 My primary research interest is geometric analysis. 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...

• Together with Zhiqin Lu we have proven that One can hear the corners of a drum.

• 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.

• Errata to dynamics of asymptotically hyperbolic manifolds, corrects a few mistakes contained in the preceding article. It turns out that the trace formula contains some rather interesting and unexpected surprises!

• On the spectral theory and dynamics of asymptotically hyperbolic manifolds, was published in Annales de l'Institut Fourier. This is primarily a survey article but also contains a few new results.

• 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.

• Zeta-regularized determinants of Laplacians on polygons, was published in Oberwolfach Reports. This is a preliminary report on joint work with Clara Aldana concerning the zeta regularized determinant of the Laplacian on Euclidean polygonal domains.

• 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!