My research interests lie in nonlinear dynamics and complex systems
and the applications of the techniques from these fields to the
physical, engineering, biological, and social sciences. Most of my
research thus far has focused on applications in physics and social
networks, but I am eager to study nonlinear behavior throughout the
sciences.
This web page describes both my projects and the student projects I
have supervised and am supervising. It should give a good idea of the
breadth of problems that interest me.
My idea of interesting and viable research is to first model a
system and then analyze that model both analytically and
computationally. This approach is highly interdisciplinary in nature,
as many of the same methods and structures arise in superficially
distinct scientific disciplines, which allows one to better understand
the structure and dynamics of the systems under study. Nonlinear
dynamics provides one of the best approaches to undertake such
research.
Starting in October 2007, I will be a faculty member (a "University
Lecturer") in the Mathematical
Institute at the University of
Oxford. I will be part of OCIAM, the Oxford Centre for
Industrial and Applied Mathematics (though I will occasionally refer
to it as a "Center" because I'm an American). As part of this job, I
will also hold a Tutorial Fellowship at Somerville College.
In June 2005, I returned to Caltech as a postdoctoral scholar in the
Department of
Physics and the Center for
the Physics of Information. I am part of the (recently)
burgeoning Caltech
condensed matter theory group. My advisor is Michael Cross. I am also
working with other people on campus (and am continuing my
collaborations with people from other institutions). After being a
physicist in math department, I am now ready to be a mathematician in
a physics department. (The truth of the matter, though, is that I
split the difference. I am an applied mathematician/theoretical
physicist---hear me roar!!!!) I am slated to stay at Tech until
around June 2007 and may spend the summer of 2007 here as well.
From April 2001 through December 2003, I was a contributing editor
for Complexity Digest, a weekly
newsletter with a lot of interesting vignettes related to complex
systems.
Since Fall 2002, I have been studying nonlinear behavior in
Bose-Einstein condensates (BECs). With Predrag Cvitanovic, I studied
spatial resonances and period-multiplied wavefunctions in BECs in
(periodic) optical lattice potentials using techniques from nonlinear
dynamics and perturbation theory. (We published an analytical
construction of period-multiplied wavefunctions in Spring 2004.
Period-doubled wavefunctions were finally reported experimentally in
April 2005.) I extended this approach to coupled BECs and co-authored
a paper with Boris Malomed and Panos Kevrekidis concerning resonant
and nonresonant coherent structures in binary and ternary BECs
consisting of different hyperfine states of the same atomic species.
With Panos Kevrekidis, I studied stable resonant modulated
amplitude waves in Bose-Einstein condensates in (periodic and
quasiperiodic) superlattice potentials. With Ricardo
Carretero-Gonzalez and Dmitri Frantzeskakis, Panos and I also examined
the dynamics and manipulation of solitary waves using "dynamical"
superlattice potentials. One of my students (Vivien Chua) used
Chirikov's overlap criterion to understand resonance overlap and
chaotic spatial dynamics in BECs in superlattices. With Martijn van
Noort, Shui-Nee Chow, and Yingfei Yi, I rigorously applied KAM and
Aubry-Mather theories to BECs in periodic lattices and superlattices
in order to explore the transition between quasiperiodic and chaotic
dynamics in these systems.
With Dmitry Pelinovsky and Maria Chugunova, I studied gap solitons
in BECs in optical lattices under the effect of Feshbach resonances.
With Hector Nistazakis, Panos Kevrekidis, Dmitri Frantzeskakis,
Alexandru Nicolin, and Jit Kee Chin (a friend of mine from Caltech who
is a graduate student in Wolfgang Ketterle's lab), I also studied
superharmonic resonances and "fractional period states" in BECs in
parametrically excited periodic lattices. I have also written a
review article (with Panos Kevrekidis, Boris Malomed, and Ricardo
Carretero-Gonzalez) concerning the connections between BECs and the
Fermi-Pasta-Ulam (FPU) problem which appeared in the March 2005 focus
issue of Chaos celebrating the 50th anniversary of the first
paper on the FPU problem.
More recently, Panos Kevrekidis, Boris Malomed, Dmitri
Frantzeskakis, and I studied BECs with spatially periodic scattering
lengths (an example of a collisionally inhomogeneous condensate). Our
work on this topic is about to appear in Physica D.
Additionally, Ryan Barnett, Gil Refael, Hanspeter Buechler, and I
just submitted a paper about vortex-lattice locking in rotating
two-component BECs.
And now for a few words about BECs...
BECs, whose existence was predicted in 1924 by Einstein and Bose, were
discovered experimentally in 1995 by the Cornell and Wieman lab at JILA, the
Ketterle group at MIT, and the Hulet group at Rice. In 2001, Ketterle,
Cornell, and Wieman received the Nobel Prize in physics for their work in
this area. To create a BEC, a magnetically-trapped, spin-polarized gas of
bosonic atoms is brought to ultracool temperatures (on the order of
nanokelvins), so that many of the particles reside in the lowest (ground)
energy state. This yields a 'macroscopic' quantum state, as each of the
bosons behave as a single, collective object.
Related to my work in Bose-Einstein condensates has been a project
I started at Caltech concerning nonlinearity management in optics.
("Nonlinearity management" is like Feshbach resonance management from
BECs; the different name reflects its more general physical context.)
This work, which involves experiments supported by theory, is in
collaboration with Panos Kevrekidis, Martin
Centurion, and Demetri
Psaltis. (Panos and I are provided the theory, and Martin and
Demetri took care of the experiments.) Here is our first paper on
this topic.
This paper has been highlighted in Physical Review Focus. It
will also be covered in Laser Focus World. The Physcal Review
Focus story is also available in Asian
languages. (Noticed that the term "nonlinearity management" could
not be translated from English.) Caltech has also posted a press
release (8/04/06) about this work. Our paper was also selected to
appear in the August 2006 issue of the Virtual
Journal of Ultrafast Science.
Quantum chaos refers to the study of the quantization of
classically chaotic systems, which exhibit fundamentally different
behavior than the quantizations of integrable (regular) systems. This
can be seen in, e.g., their spectral statistics, scarring/antiscarring
in their wavefunction amplitudes, etc. Much of the research in
quantum chaos is concered with the behavior of quantum chaotic systems
in various semiclassical regimes in order to consider correspondence
with the underlying classical dynamics. In most cases, one uses a
canonical semiclassical limit (in which an effective value of Planck's
constant goes to zero), but more generally a "semiclassical parameter"
goes to zero, so this means more generally that stationary phase
expansions, WKB expansions, periodic orbit expansions, or the like are
coming into play in some way or another. (In small molecular systems,
for example, this parameter is the square root of the ratio between
the masses of the electronic and nuclear subsystems. In this context,
one builds from Born-Oppenheimer schemes.) One can also find
interestinbg insights by studying the quantum dynamics directly.
As a Ph.D. student in the Center for Applied Mathematics at Cornell
University, I studied quantum chaos under Richard Liboff of the Electrical
and Computer Engineering and Applied and Engineering Physics departments.
The other members of my thesis committee were Steve Strogatz (T & AM
department), Greg Ezra (chemistry department), and John Guckenheimer (math
department).
Ph.D. Thesis: Quantum Chaos in Vibrating Billiard Systems [May 2002].
My thesis concerned the semiquantum dynamics of quantum billiards
with time-dependent boundaries ("vibrating quantum billiards"). A
vibrating quantum billiard can be related to small molecular systems.
Its boundaries (yielding the "nuclear" degrees of freedom) are slow
variables and are treated classically, while its confined particle
(yielding the "electronic" degrees of freedom) is treated
quantum-mechanically. My research resulted in several publications,
which are listed below in reverse chronological order. Two of them
constituted the cover story of the 9/01
issue of International Journal of Bifurcation.
Porter, Mason A. and Liboff, Richard L. [2002] A Galerkin Approach
to Electronic Near-Degeneracies in Molecular Systems. Physica
D, Vol. 167, No. 3-4: 218-247.
Porter, Mason A. [2001] Nonadiabatic Dynamics in Semiquantal
Physics. Reports on
Progress in Physics, Vol. 64, No. 9: 1165-1189.
Porter, Mason A. and Liboff, Richard L. [2001] The Radially Vibrating
Spherical Quantum Billiard. Discrete and Continuous Dynamical Systems,
310-318. Y2K International Conference on Differential Equations and
Dynamical Systems (May, 2000).
Liboff, Richard L. and Porter, Mason A. [2000] Quantum Chaos for
the Radially Vibrating Spherical Billiard. Chaos, Vol. 10, No. 2:
366-370.
In addition to my thesis work, I wrote an expository piece (with Richard
Liboff) that was the cover story of the November-December 2001 issue of
American Scientist. On 2003, this article has since been reprinted in
an online compilation (PowerWeb: Conceptual Physics). During the
same year, it was translated into German and Spanish and appeared
(respectively) in the versions of Scientific American that are
published in Germany and Spain. An Italian version was also licensed,
but I don't know if it was ever published.
In summer 2006, Caltech freshman William ("Austin") Webb studyied
the spectral statistics of quantum mushroom billiards. Classical
mushrooms have divided phase space (see the projects of Steven Lansel
and Kris Kazlowski below), so one can compute their spectral
statistics and compare the results with the Berry-Robnik distribution
that interpolates between Poisson statistics (describing integrable
systems) and Wigner statistics (describing fully chaotic systems). It
is also desirable to study other phenomena such as scarring using
these systems.
In summer 2006, Caltech sophomore Tom Mainiero, a physics major,
studied the quantization of a free particle interacting linearly with
a harmonic oscillator. The classical analog of this system, first
investigated in a 2005 paper in Physica D (by De Bievre and
coauthors) provides a clean example of a system with mixed
regular-chaotic dynamics, so studying its quantization will yield
insight into the quantization of mixed systems. In his plot, Tom
focused especially on Husimi distributions and the characterization of
the quantum signatures of the classical dynamics using sharp and broad
avoided level crossings.
Tom's SURF
report
We also recently submitted an article based on this project:
I am interested in classical billiard systems as well as quantum
ones. In a classical billiard, one has a particle (usually given by a
point) confined by a boundary of some shape and colliding perfectly
elastically against it. The trajectories describing the particle
dynamics are thus given by unions of specular reflection and free
(straight-line) motion. In quantum billiards, one studies the
Schrodinger equation with homogeneous Dirichlet boundary conditions
(i.e., the wavefunction vanishes on the boundary). For classical
billiards that behave chaotically (or exhibit mixed regular-chaotic
dynamics), the study of their quantizations is very important in the
field of quantum chaos.
During the summer 2003 and spring 2004, I advised a student (Steven
Lansel) on creating a GUI (graphical user interface) to simulate
billiard systems in Matlab. Steven did a really good job and is now
an expert at designing GUIs. This program includes several standard
examples (such as the stadium and Sinai billiards), some less
standard examples (such as mushroom and limacon billiards), and the
ability to draw one's own table to simulate. As things stand, it is
extremely useful for producing figures for papers and somewhat useful
for some teaching purposes. The current version of this program is
now ready to be used as a research tool, although I still consider it
to be in beta. This program is really spiffy; it can even make .avi
movies!
If you use this program to produce figures for some publication (which I
highly encourage!), please mention this in the acknowledgements. Feel free
to make improvements to the program on your own if you wish. If you do so,
however, please give me a copy of what you've done. I want this program to
be as good as possible, and I really appreciate such efforts and wish
to disseminate the best possible version of this program.
After completing his GUI billiard simulator, Steven Lansel worked
with me (jointly with Leonid Bunimovich, who I recruited to
collaborate on this project once the GUI was ready in Spring 2004)
during the Fall 2004 and Spring 2005 semesters to study one-particle
and few-particle billiards. First, he used his program to study
mushroom billiards with elliptical caps (and other generalizations of
mushroom billiards). Steven also studied finite-size particles in
billiards shaped like mushrooms and other geometries, showing, for
example, that one sees signatures of integrability for two interacting
confined particles in a circle despite the fact that the system is
completely chaotic. Hence, the confining geometry matters not only
for noninteracting particles but also for interacting ones. Below are
links to our research and expository articles concerning this work.
(The latter was the cover story in The Notices.)
During Summer 2006, Caltech freshman Kris Kazlowski studied
periodic orbits in generalized mushroom billiards. The hope is that
we can eventually build on this work and describe the dynamics of
mushroom billiards by developing an appropriate symbolic dynamics.
The insights of studying periodic orbits will also eventually be
useful for the study of scarring in quantum mushroom billiards.
Also don't forget to download our Graphical User
Interface to simulate billiards, for which Kris provided updates.
(The original version was written by Steven Lansel, another of my
students.)
Another subdiscipline of nonlinear dynamics that interests me
greatly is pattern formation and spatio-temporal chaos in spatially
extended systems. Really cool patterns pervade nature and everyday
life---they occur in clouds, snowflakes, sands, leaves, dripping
faucets, and myriad other places. Spatiotemporal chaos, which refers
to dynamics that are chaotic in both space and time, is also
particularly fascinating. I have helped advise a couple student
projects in these areas and am actively seeking to do further research
and mentoring in this area.
During the summer of 2003, I advised two students (Jeremy Corbett and Behram
Misree) jointly with Shui-Nee Chow on using continuous coupled maps (CCMs) to
model pattern formation in periodically forced granular media. (Jeremy is
an undergraduate at Georgia Tech and Behram started his frosh year
at MIT in Fall 2003.) We are building on previous work by Shankar
Venkataramni and Ed Ott and are particularly interested in forcing at
multiple frequencies and the patterns (and quasi-patterns) that can result.
In Fall 2004 and Spring 2005, I co-advised an undergraduate
student (Jennifer Rieser) with Slaven Peles and Mike
Schatz of the physics department. Jennifer, a very talented
Junior physics major, worked on a project pertaining to
pattern formation due contact line instability. Jennifer brought a lot
to the table, as she spent summer 2004 doing experimental work with
Mike Schatz on this project.
Imagine a thin film of liquid that is spreading along a surface.
Consider, for example, paint on a wall, whose spread---and ensuing
contact line instability---is governed by gravity. There is an
instability in the contact line (that is, the leading edge of the
film) that leads to the initially straight contact line to become
distorted as 'fingers' form. In Jennifer's experimental work, the
fluid flow is mediated by an imposed surface tension gradient (which
is achieved by imposing a light gradient). Jennifer measured
transient amplification experimentally, and her current goal is to
incorporate her experimental data into a theoretical model.
During summer 2006, Caltech freshman Tatjana Wiese studied Faraday
patterns in parametrically forced Bose-Einstein condensates. After
duplicating the results of other researchers, she started considering
the superlattice patterns that can occur with multiple-frequency
forcing.
With Chiara Daraio (a professor in Caltech's Aeronautics and
Applied Physics departments), Panos Kevrekidis, and Eric Herbold (a
graduate student in UCSD's Materials Science department), I am
examining solitary wave propagation in chains of beads ("phononic
crystals"). Our first paper looks at dimer
chains, which consistent of periodically alternating beads of two
different types. We are about to start generalizing this work to
trimers and inhomogeneous (or "randomized") chains of beads. This
research includes analytical, numerical, and experimental components.
Complex networks is a subset of applied math (several components of
which have, strangely enough, been adopted by the applied dynamical
systems community) in which techniques from subjects such as
statistical mechanics are employed to study graphs that are supposed
to represent real-world phenomena. This includes idealized graphs
like the Watts-Strogatz network and scale-free networks and also
networks constructed based on real data. Research on these topics is
relevant to a number of situations---including the World Wide Web,
Congress, college football, phylogenetic (evolutionary) networks, worm
propagation on servers, and many other applicants.
This work is joint with Peter Mucha and Thomas
Callaghan, a very talented Georgia Tech undergraduate. Thomas did all
the real work.
If you want to learn more about how well "monkeys"
(random walkers) can rank football teams, we have posted a fairly
detailed commentary on
this subject, which includes our rankings for 2003--2006. (You'll see
that they do remarkably well!) We also have written two papers on
this subject:
Our research on ranking football teams was featured in the 11/10/03
issue of ESPN: The Magazine (in a section called "The Spin",
see page 34). An article about this project also appeared on 11/14/03
in Nature
Science Update. Georgia Tech issued both long and (relatively)
short press releases (11/18/03) concerning this project. Ken
Massey now includes our rankings under the name "Random Monkeys"
("RM") on his website comparing the ranks of different computer
algorithms. It will be featured in an article dated 11/28/03 in
The Chronicle of Higher Education. This project was also
featured in Headlines and Deadlines from the American Mathematical
Society (11/21/03) and the ACM
News Service (11/19/03). It also was described in the 1/04 issue
of a French magazine called La
Recherche. This project was featured on CNN's Headline News on
12/30/03 (which I still have yet to see). It was also featured
in the 5/24/04 issue of The Atlanta Journal-Constitution (and
was posted online on 5/23), an American
Mathematical Society press release (8/4/04), and even showed up in
some
Singaporean online newspaper. Most recently, the story was picked
up on 9/4/04 by MathTrek
on Science News online (Vol. 166, No. 10). This was also picked
up by the MAA#s
MathTrek column. Most recently, our work was discussed in an
opinion piece in The
Washington Post (12/10/05). The Washington Post article
has been picked up by a number of other places.
As a Caltech alum, I highly recommend the following Foxtrot comic strip
(which appear Saturday 1/3/04): Caltech and the
BCS.
With Peter Mucha, I advised Georgia Tech undergraduate Casey Warmbrand
on a project concerning community structures formed by the
Congressional committee (and subcommittee) assignment network in the
United States House of Representatives. To analyze this network, we
used properties such as community structure, hierarchical clustering,
etc. We also incorporated political spectra to see how they
correlate with the network structure. After Casey graduated, Peter
and I recruited network theory expert Mark Newman to work with us on this
project. We have published one paper on this topic in PNAS and we
have recently submitted an archival sequel to the aforementioned paper
to Social Networks. This latter paper includes the work of another
undergraduate student (AJ Friend) on local community detection in
networks. We are planning to build on this research further by
collaborating with political scientist James Fowler of UC Davis.
At the 2006 American Physical Society March Meeting, our poster on
this topic was named a winner in the 3rd annual Gallery of Nonlinear
Images. This was published in Chaos (see the citation and link below).
Since spring 2005, Georgia Tech undergraduate discrete math major
AJ Friend has worked with Peter and me on a project concerning local
methods for determining community structure and their application to,
for example, phylogenetic (evolutionary) networks and the House of
Representatives committee assignment network. In the process, AJ
generalized a arXiv paper by Bagrow and Bollt (on local methods for
community detection) to work with weighted networks. Using this
method and others (single-linkage clustering and betweeenness-based
methods), we were able to refine our work on clique formation in the
House and also to gain further insight into structural changes
following the so-called "Republican Revolution" of 1994. We recently
submitted this work (see above) to Physica A.
In summer 2006, Caltech sophomore Yan Zhang (a mathematics major),
examined the community structure of the legislation cosponsorship
network and compared his results with the communities in the committee
assignment network. Yan focused on modularity-maximization techniques
for hierarchical clustering. The other team members for this project
(besides Yan and me) are undergraduate A.J. Friend of Georgia Tech,
undergraduate Mandi Traid of UNC Chapel Hill, political science
professor James Fowler of UCSD, and mathematics professor Peter Mucha
of UNC Chapel Hill. We submitted an article for publication in June
2007:
Zhang, Yan; Friend, A. J.; Traud, Amanda L.; Porter, Mason A.;
Fowler, James H.; and Mucha, Peter J. [2007]. Community
Structure in Congrssional Cosponsorship Networks, submitted to
Physica A.
Also check out Yan's SURF report from summer 2006:
In Summer 2007, I am advising Caltech freshman physics major Liuyi
("Ye") Pei on a project concerning community detection in networks
formed by roll call voting in the Senate and House of Representatives.
We have the data for the entire history of the U.S., so the plan is to
look at historical questions such as party realignments. This
complements previous studies of the same data set by Poole and
Rosenthal that use data mining methods. It also complements previous
work my my collaborators and I on community detection in the committee
assignment and legislation cosponsorship networks. With three
different data sets describing social relations among similar groups
of people, we hope to gain additional insight into the U.S. Congress
as a social network.
In summer 2005, Caltech physics major Eric Kelsic worked with me to
study the community structure of the social network of friendships in
data from Facebook, an online social network (organized primarily by
university affiliation) with self-identified associations. Eric used
the Caltech social network as an example to springboard to comparitive
analyses of the social structures of different universities. He also
started to consider things like the overlapping of communities in
hierarchical structures, which is known to occur in typical social
networks but is typically ignored at the outset in studies of
hierarchical clustering.
In Summer 2007, I am advising Caltech sophomore Olga Mandelshtam on
a project concerning preferential attachment in online social networks
such as Facebook, Friendster, and so on. Motivated by the different
mechanisms in the various online networks, Olga will investigate how
such microscopic differents lead to different ("macroscopic")
statistics in the networks. She will compare her results with data
from Facebook (and hopefully also LinkedIn). This project is in
conjunction with the Caltech Alumni Association. Eventually (though
likely not in the first summer), we hope to start examining the
inverse problem: given some macroscopic statistics in an online social
network, how does one design a microscopic mechanism (or a family of
mechanisms) that produces it.
In Summer 2004, I co-advised (with Shui-Nee Chow) two students
(Stephanie Chung and Caroline Seabrook) on a project concerning
singular value decompositions (SVDs) and information theory. We
continued this project during the Fall 2004 and Spring 2005 semesters,
although discussions were more sporadic once the school year started.
The data we used were the grade distributions and course survey
results from Georgia Tech's math department. Graduate students
Alexander Grigo and Ying Wang were also involved in this project.
Towards the end, we spent some time using hierarchical clustering
techniques to explore this data from a network theory perspective.
The intent was to examine research cliques in the math department
based on which courses professors teach, although we did not obtain
any conclusive results in this direction.
I studied complex systems a bit as a graduate student as well. While a
graduate student, I worked on a small project that involved looking at whale
culture using a complex systems perspective. This was joint wotk with
Gottfried Mayer-Kress of Pennsylvania State University.
"Remarks on Whale Culture from a Complex Systems Perspective" (Behavior
and Brain Sciences, 4/01, Vol. 24, No. 2, p. 344-+)
Synchronization in coupled oscillators occurs when they start to
move together in some way -- for example, phase locking in interacting
phase-only oscillators. (More intricate forms of synchronization are
also possible.) I should put in a longer description, but I am
feeling lazy.
With Mike Cross (and, to a lesser extent, Jeff Rogers and Ron
Lifshitz), I am co-advising two Caltech undergraduates on
synchronization problems in nanomechanical oscillators (which have
both phase dynamics and amplitude dynamics in their description) in
Summer 2007. The students in question are freshman Applied &
Computational Mathematics major Sherry Chen and sophomore Physics
major Matt Grau. Sherry is examining "antiferromagnetic"
synchronization, so that both positive and negative couplings are
present. Matt is going to focus on small systems of oscillators in
the hopes of doing lots of analytical work. Mike, Jeff, and Ron have
done extensive prior work on this topic.
In Summer 2007, I am advising Caltech sophomore Applied &
Computational Mathematics major Natasha ("Alex") Cayco Gajic on a
study of basins of attraction of synchronization states (and other
states) in coupled phase oscillators. She will be building on a 2006
Chaos paper by Dan Wiley, Steve Strogatz, and Michelle Girvan
that utilized a Kuramoto model consisting of coupled identical
oscillators by considering oscillators that are not identical. She
will start from the situation in which they are almost identical and
proceed from there.
This project, which has been well received by the behavioral
scientists with whom we've communicated, entails using Lienard
oscillators to (phenomenologically) model the mood swings of people
afflicted with bipolar disorder. I supervised a group of students
(Darryl Daugherty, John Urrea, and Tairi Roque) on this project at
MTBI in the summer of 2002 and mentored another student (Jessica
Snyder) on a continuation of this project in the summer of 2003.
Steve Wirkus of Cal Poly Pomona is also involved with this project.
Below are the write-ups pertaining to this work, including one paper
that we posted to the arXiv. In the future, we hope to find students
to build on this work---it is especially essential to incorporate real
data into our model. Ideally, this work will be done in conjunction
with behavioral scientists.
Daugherty, Darryl; Roque-Urrea, Tairi; Urrea-Roque, John; Snyder,
Jessica; Wirkus, Stephen; and Porter, Mason A. [2004] Mathematical
Models of Bipolar Disorder (preprint).
In the summer of 2004, I co-advised---with Christopher
Klausmeier of Georgia Tech's biology department and Leonid Bunimovich of
Georgia Tech's math department---two students (Alexei Dachevski,
Electrical and Computer Engineering and Julie Bjornstad, Discrete
Mathematics) in the study of the dynamics of plankton food chains in
the presence of seasonal variations and fluctuations in resource
availability. This system includes a predator (zooplankton), prey
(phytoplankton), and a resource (light). The students are considering
ordinary differential equation models of such systems. This project
continued during the Fall 2004 and Spring 2005 semesters. The work
done for this project is currently in our queue to write up and submit
to a theoretical ecology journal.
In summer 2005, Caltech mathematics major Sean Li worked on
extending this work. He used normal form theory and looked, in
particular, at the case of intermediate time-scale fluctuations.
One of my major research interests is Hamiltonian systems. In
fact, several of my papers on BECs have employed the perspective of
classical Hamiltonian dynamics. One of my students, Vivien Chua, also
used such a perspective to study BECs. This followed up work she did
on cubic-quintic Duffing oscillators. Another student, Adrianne
Stroup, is studied the dynamics of triple pendula.
Using the skills she learned in Fall 2003 in her study of
cubic-quintic Duffing oscillators (see below), Georgia Tech
undergraduate Vivien Chua used Chirikov's overlap criterion to study
the interactions between spatial resonances of BECs in superlattices.
Vivien started this work in Spring 2004 and finished it during the
Fall 2004 semester. We have a paper in press that details Vivien's
work:
Duffing oscillators constitute a class of the canonical examples of
low-dimensional Hamiltonian systems. In the fall of 2003, I advised a
Georgia Tech undergraduate student (Vivien Chua) on a project concerning a
generalization of Duffing oscillators that includes a quintic contribution.
Duffing oscillators occur, for example, when one applies a standing wave
ansatz to Nonlinear Schrodinger (NLS) equations (which I have exploited in my
research on BECs). Cubic-quintic Duffing oscillators arise via a similar
ansatz when applied to cubic-quintic NLS equations. Duffing oscillators are
important in other applications as well, including beam theory and electrical
circuits (which is supposedly the original application studied by Duffing).
In Summer 2004, I advised (through Caltech's SURF [undergraduate
research] program) Adrianne Stroup, who is a frosh at Caltech starting
Fall 2004. Adrienne studied single (forced and unforced) and double
pendula both analytically and numerically. She learned about
nonlinear dynamics, Hamiltonian systems, and numerical integration.
Although Adrianne didn't get to working on original results, she did
quite well---especially considering that she had just graduated from
high school!
Predrag Cvitanovic and I wrote the following paper on transition
state theory. It describes work by Martin Lo, Jerry Marsden, Shane
Ross, Turgay Uzer, and others.
When it was published, the American Mathematical Society issued a
press release. The story was picked up by the MathTrek
section of Science News Online on 9/9/05. Here is the National
Science Foundation version (9/29/05) of this press release. Georgia
Tech posted a version (9/28/05) of this as well. Sources that
have picked up these various press releases include PhysOrg.com (9/28/05), Newswise
(9/30/05), 50
Connect, Centauri
Dreams (9/30/05), Innovations
Report (9/29/05), Space.com (9/27/05),
and SpaceRef.com
(9/30/05). There is also an article in Spanish about our paper that
appeared in Tendencias
Cientificas (10/21/05). (There have also been various other
random blog and web articles, such as a West Virgina University press
release.) Here's a Spanish version that appeared on yucatan.com.mx
(11/17/05). It also appeared in Spanish in Fisica
y Sociedad (11/03/05), Espacio
Para Todos (11/07/05), El
Pais (11/02/05), and Noticias
de Informatica Matematica (Oct-Nov 2005; also, I'm ignoring
accents in some places because I'm too lazy to look up the proper html
:) ). More recently, this article was discussed in a short vignette
called "Tube Route" in Science
(11/18/05). A French article covering our paper can be found here.
Another Spanish article (maybe derived from one of the previous ones?
I didn't check) appeared in Explora
la ciencia.
In Fall 2004 and Spring 2005, I co-advised (with Slaven Peles) a
project concerning hopf bifurcations near the flutter speed in
airfoils. The student who undertook this project, Udbhav ("Woody")
Sharma, was an aerospace engineering undergraduate at Georgia Tech.
Aeroelasticity is the study of elastic deformations of an object (such
as an airfoil) in a fluid such as air. When an airfoil (or other
lifting surface) falls below a critical speed called the flutter
speed, oscillations become damped and decrease in magnitude
exponentially. Above this speed, oscillations increase exponentially,
so this system exhibits a Hopf bifurcation. This research problem
initially entailed studying the dynamics near this bifurcation for
three degree-of-freedom airfoils, though we ended up simplifying
things (gee, that never happens in theoretical studies...).
I've included some of my old projects below to provide some more ideas
about what types of things have interested me over the years. Some of the
topics below are things that I'd be willing to pick up again with an
interested student.
This particular side project arose from a question that I asked a guest
speaker (Stephen Weinstein of Kodak) in Chemical Engineering 753, a
bifurcation and continuation course taught by Paul Steen in Fall 1999. We
did not pursue this in detail, but it did help foster collaboration between
Weinstein and Steen, so some good definitely came from this.
I wrote this manuscript as part of a celestial mechanics
course (T & AM 672) taught by Joe Burns in Spring 1999.
I've taken down the link for now because my Georgia Tech math
account is now dead and my space on Caltech's server is limited. I'll
repost this one day.
I worked on this project in the fall semester of 1998 as part of a
course (Theoretical and Applied Mechanics 776) taught by John
Guckenheimer. Although I was able to duplicate some results that
were already known, I was unable to discover any original ones.
Nevertheless, I've included this as an indication of my research
interests.
I used to have the abstract posted here. I've taken it down but
will eventually put it back.
I worked on this projected on and off from the winter of 1998
until the summer of 1998. Charles Plott of Caltech's Economics
department oversaw this project, and I also worked closely with
Ken-Ichi Shimomura. We confirmed a result concerning instability in
Scarf's system and conducted an experiment to see if this instability
existed under "ideal" conditions or if it was simply a phenomenon of
the model. The experiment seemed to suggest that the instability is
not merely a phenomenon of the model. According to Plott, this is the
only known experimental result displaying such behavior, but I really
don't know whether that is an accurate statement. My involvement with
the project ended when I left Southern California for Cornell
University in August, 1998. However, I know that Plott, Shimomura,
and their students have continued to work on this project, and I
anticipate that they've gotten some very exciting results with their
work.
I used to have the abstract posted here. I'll eventually put it
back.
I used to have the paper posted here. I'll eventually put it back.
This paper was the result of a 1996 SURF project under Jerry Marsden. It
convinced Caltech's Mathematics department to award me the Bell Prize in
Mathematics Research during my Junior year. It is also the reason I was
invited to join Sigma Xi as an associate member the same year. This paper
was unfortunately never published.
My initial interest in nonlinear dynamical systems arose from a
childhood fascination with patterns. The sketches that I began
drawing when I was 3 years old included many such displays of
contrasting color. (I glanced through these sketches a couple years
ago, and several of them look remarkably similar to patterns that
occur in nonlinear science. This is a statement either of how little
I've progressed since then or of how I was born to study nonlinear
science. I'm not really sure which...)
In high school, I noticed that fractals could produce colorful
patterns in the same vein as what I liked to draw, which led to my
interest in them. In college, I discovered that I was more
interested in continuous systems than in discrete ones, although I
still retain an interest in the latter. Since then, my interests
have branched out into several fields of science that can be studied
using these methods. This origin of my research interests is also
encompassed in the theme of the personal portions of my Web
site---the backgrounds on my Web pages are a metaphorical description
of my interest in nonlinear dynamics. I wish to study patterns, and
problems in dynamical systems produce the ones that interest me the
most.
In short, my academic interests arose largely from what I consider
artistically appealing.
Steven Lansel,
undergraduate student, Electrical & Computer Engineer and Applied
Mathematics, Georgia Tech (now a Ph.D. student in electrical
engineering at Stanford University)
Richard
L. Liboff, Physics, University of Central Florida (emeritus in
Electrical & Computer Engineering, Cornell University)
Thomas Mainiero, undergraduate student, Physics, Caltech
Boris A. Malomed,
Interdisciplinary Studies, Tel Aviv University
Julie Bjornstad, Discrete Math, Georgia Tech (with Christopher
Klausmeier and Leonid Bunimovich): Summer 2004, Fall 2004, Spring
2005. Since Fall 2006, Julie has been enrolled in the Masters program in
urban/regional planning at University of North Carolina at Chapel
Hill.
Thomas S. Callaghan, Applied Math, Georgia Tech (with Peter Mucha):
Summer 2003, Fall 2003, Spring 2004, Fall 2004, Spring 2005. Thomas
was awarded a Goldwater Fellowship during his junior year (2003-2004).
Thomas also participated in the summer 2005 Park City program on
mathematical biology. Since Fall 2005, Thomas has been enrolled in
the Ph.D. program in Stanford University's Institute for Computational
Mathematics and Engineering.
Vivien Chua,
Electrical and Computer Engineering, Georgia Tech: Fall 2003, Spring
2004, Fall 2004. In Spring 2005, Vivien won an award in Georgia
Tech's Student Paper Competition as a result of our joint paper (which
was published by IJBC). Since Fall 2005, Vivien has been enrolled in
the Ph.D. program in Stanford University's Institute for Computational
Mathematics and Engineering. Her Ph.D. thesis project, advised by
Oliver Fringer and Margot Gerritsen, is to study high resolution
transport schemes (predominantly on finite-volume methods) for coastal
simulation with applications to the San Francisco bay. This is part
of SUNTANS (Stanford Unstructured Nonhydrostatic Terrain-following
Adaptive Navier-Strokes Simulator).
Stephanie Chung, Applied Math, Georgia Tech (with Shui-Nee Chow):
Summer 2004, Fall 2004, Spring 2005. Stephanie is currently working
as an actuary at Safeco in Seattle. She is in the personal lines auto
pricing department.
Jeremy Corbett, Applied Math, Georgia Tech (with Shui-Nee Chow):
Summer 2003. Starting late in 2007, Jeremy will be spending a year
teaching English in Korea.
Alexei ("Leo") Dachevski, Electrical and Computer Engineering,
Georgia Tech (with Christopher Klausmeier and Leonid Bunimovich):
Summer 2004, Fall 2004, Spring 2005. Since Fall 2006, Leo has been
enrolled in Georgia Tech's Ph.D. program in Algorithms, Combinatorics,
and Optimization (ACO).
A.J. Friend, Discrete Mathematics, Georgia Tech (with Peter Mucha):
Spring 2005, Summer 2005. AJ also participated in the summer 2005
Park City program on mathematical biology and in Penn State's MASS
program for Summer/Fall 2006. Additionally, AJ was awarded a
Goldwater Fellowship during his sophomore year (2005-2006). In summer
2007, AJ will be participating in the Park City program on statistical
mechanics.
Eric Kelsic,
Physics, Caltech: Summer 2005. Eric has been accepted into Yale
University's doctoral program on systems/quantitative biology,
Harvard's doctoral program in systems biology, Princeton's doctoral
program in quantitative biology, MIT's doctoral program in
quantitative biology, and the Berkeley/UCSF doctoral program in
bioengineering. Eric has also been accepted by a program that will
allow him to spend a year teaching in China. He'll do that for a year
starting fall 2007 and will start his doctoral program in fall 2008.
Eric has decided to enroll in Harvard's systems biology program.
Steven Lansel, Electrical
& Computer Engineering and Applied Math, Georgia Tech: Summer 2003,
Spring 2004, Fall 2004, Spring 2005. Named one of four Georgia Tech
nominees for Goldwater Fellowship (2003-2004 academic year). One of 2
Georgia Tech students awarded Georgia Tech's Sigma Xi award for
undergraduate research (Spring 2005). Since January 2006, Steven has
been enrolled in the Electrical Engineering Ph.D. program at Stanford.
Sean Li, Mathematics, Caltech: Summer 2005.
Tom Mainiero, Physics, Caltech: Summer 2006. He was a Semifinalist
in Caltech's Perpall speaking competition in Fall 2006. He applied
for a quantum information theory SURF project (Summer 2007) under Dave
Poulin.
Olga Mandelshtam, Caltech: Summer 2007.
Behram Mistree, MIT (with Shui-Nee Chow): Summer 2003. Has since decided
to double major in mathematics and electrical engineering.
Liuyi ("Ye") Pei, Physics, Caltech: Summer 2007.
Jennifer Rieser, Physics, Georgia Tech (with Slaven Peles and
Michael Schatz): Fall 2004, Spring 2005. Jennifer also participated
in the University of Maryland's physics REU program in summer 2005.
Starting in fall 2006, Jennifer will be enrolled in the Ph.D. program
in physics at Cornell University. She has been awarded an IGERT
Fellowship in nonlinear systems, although she is deferring that award
for a year and will be earning her keep by TAing that year. Jennifer
may be doing her dissertation on a topic in soft condensed matter
physics.
Caroline Seabrook, Applied Math, Georgia Tech (with Shui-Nee Chow):
Summer 2004, Fall 2004, Spring 2005. Since Fall 2005, Caroline has
been enrolled in the statistics Ph.D. program at North Carolina State
University. She will earn her Masters degree at the end of the
Spring 2007 semester and will then be selecting her advisor and
starting her research towards her doctorate. She will concurrently
be working as a graduate assistant at SAS Institute Inc. (which is
associated with NC State and developed a statistical software
package), where her job will be to analyze data that SAS gets from
school districts nationwide (these statistics are needed for the No
Child Left Behind act).
Udbhav ("Woody") Sharma, Aerospace Engineering, Georgia Tech (with
Slaven Peles): Fall 2004, Spring 2005. In June 2005, Woody started
working at the Liquid Propulation Systems Center of the Indian Space
Research Organization.
Jessica Snyder, Applied Math, Georgia Tech: Summer 2003. She
participated in the MTBI REU program at Los Alamos in summer 2004.
Jessica got her Bachelor's degree in 2005. She currently works at
Science Applications International Corporation (SAIC) in Huntsville,
Alabama. She is planning on simultaneously getting a Masters' degree
at University of Alabama at Huntsville.
Adrianne Stroup, Caltech: Summer 2004. She was a Semifinalist in
the Perpall speaking competition in Fall 2004. In Summer 2006, she
participated in an REU in Germany at the Technische Universitat
Darmstadt (by winning a Monticello Summer Internship award from
Caltech).
Casey Warmbrand, Discrete Math, Georgia Tech (with Peter Mucha):
Summer 2003, Fall 2003. He started graduate school in Mathematics at
University of Arizona in Fall 2004 and is working on a dissertation in
random matrix theory under the supervision of Ken McLaughlin. Casey
has successfully written and defended his Masters thesis. (The topic
of the thesis concerned partitions given by the Plancherel measure and
an asymptotic analysis analogous that of random matrix theory and the
Wigner semicircle law.) His doctoral thesis will focus on domino
tilings of the aztec diamond (and possibly tilings of a hexagon with
rhombi) and the use of orthogonal polynomials to relate the tilings
(and the non-intersecting paths that describe them) to probability
distributions related to random matrix theory via the asymptotic
analysis of orthogonal polynomials. (Got all that? Go ask Percy
Deift if that doesn't make any sense to you...)
William ("Austin") Webb, Applied Mathematics, Caltech: Summer 2006.
In Summer 2007, Austin will be doing a quantum information theory SURF
project under the supervision of Alexei Kitaev.
Tatjana Wiese, Caltech: Summer 2006. She applied for a number
theory SURF project under Eric Waumbach for Summer 2007.
