Research interests

Publications:

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)

A few working papers:

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

Talks

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)

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