Georgia Institute of Technology College of Sciences

The tropical cycle associated to a subvariety of a torus is the support of a weighted polyhedral complex that that records information about the original variety and its compactifications. In a recent preprint Jeff and Noah Giansiracusa introduced a notion of scheme structure for tropical varieties, and showed that the tropical variety as a set is determined by this tropical scheme structure. I will outline how to also recover the tropical cycle from this information. This involves defining a variant of Grobner theory for congruences on the semiring of tropical Laurent polynomials. The lurking combinatorics is that of valuated matroids. This is joint work with Felipe Rincon.

From bird flocks to ungulate herds to fish schools, nature abounds with examples of biological aggregations that arise from social interactions. These interactions take place over finite (rather than infinitesimal) distances, giving rise to nonlocal models. In this modeling-based talk, I will discuss two projects on insect swarms in which nonlocal social interactions play a key role. The first project examines desert locusts. The model is a system of nonlinear partial integrodifferential equations of advection-reaction type. I find conditions for the formation of an aggregation, demonstrate transiently traveling pulses of insects, and find hysteresis in the aggregation's existence. The second project examines the pea aphid. Based on experiments that motion track aphids walking in a circular arena, I extract a discrete, stochastic model for the group. Each aphid transitions randomly between a moving and a stationary state. Moving aphids follow a correlated random walk. The probabilities of motion state transitions, as well as the random walk parameters, depend strongly on distance to an aphid’s nearest neighbor. For large nearest neighbor distances, when an aphid is isolated, its motion is ballistic and it is less likely to stop. In contrast, for short nearest neighbor distances, aphids move diffusively and are more likely to become stationary; this behavior constitutes an aggregation mechanism.

Emory University, Georgia Tech and Georgia State University, with support from the National Science Foundation and the National Security Agency, will continue the series of mini-conferences and host a series of 9 new mini-conferences from 2013-2016. The first new and 10th overall of these mini-conferences will be held at Emory University on November 2-3, 2013. The conferences will stress a variety of areas and feature one prominent researcher giving 2 fifty minute lectures and 4 outstanding researchers each giving one fifty minute lecture. There will also be several 25 minute lecturers by younger reseachers or graduate students.

The problem of extending results in extremal combinatorics to sparse random and pseudorandom structures has attracted the attention of many researchers in the last decades. The breakthroughs due to several groups in the last few years have led to a better understanding of the subject, however, many questions remain unsolved. After a short introduction into this field we shall focus on some results in extremal (hyper)graph theory and additive combinatorics. Along the way some open problems will be given.