Georgia Institute of Technology College of Sciences

The central curve of a linear program is an algebraic curve specified by a hyperplane arrangement and a cost vector. This curve is the union of the various central paths for minimizing or maximizing the cost function over any region in this hyperplane arrangement. I will discuss the algebraic properties of this curve and its beautiful global geometry, both of which are controlled by the corresponding matroid and hyperplane arrangement.

Although fungi are the most diverse eukaryotic organisms, we have only a very fragmentary understanding of their success in so many niches or of the processes by which new species emerge and disperse. I will discuss how we are using math modeling and perspectives from physics and fluid mechanics to understand fungal life histories and evolution: #1. A growing filamentous fungi may harbor a diverse population of nuclei. Increasing evidence shows that this internal genetic flexibility is a motor for diversification and virulence, and helps the fungus to utilize nutritionally complex substrates like plant cell walls. I'll show that hydrodynamic mixing of nuclei enables fungi to manage their internal genetic richness. #2. The forcibly launched spores of ascomycete fungi must eject through a boundary layer of nearly still air in order to reach dispersive air flows. Individually ejected microscopic spores are almost immediately brought to rest by fluid drag. However, by coordinating the ejection of thousands or hundreds of thousands of spores fungi, such as the devastating plant pathogen Sclerotinia sclerotiorum are able to create a flow of air that carries spores across the boundary layer and around any intervening obstacles. Moreover the physical organization of the jet compels the diverse genotypes that may be present within the fungus to cooperate to disperse all spores maximally.

Consider a hyperbolic basic set of a smooth diffeomorphism. We are interested in the transitivity of Holder skew-extensions with fiber a non-compact connected Lie group. In the case of compact fibers, the transitive extensions contain an open and dense set. For the non-compact case, we conjectured that this is still true within the set of extensions that avoid the obvious obstructions to transitivity. Within this class of cocycles, we proved generic transitivity for extensions with fiber the special Euclidean group SE(2n+1) (the case SE(2n) was known earlier), general Euclidean-type groups, and some nilpotent groups. We will discuss the "correct" result for extensions by the Heisenberg group: if the induced extension into its abelinization is transitive, then so is the original extension. Based on earlier results, this implies the conjecture for Heisenberg groups. The results for nilpotent groups involve questions about Diophantine approximations. This is joint work with Ian Melbourne and Viorel Nitica.

The 5th in a series of 9 mini-conferences features Jacob Fox as the prominent researcher who will give 2 fifty-minute lectures and 4 other outstanding researchers each giving one fifty-minute lecture. There will also be several 25-minute lectures by young researchers and graduate students. The lectures will begin at 1 PM on Saturday, February 25 and conclude at at noon on Sunday, February 26.To register, please send an email to rg@mathcs.emory.edu and for complete details, see the website. Registration is free.