This picture illustrates the ``resolution blowup,'' a
technique which can be used to resolve conical
singularities. This technique was introduced in my doctoral dissertation,
completed in 2006 at Stanford University under the
supervision of Professor Rafe Mazzeo.
What is "geometric analysis?" It is an area of mathematics which
incorporates both differential geometry and analysis. The
mathematical models which describe our universe and its natural
phenomena often involve both geometry and analysis, and the
mathematical theory of geometric analysis helps us to better
understand these models. For example, on the one hand we use
partial differential equations (analysis) to understand and hence
predict the behavior of heat, light, and waves. More complicated
pde's can also describe a variety of natural phenomena including
turbulent water, mixing gases, and erosion. On the other hand, we
use differential geometry to describe shapes. How does an object's
shape affect its physical properties? This question motivates much
of my work in geometric analysis.
My research in geometric analysis focuses on the following topics:
geometric analysis on singular spaces, dynamics and mathematical
physics, spectral theory, and interdisciplinary collaboration.
Articles are organized below according to topic.
Geometric Analysis on Singular Spaces
Spectral geometry and aysmptotically
conic convergence, was published in Communications in
Analysis and Geometry. This is a refinement of the work in my
Ph.D. thesis. In these works, I constructed a parametrized
parabolic operator calculus as a technical tool to study the
spectral geometry of manifolds with isolated and iterated conic
singularities in all dimensions. There are a few technical
details omitted from the proof of the first theorem, but this
little note explains
what's missing. It is unsettling that errors inevitably occur in
mathematics research, but if we face them honestly and try to
learn from them, they can lead to interesting discoveries.
Conformal deformations of
conic metrics to constant scalar curvature, with T. Jeffres is
published in Mathematical Research Letters. We considered
conformal deformations within a class of incomplete Riemannian
metrics which generalize conic orbifold singularities. Within
this class of ``conic metrics,'' we determined obstructions to
the existence of conformal deformations to constant scalar
curvature of any sign (positive, negative, or zero). For conic
metrics with negative scalar curvature, we determined sufficient
conditions for the existence of a conformal deformation to a
conic metric with constant scalar curvature -1, and we showed
that this metric is unique within its conformal class of conic
metrics. This work is in dimensions 3 and higher. It is closely
related to the work of Kazuo
Akutagawa and Boris Botvinnik
for orbifolds, and to Kazuo's work with
Gilles Carron and Rafe Mazzeo.
Rafe Mazzeo and I worked together on A heat trace anomaly on polygons,
published in Mathematical Proceedings of the Cambridge
Philosophical Society. In the 1960s, McKean, Singer, Ray, Kac,
Fedosov (and others) observed that the t^0 coefficient in the
small time heat trace expansion for polygonal domains includes a
mysterious contribution from the angles at the corners. The heat
trace anomaly in the title of our paper refers to the fact that
at least one heat invariant is not continuous with respect to
Lipschitz convergence of domains. In our work, using some of the
tools developed in my thesis, we uncovered the cause of this
anomaly...
Zhiqin and I also wrote a piece for the Oxford University
Press Blog which is intended for a general, non-specialist
audience. Check it out
here .
More recently, my PhD student, Medet Nursultanov and I,
together with David Sher generalized these results to the
Neumann and Robin boundary conditions. This required proving a
locality principle for the heat equation with those boundary
conditions, which may be of independent interest. The
paper is published in the Matrix Annals here!
Zhiqin and I teamed up with Hamid Hezari
to investigate The
Neumann Isospectral Problem for Trapezoids. We prove
that if two trapezoidal domains in the plane have the same
spectrum for the Laplacian with Neumann boundary condition, then
they are congruent up to rigid motions of the plane. The proof
requires a careful investigation of the wave trace, and relies
heavily on the work of
Luc Hillairet. It turns out that the Neumann wave trace
is more singular than the Dirichlet wave trace, and this plays a
key role in the proof.
The most singular setting I have encountered to date is that
of rough Riemannian manifolds with boundary. Together
with Lashi
Bandara and Medet Nursultanov, we have investigated the
eigenvalues of certain weighted Laplace equations on rough
Riemannian manifolds. Using techniques of Soviet
mathematicians, Birman and Solomjak, we are able to obtain Eigenvalue asymptotics
for weighted Laplace equations on rough Riemannian manifolds
with boundary. This paper goes back to 2013 or 2014,
when Lashi and I met while I was visiting ANU. We were
interested in finding a geometric counterexample to the Kato
square root problem... We had some ideas, but were not
able to construct a counter-example. In 2015, Lashi proved
that the Kato square root problem can be solved on *any* closed
manifold for *any* rough metric (measureable coefficient,
bounded above and below locally): Rough metrics on
manifolds and quadratic estimates. This is what
motivated us to study the spectrum for the Laplacian associated
to such a metric. It was not clear how to approach this
problem initially, because these spaces are so singular, they do
not even have a well-defined notion of distance between
points(!) However, Grigori Rozenblum suggested that the
powerful techniques of Birman and Solomjak might be
helpful. Indeed, they were, although there was a
nontrivial amount of work needed to be able to apply their
techniques. Nonetheless, our work here can be considered
as a tribute to their achievements and a popularization of their
results!
Dynamics and Mathematical Physics
Dynamics of asymptotically
hyperbolic manifolds was published in the Pacific
Journal of Mathematics. This article was my first step to
generalize and extend results for the spectral and dynamical
theory of infinite volume hyperbolic manifolds (in all
dimensions) to similar results for conformally compact manifolds
with sectional curvatures asymptotic to -1. The main result is a
prime orbit theorem for the geodesic flow. Unfortunately, the
referee(s) and I both missed a few details which have since been
corrected.
On the discrete spectrum of quantum
layers, with Zhiqin Lu, has been published by the
Journal of Mathematical Physics. This work is motivated by
mesoscopic physics which describes physical phenomena at
sufficiently small length scales such that quantum effects are
observed, for example the length scales of nanotechnology. A
quantum layer is a uniform-width layer over a curved surface. We
assume that the surface is: embedded in three dimensional
Euclidean space, complete, non-compact and asymptotically flat
but not totally geodesic. The mathematical model describes a
quantum particle trapped between hard walls. We investigated
geometric conditions which are sufficient to imply the existence
of discrete spectrum, which is mathematically equivalent to the
existence of bound states in the physical model whose energies
lie beneath the essential spectrum.
Dynamics and zeta functions on
conformally compact manifolds, with P. Suarez Serrato
and
S. Tapie has been published in Transactions of the
American Mathematical Society. In this work, we generalized my
results on asymptotically hyperbolic manifolds to conformally
compact manifolds and proved further results for certain
weighted dynamical zeta functions, weighted prime orbit theorems
and relationships between the geometry and spectrum of these
manifolds.
It turned out that several technical difficulties must be
overcome to achieve the goals mentioned in the previous Report.
Clara and I have tackled the first of these in
A Polyakov formula for sectors. We determine the effect
of varying the opening angle of a Euclidean sector on the
zeta-regularized determint of the Laplacian.
With Ksenia Fedosova and Genkai Zhang, we investigated the
Selberg zeta function, and a few of its close relatives, on
Teichmüller space in, Second variation of
Selberg zeta functions and curvature asymptotics. By
obtaining an explicit formula for the second variation of the
Selberg zeta function, we study its asymptotics for large
arguments. As an application, we show that the Ricci curvature
of the Hodge bundle agrees with the Quillen curvature up to a
term which is exponentially decaying.
Spectral Theory
The fundamental gap and one
dimensional collapse, with Zhiqin Lu, has been published
in Proc. of the C.R.M. We investigated the gap between the first
two eigenvalues of the Dirichlet Laplacian, known as the
fundamental gap, on two and higher dimensional convex domains.
We showed that generically if a domain undergoes one dimensional
collapse, the fundamental gap tends toward infinity.
Eigenvalues of collapsing domains
and drift Laplacians, with Zhiqin Lu, has been published
in Math. Res. Letters. This work contains results which relate
the Dirichlet problem in \R^n to a certain Neumann problem in
\R^{n+1} and more general results for drift (also known as
Bakry-Emery) Laplacians. Based on a new maximum principle we
showed that all results based on the gradient estimate method of
Li and Yau (among others) also hold for Bakry-Emery manifolds.
La geometrie de Bakry-Emery et l'ecart
fondamental, was published in the Actes de la Seminaire
de Theorie Spectrale et Geometrie. It is a brief survey (in
French) of recent results culminating in the proof of the
fundamental gap conjecture by Andrews and Clutterbuck.
The fundamental gap of simplices,
with Zhiqin Lu, has been published in Communications in
Mathematical Physics. We focus on the moduli spaces of simplices
in all dimensions, and later specialize to the moduli space of
Euclidean triangles. Our first theorem is a compactness result
for the gap function on the moduli space of simplices in any
dimension. Our second main result verifies a a conjecture of
Antunes-Freitas: the fundamental gap of any non-equilateral
Euclidean triangle with unit diameter is strictly larger than
that of the unit equilateral triangle.
Zhiqin Lu and I teamed up with
Nelia Charalambous to investigate Eigenvalue estimates on Bakry-Émery
manifolds, which has been published in the Springer
Proceedings in Mathematics and Statistics, 119, in the volume,
Elliptic and parabolic equations. We proved lower bounds for the
eigenvalues of the Bakry-Emery Laplacian on manifolds with and
without boundary.
Zhiqin Lu and I wrote The Sound of
Symmetry, which presents many of the general techniques
used in the study of isospectral questions. We applied these
techniques to prove a small collection of positive isospectral
results
What is the Weyl Law for a Bakry-Emery manifold? One might
expect the weighted volume to appear in the formula. However, it
turns out that this is not the case. Nelia Charalambous and I
have investigated the heat trace and its short time asymptotic
behavior, to determine the Weyl law in this setting, as well as
demonstrate a few isospectral results, in The heat trace for
the drifting Laplacian and Schrödinger operators on manifolds.
Applications of Geometric Analysis
Mathematical models for erosion
and the optimal transportation of sediment, with B. Birnir was
published in the International Journal of Nonlinear Sciences and
Numerical Simulation. We proved existence of entropy solutions
and uniqueness of weak solutions to the nonlinear PDEs
demonstrated by Birnir, Hernandez, Merchant, and Smith as a
deterministic model for erosion. We then expressed the flow of
sediment as an optimal transport problem and proved that a
particular class of weak solutions actually implement the
optimal transportation of the sediment flow.
Love Games: A Game Theory Approach to Compatibility,
with K. Bever, is joint work based on Kerstin's bachelor thesis
at the University of Goettingen. Have you ever read or taken a
compatibility quiz in a popular magazine? We thought it would be
interesting to create a quiz based on actual rigorous
mathematics. Check it out!
Many Ways to Stay in the Game: Individual Variability
Maintains High Biodiversity in Planktonic Microorganisms,
with Susanne
Menden-Deuer is the first paper in our ongoing
collaboration to increase the understanding of both zooplankton
and phytoplankton. These tiny creatures are at the bottom of the
food chain and have a crucial role in the health of our entire
planet. In this paper we have discovered what appears to be the
key to solving the so-called ``paradox of the plankton.'' The
paradox is that the biodiversity of planktonic microorganisms
would appear to contradict competition theory and the
competitive exclusion principle. However, we have found that
with an appropriate competition model which incorporates
varibility among individual organisms, which has been observed
and recorded in numerous biological references, it turns out
that arbitrarily high numbers of species may coexist. The key
mathematics underlying this work is what I like to think a
rather novel application of geometric measure theory and real
algebraic geometry to game theory.
The level sets of typical games
contains a characterization of, as the title would suggest, the
level sets of the payoff functions for most non-cooperative
games. The main result was certainly known to J. Nash and
interestingly shows the connection between his work on
non-cooperative game theory and real algebraic geomerty.
Susanne and I have continued our work towards a better
understanding of plankton in particular and microbes in
general. Check out The
Theory of Games and Microbe Ecology!
If you have trouble accessing any of these articles or would like
further information regarding my research, please send me an email!
I also review articles for the American Mathematical Society and
Zentralblatt Math. Check out MathSciNet for the
AMS, or for
Zentralblatt. You can find my reviews by searching for
"Rowlett" in the "Reviewer" search-field. Lately I have been too
busy for much reviewing, but I hope to get back to it soon!