Georgia Institute of Technology College of Sciences

Topological quantum field theory associates to a surface a sequence of vector spaces and to curves on the surface, sequence of operators on that spaces. It is expected that these operators are Toeplitz although there is no general proof. I will state it in some particular cases and give applications to the asymptotics of quantum invariants like quantum 6-j symbols or quantum invariants of Dehn fillings of the figure eight knot. This is work in progress with (independently) L. Charles and T. Paul.

If $X_1,...,X_n$ are a random sample from a density $f$ in $\mathbb{R}^d$, then with probability one there exists a unique log-concave maximum likelihood estimator $\hat{f}_n$ of $f$. The use of this estimator is attractive because, unlike kernel density estimation, the estimator is fully automatic, with no smoothing parameters to choose. We exhibit an iterative algorithm for computing the estimator and show how the method can be combined with the EM algorithm to fit finite mixtures of log-concave densities. Applications to classification, clustering and functional estimation problems will be discussed, as well as recent theoretical results on the performance of the estimator. The talk will be illustrated with pictures from the R package LogConcDEAD. Co-authors: Yining Chen, Madeleine Cule, Lutz Duembgen (Bern), RobertGramacy (Cambridge), Dominic Schuhmacher (Bern) and Michael Stewart

Interactions between trophic levels are influenced not only by species abundances, but also by the behavioral, life history, morphological traits of the interacting species as well. Adaptive changes in these traits can be heritable or plastic in nature and both yield phenotypic change that occurs as fast as changes in population abundances. I present how fast-slow systems theory can be used to understand the effects rapid adaptation has on community dynamics in predator-prey systems. This analysis emphasizes that heritable and plastic traits have different effects on community dynamics.

I will survey the program of realizing various versions of Floer homology as a theory of geometric cycles. This involves the description of infinite dimensional manifolds mapping to the relevant configuration spaces. This approach, which goes back to Atiyah's address at the Herman Weyl symposium, is in some ways technically simpler than the traditional construction based on Floer's version of Morse theory. In addition, it opens up the possibility of defining more refined invariants such as bordism andK-theory.