| PROPOSED REU PROJECTS The following are sample REU projects proposed by the faculty of the Department of Mathematics and Statistics at UNM. They serve as guidelines for choosing a project. Interpolation using polynomial of 2 variables, mentored by Michael Nakamaye. Lagrange interpolation gives the equation of a function of the form y = f(x), where f(x) is a polynomial, passing through specified points in the Euclidean plane. If one looks instead for a more general polynomial of the form p(x,y) = 0, and in addition looks for higher multiplicities at the specified points, much less is known. This problem provides a nice mix of algebra and geometry and plent of room for creative effort! Self--accelerating flows for the Navier--Stokes equations, mentored by Jens Lorenz. Prof. Lorenz and Dr. Randy Ott recently succeeded in constructing an incompressible velocity field $u(x)$ which accelerates under its own pressure. (The phenomenon of self--acceleration is necessary, but not sufficient, for blow--up to occur. The blow--up problem for Navier--Stokes is one of the Clay prize problems.) One can introduce parameters in the self---accelerating flow and ask: How large can one make the acceleration when certain norms of $u(x)$ (e.g., the energy) remain fixed? A good visualization of the self--accelerating flow is also of interest. Simulation of a 1D polymer system, mentored by Daniel Appelo. The aim of this project is to simulate a basic polymer problem such as the self-assembly of a spherical micelle or the phase transition in a oil-water mixture. Such systems are be described by the so called self-consistent-field equations which can be solved numerically by a combination of solution of non-linear equations, solution of linear parabolic partial differential equations and computation of integrals using quadrature. As an extension the detailed description of the pathway between states in the system can be modeled by the so called string method. This requires the solution of ordinary differential equations and interpolation between system states.
Discrete and dyadic harmonic analysis, mentored by Cristina Pereyra.
Advanced undergraduate students who have
been exposed to basic ideas of Fourier analysis and are acquainted
with the Haar basis, can start exploring
a number of important concepts in harmonic analysis in the discrete
(purely linear algebra) and in the dyadic setting,
where the theory is much cleaner and transparent,
see Pereyra & Ward (to appear). One can analyze
the finite dimensional cases and study, analytically or numerically,
what happens as the dimension
goes to infinity, and justify the appearance of certain
functional analysis tools.
One can also explore and justify very powerful algorithms, based
on subtle time/frequency
arguments (for example the non-standard form
introduced by Beylkin, Coifman and Rokhlin (1991), which requires
good understanding of the multiresolution analysis, and almost orthogonality
arguments), in fact students can be lead
to discover by themselves these algorithms in simple case studies.
3d vortex filament motion, mentored by M. Nitsche. Analisa Calini used local induction approximation to obtain knotted steady solutions of vortex filaments. We will compute 3d simulations of regularized filament evolution and compare the results to analytical solutions.
Evolution of elliptical vortex sheets, mentored by M. Nitsche.
An initially flat circular vortex sheet inducing potential flow
past a disk evolves under its self-induced motion, and rolls
up to form one vortex ring (Krasny & Nitsche 2002).
An initially spherical vortex sheet inducing potential flow
past a sphere also rolls up into a vortex ring. However, unlike the flat
sheet case, this ring sheds an infinite set of self-similar vortices
(Nitsche 2001).
We will investigate the evolution of a family of initially elliptical
sheets spanning the range between the disk and the sphere and
determine how the vortex ring shedding depends on the initial ellipticity.
Discrete-time modeling of mosquito-borne disease and control, mentored by Helen Wearing. Extend the simple discrete-time model of infectious disease to include human and mosquito populations and sources of stochasticity, using the mosquito-borne disease, dengue, as a case study. Conduct sensitivity analysis of the model and investigate the impacts of mosquito control. Potentially estimate parameters by fitting the model to epidemic data. Lift and drag of airfoils, mentored by Peter Vorobieff. The undergraduate research project will involve conducting experiments with a wind and water tunnel, analyzing their results to determine the dependence of lift and drag on the flow speed and angle of attack, where applicable, for several bodies: a symmetric airfoil, an asymmetric airfoil, a cylinder, and a sphere. In addition, the students will be required to produce flow-visualization images with the digital camera and to comment on the transient patterns. Calculus in the ring of integers, mentored by Alexandru Buium and Santiago Simanca. There is an arithmetic analogue of differential calculus in which functions of a real variable are replaced by integer numbers and in which the derivative operator on functions is replaced by a Fermat quotient operator $\delta: Z \rightarrow Z$, $\delta x=(x-x^p)/p$. Cf. A. Buium, Inventiones Math., 122, 2 (1995), 309-340. This can be generalised to integers in number fields with bounded ramification. Accordingly there is a notion of (non-linear) differential operator acting on such integers. One possible project would be to extend this theory to integers in fields with unbounded ramification. This might have applications to diophantine problems (such as Mazur's question on effective bounds for the torsion of curves in Jacobians; such applications are known in the unramified case, cf. A.Buium, Duke Math. J. 82, 2 (1996), 349-367.) Another possible project would be to express various well-known arithmetically flavored functions (such as power residue symbols, Hecke characters, e.t.c.) as (non-linear) ``differential operators'' in this arithmetic theory. This would be part of a more general conjectural picture explained in: A. Buium, Trans. AMS 357, 3 (2005), 901-964. Finally one could investigate arithmetic analogues of various classical ordinary or partial differential equations and/or techniques related to such equations (along the lines of the papers by A. Buium and S. Simanca, Arithmetic partial differential equations I and II, Advances in Math, 225 (2010).
Investigation of fluid cavitation, mentored by Deborah Sulsky.
Cavitation was first studied by Lord Rayleigh in the late 19th century.
The collapse of cavities is a relatively low energy event, but it is highly
localized and therefore can erode metals and shorten the useful life of
pumps or propellors. However, cavitation can also be harnessed for
productive purposes such as ultrasonic cleaning, or the breakdown of
suspended particles in paints or milk. Cavitation also plays a role in the
destruction of kidney stones in shock wave lithotripsy.
Spectral shift functions, mentored by A. Skripka.
In problems of quantum mechanics (for instance, in computation of the energy of a
system or time evolution of a system), one encounters linear operators [that is,
infinite-dimensional generalizations of matrices] that arise as functions of other
operators. It can be important to know how a small perturbation of an operator effects
the change of a function of the operator. Some aspects of this change are reflected in
spectral shift functions, which originate from I.M. Lifshits's work on theoretical
physics of 1947 - 1952. Existence of higher order spectral shift functions, which
allow to encompass more general perturbations, has recently been proved by D. Potapov,
A. Skripka, and F. Sukochev, but properties of these functions are yet to be
understood. Testing/discovering (and, later, proving) properties of these functions
should start with finite matrices (of low dimensions). To be ready to start working on
a project, a student should possess knowledge of basic linear algebra. Familiarity
with mathematics software (Matlab, Maple, or Mathematica) would also be useful.
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