Research
[click to expand/collapse]
The links in
green are notes from conference proceedings.
Please contact me for a copy of any paper you cannot
download.

Distinguishing graphs with zeta functions
and generalized spectra,
with Christina Durfee
Submitted. (Revised: Oct 17, 2014)
A fundamental problem in graph theory is: when is a graph determined by its
spectrum? We investigate analogues of this question with zeta functions
in place of spectrum. Our work suggests that zeta functions are more
effective at
distinguishing graphs than the usual types of spectra studied.

Test vectors and central Lvalues for GL(2),
with Daniel File and Ameya Pitale
Submitted.
We extend work of Gross and Prasad on test vectors for GL(2) to cases of joint
ramification, and use this to generalize the Lvalue formula of my
IMRN paper with Whitehouse, an averagevalue formula of
FeigonWhitehouse, and a nonvanishing mod p result of
MichelRamakrishnan.

Periods and nonvanishing of central Lvalues for GL(2n),
with Brooke Feigon and David Whitehouse
Submitted. (Revised: Apr 16, 2014)
Under some local hypotheses, we prove a relation between the nonvanishing
of twisted central Lvalues for GL(2n) and periods over
GL(n, E), where E is a quadratic extension.
We also deduce analogous local results for supercuspidal representations.

How often should you clean your room?
with Krishnan (Ravi) Shankar
Submitted. (Revised: Aug 31, 2014)
We introduce and study a combinatorial optimization problem motivated by
the question, "How often should you clean your room?"

Local root numbers, Bessel models, and a conjecture of Guo and Jacquet,
with Masaaki Furusawa
Journal of Number Theory, Special Issue in Honor of
Steve Rallis, Vol 146
(2015), pp. 150170.
We make a conjecture about the transfer of global SO(2)Bessel periods on
SO(2n+1) to GL(n, E) periods on GL(2n), where
E is the quadratic extension associated to the relevant form of SO(2),
and prove this when n = 2.

On central critical values of the degree four Lfunctions for GSp(4):
a simple trace formula,
with Masaaki Furusawa
Mathematische Zeitschrift, Vol 277, No. 1 (2014), pp. 149180.
As an application of the Fundamental Lemma I and III papers, we
prove a global Bessel identity for cuspidal automorphic representations of
GSp(4) which are supercuspidal at some component (plus some other local
hypotheses).
In particular, one obtains the global GrossPrasad Conjecture (a nonvanishing
theorem) for such representations.

On central critical values of the degree four Lfunctions for GSp(4): the fundamental lemma III,
with Masaaki Furusawa and Joseph Shalika
[preprint version]
Memoirs of the AMS, Vol 225, No. 1057 (2013), x+134pp.
We extend the fundamental lemma from our
American Journal paper below, as well as one due to FurusawaShalika,
to the full Hecke algebra.
 Nonunique factorization and principalization
in number fields
Proceedings of the AMS, Vol. 139, No. 9 (2011), pp. 30253038.
We describe the number and structure of irreducible factorizations of
an algebraic integer in the ring of integers of a number field, using
what were essentially Kummer's ideas.
 A relative trace formula for a compact Riemann surface, with
Mark McKee and Eric Wambach (Errata to
published version)
[Corrected version (Feb. 2, 2012)]
International Journal of Number Theory, Vol. 7, No. 2 (2011), pp. 389429.
We interpret a relative trace formula on a hyperbolic compact Riemann
surface as a relation between the period spectrum and ortholength
spectrum of a given closed geodesic. This leads to various asymptotic
results on periods and ortholengths, as well as some simultaneous nonvanishing
results for two different periods.
 On central critical values of the degree four
Lfunctions for GSp(4): the fundamental lemma II, with Masaaki
Furusawa [preprint version]
American Journal of Mathematics, Vol. 133, No. 1 (2011), pp. 197233.
We propose a different kind of relative trace formula than FurusawaShalika
to relate central spinor Lvalues to Bessel periods, and prove the
corresponding fundamental lemma. This relative trace formula has several
advantages over the previous ones.
 Central Lvalues and toric periods
for GL(2), with David Whitehouse
International Mathematical Research Notices (IMRN) 2009, No. 1 (2009), pp. 141191.
Using Jacquet's relative trace formula, we get a formula for the central value
of a GL(2) Lfunction, refining results of Waldspurger.
[Old version (Nov. 13, 2006). This uses a simpler trace formula but is much less general.]
 Central
Lvalues and toric periods for GL(2)
Automorphic Represenations, Automorphic
Forms, Lfunctions and Related Topics, Jan. 2125, 2008, RIMS, Kyoto,
Conference Proceedings.
This is basically an extended introduction to the above paper, ending with
an outline of the relative trace formula approach to proving special value
formulas.
 Shalika periods on GL(2,D) and GL(4),
with Herve Jacquet
[preprint version]
Pacific Journal of Mathematics, Vol. 233, No. 2 (2007), pp. 341370.
Here we use a relative trace formula
to study period integrals, which yield results about exteriorsquare Lfunctions, and thus about transfer to GSp(4).

Transfer from GL(2,D) to GSp(4)
Proceedings of the 9th Autumn Workshop on Number Theory,
Hakuba, Japan (2006).
These are notes from a talk explaining an application of my work with Jacquet
(above) to the question of transferring representations to GSp(4).
 Fourdimensional Galois representations of solvable type and automorphic forms
[abstract]
Ph.D. Thesis, Caltech, 2004.
This contains the results in the two papers below, as well as a
classification of representations into GSp(4,C) of solvable type and
minor additional modularity results.
I wrote an informal note about
my thesis for
the layman
(by which I mean the mathematically or scientifically minded layman).
 Modularity of hypertetrahedral
representations
Comptes Rendus Mathematique, Vol. 339, No. 2 (2004), 99102.
This proves a new case of modularity
for fourdimensional Galois representations induced from a nonnormal
quartic extension. In particular, one obtains examples of modular
representations which are not essentially selfdual.

A symplectic case of Artin's conjecture
Mathematical Research Letters, Vol. 10, No. 4 (2003), 483492.
This gives a new case of Artin's conjecture in GSp(4,C) by establishing
the more general Langlands' reciprocity law in this case.
Undergraduate Research Supervised
