Research interests (amongst other things):

• Computational aspects of automorphic forms:
  • Maass waveforms (non-holomorphic automorphic forms)
  • Harmonic weak Maass forms
  • Weak modular forms
  • Vector-valued modular forms 
  • Non-congruence modular forms
• Weil representations 
• Computational spectral theory for Fuchsian groups
• Mathematical aspects of Quantum Chaos
• Dynamical systems,  transfer operators and zeta functions 

Detailed research statement

Download research statement.

Other research activities

I am  a senior member of the NSF FRG (Focused Research Group) project "L-functions and Modular Forms" which is large project in the computational aspects of modular forms (holomorphic as well as Maass) and L-functions. The PI of the FRG is William Stein and co-PI's are: A. Booker, B. Conrey, N. Elkies, M. Rubinstein and P. Sarnak.

I am an editor of the L-functions and modular forms database project (www.lmfdb.org). This project is a joint effort of a large number of researchers in modular forms and L-functions. The goal is to make information (theoretical and numerically computed) related to modular forms and L-functions available to a larger mathematical audience (i.e. not only to experts).  The website is also meant to serve as a platform to make new data   available.

Master projects

If you are interested in number theory, modular forms or related topics (in particular involving computations) and want a topic for for a Bachelor, Master or Diploma thesis please contact me and describe your interest and background.

• Recent projects:
  • On statistics of Hecke eigenvalues, Diploma thesis by J.-L. Landvogt. 2010.

Previous Positions  

04/2008--03/2012

 Research assistant (Wissenschaftlicher Mitarbeiter) with Prof. Jan-Hendrik Bruinier at TU-Darmstadt.

06/2005--03/2008

 Research assistant (Wissenschaftlicher Mitarbeiter) with Prof. Dieter Mayer at TU-Clausthal.

02/2005--05/2005

Researcher (50%) at the Department of Mathematics, Uppsala University.

09/2002--12/2004

Lecturer (50%) at the Department of Mathematics, Uppsala University.

01/1999--09/2002

PhD student with a position at the Department of Mathematics, Uppsala University.

01/1999--09/2002

PhD student with stipend at the Department of Mathematics, Uppsala University.

    My Phd was obtained at Uppsala University (Sweden) and my advisor was prof. Dennis A. Hejhal (U. Uppsala and U. Minnesota MN.). Other members of the research group there are Andreas Strömbergsson, Pierre Bäcklund, Andreas Juhl and Andreas Södergren. A former member is Helen Avelin. See also the research group in analytic number theory. The research group in Uppsala was (is) a member of the EC-funded Research Training Network Mathematical Aspects of Quantum Chaos.  

Publications

• Weil representations associated with finite quadratic modules. Accepted for publication in Math. Z. 2012, (20p.)
• Newforms and Spectral Multiplicities for Γ0(9), arXiv:1106.5741. To appear in Proc. of the London Math. Soc. (30p.)
• Computation of harmonic weak Maass forms, joint with J.-H. Bruinier, Exp. Math., Vol 21 (2), 2012, 117--131. (arXiv:1101.3190.).
• On the Rankin convolution of Siegel Cusp Forms of degree two (joint with N.-P. Skoruppa and N. Ryan). To appear in Math. Comp., 2011 (16p.) 
• The Transfer Operator for the Hecke Triangle Groups, (j/w Dieter Mayer and Tobias Mühlenbruch), Discrete Contin. Dyn. Syst, Vol. 32, No. 7, 2012, 2453--2484. ( arXiv:09.12.2236v2)
• Symbolic dynamics for the geodesic flow on Hecke surfaces. Journal of Modern Dynamics, joint with D. Mayer. Volume 2, No. 4, 2008, 581--627 (arXiv:0801.3951)
• Computation of Maass Waveforms with Non-Trivial Multiplier Systems. Math. Comp., 77 (164), pp.2375-2416 (2008)
• Hecke Operators for Maass Waveforms on PSL(2,Z) with Integer Weight and Eta Multiplier, IMRN Vol. 2007, Art. ID rnm062 (25p.)

Book Chapter

• Maass waveforms on (Γ0(N),χ) (Computational aspects), Ch. 6 in Hyperbolic Geometry and Applications in Quantum Chaos and Cosmology, LMS Lecture Notes Series, No. 397. 2011. (preprint version: pdf ps)

Theses

 If requested I can also send out copies by email (in pdf-format) or paper.    

• PhD thesis: Computational Aspects of Maass waveforms  (Official persistent urn and errata to the printed version).  
•  On zeros of linear combinations of K-Bessel Functions (licentiate thesis)
• Iterated Function Systems, The Chaos Game And Invariant Measures (MSc. thesis) (as .ps.gz)  

Preprints

(If any mistakes are found in the preprints, please let me know.) 

• On the Weil representation for finite quadratic modules,  arXiv:1108.0202

• Computation of Selberg Zeta Functions on Hecke Triangle Groups, ArXiv:0804.4837v1.

• Symbolic dynamics for the geodesic flow on Hecke surfaces. Published by the Journal of Modern Dynamics. 
• Computation of Maass Waveforms with Non-Trivial Multiplier Systems. Published in Math. Comp.  

Invited Lecture Series

• “Arithmetic Quantum Chaos” (4 lectures) at the “Expository Quantum Lecture Series 5”, January 2012, Universiti Putra Malaysia, Serdang, Selangor, Malaysia. 
• “Maass waveforms for SL(2,Z)  and subgroups, from a computational point of view” (4 lectures) at the 2009 SMS Summerschool “Automorphic Forms and L-functions: Computational Aspects”, CRM, Montreal, Canada.
• “Spectral theory and automorphic forms for modular groups” (4 lectures) at “A Workshop on Modular Forms and Related Topics”, February 2012, American University of Beirut, Beirut, Lebanon. 

Invited Talks

• • Dimension formulas for vector-valued Hilbert modular forms
    •  09/08/12, Building Bridges, 1st EU-US conference on Automorphic Forms, Aachen, Germany.
    •   06/06/12, Workshop on automorphic forms and L-functions, IMAR, Bucharest, Romania.
• On multiplicities and newforms for Γ0(9)
  • TU Clausthal, February 24, 2012.
  • University of Bristol, November 23, 2011.
  • Universität Köln, July 3, 2011. 
  • Lille, May 6, 2010.
  • Humboldt Universität zu Berlin, January 19, 2010.
• On theoretical and computational aspects of the Weil representation. Kolloquium, Siegen, September 28, 2010.
• On some liftings from half-integral weight, scalar-valued modular forms to vector-valued modular forms. Durham Days on Modular Forms, 30th August - 3rd September 2010, Durham, UK. 
• Computational aspects of vector-valued Poincaré series for the Weil representation 
  • AKLS seminar on automorphic forms, 29th September 2009, University of Siegen, Siegen, Germany.
  •  23rd Annual Workshop on Automorphic Forms and Related Topics, 10th-13th March 2009, Bucknell University, Lewisburg, Pennsylvania. 
• A dynamical approach to the Selberg zeta function, Workshop:  DSG Workshop on Dynamical Systems and Quantum Mechanics, 25th March 2009, Queen Mary, University of London, UK.  
• Computations of (non-holomorphic) automorphic forms on GL2, Workshop:  Computations with Modular Forms, 18 August - 22 August 2008, Bristol, UK.
•    Computations of Selberg Zeta functions for Hecke Triangle Groups, MPIM Bonn, 2nd Japanese-German Workshop, 21 February 2008.  
  (A very similar talk was held at Kolloquium Mathematische Physik in Clausthal on 8 February).  
•    Symbolic Dynamics for the Geodesic Flow on Hecke Triangle Surfaces, Ryukyu University (Okinawa), 27 September 2007.  
•    Hecke Operators for PSL(2,Z) with integer weight and non-trivial multiplier
  •  AKLS-seminar, Aachen, 4 April 2007.
  • minisymposium at the DMV-Tagung in Berlin, 27 March 2007.
•    Transfer Operators For Hecke Triangle Groups 
  • DNA seminar, Uppsala, 4 September 2006.  
  • "Arithmatrics" conference, Bordeaux, 26 April 2006. 
• Computational Aspects of Maass waveforms with non-trivial multiplier systems
  • TU-Clausthal, 18 May 2005.
  • DNA Miniworkshop, KTH, 14 April 2005. 
• Presentation at my thesis defence.

Pictures

• Figures of zeros of Holomorphic cusp forms (computed by me) appear in 
  • Dana McKenzie, What's Happening in the Mathematical Sciences, Volume 8
  •  Peter Sarnak, Recent progress on the quantum unique ergodicity conjecture. Bull. Amer. Math. Soc. (N.S.) 48 (2011).
  • Amit Ghosh and Peter Sarnak, Real zeros of holomorphic Hecke cusp forms. Preprint. arXiv:1103.3262v1.
• Plots of Maass waveforms appear in:
  • Front page of: Nicolas Bergeron, Le Spectre des Surfaces Hyperbolique, CNRS Éditions, EDP Sciences, 2011.
• Misc. Figures:
  • Gallery of Maass Waveforms, and 
  • a few unsorted pictures.

Programs

As is the nature of these kind of things, most programs listed as available available are also under (perpetual) development. 

• Ongoing projects in Sage/Page: 
  • Efficient algorithms for non-congruence subgroups (enumerate, classify up to conjugacy etc.)
  • Holomorphic modular forms for non-congruence subgroups.
  • Maass forms for subgroups of the modular group.
  • Harmonic weak Maass forms for subgroups of the modular group.
  • Vector-valued modular forms and harmonic weak Maass forms for the Weil representation.
  • The Selberg zeta function (is currently available for Hecke triangle groups)
• My current Sage projects (most) are available from my psage clone: http://code.google.com/r/fredrik314-psage/
• A Sage program for the Weil representation associated to arbitrary finite quadratic modules (over the rationals) is included in the fqm package of Nils Skoruppa et. al and can be obtained from: http://hg.countnumber.de/fqm-devel
• Available for separate download: 
  • Harmonic weak Maass forms for the Weil representation of (ZZ, x->Nx^2). Implementation in Sage.
  • Holomorphic Poincaré series for the weil representation of (ZZ, x->Nx^2). Implementation in Sage.
  • Computing Selberg Zeta for Hecke triangle groups in MuPAD SelbergZHecke.mu
     
  •   Compute Maass waveforms in Fortran 90, maasswf-src-0.99v2.tar.gz.  README.txt
  • Multiprecision implementation of LAPACK in Fortran 90, using ARPREC. mplapack.
  • The Whittaker W-function (efficient for large parameter values). Implementation in Fortran 90, whittakerw.
• Previous projects in Fortran. These are no longer being developed but the code is available by request:
  • Computation of Harmonic weak Maass forms for the Weil representation.
  • Computation of Harmonic weak Maass forms for congruence subgroups.
• Fourier coefficients for Gamma_0(229) and R=0.