well, what can cause them to fail, and when they seem to produce accurate results even though their hypotheses are violated fantastically. I'll also present a new model for multi-stage complex contagions--in which fanatics produce greater influence than mere followers--and examine dynamics on networks with hetergeneous correlations. (This talk discusses joint work with Davide Cellai, James Gleeson, Sergey Melnik, Peter Mucha, J-P Onnela, Felix Reed-Tsochas, and Jonathan Ward.)