Research interests

Briefly: applying harmonic analysis of automorphic forms to number theory.

Publications:

Global automorphic Sobolev theory and the automorphic heat kernel.   Illinois J. Math. 65 (2021), no. 2, 261-286
Construct an automorphic heat kernel via its automorphic spectral expansion in terms of cusp forms, Eisenstein series, and residues of Eisenstein series; prove uniqueness as an application of operator semigroup theory; and prove the smoothness by proving that the automorphic spectral expansion converges in the C-infinity-topology
Constructing Poincare series for number theoretic applications.  New York J. Math. 22 (2016), 1221-1247.
Designing higher rank Poincare series for applications, e.g. for lattice point counting, moments of L-functions on higher rank groups, constructing eigenfunctions for pseudo-Laplacians; construct Poincare series by winding up solutions to differential equations on symmetric spaces; develop necessary global zonal spherical Sobolev theory
Branching of automorphic fundamental solutions.  Michigan Math. J. 64 (2015), no. 2, 263-277.
Pathwise meromorphic continuations may differ by an Eisenstein series
An exact formula relating lattice points in symmetric spaces to the automorphic spectrum.   Illinois J. Math. 56 (2012), no. 3, 805-823.
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
On graphs for which every planar immersion lifts to a knotted spatial embedding.  Involve 1 (2008), no. 2, 145-158.
(With Joel Foisy, Chad Versace, Alice Wilson) Topological graph theory: intrinsically linkable vs intrinsically linked graphs, intrinsically knottable vs intrinsically knotted graphs

A few working papers:

Notes on Jorgenson and Lang's Section 1.5, "Characters on the parabolics" (updated February 2020)
explicate and clarify Jorgenson and Lang's discussion in Section 1.5 of their book on heat Eisenstein series on SL(n,C), addressing some ambiguities and inaccuracies
Unbounded Operators on Hilbert Spaces (updated August 2016)
describe constructions (Friedrichs, Von Neumann) of self-adjoint extensions
Fundamental solution for (Delta - lambda_z)^n on a symmetric space G/K (updated June 2012)
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, now subsumed into paper on designing Poincare series, above)
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 (Updated July 2013)
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

Invited Research Talks

The automorphic heat kernel from a geometric perspective. (Nov 6, 2019)
University of Minnesota Automorphic Forms Seminar, Minneapolis, Minnesota.
The Automorphic Heat Kernel: Spectral and Geometric Points of View. (Oct 5, 2019)
Maine-Quebec Number Theory Conference, Orono, Maine.
Semigroups and uniqueness of solutions of differential equations in automorphic forms IV. (Nov 19, 2018)
University of Minnesota Automorphic Forms Seminar, Minneapolis, Minnesota.
Semigroups and uniqueness of solutions of differential equations in automorphic forms III. (Nov 12, 2018)
University of Minnesota Automorphic Forms Seminar, Minneapolis, Minnesota.
Semigroups and uniqueness of solutions of differential equations in automorphic forms II. (Nov 5, 2018)
University of Minnesota Automorphic Forms Seminar, Minneapolis, Minnesota.
Semigroups and uniqueness of solutions of differential equations in automorphic forms I. (Oct 29, 2018)
University of Minnesota Automorphic Forms Seminar, Minneapolis, Minnesota.
Automorphic Spectral Decomposition for GL(3,Z). (Nov 2017)
University of Minnesota Automorphic Forms Seminar, Minneapolis, Minnesota.
Automorphic Heat Kernels. (Sep 2017)
University of Minnesota Automorphic Forms Seminar, Minneapolis, Minnesota.
Global Automorphic Sobolev Spaces and Automorphic Heat Kernels (Sep 2017)
AMS Special Session on Automorphic Forms and L-functions, Fall Eastern Sectional Meeting, State University of New York at Buffalo, Buffalo, New York.
Applications of Modern Analysis to Automorphic Forms and Analytic Number Theory. (Oct 2016)
AMS Special Session on Representation Theory, Automorphic Forms, and Related Topics, Fall Central Sectional Meeting, University of St. Thomas, Minneapolis, Minnesota.
Zeros of Zeta Functions and Eigenvalues of Pseudo-Laplacians (Sep 2014) Narrative, Slides
AMS Central Sectional Meeting, University of Wisconsin-Eau Claire
Designing Poincare series for number theoretic applications (Mar 2014)
Lie Theory Seminar, University of Minnesota
Automorphic Spectral Identities from Differential Equations (Apr 2013)
Joint seminar: Collaborative Number Theory Seminar and Representation Theory Seminar, CUNY
Automorphic Spectral Identities from Differential Equations (Apr 2013)
Automorphic Forms Seminar, University of Minnesota
Automorphic spectral identities, integral moments of L-functions, and lattice points in a symmetric space (Feb 2012)
Group, Lie, and Number Theory Seminar, University of Michigan
Integral moments for Rankin-Selberg convolutions for GL(n) × GL(n) over a totally complex number field (Jan 2012)
Purdue University Automorphic Forms Seminar
Explicit 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
Automorphic Spectral Theory and Number Theoretic Applications (Oct 2010)
colloquium talk at Reed College
Relative trace formulas for GL(3), an alternate prescription for spectral identities (Nov 2008)
Report from AIM Workshop on GL(3), University of Minnesota

Contributed Research Talks

Eigenvalues of pseudo-Laplacians and Compact Periods of Eisenstein Series. (August 2015)
Illinois Number Theory Conference, University of Illinois at Urbana-Champaign
Branching of Automorphic Fundamental Solutions (July 2014)
Building Bridges: 2nd EU/US Workshop on Automorphic Forms and Related Topics, University of Bristol
Branching of Automorphic Fundamental Solutions (June 2014)
2014 Midwest Number Theory Conference for Graduate Students and Recent PhDs, University of Illinois at Urbana-Champaign
Designing Poincare Series for Number Theoretic Applications (Jan 2014)
Joint Mathematics Meetings, Baltimore, Maryland
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)

Colloquia and Other Presentations

Large Primes: Diamonds, Keys, and a One Million Dollar Question (Jul 2021)
REU, SUNY Potsdam
Large Primes: Diamonds, Keys, and a One Million Dollar Question (Feb 2019)
Mathematics Colloquium, Hamline University
Number Theory and Mathematical Physics: Applying applied math to the purest of pure math (Mar 2017)
MCS (Mathematics, Computer Science, and Statistics) Seminar, Gustavus Adolphus College
Prime Numbers, Quantum Chaos, and Pseudo-Laplacians (Sep 2015)
Center for Applied Mathematics Colloquium, University of St. Thomas
Prime Numbers, Quantum Chaos, and Pseudo-Laplacians (Nov 2014)
Mathematics & Statistics Colloquium, St. Cloud State University
Open Conjectures in Number Theory (May 2014)
Math and Actuarial Science Club, University of St. Thomas
Blending Evaluative and Formative Assessment (Jan 2014)
Joint Mathematics Meetings, Baltimore, Maryland
What is a Number? (Sept 2013)
University of Minnesota Talented Youth Mathematics Program Seminar
What is a Number? (Oct 2011)
Mathematics and Computer Science Colloquium, Wabash College
Pythagorean Triples and Fermat's Last Theorem (May 2011)
Science Speakers Series talk at Goshen College (expository, for a mixed audience)

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