[expand all] [collapse all]

Collections [click to expand/collapse]

Research [click to expand/collapse]

Topics: My primary interests lie at the intersection of number theory and algebra, particularly understanding and discovering algebraic structures in arithmetic, and connecting different types of structures. Much of this is done with harmonic analysis. I've also done some things with graph theory, combinatorial optimization and spectral geometry.

Links in green are from conference proceedings. Please contact me for a copy of any paper you cannot download.

  1. The Jacquet-Langlands correspondence, Eisenstein congruences, and integral L-values in weight 2
    Mathematical Research Letters, to appear.
    We use the Jacquet-Langlands correspondence to generalize congruence results of Mazur to non-prime level and to Hilbert modular forms.

  2. Periods and nonvanishing of central L-values for GL(2n), with Brooke Feigon and David Whitehouse
    Submitted (revised Nov 4, 2015).
    Under some local hypotheses, we prove a relation between the nonvanishing of twisted central L-values for GL(2n) and periods over GL(n, E), where E is a quadratic extension. We also deduce analogous local results for supercuspidal representations.

  3. Test vectors and central L-values 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 L-value formula of my IMRN paper with Whitehouse, an average-value formula of Feigon-Whitehouse, and a nonvanishing mod p result of Michel-Ramakrishnan.

  4. A comparison of automorphic and Artin L-series of GL(2)-type agreeing at degree one primes, with Dinakar Ramakrishnan
    Contemporary Mathematics 664, Advances in the Theory of Automorphic Forms and their L-functions (Cogdell volume) (2016), 339-350.
    We show that if a 2-dimensional Artin representation corresponds to an automorphic representation outside of a density 0 infinite set of places of a certain form, then they correspond everywhere.

  5. Strong local-global phenomena for Galois and automorphic representations
    RIMS Kôkyûroku 1973, Modular forms and automorphic representations (2015), 120-130.
    An exposition of Strong Multiplicity One type results and refinements, with an aim to explain the results of my Contemp. Math. paper with Ramakrishnan.

  6. Distinguishing graphs with zeta functions and generalized spectra, with Christina Durfee [arXiv version]
    Linear Algebra and its Applications 481 (2015), 54-82.
    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.

  7. How often should you clean your room? with Krishnan (Ravi) Shankar
    Discrete Mathematics & Theoretical Computer Science, Vol. 17, No. 1 (2015), 413-442.
    We introduce and study a combinatorial optimization problem motivated by the question, "How often should you clean your room?" See also popular write-ups by Francis Woodhouse and Jon Kujawa.

  8. 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), 150-170.
    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.

  9. On central critical values of the degree four L-functions for GSp(4): a simple trace formula, with Masaaki Furusawa
    Mathematische Zeitschrift, Vol. 277, No. 1 (2014), 149-180.
    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 Gross-Prasad Conjecture (a nonvanishing theorem) for such representations.

  10. On central critical values of the degree four L-functions 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 Furusawa-Shalika, to the full Hecke algebra.

  11. Nonunique factorization and principalization in number fields
    Proceedings of the AMS, Vol. 139, No. 9 (2011), 3025-3038.
    This describes 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.

  12. A relative trace formula for a compact Riemann surface, with Mark McKee and Eric Wambach [errata, corrected version]
    International Journal of Number Theory, Vol. 7, No. 2 (2011), 389-429.
    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.

  13. On central critical values of the degree four L-functions for GSp(4): the fundamental lemma II, with Masaaki Furusawa [preprint version]
    American Journal of Mathematics, Vol. 133, No. 1 (2011), 197-233.
    We propose a different kind of relative trace formula than Furusawa-Shalika to relate central spinor L-values to Bessel periods, and prove the corresponding fundamental lemma. This relative trace formula has several advantages over the previous ones.

  14. Central L-values and toric periods for GL(2), with David Whitehouse
    International Mathematics Research Notices (IMRN) 2009, No. 1 (2009), 141-191.
    Using Jacquet's relative trace formula, we get a formula for the central value of a GL(2) L-function, refining results of Waldspurger.
    [Old version (Nov. 13, 2006). This uses a simpler trace formula but is much less general.]

  15. Central L-values and toric periods for GL(2)
    RIMS Kôkyûroku 1617, Automorphic Representations, Automorphic Forms, L-functions and Related Topics (2008), 126-137.
    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.

  16. Shalika periods on GL(2,D) and GL(4), with Hervé Jacquet [preprint version]
    Pacific Journal of Mathematics, Vol. 233, No. 2 (2007), 341-370.
    Here we use a relative trace formula to study period integrals, which yield results about exterior-square L-functions, and thus about transfer to GSp(4).

  17. Transfer from GL(2,D) to GSp(4)
    Proceedings of the 9th Autumn Workshop on Number Theory, Hakuba, Japan (2006), 10pp.
    These are notes from a talk explaining an application of my work with Jacquet (above) to the question of transferring representations to GSp(4).

  18. Four-dimensional Galois representations of solvable type and automorphic forms [abstract]
    Ph.D. Thesis, Caltech, 2004, 81pp.
    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).

  19. Modularity of hypertetrahedral representations [preprint version]
    Comptes Rendus Mathematique, Vol. 339, No. 2 (2004), 99-102.
    This proves a new case of modularity for four-dimensional Galois representations induced from a non-normal quartic extension. In particular, one obtains examples of modular representations which are not essentially self-dual.

  20. A symplectic case of Artin's conjecture
    Mathematical Research Letters, Vol. 10, No. 4 (2003), 483-492.
    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

Notes [click to expand/collapse]

Course notes → Courses page

Notes on Number Theory and Representation Theory

Notes on Graph Theory and Algebraic Combinatorics

Links [click to expand/collapse]

[expand all] [collapse all]

Best viewed in Lynx