Georgia Institute of Technology College of Sciences

Wajnryb showed that the mapping class group of a surface can be generated by two elements, each given as a product of Dehn twists. We will discuss a follow-up paper by Korkmaz, "Generating the surface mapping class group by two elements." Korkmaz shows that one of the generators may be taken to be a single Dehn twist instead. He then uses his construction to further prove the striking fact that the two generators can be taken to be periodic elements, each of order 4g+2, where g is the genus of the surface.

This talk contains two parts. First I will discuss our work related to causal modeling in hybrid systems. The idea is to model jump conditions as caused by impulsive inputs. While this is well defined for linear systems, the notion of impulsive inputs poses problems in the nonlinear case. We demonstrate a viable approach based on nonstandard analysis. The second part deals with dynamical systems with delays. First I will show an application of the maximum principle to a delayed resource allocation problem in population dynamics solving a problem in the model of a bee colony cycle. Next I discuss some problems regarding causality in systems with varying delays. These problems relate to the well-posedness (existence and uniqueness) and causality of the mathematical models for physical phenomena, and illustrate why one should consider the physics first and then the mathematics. Finally, I consider the post Newtonian problem as a problem with state dependent delay. Einstein’s field equations relate space time geometry to matter and energy distribution. These tensorial equations are so unwieldy that solutions are only known in some very specific cases. A semi-relativistic approximation is desirable: One where space-time may still be considered as flat, but where Newton’s equations (where gravity acts instantaneously) are replaced by a post-Newtonian theory, involving propagation of gravity at the speed of light. As this retardation depends on the geometry of the point masses, a dynamical system with state dependent delay results, where delay and state are implicitly related. We investigate several problems with the Lagrange-Bürman inversion technique and perturbation expansions. Interesting phenomena (entrainment, dynamic friction, fission and orbital speeds) not explainable by the Newtonian theory emerge. Further details on aspects of impulsive systems and delay systems will be elaborated on by Nak-seung (Patrick) Hyun and Aftab Ahmed respectively.

We study the problem of estimation of a linear functional of the eigenvector of covariance operator that corresponds to its largest eigenvalue (the first principal component) based on i.i.d. sample of centered Gaussian observations with this covariance. The problem is studied in a dimension-free framework with its complexity being characterized by so called "effective rank" of the true covariance. In this framework, we establish a minimax lower bound on the mean squared error of estimation of linear functional and construct an asymptotically normal estimator for which the bound is attained. The standard "naive" estimator (the linear functional of the empirical principal component) is suboptimal in this problem. The talk is based on a joint work with Richard Nickl.

In a recent conjecture by Tian Yang and Qingtao Chen, it has been observedthat the log of Turaev-Viro invariants of 3-manifolds at a special root ofunity grow proportionnally to the level times hyperbolic volume of themanifold, as in the usual volume conjecture for the colored Jonespolynomial.In the case of link complements, we derive a formula to expressTuraev-Viro invariants as a sum of values of colored Jones polynomial, andget a proof of Yang and Chen's conjecture for a few link complements. Theformula also raises new questions about the asymptotics of colored Jonespolynomials. Joint with Effie Kalfagianni and Tian Yang.