My research interests
I have been working on Representation theory of p-adic groups
and motivic
integration (these interests started with my Ph.D.
under the direction of
Thomas C. Hales
at the
University of Michigan ). Recently, I started getting interested in
the Trace
Formula and its applications.
Papers and Preprints
- Orbital integrals and normalizations of
measures (with appendix by Matthew Koster).
Lecture notes for the
winter
school at the
National University of Singapore (2018/19).
-
Counting abelian varieties over finite fields via Frobenius densities
(with Jeff Achter , S. Ali Altug ,
and Luis Garcia , and
with an appendix by Wen-Wei Li and
Thomas Rud ).
Algebra and Number Theory, to appear.
- Uniform analysis on local
fields and applications to orbital integrals ,
(with
R. Cluckers and
I. Halupczok ).
Trans. Amer. Math. Soc. Ser. B, Vol. 5, 125166. (2018).
- Elliptic curves, random
matrices and orbital integrals
(with Jeff Achter and with
an appendix by S. Ali Altug ).
Pacific J. Math. 286 no. 1 (2017), pp 1-24.
- The canonical measure on a
reductive p-adic group is motivic (with David Roe ) (2015).
Ann. Sci. Ec. Norm. Sup. 50, no. 2, 345-355, 2017.
This is a companion paper to
this one , removing an unnecessary technical difficulty about
motivic-ness of Haar measure on general reductive groups.
- Endoscopic transfer of
orbital intgerals in large residual characteristic
(with
T.C. Hales ).
Amer. J. Math. Volume 138, Number 1, February 2016 pp. 109-148
- Shalika germs for sl(n) and sp(2n) are motivic (with Lance Robson and
Sharon Frechette ).
In the proccedings of WIN-Europe, 2013 .
- Transfer principles for bounds of motivic exponential functions
(with
R. Cluckers and
I. Halupczok ).
In the proceedings of
Simons symposium on Families of Automorphic forms and the Trace Formula
(2014) .
- Motivic functions,
integrability, and uniform in p bounds for orbital integrals
(with
R. Cluckers and
I. Halupczok ).
Electronic Math. Research Announcements in Math. Sciences, volume 21, pp. 137-152, 2014.
published version .
This is an announcement (and summary of all the results) of the three
papers below, and related work.
- Appendix B to Sato-Tate
theorem for families and low-lying zeros of automorphic L-functions
by Sug Woo Shin and Nicolas
Templier.
(with
R. Cluckers and
I. Halupczok ).
Published version in Invent. Math.
In the appendix, we prove a uniform in p bound for normalized orbital
integrals.
-
Local integrability
results
in harmonic analysis on reductive groups in large positive characteristic.
(with
R. Cluckers and
I. Halupczok ).
Ann. Sci. Ec. Norm. Sup., Volume 47, No. 6 (2014), 1163-1195.
-
Integrability of oscillatory functions on local fields: transfer principles.
(with
R. Cluckers and
I. Halupczok ).
Duke Math J., vol.163 No. 8, pp. 1549-1600, (2014).
- Transfer to
characteristic zero -- appendix to
"The fundamental lemma of
Jacquet-Rallis in positive characteristics" by Zhiwei Yun.
Duke Math J. 156 No. 2 (2011), 220-227.
- On the computability
of some positive-depth characters near the identity (with R.
Cluckers ,
C. Cunningham , and
L. Spice ).
Represent. Theory 15 (2011), 531-567.
- An overview of arithmetic
motivic integration. (with Y. Yaffe)
in "Ottawa Lectures on p-adic Groups",
C. Cunnigham and M. Nevins, Eds., Fields Institute Monograph Series, 2009.
-
Motivic proof of a character formula for SL(2) (with
Clifton Cunningham ).
Experiment. Math. 18 No.1 (2009), pp.11-44.
-
Motivic Haar Measure
on Reductive Groups .
Canadian J. Math. 58 (2006), No. 1, 93--114.
- Motivic nature of
character values of depth-zero representations.
IMRN 2004, no. 34, 1735 - 1760.
-
Virtual Transfer
Factors ,
(with
T.C. Hales ).
Represent. Theory 7 (2003), 81--100 (electronic).
-
Common hypercyclic
vector for the multiples of backward shift
, (with E.V. Abakumov).
J. Funct. Anal. 200 (2003), no. 2, 494--504.
-
Composition operators on the space of Dirichlet series
with square summable coefficients, (with
Haakan Hedenmalm ).
Michigan Math. Journal, 46 (1999), no. 2, 313--329.
Motivic integration in representation theory
Here are some older papers by
my advisor and
others
showing that various constructions arising in representation theory
of p-adic groups can be expressed geometrically (for almost all p)
by means of motivic integration. The most spectacular application of this approach so far is the transfer priciple for the Fundamental Lemma.
These papers use the theory of motivic integration developed by
J. Denef, F. Loeser and R. Cluckers.
There is also a different (though related) theory of motivic integration,
developed by E. Hrushovski and D. Kazhdan. Here are some related papers (a
very incomplete list):
Motivic Integration links and resources
Representation Theory links and resources
Acknowledgment
Since 2006, my research has been supported by NSERC.