Georgia Institute of Technology College of Sciences

The point-pushing subgroup of the mapping class group of a surface with a marked point can be considered topologically as the subgroup that pushes the marked point about loops in the surface. Birman demonstrated that this subgroup is abstractly isomorphic to the fundamental group of the surface, \pi_1(S). We can characterize this point-pushing subgroup algebraically as the only normal subgroup inside of the mapping class group isomorphic to \pi_1(S). This uniqueness allows us to recover a description of the outer automorphism group of the mapping class group.

Biology is becoming increasingly quantitative, with large genomic datasets being curated at a rapid rate. Sound mathematical modeling as well as data science approaches are both needed to take advantage of these newly available datasets. I will describe two projects that span these approaches. The first is a Markov chain model of naturalselection acting at two scales, motivated by the virulence-transmission tradeoff from pathogen evolution. This stochastic model, under a natural scaling, converges to a nonlinear deterministic system for which we can analytically derive steady-state behavior. This analysis, along with simulations, leads to general properties of selection at two scales. The second project is a bioinformatics pipeline that identifies gene copy number variants, currently a difficult problem in modern genomics. This quantificationof copy number variation in turn generates new mathematical questionsthat require the type of probabilistic modelling used in the first project.

I will discuss applications of the theory of gradient flows to the dynamics of evolution equations. First, I will review how to obtain convergence rates towards equilibrium in the strictly convex case. Second, I will introduce a technique developed in collaboration with Moon-Jin Kang that allows one to obtain convergence rates towards equilibrium in some situations where convexity is not available. Finally, I will describe how these techniques were useful in the study of the dynamics of homogeneous Vicsek model and the Kuramoto-Sakaguchi equation. The contributions on the Kuramoto-Sakaguchi equation are based on a joint work with Seung-Yeal Ha, Young-Heon Kim, and Jinyeong Park. The contributions to the Vicsek model are based on works in collaboration with Alessio Figalli and Moon-Jin Kang.

We show that multilinear dyadic paraproducts and Haar multipliers, as well as their commutators with locally integrable functions, can be pointwise dominated by multilinear sparse operators. These results lead to various quantitative weighted norm inequalities for these operators. In particular, we introduce multilinear analog of Bloom's inequality, and prove it for the commutators of the multilinear Haar multipliers.

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.

The multi-server queue with non-homogeneous Poisson arrivals and customer abandonment is a fundamental dynamic rate queueing model for large scale service systems such as call centers and hospitals. Scaling the arrival rates and number of servers gives us fluid and diffusion limits. The diffusion limit suggests a Gaussian approximation to the stochastic behavior. The fluid mean and diffusion variance can form a two-dimensional dynamical system that approximates the actual transient mean and variance for the queueing process. Recent work showed that a better approximation for mean and variance can be computed from a related two-dimensional dynamical system. In this spirit, we introduce a new three-dimensional dynamical system that estimates the mean, variance, and third cumulant moment. This surpasses the previous two approaches by fitting the number in the queue to a quadratic function of a Gaussian random variable. This is based on a paper published in QUESTA and is joint work with Jamol Pender of Cornell University.

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.

The first meeting of our Spring 2017 professional development seminar for postdocs and other interested individuals (such as advanced graduate students). A discussion of the triumvirate of faculty positions: research, teaching, and service.

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.