Luke Rogers
Department of Mathematics
196 Auditorium Rd,
University of Connecticut, U3009
Storrs, CT 062693009
U.S.A.
rogers (the usual symbol) math (dot) uconn
(dot) edu


Quick Links

About Me
I am an Assistant Professor in the Mathematics Department at the
University of Connecticut. In the past I have been an H.C. Wang
Postdoc at Cornell University, a Lecturer in the Department of
Mathematics at the State University of New York at Stony Brook,
and a
graduate student at Yale University. My mathematical interests
include analysis on fractals, Sobolev spaces and quasiconformal
mappings These have connections to harmonic analysis, potential
theory, complex analysis and geometric measure theory. In addition
to
my academic work I am an avid rockclimber, occasional cyclist and
hiker, and allaround outdoor enthusiast.
Research
My research interests are mostly related to analysis on spaces which
lack smoothness properties. These spaces include Euclidean domains
(and pieces of manifolds) which have highly irregular boundaries as
well as spaces for which the intrinsic structure is very
nonEuclidean.
There are several approaches to analysis to metricmeasure spaces
that do not have a Euclidean structure. In order to get
started one usually makes some assumptions that provide a large
class of "wellbehaved" functions. Most of the work I have
done recently is in what is usually called "Analysis on Fractals",
though parts of it should perhaps more accurately be called
"Analysis on MetricMeasureDirichlet spaces"or "Analysis on
selfsimilar spaces". In this approach one either constructs
or assumes the existence of a Dirichlet form, which one should think
of as being an abstract version of the L^2 norm of the Euclidean
gradient, and therefore (by general considerations) a Laplacian
operator. The "wellbehaved functions" one considers are those
in the domain of the Dirichlet form (finite energy functions  a
sort of Sobolev space), or in the domain of the Laplacian
("differentiable functions") or some power of the Laplacian ("smooth
of some finite order"). One can then try to build a theory
that parallels the usual calculus in Euclidean spaces for this class
of functions and operators, as well as studying the associated
differential and partial differential equations. Eventually
one would like to be able to analyze the behavior of solutions on
spaces that approximate structures which occur in nature. To
give just one example, one could ask what solutions of the wave
equation look like on a percolation network (such as a distribution
of oil or gas in a rock formation); I emphasize that we are a long
way from being able to give a good answer to this question!
This type of analysis involves a blend of harmonic analysis,
potential theory, functional analysis and probability theory.
To get started one needs the Dirichlet form. In the cases
where the resulting potential theory will give point sets positive
capacity (and therefore finite energy functions will be continuous)
the form can be constructed as a limit of forms on graphs, which
often gives a concrete way to compute with interesting
functions. This approach is especially useful on selfsimilar
sets because selfsimilarity gives a relationship between the local
and global analytic structures which is a little like that in
Euclidean spaces (which are about as selfsimilar as it is possible
to be!). There are quite a lot of analytic results where some
sort of selfsimilarity plays an important role, and several of my
papers are on results of this type. Once one leaves the
selfsimilar setting many things become more difficult. There
are some things that can be done by purely functional analytic
methods if one knows strong estimates for the heat kernel associated
to the Dirichlet form (for example my paper with Strichartz and
Teplyaev on Smooth Bumps contains a result of this type) but
constructing Dirichlet forms and giving explicit descriptions of
finite energy functions is much more difficult.
There are a lot of interesting problems in this area, so I will not
try to mention all of them. Among the things I am thinking
about are geometric structures on metricmeasureDirichlet spaces
(in particular Riemannian structures associated to the form, as in
my paper with Ionescu and Teplyaev on Derivations and Dirichlet
forms) and the associated question of developing noncommutative
geometries and eventually quantum field theories on selfsimilar
spaces. I am looking at several open problems about smooth
functions and their properties on selfsimilar sets with
resistancetype Dirichlet forms (and on their product spaces and on
fractafolds constructed therefrom). At the same time I am
investigating the behavior of the metric in harmonic coordinates,
existence of embeddings via eigenfunction coordinates (which is
related to some questions in applied mathematics) and the nature of
the maps between the resistance metric and harmonic coordinate
metric (when it exists). The overall goal here is to
understand the extent to which analysis of this type is similar to
the study of SturmLiouville problems associated to singular
measures. Finally I am working on some methods for the
construction of Dirichlet forms on sets which do not fit within the
existing theory.
For completeness, I should return to a point mentioned earlier,
which is that there are other approaches to analysis on metric
measure spaces. One important and well known one is introduced in
the book "Analysis on metric spaces" by Juha Heinonen. In this
approach one assumes existence of a large class of rectifiable
curves, from which it follows that the Lipschitz functions are a
rich and interesting collection. Pursuing this idea leads to a
sort of first order calculus on the space. In general this is a
different kind of theory than that in analysis on fractals, where
there are often no rectifiable curves and the Lipschitz functions
are not the natural class to consider. However there are some
results in the intersection between these two theories, and I
believe that many more natural connections will become clear over
time. I have not worked in this area recently, but my thesis work on
Sobolev extension theorems is closely connected to it.
Some results
Sobolev Spaces and other spaces of "smooth" functions
My thesis work was about universal extension operators for
Sobolev
spaces, which are operators that extend functions from any Sobolev
space on a domain (locally they are defined on any function that
is
locally integrable) to the corresponding Sobolev space on the
ambient
Euclidean space, with estimates. The main result of my thesis
extended methods of E. Stein and P.W. Jones to show that a
universal
extension operator exists for Sobolev spaces on locallyuniform
domains. If you are familiar with the "conecondition" for
boundary points of a domain, which is often assumed when studying
boundary value problems, then you can think of locally uniform
domains as a generalization in which there is a "twisting cone"
at every boundary point. The basic example of a twisting cone is
the
region between two logarithmic spirals.
Subsequently I have generalized the results of my thesis to
consider Sobolev spaces on domains satisfying a weaker condition
that
is more measuretheoretic than geometric. The condition is usually
called Ahlfors (or AhlforsDavid) regularity. It says that if we
take
a ball of radius r (for r between 0 and 1) around any point in the
domain, then the intersection of the ball and the domain has
measure
at least Cr^n. This condition has been shown to be necessary for
the
existence of a bounded linear extension operator by Hajlasz,
Koskela
and Tuominen. An earlier result of Rychkov shows that this is
sufficient for the construction of Sobolev extensions of fixed
order.
Analysis on fractals
Much of my work in this area is joint with Bob
Strichartz (Cornell), and different projects have involved a
number
of other people, including Kasso Okoudjou (University of
Maryland),
Erin Pearse (U. of Iowa), Huojun Ruan ( Zhejiang U.), Alexander
Teplyaev (U. of Connecticut) and Marius Ionescu (U. of
Connecticut)
as well as my undergraduate advisees Michael Barany (formerly at
Cornell), Matthew Begue (formerly at UConn), Jessie DeGrado
(formerly at Cornell), Tyler Reese (UConn).
One of the main problems we have addressed is related to the
structure of smooth functions on certain fractal sets. This
structure
is very different to that of smooth functions on Euclidean spaces,
not least because on fractals the product of smooth functions is
almost never smooth! What we have been doing is constructing
analogues of some tools of classical analysis (including smooth
bump
functions, partitions subordinate to open covers, distributions,
etc.) in the fractal setting. These should be useful for studying
differential equations on fractal structures. So far we have
quite complete
results for the existence of smooth bump functions on metric
measure
spaces, and have a solution to the smooth partitioning problem in
the
postcritically finite (p.c.f.) case. This lets us define
distributions on p.c.f. fractals and establish their basic
properties,as well as study pseudodifferential operators of
various kinds. The tools involved in the proofs are both analytic
and
probabilistic.
A group of us also investigated a constructive approach to the
resolvent kernel for the Laplacian on p.c.f. selfsimilar sets. We
were able to give a series description of this kernel in which the
terms are rescaled and localized copies of functions that satisfy
an
eigenfunction equation on the interior of the fractal. This work
is
related to earlier results I obtained with Jessie DeGrado and Bob
Strichartz, which allow the computation of the harmonic gradients
introduced by Teplyaev in the case of the Sierpinski Gasket.
I subsequently proved some estimates which allow this approach to
be applied to infinite blowups of these fractals and which
generalize known bounds for the resolvent to the complex plane
with the negative real axis removed.
I have a number of
other papers on related topics, including
Modulation Spaces, multipliers and PDE
Modulation spaces are spaces of functions with a some phase space
localization measured by the modulation norm. As such they are
well
adapted to studying the evolution of the phase space structure of
solutions of partial differential equations. It is well known that
L^2 quantities can be used to describe energies, and many
wellknown
PDE (eg the wave equation) have a conservation of energy property
expressible in these terms. On the other hand, L^p properties for
p
different than 2 are not usually conserved. Instead one may wish
to
look at phasespace localization as expressed using the modulation
space norm. The results for unimodular multipliers (including
those
for the wave and Schroedinger equations) are in a paper I
collaborated on with Arpad Benyi, Kasso Okoudjou and Karlheinz
Grochenig. This is the initial step for a number of projects
related
to modulation spaces and PDE.
Teaching
Current courses
This semester I am teaching Math
5110: Introduction to Modern Analysis and Math 3094: Analysis on Fractals.
Past courses
 Spring 07. Math 424, Fourier Analysis and Wavelets. An
undergraduate course for math, sciences and engineering
students.
 Fall 06. Math 413, Honors Introduction to Analysis. Honors
version of the introductory real analysis course.
 Fall 06. Math 103: Math Explorations. Mathematical thinking
for real life problems, aimed at students who do not intend to
continue in mathematics.
 Spring 06. Math 712 Planar harmonic measure. An advanced
graduate course.
 Fall 05. Math 621: Measure and Integration. An introduction
for graduate students.
 Fall 05. Math 311: Introduction to Analysis. Basics of real
analysis, aimed at undergraduate majors and minors.
 Spring 05. Math 311: Introduction to Analysis. Basics of real
analysis, aimed at undergraduate majors and minors.
 Fall 04: Math 191: Calculus for Engineers. (Two Sections)
Single variable calculus for engineering freshmen.
 Mat 342: Applied Complex Analysis. Introduction to complex
analysis with some applications (eg to physics).
 Mat 125: Calculus A. Single variable differential calculus.
Over several semesters I taught the entire calculus sequence at
Yale, some courses more than once.
 Math 112: Differential Calculus of one variable.
 Math 115: Integral Calculus of one variable.
 Math 120: Calculus of several variables.
Other People