My Research

© Helmut Rüb, Kinder Uni Bonn, 2011
My research is in geometric 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.

The new stuff (pre-prints and to-appear)

Fourier expansions of vector-valued automorphic functions with non-unitary twists, with K. Fedosova and A. Pohl, (2022).

Polyakov formulas for conical singularities in two dimensions with C. L. Aldana and K. Kirsten (2020).

The heat kernel on curvilinear polygonal domains in surfaces with Medet Nursultanov and David Sher, pre-print submitted in 2019. Reviewing such a long paper will probably take some time...

Research Publications

The isospectral problem for flat tori from three perspectives, with E. Nilsson and F. Rydell, published online in the Bulletin of the American Mathematical Society, September 30, (2022).

Diversity strengthens competing teams, with C. J. Karlsson and M. Nursultanov, Royal Society Open Science, August 10, (2022).

Eigenvalue estimates for weighted Laplace equations on rough Riemannian manifolds with boundary with Lashi Bandara and Medet Nursultanov, Annali SNS (2021).

Donut choirs and Schiemann's symphony, Oberwolfach Reports, Workshop 2123, June 2021.

Biodiversity of marine microbes is safeguarded by phenotypic variability in ecological traits, with S. Menden-Deuer, M. Nursultanov, S. Collins, and T. Rynearson, PLOS One (2021).

Crystallographic groups, strictly tessellating polytopes, and analytic eigenfunctions with Max Blom, Henrik Nordell, Oliver Thim, and Jack Vahnberg, American Math Monthly (2021).

The Dirichlet isospectral problem for trapezoids with Hamid Hezari and Zhiqin Lu, Journal of Mathematical Physics (2021).

Decisions and disease: a mechanism for the evolution of cooperation with Carl-Joar Karlsson, in Nature's Scientific Reports (2020).

The new normal? Mathematics Indicates That an HIV-Style Strategy Could Be Applied to Manage the Coronavirus, in Springer's Mathematics Online First Collection, Math in the time of Corona, (2020).

The heat trace for the drifting Laplacian and Schrödinger operators on manifolds, with Nelia Charalambous, Asian Journal of Mathematics, (2019).

Second variation of Selberg zeta functions and curvature asymptotics, with Ksenia Fedosova and Genkai Zhang, Annals of Global Analysis and Geometry, (2019).

How to hear the corners of a drum with Medet Nursultanov and David Sher, Matrix Annals, (2019).

The Theory of Games and Microbe Ecology with Susanne Menden-Deuer, Theoretical Ecology, (2018).

A Polyakov Formula for Sectors, with Clara Aldana, Journal of Geometric Analysis (2018), 
and its Erratum (2019)

The Neumann Isospectral Problem for Trapezoids, with Hamid Hezari and Zhiqin Lu, Annales de l'Institut Henri Poincaré, (2017).

One can hear the corners of a drum, with Zhiqin Lu, Bulletin of the London Math Society, (2015).

The Sound of Symmetry, with Zhiqin Lu, American Math Monthly, (2015).
Erratum: a very careful reader found a few misprints in the article, which we have corrected here.

Eigenvalue estimates on Bakry-Émery manifolds, with Nelia Charalambous and Zhiqin Lu, Springer Proceedings, (2015).

A heat trace anomaly on polygons, with Rafe Mazzeo, Mathematical Proceedings of the Cambridge Philosophical Society, (2015).

Dynamics and zeta functions on conformally compact manifolds, with Pablo Suarez-Serrato and Samuel Tapie, Transactions of the AMS, (2015).

Love Games: A Game Theory Approach to Compatibility, with Kerstin Bever, J. Humanistic Math, (2015).

The fundamental gap and one dimensional collapse, with Zhiqin Lu, Proceedings of the C.R.M. (2014).

The level sets of typical games, Notices of the AMS, (2014).

Many Ways to Stay in the Game: Individual Variability Maintains High Biodiversity in Planktonic Microorganisms, with Susanne Menden-Deuer, Journal of the Royal Society Interfaces, (2014).

The fundamental gap of simplices, with Zhiqin Lu, Communications in Mathematical Physics, (2013).

Mathematical models for erosion and the optimal transportation of sediment with Bjorn Birnir, International Journal of Nonlinear Sciences and Numerical Simulation, (2013).

Zeta-regularized determinants of Laplacians on polygons, Oberwolfach Report, no. 25, doi:10.4171/owr/2012/25, (2012).

Eigenvalues of collapsing domains and drift Laplacians, with Zhiqin Lu, Math Research Letters, (2012).

On the discrete spectrum of quantum layers, with Zhiqin Lu, Journal of Mathematical Physics, (2012).

On the spectral theory and dynamics of asymptotically hyperbolic manifolds, Annales de l'Institut Fourier, (2010).

La geometrie de Bakry-Emery et l'ecart fondamental, Actes de la Seminaire de Theorie Spectrale et Geometrie, (2009-2010), en français.

Conformal deformations of conic metrics to constant scalar curvature, with Thalia Jeffres, Math Research Letters, (2010).

Dynamics of asymptotically hyperbolic manifolds, Pacific Journal of Mathematics (2009), and its Errata to dynamics of asymptotically hyperbolic manifolds, PJM (2014).

Spectral geometry and aysmptotically conic convergence, Communications in Analysis and Geometry, (2009), and a small correction here.

Hook a book - for free!

Blast into Math! A fun and rigorous introduction to pure mathematics. Bookboon, (2013).

Popular Science Publications

La dilemme de la maladie, Publié le 16 février 2021 par binaire, dans Le Monde.

听鼓的声音, by 陆志勤 和 罗茱莉 published in 數理人文, (2019). (My Chinese name is 罗茱莉).

Can one hear the corners of a drum? Well, yes! With Zhiqin Lu. Oxford University Press Blog, (2015).

PhD Thesis

Spectral geometry and asymptotically conic convergence, Stanford University, (2006).

Lectures and interviews

The mathematics of hearing the shape of a drum, on June 7th, 2022, for the Seminars on Inverse Problems Theory and Applications.

Trigonometric eigenfunctions, strictly tessellating polytopes, and crystallographic groups, is based on joint work with M. Blom, H. Nordell, O. Thim, and J. Vahnberg. This talk was happily a part of the conference Microlocal and Global Analysis, Interactions with Geometry held digitally from the University of Potsdam in February 2022.

Sounds of Symmetry, a short talk as part of the Mathematical Congress of the Americas Special Session on Spectral Geometry, (July 2021).

Spectral geometry in the clouds talk on joint work with C. Aldana and K. Kirsten: SE Play (k) 0:00 / 52:03 Same-same but different: Polyakov formulas via microlocal analysis and mathematical physics (July 2020).

5 Minute Talk, Mathematical Sciences Research Institute, Microlocal Analysis Semester (Fall 2019).

Geometric Analysis, Mathematical Sciences Research Institute Connections for Women Microlocal Analysis Workshop (2019).

The sound of a singularity? based on joint work with M. Nursultanov and D. Sher Banff International Research Station Workshop Probing the Earth and the Universe with Microlocal Analysis, (2019).

Public lecture titled The Spectrum: Incomputable yet physically tangible numbers, The University of Queensland Winter School, sponsored by Australian Mathematical Sciences Institute, (2018).

Eigenvalue and heat trace asymptotics for drifting Laplacians, a talk based on joint work with Nelia Charalambous given at Casa Matematica in Oaxaca, Mexico (2016).

Susanne Menden-Deuer speaking about our joint research on plankton, at the Conference on the Mathematics of Planet Earth, supported by the Society for Industrial and Applied Mathematics (SIAM) (2016).

A Polyakov formula for sectors, a talk based on joint work with Clara Aldana given at the conference, Evolution equations on singular spaces, at CIRM in Luminy, France (2016).

Bjorn Birnir speaking about our joint work on erosion, at the Kavli Institute for Theoretical Physics in Santa Barbara, (2013).

Interview at the Universite de Nantes (in French) following the conference Exposants critiques et groupes kleiniens, April 13-15, (2011) in Nantes, France. The interview was conducted by Leo Larroche, a member of the theater group Les n+1, of the company Les ateliers du spectacle, who were preparing a theatrical performance piece on mathematics.