May

21

2018

Starting with results by Moser, Nash, and DeGiorgi in the late 50's-early 60's, the regularity theory of elliptic and subelliptic equations with rough coefficients has been thoroughly developed. There is also a smaller theory still in its infancy of infinitely degenerate elliptic equations with smooth data, beginning with work of Fedii and Kusuoka-Strook, and continued by Morimoto and Christ. In this talk I will present our results that initiate a regularity theory in the context of equations that are both infinitely degenerate elliptic and have rough data.

May

22

2018

We report on a preprint of G. Panti. Panti gives a direct proof using continued fraction-type maps which decrease heights (for appropriate sets of algebraic numbers) to reprove the result that any triangle group whose invariant trace field is a quadratic number field acts transitively on the elements of this field. Compare this result with the fact that the rational numbers form a single orbit of the action of PSL(2,Z). Earlier proofs have used a mixture of techniques, in particular results from the theory of translation surfaces.

May

22

2018

The Tokunaga condition is an algebraic rule that provides a detailed description of the branching structure in a self-similar tree. Despite a solid empirical validation and practical convenience, the Tokunaga condition lacks a theoretical justification. Such a justification is suggested here. We define a geometric branching processes *G(s)* that generates self-similar rooted trees. We establish the equivalence between the invariance of *G(s)* with respect to a time shift and a one-parametric version of the Tokunaga condition.

May

23

2018

May

23

2018

Agrobacteria represent a group of Gram-negative bacteria that have the unusual ability to execute inter-kingdom gene transfer. This process is dependent on oncogenic plasmids and is critical for the bacteria to establish a nutrient-replete niche. This process has also been modified as a biotechnology tool for addressing questions in fundamental plant biology and for adding traits to crop plants. However, few lineages of bacteria and plasmids have been studied and the variation in traits that could affect virulence is poorly understood.