Robert G. Niemeyer

Assistant Professor

About Me

Photograph of Robert G. Niemeyer

I am a term assistant professor in the Department of Mathematics & Statistics at the University of Maine in Orono, ME and a member of CompuMAINE. My research interests mostly reside in the intersection of fractal geometry and mathmatical billiards. Recent projects have allowed me to branch out into number theory and physically-inspired billard problems. If you would like a copy of my research statement, please send me an e-mail.

After receiving my PhD from the University of California, Riverside under the supervision of M. L. Lapidus, I went to the Unviersity of New Mexico as an NSF MCTP Postdoctoral Fellow (2012-2015). Now at the University of Maine, I can say I've certainly seen everything in between, from the desert to the green.

In the fall of 2016, I will be applying for jobs, traveling to conferences and teaching real analysis and a general education course on the nature and language of mathematics. Taking the opportunity to advance my skills in the classroom, I've been learning to use iClickers and flip the classroom. Additionally, I've been taking advantage of online forums (Piazza and Blackboard) to accommodate students with difficult work schedules.

Curriculum Vitae Teaching Statement Diversity Statement

Robert G. Niemeyer
University of Maine
Dept. of Mathematics & Statistics
5752 Neville Hall
Orono, Maine 04469 Office: 332
niemeyer "at" math.umaine.edu

Research

I am a dynamicist with specific research interests in topological dynamical systems, both pure and applied. Below I give brief details about my own work and projects with collaborators. The central theme of my research in dynamical systems is computer experiment as a means of supporting and proving conjectures.

C. Johnson and I are currently investigating the behavior of an infinite interval exchange transformation on a fractal flat surface. Related to this is another joint project concerning the existence of a saddle connection on a fractal billiard table that connects two elusive points of the fractal billiard table. The definition of a fractal flat surface is still not concretely stated, but in specific instances can be formally defined, this being an outgrowth of the description of an infinite interval exchange transformation.

My joint work with Joe P. Chen on a project that extends the work of Jeremy Tyson and Estibalitz Durand-Cartagena [Du-CaTy] has recently appeared in the Journal of Mathematical Analysis & Applications. The project involves describing periodic orbits of a self-similar Sierpinski carpet billiard table.

I have worked with M. L. Lapidus on various papers on the topic of fractal billiards. Specifically, a recent article with M. L. Lapidus and R. L. Miller consists of determining periodic orbits of the T-fractal billiard table. An eventual goal is understanding whether or not there are equidistributed orbits on the T-fractal billiard table. An initial step in this direction is to understand whether or not every orbit with an irrational direction is recurrent in the T-fractal billiard.

Mathematical Andreev reflection is retro-reflection with parity. By this, we mean that when a pointmass intersects a portion of a billiard table not obeying the standard Law of Reflection, the pointmass returns upon the incoming path.

In addition to this, the ball is assigned a parity and such parity changes with each intersection with said side. The side for which the billiard ball experiences Andreev reflection is call the Andreev subset of the billiard table boundary. We begin building a mathematical foundation for dynamics in an Andreev billiard table, as it is discussed within the context of physics. To such end, we have begun simulating billiard trajectories in a 2D rectangular billiard table with the base of the billiard table being the Andreev subset. Andreev reflection, as it is discussed above, is meant to ideally model the dynamics of an electron in a nanowire lying on a superconductor plate. Elementary results in mathematical billiards are quickly applied to classify dynamics in such a model. We also perturb the boundary of the 2D model so as to introduce explanations for observed behavior not fitting existing models for electron-hole dynamics in a nanowire.

In collaboration with R. E. Niemeyer, we examine the effect of particular parameter changes in a system of ordinary differential equations modeling the interaction between people of a nation-state, the amount of internal strife and the amount of resources within the system.

Peter Turchin et al, construct a system of ordinary differential equations that accurately models the rise and fall of particular empires. However, they do not fully examine the robustness of their model, nor do they fully justify the simplification of a 3D system to a 2D system. While their work is empirically sound, we argue that one does not need to reduce the complexity of the system by collapsing the state resources variable S and the internal strife variable W into a new variable I, called the instability index. If one maintains the conjecture that all three variables are important in describing the rise and fall of an empire, then other dynamical systems may provide better explanatory power and insight into how to predict the effect of diminishing resources in a city-state and an increase in internal strife.

Additional work has focused on computational modeling of number-theoretic problems so as to gain insight into the behavior of particular systems. Specifically, a past project has focused on understanding the Collatz conjecture. Recent joint work with D. Bradley, A. Khalil and E. Ossanna has focused on finding a rigorous connection between a matrix representation of Pascal's triangle modulo a square-free integer and a planar representation of such a matrix, with the immediate goal being to calculate the box-counting of said planar representation. For the sake of brevity, further discussion on these topics will not be given here.

Publications · Papers · Notes

Publications

  • Towards the Koch snowflake fractal billiard--Computer experiments and mathematical conjectures (with M. L. Lapidus), in "Gems in Experimental Mathematics", T. Amdeberhan, L. A. Medina and V. H. Moll (Eds.), Contemporary Mathematics, Amer. Math. Soc., Providence, R. I., 517 (2010), 231-263.
  • Sequences of compatible periodic hybrid orbits of prefractal Koch snowflake billiards (with M. L. Lapidus), Discrete and Continuous Dynamical Systems Ser. A Vol 33, No 8 (2013), 3719-3740.
  • The current state of fractal billiards (with M. L. Lapidus), in: ``Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II: Fractals in Applied Mathematics'' (D. Carfi, M. L. Lapidus, M. van Frankenhuijsen, E. P. J. Pearse, eds.), Contemporary Mathematics, Amer. Math. Soc., Providence, RI, 601 (2013), pp. 251-288.
  • Periodic billiard orbits of self-similar Sierpinski carpets (with J. P. Chen), J. Math. Anal. Appl. 416 (2014), 969--994.
  • Nontrivial paths and periodic orbits of the T-fractal billiard (with M. L. Lapidus and R. L. Miller), Nonlinearity 29 (2016), 2145--2172.

Papers Submitted for Publication

  • Properties of the flow on a polygonal Andreev billiard, 2016.

Papers in Preparation/Progress

  • An interval exchange transformation on a fractal flat surface (with C. C. Johnson), 2016.
  • A fractal flat surface for the Koch snowflake fractal billiard (with C. C. Johnson), 2016.
  • A computational investigation of a particular nontrivial path (with C. C. Johnson), 2016.
  • Generating upward sweeps in population using the Turchin--Korotayev model (with R. E. Niemeyer), 2016.
  • Fractal dimension of residue sets within Pascal's Triangle under square-free moduli (with D. Bradley, A. Khalil and E. Ossanna), 2016.

Other Manuscripts not Intended for Publication

  • Families of periodic orbits of the Koch snowflake billiard (with M. L. Lapidus), preprint (2011), 63 typed pages. [E-print: arXiv:math.DS:1105.0737v1, 2011.] (Pieces of this were taken out and put elsewhere.)

Courses & Mentoring

Simply put, I believe in challenging my students and making it known that I care about their success and academic well-being. My interest in mentoring is rooted in a strong belief that it is not enough to teach, but one must help others along their path.

In the spring of 2017, I will be teaching Calculus III -- Multivariable Calculus and serving on a committee tasked with developing learning outcomes for Calculus I.

Math 228: Calculus III

For students of mathematics, engineering and the sciences. Vector algebra, geometry and calculus; multivariable differential and integral calculus, including the theorems of Gauss, Green and Stokes.

Mentoring

While an MCTP Postdoctoral Fellow at the University of New Mexico, I organized various discussions for graduate students. Slides from two of this discussions are below. I continue to organize seminars and information sessions for students and recent hires in an attempt to pass on my own knowledge.

Texts that I've found useful: suggested texts

Student Projects

I encourage interested students to contact me about doing either an undergraduate thesis or a graduate thesis. Below are some topics that can be modified or expanded upon to suit a student's interest and abilities.

The Koch snowflake fractal billiard table is a fractal billiard table with nowhere differentiable boundary. A priori, reflection at any of the points is not well-defined. Building on previous works, you will be asked to simulate orbits with irrational directions and discuss your experimental results in the context of the more general theory for rational billiard tables. Programming experience in Maple or Mathematica is essential, as the interested student will be learning about computer algebra systems and their effectiveness in providing insight into fractal billiard systems.

The T-fractal billiard table is a somewhat nicer fractal billiard table in that a fair amount of the boundary of the billiard table yields a well-defined tangent. You will be asked to simulate the orbits with irrational directions and discuss your experimental results in the context of the more general theory for rational billiard tables. Programming experience in Maple or Mathematica is essential, as the interested student will be learning about computer algebra systems and their effectiveness in providing insight into fractal billiard systems.

For those more interested in the broader subject of dynamics, one can also compare and contrast the notions of ergodic, weak mixing and strong mixing. This project is best suited for an undergraduate student finished with advanced calculus and interested in learning about measure theory. In preparing the student for the senior thesis, one will first be asked to read and understand the proof of the Poincare Recurrence Theorem and the definitions involved.

© 2016 Jennifer Kwan