|
Research
Interests
I have developed a diverse set of interests in my research
program which includes PDEs, fluid dynamics, mathematical biology,
mathematical physics, complex
dynamics, and combinatorics. A detailed research statement can be
found here.
Below you will find brief descriptions of my research projects along
with links to any publications/preprints associated to
each project. They are presented in roughly chronological
order.
On Soccer Balls and
Linearized Inverse Statistical Mechanics
w/ J. von Brecht (UCLA)
Left: The "soccer ball''
ground state using a
designer potential. Right: An approximation of the
steady state using the linear theory for continuum spherical solutions.
|
The classical inverse
statistical mechanics question involves inferring properties of
pairwise interaction potentials from exhibited ground states. For
patterns that concentrate near a sphere, the ground states can range
from platonic solids for small numbers of particles to large systems of
particles exhibiting very complex structures. In this setting, our
previous work allows us to infer that the linear instabilities of
the pairwise potential accurately characterize the resulting nonlinear
ground states. Potentials with a small number of spherical harmonic
instabilities may produce very complex patterns as a result. This leads
naturally to the linearized inverse statistical mechanics question:
given a finite set of unstable modes, can we construct a potential that
possesses precisely these linear instabilities? If so, this would allow
for the design of potentials with arbitrarily intricate spherical
symmetries in the ground state. In this paper, we solve our
linearized inverse problem in full, and present a wide variety of
designed ground states. Submitted, and a preprint will be available
soon. |
Predicting pattern
formation in particle interactions
w/ J. von Brecht (UCLA), T. Kolokolnikov (Dolhousie U.) A.
Bertozzi (UCLA)
Minimizing patterns w/
Force law F(r) = tanh((1-r)a)+b in Three dimensions.
|
Large systems of particles
interacting pairwise in d-dimensions give rise to extraordinarily
rich patterns. These patterns generally occur in two types. On one
hand, the particles may concentrate on a co-dimension one manifold such
as a sphere (in 3D) or a ring (in 2D). Localized, space-filling,
co-dimension zero patterns can occur as well. In this paper, we utilize
a dynamical systems approach to predict such behaviors in a given
system of particles. More specifically, we develop a non-local linear
stability analysis for particles uniformly distributed on a d-1
sphere. Remarkably, the linear theory accurately characterizes the
patterns in the ground states from the instabilities in the pairwise
potential. This aspect of the theory then allows us to address
the issue of inverse statistical mechanics in self-assembly: given a
ground state exhibiting certain instabilities, we construct a potential
that corresponds to such a pattern.
J. von Brecht, D. Uminsky, T.
Kolokolnikov, A. Bertozzi. Predicting
pattern
formation
in
particle
interactions. To appear in Mathematical
Models
&
Methods
in Applied Sciences, 2011.
|
A multi-moment vortex method for
2D viscous fluids
w/ C.E. Wayne (Boston U.) and A. Barbaro (UCLA)
Course grid simulation of
tripole relaxation using MMVM.
|
In this paper we introduce
simplified, combinatorially exact formulas that arise in the vortex
inteaction model found in [R. Nagem, G. Sandri, D. Uminsky, C.E.
Wayne. SIADS 8(1),
2009, 160-179.]. These combinatorial
formulas allow for the efficient implementation and development of a
new multi-moment vortex method (MMVM) using a hermite expansion for
each vortex element to
simulate 2D vorticity. The method naturally allows the particles to
deform and become highly anisotropic as they evolve without the added
cost of computing the Biot-Savart integral. Here we will
focus much of our attention on the implementation of a single particle,
large number of Hermite moments case, in the context of quadrupole
perturbations of the Lamb-Oseen vortex. At smaller perturbation values,
we show the method captures the shear diffusion mechanism and the
rapid
relaxation (on Re^1/3 time scale) to an axisymmetric state. In addition
we numerically investigate the spatial convergence of the single
particle method and show that the method converges faster than
polynomial. Finally we will end with several computed examples of full
multi-moment vortex method and discuss the results in the context of
classic vortex methods.
D. Uminsky, C.E. Wayne, A. Barbaro. A
multi-moment vortex method for
2D viscous fluids. To appear in the Journal
of Computational Physics, 2011.
|
Stability of ring
patterns arising from 2D particle interactions
w/ T. Kolokolnikov (Dalhousie U.) H. Sun (UCLA) and
A. Bertozzi (UCLA)
Minimizing patterns w/
Force law F(r) = tanh((1-r)a)+b.
|
Pairwise particle
interactions arise in diverse physical systems ranging from insect
swarms to self assembly of nanoparticles. This letter develops a
fundamental theory for the morphology of patterns in two dimensions -
which can range from ring and annular states to
more complex spot patterns with N-fold symmetry. Many of these patterns
have been observed in nature although a general theory has been
lacking, in particular how small changes to the interaction potential
can lead to large changes in self-organized state. Emergence of these
patterns is explained by a stability analysis of a ring solution. This
analysis leads to analytic formulae involving the interaction potential
that provide detailed information about the structure of complex
equilibria.
T. Kolokolnikov, H. Sun, D. Uminsky, A.
Bertozzi. Stability of ring
patterns arising from 2D particle interactions. Physical Review E, Rapid
Communications, 84(1), 015203(R), 2011.
|
Generalized
Birkhoff-Rott equation for 2D active scalar problems
w/ H. Sun (UCLA) and Andrea Bertozzi (UCLA)
Vortex density sheet solutions at t=1 with varying amounts of
incompressible and potential contributions to the kernel.
|
In this paper we
derive evolution equations for the 2D active scalar problem
when the solution is supported on 1D curve(s). These equations
are a generalization of the Birkhoff-Rott equation when vorticity is
the active scalar. The formulation is Lagrangian and is valid for
nonlocal kernels $K$ that may include both a gradient and an
incompressible term. We develop a numerical method for implementing the
model which achieves second order convergence in space and fourth order
in time. We verify the model by simulating classic active scalar
problems such as the vortex sheet problem (in the case of inviscid,
incompressible flow) and the collapse of delta ring solutions (in the
case of pure aggregation), finding excellent agreement. We then
study two examples with kernels of mixed type - i.e.,
kernels that contain both incompressible and gradient flows. The first
example is a vortex density model which arises in
superfluids. We analyze the effect of the added gradient
component on the Kelvin-Helmholtz instability. In the second example,
we examine a nonlocal biological swarming model and study the dynamics
of density rings which exhibit complicated milling behavior.
H. Sun, D. Uminsky, A. Bertozzi. Generalized
Birkhoff-Rott
equation
for 2D active scalar problems. To appear
in SIAM Journal on Applied
Mathematics, 2011.
|
Hydrodynamic origins
of whale flukeprints
w/ R. Levy (Harvey Mudd College) A. Park (Harvey Mudd College), J.
Calambokidis (Cascadia Research), and
Germain Rousseaux (Université de Nice)
|
Whale flukeprint created
by a cruising humpback whale.
|
Whale flukeprints are an
often observed, but poorly understood,
phenomenon. Used by whale researchers to locate whales, flukeprints
refer to a strikingly smooth oval-shaped water patch which forms behind
a swimming or diving whale on the surface of the ocean and persists up
to several minutes. In this paper we provide a description of
hydrodynamic theory and related experiments explaining the creation and
evolution of these “whale footprints.” The theory explains that the
motion of the fluke provides a mechanism for shedding of vortex rings
which subsequently creates a breakwater that damps the short wavelength
capillary waves. The theory also suggests that the role of natural
surfactants are of secondary importance in the early formation of these
footprints.
R. Levy, D. Uminsky, A. park, J. Calambokidis. A
theory
for
the
hydrodynamic origins of whale flukeprints. International Journal of
Non-Linear Mechanics, Volume 46, Issue 4, May
2011, Pages 616-626.
R. Levy, D. Uminsky. Formation of ocean surface patterns by cetacean
fluke oscillations. To appear in IMA Volume on Natural Locomotion in
Fluids and on Surfaces: Swimming, Flying, and Sliding, 2011.
|
The
QCD
β-function from global solutions to Dyson–Schwinger
equations
w/ G. Van
Baalen (Boston U.), D. Kreimer (Boston U.) and K.Yeats (Simon Fraser U.)
The set of vertices and
propagators in the background field gauge.
|
We study quantum chromodynamics from the viewpoint of
untruncated Dyson–Schwinger equations turned to an ordinary
differential equation for the gluon anomalous dimension. This
non-linear equation is parameterized by a function P(x)
which is unknown beyond perturbation theory. Still, very mild
assumptions on P(x) lead to stringent restrictions for
possible solutions to Dyson–Schwinger equations. We establish that the
theory must have asymptotic freedom beyond perturbation theory and also
investigate the low energy regime and the possibility for a mass gap in
the asymptotically free theory.
G. Van Baalen, D. Kreimer, D. Uminsky,
K. Yeats. The
QCD
beta
function
from
global
solutions
to
Dyson-Schwinger
equations. Annals
of
Physics, Volume 325, Issue 2, February 2010, Pages 300-324.
|
Generalized Helmholtz/Kirchoff Model for
Two-Dimensional Distributed Vortex Motion
w/ R. Nagem (Boston U.), G. Sandri (Boston U.), and C.E.
Wayne (Boston U.)
The two-dimensional Navier–Stokes equations are rewritten
as a system of coupled nonlinear ordinary differential equations. These
equations describe the evolution of the moments
of an expansion of the vorticity with respect to Hermite
functions and of the centers of vorticity concentrations. We prove
the convergence of this expansion and show that in the zero
viscosity and zero core size limit we formally recover the
Helmholtz–Kirchhoff model for the evolution of point vortices. The
present expansion systematically incorporates the effects of both
viscosity and finite vortex core size. We also show that a low-order
truncation of our expansion leads to the
representation of the flow as a system of interacting
Gaussian (i.e., Oseen) vortices, which previous
experimental work has shown to be an accurate approximation
to many important physical flows [P. Meunier, S. Le
Dizès, and T. Leweke, C. R. Phys., 6 (2005), pp.
431–450].
R. Nagem, G. Sandri, D. Uminsky, C.E.
Wayne. Generalized
Helmholtz/Kirchoff
Model
for
Two-Dimensional
Distributed
Vortex
Motion. SIAM
Journal on Applied Dynamical Systems, 8(1),
2009, 160-179.
The QED β-function from global solutions
to Dyson–Schwinger equations
w/ G. Van Baalen (Boston U.), D. Kreimer (Boston U.) and K.Yeats (Simon
Fraser U.)
We discuss the structure of beta functions as determined by the
recursive nature of Dyson–Schwinger equations turned into an analysis
of ordinary differential equations, with particular emphasis given to
quantum electrodynamics. In particular we determine when a separatrix
for solutions to such ODEs exists and clarify the existence of Landau
poles beyond perturbation theory. Both are determined in terms of
explicit conditions on the asymptotics for the growth of skeleton
graphs.
D. Kreimer, D. Uminsky, G. Van Baalen,
K. Yeats. The
QED
beta
function
from
global
solutions
to
Dyson-Schwinger
equations. Annals
of
Physics, Volume 324, Issue 1, January 2009, Pages
205-219.
A Mathematical Model
of Crocodilian Population Using Delay-Differential Equations
w/ A. Gallegos (Occidental College), T. Plummer*,
C. Vega*, C. Wickman*, and M. Zawoiski*
*
Undergraduate
research
students participating in Applied
Mathematical Sciences Summer Institute (AMMSI)
2007.
The fully mature american
crocodile.
|
The crocodilia have
multiple interesting characteristics that affect their population
dynamics. They are among several reptile species which exhibit
temperature-dependent sex determination (TSD) in which the temperature
of egg incubation determines the sex of the hatchlings. Their life
parameters, specifically birth and death rates, exhibit strong
age-dependence. We develop delay-differential equation (DDE) models
describing the evolution of a crocodilian population. In using the
delay formulation, we are able to account for both the TSD and the
age-dependence of the life parameters while maintaining some analytical
tractability. In our single-delay model we also find an equilibrium
point and prove its local asymptotic stability. We numerically solve
the different models and investigate the effects of multiple delays on
the age structure of the population as well as the sex ratio of the
population. For all models we obtain very strong agreement with the age
structure of crocodilian population data as reported in Smith and Webb
(Aust. Wild. Res. 12, 541–554, 1985). We also obtain reasonable
values for the sex ratio of the simulated population.
A. Gallegos, T. Plummer, D. Uminsky,
C. Vega, C. Wickman, M. Zawoiski. A
Mathematical
Model
of
a
Crocodilian
Population
Using
Delay-Differential
Equations. Journal of Mathematical Biology. 57(5),
2008, 737-754.
|
Singular Perturbations
of zn
w/ R. Devaney (Boston U.), M. Holzer (U. of Minnesota) , D. Look (St. Lawrence U.), M.
Moreno Rocha (Centro de Investigaci´on en
Matem´aticas)
A Julia set for a
singularly
perturbed function of the form
z3 + C/z3 .
|
In this paper we give a
survey (with some proofs in elementary cases) of the vast array of
different phenomena that occur in the family of rational maps
Fc(z) = zn + c / zd,
especially in the case where the parameter c is small.
R. Devaney, M. Holzer, D. Look, M.
Moreno Rocha, D. Uminsky. Singular
Perturbations of zn. In Transcendental Dynamics and
Complex Analysis. eds. P. Rippon and G. Stallard. Cambridge
University Press, 2008, 111-137.
|
Vorticity dynamics and
sound generation in two-dimensional fluid flow
w/ R. Nagem (Boston U.) and G.
Sandri (Boston U.)
An approximate solution to the two-dimensional incompressible fluid
equations is constructed by expanding the vorticity field in a series
of derivatives of a Gaussian vortex. The expansion is used to analyze
the motion of a corotating Gaussian vortex pair, and the spatial
rotation frequency of the vortex pair is derived directly from the
fluid vorticity equation. The resulting rotation frequency includes the
effects of finite vortex core size and viscosity and reduces, in the
appropriate limit, to the rotation frequency of the Kirchhoff point
vortex theory. The expansion is then used in the low Mach number
Lighthill equation to derive the far-field acoustic pressure generated
by the Gaussian vortex pair. This pressure amplitude is compared with
that of a previous fully numerical simulation in which the Reynolds
number is large and the vortex core size is significant compared to the
vortex separation. The present analytic result for the far-field
acoustic pressure is shown to be substantially more accurate than
previous theoretical predictions. The given example suggests that the
vorticity expansion is a useful tool for the prediction of sound
generated by a general distributed vorticity
field.
R. Nagem, G. Sandri, D. Uminsky. Vorticity
Dynamics
and
Sound
Generation
in
Two-Dimensional
Incompressible
Fluid
Flow. The Journal of the Acoustical Society of America. 122(1),
July
2007.
Unbounded regions of Infinitely Logconcave Sequences
w. K. Yeats (Simon Fraser U.)
We study the properties of a logconcavity operator on a symmetric,
unimodal subset of infinite sequences. In doing so we are able to prove
that there is a large unbounded region in this subset that is infinite
logconcave. This problem was motivated by the conjecture of Boros and
Moll that the binomial coefficients are infinite logconcave.
D. Uminsky, K. Yeats. Unbounded
Regions of Infinitely Logconcave Sequences. Electronic Journal
of Combinatorics. 14(1), November 2007.
Blowup Points and Baby
Mandelbrot Sets for Singularly Perturbed Families of Rational Maps
w/ R. Devaney (Boston U.) and M.
Holzer (U. of Minnesota)
The
parameter plane for
F(z) =C(z + 1 / z).
|
In this paper we describe
the structure of the parameter plane for the family of rational maps
given by C(z + 1 / z). We show that in any neighborhood of C=0
there are infinitely many parameters for which the Julia set is the
entire plane (blowup parameters) as well as infinitely many copies of
small Mandelbrot sets.
R. Devaney, M. Holzer, D. Uminsky. Blowup Points
and Baby Mandelbrot Sets for Singularly Perturbed Families of Rational
Maps. In Complex Dyanamics: Twenty-Five Years after
the Appearance of the Mandelbrot Set. Eds. R. Devaney and L. Keen.
American Mathematical Society, 2006, 51-62.
|
Sierpinski Carpets and
Gaskets as Julia Sets of Rational Maps
w/ P. Blanchard (Boston U.), R. Devaney (Boston U.), D. Look (St. Lawrence U.), M.
Moreno Rocha (Centro de Investigaci´on en
Matem´aticas), P. Seal (Boston U.), S. Siegmund (TU Dresden)
The Parameter plan for the
family of functions z2 +c/z2.
|
In this paper we give a
survey of the dynamical and parameter planes for the family of rational
maps
Fc(z) = z2 + c/z2.
|
Spectral Analysis of
the Supreme Court
w/ B. Lawson (Santa Monica
College) and M. Orrison (Harvey Mudd College)
Generalized Spectral analysis of approval voting data uses
representation theory and the symmetry of the data to project the
approval voting data into orthogonal and interpretable subspaces. In
this paper we use generalized spectral analysis to study the voting
behavior in the Rehnquist Court (1994-1998) which had 192 non-unanimous
cases and correctly observe the well known partnerships and coalitions
in the voting behavior of the Supreme Court justices.
B. Lawson, M. Orrison, D.
Uminsky. Spectral
Analysis of the Supreme Court. Mathematics Magazine. 79(5),
2006,
340-346.
The Escape Trichotomy
for Singularly Perturbed Rational Maps
w/ R. Devaney (Boston U.) and D.
Look (St. Lawrence U.)
An example of a Julia set
that is cantour set ofJordan curves.
|
In this paper we consider
the dynamical behavior of the family of complex rational maps given by Fµ(z)
=zn + µ/zd
where n \ge 2, d \ge 1. Despite the high degree of these maps, there is
only one free critical orbit up to symmetry. Also, the point at 1 is
always a superattracting fixed point. Our goal is to consider what
happens when the free critical orbit tends to 1. We show that there are
three very different types of Julia sets that occur in this case.
Suppose the free critical orbit enters the immediate basin of
attraction of 1 at iteration j. Then we show:
(1) If j = 1, the Julia set is a Cantor set;
(2) If j = 2, the Julia set is a Cantor set of simple closed curves;
(3) If j > 2, the Julia set is a Sierpinski curve.
R. Devaney, D. Look, D. Uminsky. The
Escape Trichotomy for Singularly Perturbed Rational Maps. Indiana
University
Mathematics
Journal, 54, Dec. 2005, 1621-1634.
|
Iteration of an
even-odd splitting map can make integration easier*
w/ E. Huerta-Sanchez (UC
Berkeley) and A. Lopez (UPR)
We study the dynamics of the map and
even odd splitting map, F, on the space of rational functions, in the
context of a new method of integration. We give a recursive formula for
the iterates of a model family of rational functions, which is closed
under the action of F. We give a class of rational functions that are
mapped to zero by two iterations of F. We prove that all polynomials
are eventually mapped to even functions by F, and we determine the
number of iterations required for a given polynomial. We use power
series representation to determine which rational functions are
eventually mapped to even functions by F
E. Huerta-Sanchez, A. Lopez, D.
Uminsky. Iteration
of
an
Even-Odd
Splitting
Map
Can
Make
Integration
Easier . Pi
Mu
Epsilon Journal. 11(5), Fall 2001, 241-250.
*Recieved the Richard V.
Andree Award for the best paper of the year, Pi Mu Epsilon Journal,
2001.
|
