- An exact formula relating lattice points in symmetric spaces to the automorphic spectrum (
*To appear, Illinois J. Math.*) - extract an exact formula for smoothed lattice-point counting in symmetric spaces from a spectral identity obtained by producing two expressions for the automorphic fundamental solution (a Poincare series) to an invariant differential operator; develop a global automorphic Sobolev theory (results from PhD thesis)

- Fundamental solution for (Delta - lambda_z)^n on a symmetric space G/K (
*Updated May 2011*) - develop global zonal spherical Sobolev theory; use harmonic analysis of bi-K-invariant functions to obtain an integral representation for the fundamental solution; evaluate the integral using Hecke's identity, producing an explicit expression, with an eye towards further applications involving the associated Poincare series (results from PhD thesis)
- SL(2) Spherical Functions from Integral Representations (
*June 2010*) - compute the SL(2) spherical functions from integral over unipotent radical
- Spherical Function as Integral over Affine (
*Dec 2009*) - compute the GL(2) spherical functions as left-average-over-K of spherical vector in principal series; use Bruhat decomposition to transform GL(3)integral over K to an integral over (affine!) unipotent radical
- SL(2) Spherical Functions from Integral Representations (
*June 2010*) - compute the SL(2) spherical functions from integral over unipotent radical
- Towards GL(3) Spherical Functions (
*Oct 2009*) - Haar measure in Cartan coordinates, Casimir on principal series, Casimir on bi-K-invariant functions, PDE for spherical functions
- Integral Representations of L-functions (
*Feb 2009*) - brief discussion of GL(n)xGL(m) L-functions, starting with Iwasasa-Tate zeta integral, also Rankin-Selberg
- Harmonic Analysis of GL(2) and GL(3) Automorphic Forms (
*Jan 2009*) - L^2 decomposition of automorphic forms; cuspforms, pseudo-Eisenstein series, maximal and minimal parabolic Eisenstein series; constant terms and functional equations of Eisenstein series

- Fundamental Solution for (Delta - lambda)^N on G/K (
*Dec 2011*) - Notre Dame Lie theory seminar talk
- Geometry, Arithmetic, and Questions about Numbers (
*Nov 2011*) - Purdue University Calumet colloquium talk
- Number theoretic applications of the automorphic spectral theory of higher rank groups (
*Oct 2011*) - Ohio State number theory seminar talk
- What is a Number? (
*Oct 2011*) - colloquium talk at Wabash College: From doubling the square to Riemann's zeta function: historical developments in the concept of a number as well as contemporary perspectives and questions.
- Pythagorean Triples and Fermat's Last Theorem (
*May 2011*) - Science Speakers Series talk at Goshen College (expository, for a mixed audience)
- Automorphic Spectral Theory and Number Theoretic Applications (
*Oct 2010*) - colloquium talk at Reed College
- Spectral identities and exact formulas for counting lattice points in symmetric spaces (
*Nov 2009*) - presentation given at the Midwest Number Theory Conference for Graduate Students (Madison, 2009)

Goshen College Math