Norbert Peyerimhoff's Home Page

University of Durham --- Department of Mathematical Sciences

Norbert Peyerimhoff

Department of Mathematical Sciences
Durham University
Science Laboratories
South Road
Durham, DH1 3LE

Phone:: 0191 334 3114
Department fax:: 0191 334 3051
e-mail:: norbert.peyerimhoff@durham.ac.uk

Research areas

Spectral Geometry
Schrödinger operators
Symmetric Spaces
Dynamical Systems
Graph Theory
Mathematical and computational aspects of X-ray crystallography

In July 2019, I gave a lecture course ''Curvature Notions on Graphs'' in the LMS Undergraduate Summer School at the University of Leeds. Please click here for the lecture notes.

EPSRC Grant 2013-2016

Principal Investigator in the joint EPSRC grant EP/K016687/1 Topology, Geometry and Laplacians of Simplicial Complexes with Stefan Dantchev and Ioannis Ivrissimtzis (both School of Engineering and Computing Sciences, Durham University) and Alina Vdovina (School of Mathematics and Statistics, Newcastle University).

Our Research Associate was Shiping Liu. In October 2016, he moved as a "Thousand Youth Talents Program" Professor (特任教授，2016“青年千人计划”) to the School of Mathematical Sciences, University of Science and Technology of China (USTC), Hefei. Click here for his new homepage.

Research interests

My research interests are centered around geometry. Of particular interest are relations between geometry and Laplace- and Schrödinger operators and relations between geometry and dynamical systems. In most cases, the spaces under considerations are non-positively curved or have at least a noncompact covering space. I am also interested in analogues and differences between the continuous setting of Riemannian manifolds and the discrete setting of graphs, in explicit constructions of expander graphs, spectral graph theory, and synthetic curvature notions in discrete spaces. Recently, I also started a joint project with OlexSys, a crystallographic software company on campus, to investigate the exciting mathematical theory behind crystallography, and we are jointly supervising a PhD student (Laura Midgley).

Edited book

Dynamics and Analytic Number Theory, London Mathematical Society Lecture Note Series 437, eds. D. Badziahin, A. Gorodnik, N. Peyerimhoff, Cambridge University Press, 2016.

Written by leading experts, this book explores several directions of current research at the interface between dynamics and analytic number theory. Topics include Diophantine approximation, exponential sums, Ramsey theory, ergodic theory and homogeneous dynamics. The origins of this material lie in the 'Dynamics and Analytic Number Theory' Easter School held at Durham University in 2014. Key concepts, cutting-edge results, and modern techniques that play an essential role in contemporary research are presented in a manner accessible to young researchers, including PhD students. This book will also be useful for established mathematicians. The areas discussed include ubiquitous systems and Cantor-type sets in Diophantine approximation, flows on nilmanifolds and their connections with exponential sums, multiple recurrence and Ramsey theory, counting and equidistribution problems in homogeneous dynamics, and applications of thin groups in number theory. Both dynamical and 'classical' approaches towards number theoretical problems are also provided.

"Metric Diophantine approximation - aspects of recent work" by Victor Beresnevich, Felipe Ramírez and Sanju Velani
"Exponents of Diophantine approximation" by Yann Bugeaud
"Effective equidistribution of nilflows and bounds on Weyl sums" by Giovanni Forni
"Multiple recurrence and finding patterns in dense sets" by Tim Austin
"Diophantine problems and homogeneous dynamics" by Manfred Einsiedler and Tom Ward
"Applications of thin orbits" by Alex Kontorovich

Preprints

Publications

Preprints

with I. Ivrissimtzis and A. Vdovina
Trivalent expanders, (Δ−Y)-transformation, and hyperbolic surfaces, 2018
with M. Täufer
Eigenfunctions and the Integrated Density of States on Archimedean Tilings , 2018
with D. Cushing, S. Kamtue, J. Koolen, Sh. Liu and F. Münch
Rigidity of the Bonnet-Myers inequality for graphs with respect to Ollivier Ricci curvature , 2018
with D. Cushing, Sh. Liu and F. Münch
Curvature calculations for antitrees , 2018
with E. Samiou
A support theorem for the X-ray transform on manifolds with plane covers, 2018
with M. Keller and Sh. Liu
A note on eigenvalue bounds for non-compact manifolds, 2017
with Sh. Liu, F. Münch and Ch. Rose
Distance bounds for graphs with some negative Bakry-Emery curvature, 2017
with Sh. Li and F. Münch
Rigidity properties of the hypercube via Bakry-Emery curvature, 2017
with M Egidi, Sh. Liu and F. Münch
Ricci curvature and eigenvalue estimates for the magnetic Laplacian on manifolds, 2016
with Sh. Liu and F. Münch
Curvature and higher order Buser inequalities for the graph connection Laplacian, 2015, MPIM preprint 2016(3)
with Sh. Liu and A. Vdovina
Signatures, lifts, and eigenvalues of graphs, 2014
with G. Knieper
Noncompact harmonic manifolds, 2013
Oberwolfach Preprints 2013-08
with I. Ivrissimtzis and A. Vdovina
Trivalent expanders and hyperbolic surfaces, 2012

Publications

Note: The papers listed here are not linked to retrievable files. If you want an offprint, let me know.

with Sh. Liu
Eigenvalue ratios of nonnegatively curved graphs
Combinatorics, Probability and Computing 27 (2018), 829-850
with J.R. Parker and K. F. Siburg
Minimizing length of billiard trajectories in hyperbolic polygons , 2016
accepted in Conformal Geometry and Dynamics
with D. Cushing and Sh. Liu
Bakry-Émery curvature functions of graphs
Canadian Mathematical Society, 53 pages, Published: 2018-07-05
with D. Bourne, D. Cushing, Sh. Liu and F. Münch
Ollivier-Ricci idleness functions of graphs
SIAM Journal of Discrete Mathematics 32 (2018), 1408-1424
with M. Täufer and I. Veselic
Unique continuation principles and their absence for Schrödinger eigenfunctions on combinatorial and quantum graphs and in continuum space
Nanosystems: Physics, Chemistry, Mathematics 8 (2017), 216-230
with Sh. Liu and F. Münch
Bakry-Emery curvature and diameter bounds on graphs
Calculus of Variations and Partial Differential Equations 57 (2018), Art. 67, 9
with M. Keller and F. Pogorzelski
Sectional curvature of polygonal complexes with planar substructures
Advances in Mathematics 307 (2017), 1070-1107.
with E. Samiou
The X-ray transform on 2-step nilpotent Lie groups of higher rank
Rendiconti Seminario Matematico Univ. Pol. Torino 74 (2016), 299-307
(special issue dedicated to the 23 - 26 February 2015 Workshop in memory of Sergio Console at the Dipartimento di Matematica, Università di Torino)
with C. Lange, Sh. Liu and O. Post
Frustration index and Cheeger inequalities for discrete and continuous magnetic Laplacians
Calculus of Variations and Partial Differential Equations 54 (2015), 4165-4196.
with N. Barker, N. Boston and A. Vdovina
Regular algebraic surfaces, ramification structures and projective planes
Beauville surfaces and groups, Springer Proc. Math. Stat., 123, 2015, pp. 15-33
with K. Leonhardt, M. Tautenhahn and I. Veselic
Wegner estimate and localization for alloy-type models with sign-changing exponentially decaying single-site potentials
Reviews in Mathematical Physics 27 (2015), 45pp
with G. Knieper
Geometric properties of rank one asymptotically harmonic manifolds
Journal of Differential Geometry 100 (2015), 507-532, see also arXiv:1307.0629
with G. Knieper
Harmonic functions on rank one asymptotically harmonic manifolds
Journal of Geometric Analysis 26 (2016), 750-781, see also arXiv:1404.4290
with N. Barker, N. Boston and A. Vdovina
An infinite family of 2-groups with mixed Beauville structure
International Mathematics Research Notices 11 (2015), 3598-3618
with E. Samiou
Integral Geometric Properties of Non-compact Harmonic Spaces
Journal of Geometric Analysis 25 (2015), 122-148
Y. Yang, N. Peyerimhoff and I. Ivrissimtzis
Linear correlations between spatial and normal noise in triangle meshes
IEEE Transactions on Visualization and Computer Graphics 19 (2013), no. 1, 45-55
with I. Ivrissimtzis
Spectral representations of vertex transitive graphs, Archimedean solids and finite Coxeter groups
arXiv:1106.2509
Groups, Geometry and Dynamics 7 (2013), no. 3, 591--615
with K. Gittins, M. Stoiciu and D. Wirosoetisno
Some spectral applications of McMullen's Hausdorff dimension algorithm
arXiv:1112.1020
Conformal Geometry and Dynamics 16 (2012), 184-203
with N. Barker, N. Boston and A. Vdovina
New examples of Beauville surfaces
Monatshefte für Mathematik 166 (2012), no. 3-4, 319-327
with A. Vdovina
Cayley graph expanders and groups of finite width
Journal of Pure and Applied Algebra 215 (2011), no. 11, 2780-2788
with S. Castle and K. F. Siburg
Billiards in ideal hyperbolic polygons
Discrete and Continuous Dynamical Systems A 29 (2011), no. 3, 893-908
with M. Keller
Cheeger constants, growth and spectrum of locally tessellating planar graphs
Mathematische Zeitschrift 268 (2011), no. 3-4, 871-886
with E. Samiou
Spherical spectral synthesis and two-radius theorems on Damek-Ricci spaces
Arkiv för matematik 48 (2010), no. 1, 131-147
with D. Lenz, O. Post and I. Veselic
Continuity of the integrated density of states on random length metric graphs
Mathematical Physics, Analysis and Geometry 12 (2009), no. 3, 219-254
with D. Lenz, O. Post and I. Veselic
Continuity properties of the integrated density of states on manifolds
Japanese Journal of Mathematics 3 (2008), Issue 1, 121-161
with G. Knieper
Ergodic properties of isoperimetric domains in spheres
Journal of Modern Dynamics 2 (2008), no. 2, 339-358
with D. Lenz, I. Veselic
Groupoids, von Neumann algebras and the integrated density of states
Mathematical Physics, Analysis and Geometry 10 (2007), no. 1, 1-41
with L. Karp
Geometric heat comparison criteria for Riemannian manifolds
Annals of Global Analysis and Geometry 31 (2007), 115-145
with O. Baues
Geodesics in non-positively curved plane tesselations
Advances of Geometry 6 (2006), no. 2, 243-263
with S. Klassert, D. Lenz, P. Stollmann
Elliptic operators on planar graphs: unique continuation for eigenfunctions and nonpositive curvature
Proc. Amer. Math. Soc. 134 (2006), no. 5, 1549-1559
Spherical means on compact locally symmetric spaces of non-positive curvature
Forum Mathematicum 18 (2006), no. 3, 391-417
with D. Lenz, I. Veselic
Integrated density of states for random metrics on manifolds
Proc. Lond. Math. Soc. (3) 88 (2004), no. 3, 733-752
with E. Samiou
The Cheeger constant of simply connected, solvable Lie groups
Proc. Amer. Math. Soc. 132 (2004), no. 5, 1525-1529
with D. Lenz, I. Veselic
Random Schrödinger operators on manifolds
Markov Processes and Related Fields 9 (2003), no. 4, 717-728
(Special issue containing contributions by participants of the conference
Mathematical aspects of random systemes and statistical mechanics, Mai 2002, CIRM-Luminy
in the honour of Leonid Pastur)
with K.F. Siburg
The dynamics of magnetic flows for energies above Mane's critical value
Israel Journal Math. 135 (2003), 269-298
Simplices of maximal volume or minimal total edge length in hyperbolic space
Journal Lond. Math. Soc. (2) 66 (2002), no. 3, 753-768
with I. Veselic
Integrated density of states for ergodic random Schrödinger operators on manifolds
Geom. Dedicata 91 (2002), 117-135
with L. Karp
Extremal properties of the principal Dirichlet eigenvalue for regular polygons in the hyperbolic plane
Arch. Math. 79 (2002), no.3, 223-231
Isoperimetric and ergodic properties of horospheres in symmetric spaces
Proc. Symp. Pure Math. 69
in: Smooth Ergodic Theory and its Applications
Amer. Math. Soc. (2001)
with O. Baues
Curvature and geometry of tesselating plane graphs
Discrete Comput. Geom. 25 (2001), no. 1, 141-159
with L. Karp
Spectral gaps of Schrödinger operators on hyperbolic space
Math. Nachrichten 217 (2000), 105-124
with L. Karp
Horospherical means and uniform distribution of curves of constant geodesic curvature
Math. Zeitschrift 231 (1999), no. 4, 655-677
with L. Karp
On the distribution of hypersurfaces equidistant from totally geodesic submanifolds in hyperbolic space
Analysis 18 (1998), no. 3, 217-225
with M. Lesch
On index formulas for manifolds with metric horns
Commun. Partial Differ. Equations 23 (1998), no. 3-4, 649-684
Areas and intersections in convex domains
Amer. Math. Monthly 104 (1997), no. 8, 697-704
with M. Hüttenhofer, M. Lesch
Mathematik in Anwendung mit C++
Algorithmen aus Analysis und Zahlentheorie
UTB für Wissenschaft, Uni Taschenbücher 1816 (1994), see below
with J. Brüning, H. Schröder
The $\overline{\partial}$-operator on algebraic curves
Commun. Math. Phys. 129 (1990), no. 3, 525-534

Authored book (German)

Mathematik in Anwendung mit C++, M. Hüttenhofer, M. Lesch, N. Peyerimhoff, Quelle & Meyer, UTB für Wissenschaft, 1994.

In diesem Buch werden ausgewählte Themen der Mathematik dargestellt, die sich besonders gut durch Algorithmen veranschaulichen lassen. Im zahlentheoretischen Teil des Buches geht es um die Teilbarkeit ganzer Zahlen und um Primzahlen -- eine Thematik, die in jüngster Zeit aufgrund moderner Verschlüsselungstechniken einen enormen Aufschwung erlebte. Polynome und die in den Naturwissenschaften bedeutsame Fouriertheorie sind Gegenstand des Analysis-Teils. Die Darstellung erstreckt sich von der Theorie bis zur kompletten Implementation in der Programmiersprache C++. Hierzu ist eine Begleitdiskette zum Buch beim Quelle & Meyer Verlag erhältlich. Eine ganze Reihe von Aufgaben und praktischen Beispielen dienen zur Vertiefung der vorgestellten Themen.

Supervision of Staatsexamens- and Diplomarbeiten in Bochum

Kerstin Danneberg (Staatsexamensarbeit, October 2002)

Annika Hass (Staatsexamensarbeit, October 2002)

Barbara Hermey (Staatsexamensarbeit, October 2002)

Thorsten Henkel (Diplomarbeit, March 2004)

Sandra Zeidler (Staatsexamensarbeit, October 2004)

Nina Maus (Diplomarbeit, October 2005)

PhD Supervision in Durham

Femke Douma (completed on April 2010), Thesis: Counting and Averaging Problems in Graph Theory

Michela Egidi (joint supervision with O. Post, completed on November 2015), Thesis: Geometry, Dynamics and Spectral Analysis on Manifolds

Supanat Kamtue, started in November 2017 (Royal Thai Government Scholarship)

Laura Midgley, started in October 2018 (European Regional Development Fund)
English translation (by Femke Douma) of the article
E. Artin Ein mechanisches System mit quasiergodischen Bahnen
Hamburger Mathematische Abhandlungen 3 (1924), 170-175

Teaching

Epiphany term 2004/05: 3H/4H Geometry

Michaelmas term 2005/06: Msc Course Measure Theory and Dynamics (see information)

Epiphany term 2005/06: 3H/4H Algebraic Geometry

Epiphany term 2005/06: 2H Contours and Hyperbolic Geometry (see information)

Michaelmas/Epiphany term 2006/07: 4H Riemannian Geometry

Epiphany term 2007/08: 3H/4H Analysis

Michaelmas term 2008/09: 3H/4H Geometry (see information)

Michaelmas term 2008/09: 4H Riemannian Geometry

Michaelmas and Epiphany terms 2009/10: 3H/4H Analysis

Michaelmas and Epiphany terms 2010/11: 4H Riemannian Geometry (see information)

Michaelmas and Epiphany terms 2011/12: 3H/4H Analysis (see information)

Michaelmas term 2012/13: 1H Core B2: Problem Solving

Michaelmas term 2012/13: 3H Differential Geometry

Michaelmas term 2013/14: 1H Core B2: Problem Solving (see information)

Epiphany term 2013/14: 3H/4H Algebraic Geometry (see information)

Michaelmas term 2014/15: 1H Core B1: Analysis

Epiphany term 2014/15: 1H Core B1: Analysis

Michaelmas term 2015/16: 1H Core B1: Analysis

Epiphany term 2015/16: 1H Core B1: Analysis

Michaelmas term 2016/17: 1H Core B1: Analysis

Michaelmas term 2017/18: 4H Riemannian Geometry

Michaelmas term 2017/18: PG Course Spectra and geometry of graphs and networks (MAGIC096)

Epiphany term 2017/18: 4H Riemannian Geometry

Michaelmas term 2018/19: 4H Riemannian Geometry

Michaelmas term 2017/18: PG Course Spectra and geometry of graphs and networks (MAGIC096)

Links:

Durham Pure Mathematics Home page
(formal page).
Durham Pure Mathematicians' page
(intended for members of the department but open to all).
Department of Mathematical Sciences home page.
The University of Durham home page