Mathematics

Macroecology for microbes: modeling patterns of phylogenetic diversity

James O'Dwyer (UIUC Plant Biology)

Abstract: Macroecological patterns are aggregated over large numbers of individuals, and often display a kind of universal behavior across different systems, independent of differences in their underlying ecological processes. Physicists benefit from several principles which can underly universal behavior: for example, laws of large numbers. Can we identify similar principles in ecology, and understand which phenomena are universal and which are more contingent on mechanism? In this talk I will put these questions in context, and present new models and data which shed light on when and why ecological systems display emergent patterns.

Complexity of finitely generated residually finite groups

Alexei Myasnikov (Stevens Institute of Technology)

Abstract: Finitely presented residually finite groups are usually thought of as nice from the algorithmic view-point, in particular, they have decidable word problem. In this talk I will address the following general questions for such groups G: how large could be the Dehn function of G? How large could be the gap between the complexity of the word problem and the Dehn functions of G? What is the time complexity of the classical McKinsey algorithm for the word problem in G (this is the only known uniform algorithm for the word problem in such groups)? How large could be the depth functions in G? The depth function measures how deep one has to go into finite index subgroups to separate a non-trivial element of a given length in G from the identity. These are joint results with O. Kharlampovich and M. Sapir. I will also discuss finitely generated recursively presented residually finite groups with really strange algorithmic properties, so called Dehn monsters. To build them we need Golod-Shafarevich construction and a forcing-type argument from logic. This is based on joint results with D. Osin and B. Khoussainov.

Some arithmetic properties of numbers of the form $\left\lfloor p^c \right\rfloor$

Zhenyu Guo (University of Missouri - Columbia)

Abstract: For fixed $c \in (1, 149/87)$, we derive an asymptotic formula for the number of primes $p \le x$ such that $\left\lfloor p^c \right\rfloor$ is squarefree, where $\left\lfloor . \right\rfloor$ is the floor function. We also prove that there are infinite many $p$ such that $\left\lfloor p^c \right\rfloor$ is an almost prime.

The Broken Circuit Complex and the Orlik-Terao Algebra of a Hyperplane Arrangement

Dinh Le Van (Universität Osnabrück)

Abstract: The Orlik-Terao algebra of an arrangement is a commutative analogue of the well-studied Orlik-Solomon algebra, which is the cohomology ring of the arrangement complement. Recently, it has attracted significant attention, mainly because it encodes subtle information missing in the Orlik-Solomon algebra: it was used by Schenck-Tohaneanu to characterize 2-fomality, a non-combinatorial property which is necessary for the arrangement to be free and for the complement space to be aspherical. Nevertheless, the Orlik-Terao algebra is a deformation of a combinatorial object, the broken circuit algebra. Exploiting this strong connection between the two algebras, I will give in this talk characterizations for the following properties of each of them: having a linear resolution, being a complete intersection, and being Gorenstein. If time permits, a generalization of Schenck-Tohaneanu's result will also be discussed.

Ancient solutions

Natasa Sesum (Rutgers)

Abstract: I will discuss ancient solutions in the context of the mean curvature flow, the Ricci flow and the Yamabe flow. I will discuss the classification result in the Ricci flow, construction result of infinitely many ancient solutions in the Yamabe flow. In the last part of the talk I will mention the most recent result about the unique asymptotics of non-collapsed ancient solutions to the mean curvature flow which is a joint work with Daskalopoulos and Angenent.