Sunflower Theorems and Convex CodesAlgebra & Discrete Mathematics
|Speaker:||Amzi Jeffs, University of Washington|
|Start time:||Mon, Apr 6 2020, 1:10PM|
In the 1970s neuroscientists O'Keefe and Dostrovsky made a groundbreaking experimental observation: neurons called "place cells" in a rat's hippocampus were active in a convex subset of the animal's environment, and thus encoded a cognitive map of the environment. In 2013, Curto et al. introduced a mathematical model of place cells and began the task of classifying the combinatorial "convex codes" that arise in this model. A key question in this field is the following: given a convex code on n neurons, what is the smallest dimension of Euclidean space in which one can find a realization using convex open sets? We will introduce some new discrete geometry theorems in the spirit of Helly and Tverberg. We will use these theorems to prove that, surprisingly, the smallest dimension mentioned above may be exponentially large in terms of the number of neurons n.
Please join us for this virtual talk at https://ucdavisdss.zoom.us/j/842986080 (room opens at 1pm)