Time & location: All talks are on Thursday in Gibson 414 at 3:30pm unless otherwise noted. Refreshments in Gibson 426 after the talk.
Comments indicating vacations, special lectures, or change in location or time are made in green.
Discrete dynamics is the study of iteration. A primary objective of dynamics is the classification of points in a set S according to their orbits under repeated application of a self-map f : S --> S. Classically, S is taken to be R^n or C^n, and real and complex dynamics are mature and thriving fields of study. But for a number theorist, it is natural to take S to be a set of arithmetic interest such as Z or Q. The past 25 years has seen the development of the field of Arithmetic Dynamics, in which one studies dynamical analogues of classical results in number theory and arithmetic geometry. Here are two illustrative problems: Let f(z) in Q(z) be a rational function. (1) There are always infinitely many complex numbers with finite forward orbit under iteration of f, but how many of these complex number can be rational numbers? (2) If we take a rational starting point alpha in Q, when is it possible for infinitely many points in the orbit of alpha to be integers? In this talk I will discuss these and other problems in arithmetic dynamics. As is typical in number theory, there are many questions that are easy to state, but difficult to solve.
Basak Gurel - University of Central Florida
A general, but not universal, feature of Hamiltonian dynamical systems is that such systems tend to have numerous periodic orbits. In fact, as the Conley conjecture asserts, for a broad class of closed symplectic manifolds, every Hamiltonian diffeomorphism (the time-one map of a Hamiltonian flow) has infinitely many simple periodic orbits. On the other hand, it is easy to see that the conjecture fails for a general symplectic manifold such as the two-dimensional sphere or the Euclidean space.
One variant of the Conley conjecture applicable to such manifolds, inspired by a celebrated theorem due to Franks, asserts that a Hamiltonian diffeomorphism with more than necessary fixed points has infinitely many simple periodic orbits. Here the threshold is usually interpreted as a lower bound arising from some version of the Arnold conjecture; for instance, it is n+1 for the n-dimensional complex projective space. For the two-sphere, the assertion is a special case of Franks' theorem: every area preserving homeomorphism of the sphere with more than two fixed points has infinitely many periodic points.
In this talk we will discuss various aspects of the existence question for periodic orbits of Hamiltonian flows, focusing on recent results obtained by Floer homological methods in the realm of higher dimensional generalizations of Franks'.
Karen Kafadar - Indiana University
Statistics has played a key role in the development and validation of forensic methods, as well as in the inferences (conclusions) obtained from forensic evidence.Further, statisticians have been important contributors to many areas of science, such as chemistry (chemometrics), biology (genomics), medicine (clinical trials), and agriculture (crop yield), leading to valuable advances that extend to multiple fields (spectral analysis, penalized regression, sequential analysis, experimental design).The involvement of statistics specifically in forensic science has demonstrated its value in the development of DNA "fingerprinting," assessment of bullet lead evidence, and the significance of findings in the U.S. anthrax investigations. I will discuss the statistical issues involved in these three situations and then suggest ways in which statisticians and mathematicians together can use their expertise to strengthen forensic evidence (experimental design, estimating error rates, developing other approaches based on less subjective interpretations, etc.), thereby raising the level of confidence in the forensic system.
Anna Lachowska - Yale University
The small quantum group u(g)_l is a finite dimensional Hopf algebra associated to the semisimple Lie algebra g and a primitive l-th root of unity. It was defined by G. Lusztig as the quantum analog of the finite dimensional (reduced) enveloping algebra in positive characteristic, and used to define quantum topological invariants of 3-manifolds (e.g. Reshetikhin-Turaev and Hennings-Kauffman-Radford inariants). These connections provide motivations for the study of the structure and representations of u(g)_l.
I will describe the known results on the structure of the center of the small quantum group, as well as possible implications and further directions of study.
The talk is partially based on a joint work with R. Bezrukavnikov.
Cory Hauck - Oak Ridge National Laboratory
Kinetic theory is a broad discipline which seeks to describe the macroscopic behavior of particle systems based on an appropriate coarse graining of the dynamics at the microscopic level. Examples of such systems include dilute gases, plasmas, neutron and photon transport, and electron transport in materials. In all of these applications, the kinetic transport equation shares a remarkably similar mathematical structure, featuring advection in phase space and collision operators that model particle interactions. In this talk, I will review the basic formalism of kinetic theory, discuss some of the main challenges of kinetic-based simulation, and finally present some of my own related work in the area.
Speaker - Institution
Yi Jiang - Georgia State University
Extracellular matrix (ECM), a fibrous material that forms a network in a tissue, significantly affects many aspects of cellular behavior, including cell movement and proliferation. Transgenic mouse tumor studies indicate that excess collagen, a major component of ECM, enhances tumor formation and invasiveness. Moreover, cell interactions with the collagen matrix result in aligned fibers that facilitate cell invasion. However, the underlying mechanisms are unclear since the properties of ECM are complex, with diverse topographies and mechanical properties depending on various biophysical parameters. We have developed a three-dimensional elastic computational fiber network model, and parameterized it with in vitro collagen tensile experiments. Using this model, we simulate mechanical testing of fiber networks and examine the mechanical properties of fiber networks with varying density, alignment, and crosslinking. The computational model and simulation results can fill in the gap between microscopic single collagen fiber studies and macroscopic collagen gel studies. This model is the first step toward a fully biomechanical cell-matrix interaction model for changes in matrix organization during cell migration and tumor invasion.
Laura Matusevich - Texas A&M University
Vadim Zipunnikov - Johns Hopkins School of Public Health
We develop a flexible framework for modeling high-dimensional functional and imaging data observed longitudinally. The approach decomposes the observed variability of high-dimensional observations measured at multiple visits into three additive components: a subject-specific functional random intercept that quantifies the cross-sectional variability, a subject-specific functional slope that quantifies the dynamic irreversible deformation over multiple visits, and a subject-visit specific functional deviation that quantifies exchangeable or reversible visit-to-visit changes. The proposed method is very fast, scalable to studies including ultra-high dimensional data, and can easily be adapted to and executed on modest computing infrastructures. The method is applied to the longitudinal analysis of diffusion tensor imaging (DTI) data of the corpus callosum of multiple sclerosis (MS) subjects. The study includes 176 subjects observed at 466 visits. For each subject and visit the study contains a registered DTI scan of the corpus callosum at roughly 30,000 voxels.
Karl Dilcher - Dalhousie University, Halifax, Canada
We derive new identities for a polynomial analogue of the Stern sequence and define two subsequences of these polynomials. We obtain various properties for these two interrelated subsequences which have 0-1 coefficients and can be seen as extensions or analogues of the Fibonacci numbers. We also define two analytic functions as limits of these sequences. As an application we obtain evaluations of certain finite and infinite continued fractions whose partial quotients are doubly exponential. In a case of particular interest, the set of convergents has exactly two limit points. (Joint work with K. B. Stolarsky).
Dan Edidin - University of Missouri
An old problem in signal processing is to recover a signal from a collection of intensity measurements. The mathematical difficulty is that information about phase must be retrieved from a priori phaseless information. I explain how techniques from algebraic geometry can be used to show that signal reconstruction is possible from surprisingly few intensity measurements.
Chongchun Zeng - Georgia Tech
In this talk, we start with the mathematical theory of wind-generated water waves in the framework of the interface between two incompressible inviscid fluids under the influence of gravity. This entails the careful study of the stability of the shear flow solutions to the interface problem of the two-phase Euler equation. Based on a rigorous derivation of the linearized equations about an arbitrary steady solution, we obtained rigorously the linear instability criterion of Miles due to the presence of the critical layer in the steady shear flows. Our analysis is valid even in the presence of surface tension and a vortex sheet (discontinuity in the tangential velocity across the air--sea interface). We are thus able to give a unified equation including the Kelvin--Helmholtz and quasi-laminar models of wave generation put forward by Miles. While the rigorous nonlinear instability proof is still missing for this problem, we are aiming at a stronger statement -- constructing local unstable manifolds of the full nonlinear system of the interface problem of the Euler equation. In the second half of the talk, we discuss the unstable manifolds of steady states of the Euler equation on fixed domains. Suppose the linearized equation at a steady state $v_*$ has an exponential dichotomy with a finite dimensional unstable subspace.
By rewriting the Euler equation as an ODE on an infinite dimensional manifold in $H^k$, $k>\frac n2 +1$, the unstable manifold of $v_*$ is constructed under certain conditions on the Lyapunov exponents of the vector field $v_*$. In particular, this leads to the desired nonlinear instability of $v_*$ in the sense that small $H^k$ perturbations can lead to $L^2$ derivation of the solutions.
Speaker - Institution
Debraj Chakrabarti - Central Michigan University
One of the most famous theorems of mathematics is the Riemann Mapping Theorem, first stated in a special case by Riemann in 1851 in his PhD thesis. As early as 1906 Poincaré showed that there is no simple analog of the RMT in several variables. Subsequently, the problem of holomorphic mapping between domains in several dimensions was studied intensively, and a breakthrough was achieved in 1976 by C. Fefferman when he showed that biholomorphic maps of strongly pseudo-convex domains extend smoothly up to the closure. After reviewing these classical results, I will talk about some recent advances which relate extension results of the Fefferman type with with non-existence of maps results of the Poincaré type.
Mathematics Department, 424 Gibson Hall, New Orleans, LA 70118 504-865-5727 firstname.lastname@example.org