Delving into Mathematical Mysteries and Spectral Correspondences.


I am a number theorist. I work on aspects of the Jacquet-Langlands correspondence in the Langlands program. To be specific: I study the discrete spectrum and the eigenfunctions of the Laplacian for a special class of “arithmetic” surfaces, namely those with constant negative curvature, finite area and a finite number of “punctures” or “cusps”. These are of fundamental interest in mathematics and physics. Such surfaces are called hyperbolic and when we “puncture” them we have essentially removed a point from the surface and pulled it away to infinity. The “arithmeticity” of the surface relates to its association with subgroups of the modular group. The eigenfunctions are called Maass waveforms and they remain a mystery some now sixty years after their discovery. No explicit construction exists for any of these functions on the full modular group. Their existence and information about their eigenvalues come mostly from the Selberg trace formula and from conjectures supported by exhaustive numerical computations. Maass wave forms, non-Euclidean sine waves, arise naturally in such seemingly diverse fields as number theory, dynamical systems, statistical mechanics, quantum chaos in theoretical physics and cosmology. I investigate spectral correspondences predicted by the Jacquet Langlands Correspondence between the spectra of some finite area arithmetic surfaces without “punctures” and a related class of finite area surfaces with “punctures”. Please see my most recent paper, in the Journal of Number Theory , with Stefan Lemurell on spectral correspondences for Maass waveforms on quaternion groups
Mathematics Education

I work on promoting diversity in mathematics and the sciences. My research is focused on the pedagogical, content, soccio-economic, cultural and structural issues which inform the low rates of mathematics and science achievement amount students of color. My work aims to increase the participation of underrepresented minorities in mathematics and science from grade school to graduate school.

I focus on:

  • Strategies which enhance the mathematical preparedness of underrepresented students for college level mathematics and those which support access to meaningful mathematics while in college;
  • Understanding and implementing strategies which foster retention and persistence in mathematics for all students but particularly those from underrepresented communities
  • Developing and understanding meaningful and innovative uses of emerging instructional technologies in support of mathematics instruction.
  • Building community/Social infrastructure that supports mathematics achievement in underrepresented communities, in particular, the challenges and opportunities for the development of institutional partnerships aimed at connecting mathematics to the community specifically the creating and sustaining of links between Middle Schools, High Schools, Two and four year College and Research Institutions in a particular geographic region.
  • Strategies to support the participation of members of underrepresented communities in research mathematics.
  • The use of number theory in the teaching and learning of mathematics and in mathematics education research.

Here are three resources that inform my thinking on mathematics and mathematics education:

Please see some recent work, with John Belcher, on the the potential of the use of music to build intuitions about deep mathematical questions and on Using a Mathematics Cultural Resonance Approach for Building Capacity in the Mathematical Sciences for African American Communities.

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