Department of Mathematics

Martin J. Mohlenkamp

Associate Professor
Department of Mathematics
College of Arts & Sciences
Ohio University

See my mathematical geneology.

Contact Information

• Solid mail: Department of Mathematics / Morton Hall 321 / 1 Ohio University / Athens OH 45701 USA
• E-mail: mjm@math.ohiou.edu
• Office: 315-B Morton Hall
• Phone: (740) 593-1259
• Fax: (740) 593-9805
• For office hours check one of the courses I am currently teaching.

Teaching

Courses

• Fall 2009: MATH 163A (also PED 113)
• Spring 2009: MATH 446/546 and MATH 649 (also PED 113)
• Winter 2009: MATH 266A and MATH 344 (also PED 113)
• 2008: Fall MATH 263A and MATH 444/544; Spring: MATH 266B and MATH 344; Winter: MATH 163A and MATH 263B
• 2007: Fall: MATH 163A and MATH 263A; Spring: MATH 344; Winter: MATH 163A and MATH 344
• 2006: Fall: MATH 163A and MATH 344; Spring: MATH 266B; Winter: MATH 266B and MATH 250
• 2005: Fall: MATH 266A and MATH 640A; Spring: MATH 266B and MATH 446/546; Winter: MATH 266A and MATH 163A
• 2004: Spring: MATH 640C and MATH 250; Winter: MATH 640B
• 2003: Fall: MATH 640A and MATH 163A; Spring: MATH 640C and MATH 263B; Winter: MATH 640B and MATH 163B
• 2002: Fall: MATH 640A

Resources

Numerical Analysis Comprehensive Exam

For PhD students planning to take the Numerical Analysis comprehensive exam, I have created a syllabus.

Wavelet Materials

I have organized some wavelet materials for a short course I taught in 2004.

Good Problems

We have developed a method to gently teach mathematical writing.
    Good Problems: teaching mathematical writing
D. Bundy, E. Gibney, J. McColl, M. Mohlenkamp, K. Sandberg, B. Silverstein, P. Staab, and M. Tearle.
University of Colorado APPM preprint #466, August 15, 2001.
Up-to-date materials through a Student's Guide.

C programming

For theoreticians who wish they could use computers more effectively, but don't know where to start, I have a virtual C programming class.

Wavelets, Their Friends, and What They Can Do for You. Martin J. Mohlenkamp and Maria Cristina Pereyra. EMS Series of Lectures in Mathematics, June 2008. (flyer; order in the Americas; order elsewhere)

Tensor Rank Visualization Tool. Martin J. Mohlenkamp. First release November 2009.

Research

General Interests

• Fast Algorithms: How to get a computer to give you the (right) answer as quickly as possible.
• Numerical Analysis: How to adapt a continuous problem (from physics, for example) into something a computer can solve (preferably with a fast algorithm).
• Applied Mathematics: How to bring the power of Mathematics to bear on problems from other fields (often using Numerical Analysis).
• (Computational) Harmonic Analysis: How to represent the world efficiently in terms of waves (and wavelets).
• Mathematics: How to see the beautiful structures all around us.

Projects and Publications

The Multiparticle Schrodinger Equation

It is notoriously difficult to compute numerical solutions to this basic governing equation in quantum mechanics. This project is big enough that it needs its own web page. See also the press release.

• Approximating a Wavefunction as an Unconstrained Sum of Slater Determinants
  Gregory Beylkin, Martin J. Mohlenkamp, and Fernando Perez
  Journal of Mathematical Physics, 49(3):032107, 2008.

(Copyright 2008 American Institute of Physics. This article may be found at http://link.aip.org/link/?JMP/49/032107. It can also be downloaded here for personal use only; any other use requires prior permission of the author and the American Institute of Physics.)

• Convergence of Green Iterations for Schrodinger Equations
  Martin J. Mohlenkamp and Todd Young
  in Recent Advances in Computational Science: Selected Papers from the International Workshop on Computational Sciences and Its Education. P. Jorgensen, X. Shen, C-W. Shu, N. Yan, editors. World Scientific, 2008.

(Preprint (.pdf))

• A Center-of-Mass Principle for the Multiparticle Schrodinger Equation
  Martin J. Mohlenkamp
  Submitted for publication August 2009.

(Preprint (.pdf))

Multivariate Regression

Regression is the art of building a function that approximately matches the data, and gives a reasonable value at new data locations. In this work we build a regression method that scales linearly with the dimension, and so can be used in high dimensions.

• Multivariate Regression and Machine Learning with Sums of Separable Functions.
  Gregory Beylkin, Jochen Garcke, and Martin J. Mohlenkamp
  SIAM Journal on Scientific Computing, 31(3): 1840-1857 (2009)

(link; preprint)

Numerical Analysis in High Dimensions

It is a common experience in numerical analysis to develop a very nice algorithm in dimension one or two, discover it is painfully slow in dimension three or above, and then give up and go work on other nice algorithms in dimension one or two. The cause of this is clear: computational costs grow exponentially with dimension. We now have a technique to bypass this Curse of Dimensionality in both low (2-4) and high (hundreds) dimensional settings.

• Numerical Operator Calculus in Higher Dimensions.
  Gregory Beylkin and Martin J. Mohlenkamp
  Proceedings of the National Academy of Sciences, 99(16):10246-10251, August 6, 2002.

(University of Colorado APPM preprint #476 August 2, 2001; Abstract and final journal version)

• Algorithms for Numerical Analysis in High Dimensions
  Gregory Beylkin and Martin J. Mohlenkamp
  SIAM Journal on Scientific Computing, 26(6):2133-2159, 2005

(University of Colorado APPM preprint #519, February 2004 (.pdf))

Trigonometric Identities

Although it seems like there should be nothing new in trigonometry, we stumbled upon some rather cute identities for sine of the sum of several variables.

• Trigonometric Identities and Sums of Separable Functions
  Martin J. Mohlenkamp and Lucas Monzon
  The Mathematical Intelligencer, 27(2):65--69, 2005.

(Preprint (.pdf))
An earlier version is available as:
    An Identity for Sine of the Sum of Several Variables
Martin J. Mohlenkamp and Lucas Monzon
University of Colorado APPM preprint #480, October 24, 2001.

Fast Algorithms for Oscillatory Matrices

From the Fall of 1999 to the Summer of 2002, I was primarily supported by a National Science Foundation Mathematical Sciences Postdoctoral Research Fellowship, to work on the Fast Application of Integral Operators with Oscillatory Kernels (.ps).

Spectral Projectors

• Fast Spectral Projection Algorithms for Density-Matrix Computations
  Gregory Beylkin, Nicholas Coult, Martin J. Mohlenkamp
  Journal of Computational Physics, 152(1):32-54, 10 June 1999. (ID jcph.1999.6215)

(University of Colorado APPM preprint #392, August 12, 1998; Final version from the ideal library)

Spherical Harmonics

My thesis was a Fast Transform for Spherical Harmonics. (Like an FFT, but for the sphere.) Completed in the spring of 1997 under the direction of R.R. Coifman at Yale University.
(Abstract, Thesis itself (.ps))

• A Fast Transform for Spherical Harmonics
  Martin J. Mohlenkamp
  Journal of Fourier Analysis and Applications, 5(2/3):159-184, 1999.

(Preprint)
A software library is available.
I have also created A User's Guide to Spherical Harmonics for those new to the area.

Resources

• Are you plagued and annoyed by chain letters? Here is an anti-chain letter that absolves you of all bad luck from not sending other chain letters. If you want to you can distribute it to others.