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Research
INTERESTS: Analytic Number Theory; Harmonic Analysis; Automorphic Forms, Representations, and L-functions; Hyperbolic Geometry and Spectral Theory; Group Actions and Ergodic Theory; Sieve and Circle Methods

LETTERS / EXPOSITORY:

PAPERS: (which may vary slightly/severely from the published version)

PhD Students and Theses

Andre Hernandez-Espiet (Rutgers PhD, ongoing).

Edna Jones (Rutgers PhD, 2022). Griffiths Postdoc at Duke U.
"The Kloosterman Circle Method and Weighted Representation Numbers of Positive Definite Quadratic Forms."

Alex Karlovitz (Rutgers PhD, 2022), Lockheed Martin.
"An Extension of Hejhal's Algorithm to Infinite Volume Fundamental Domains."

Brooke Ogrodnik (Rutgers PhD, 2021), Metron.
"On the Local-Global Conjecture for Commutator Traces." J. Number Theory, 2022.

Katie McKeon (Rutgers PhD, 2018). IDA-CCR.
"Low-Lying Geodesics in an Arithmetic Hyperbolic Three-Manifold."

Arseniy Sheydvasser (Yale PhD, 2018). Postdoc at CUNY Grad Center, The Technion.
"A Corrigendum to Unreasonable Slightness", Math Monthly 2016;
"Orders of Quaternion Algebras with Involution";
"Quaternion Orders and Superpackings", JNT 2019.

Liyang Zhang (Yale PhD, 2017). Postdoc at Rutgers/Boston College; tenure-track Asst Prof at Fairfield University.
"Quantum unique ergodicity of degenerate Eisenstein series on GL(n)", Comm Math Phys, 2018.

Max Ehrman (Yale PhD, 2017). NSERC Postdoc, U Toronto Washington Nationals (MLB)
"Almost Primes in Thin Orbits of Pythagorean Triangles", IMRN 2017.

Shinnyih Huang (Yale PhD, 2015), Deutsche Bank.
"An Improvement On Zaremba's Conjecture", GAFA 2015.

Xin Zhang (Stony Brook PhD 2014), postdoc at Tel Aviv/UIUC/IHES, tenure-track Asst Prof at Hong Kong University.
"On the Local-Global Principle for Integral Apollonian-3 Circle Packings", Crelle 2015.

Ilya Vinogradov (Princeton PhD 2012), postdoc at Bristol/Princeton. Now: Two Sigma Investments.
"Effective bisector estimate with application to Apollonian circle packings", IMRN 2013.

REU Projects

  • 2022 Rutgers REU:

    • Enver Aman (Rutgers). Strong Approximation for the Markoff Surface

    Emma Hasson (Bard College at Simon's Rock). Analytic Number Theory in Lean

    Max Lind (Princeton). Maass Forms and Computations

    Archana Mohandas (MIT). Inversive Geometry in Lean


  • 2020 Rutgers REU:

    • Colin Fan (Rutgers), Weill Scholarship and Tilla Weinstein Award, now math PhD student at Northwestern.

    Brittany Gelb (Muhlenberg), now math PhD student and Presidential Fellow at Rutgers.

    Brandon Gomes (Rutgers), now software engineer.

    Saket Shah (Princeton), now math PhD student a U Michigan.

    Kai Shaikh (UC Berkeley), now math Masters student at Oxford, then math PhD student at U Toronto.


  • 2018 Rutgers REU: Taxonomy of Crystallographic Sphere Packings Paper is available here, to appear, J Number Theory, 2019.

    • Devora Chait (Queens College) Goldwater Scholarship winner 2019, now CS PhD student at NYU

    Alisa Cui (Yale)

    Gabriel Eiseman (Rutgers), now CS PhD student at Georgia Tech

    Zack Stier (Princeton), now NSF grad research fellow, math PhD student at UC Berkeley


  • 2015 Rutgers REU:

    • Peter Cohen (Bowdoin): Goldwater Scholarship honorable mention 2016, now: Operations Research PhD student at MIT.
    "An Analytic Heuristic for Multiplicity Computation for Zaremba's Conjecture", Journal of Number Theory Volume 184, March 2018, Pages 179--191.

    Jenny Crawford (Rowan)

    Lolly Kenigsberg (Stony Brook). Goldwater Scholarship winner 2016, now: math PhD student at Columbia

    Matt Kownacki (Rutgers), now: math PhD student at Boston College
    "On the Distribution of Witnesses in the Miller-Rabin Test"

    Organization

    I co-organize the Number Theory Seminar and the Colloquium at Rutgers.

    I co-organize (with Kolya Bogachev and Sasha Kolpakov) the Vinberg Distinguished Lecture Series.

    I co-organized (with Alla Detinko, Dane Flannery, Alexander Hulpke, and Misha Kapovich) the Collaborate@ICERM Group on Practical Computation with Infinite Linear Groups, Jun 6 - 10, 2022.

    I co-organized (with Alice Mark and Daniel Allcock) of the AIM workshop Arithmetic reflection groups and crystallographic packings, Mar 7 -11, 2022.

    I co-organized (with Alla Detinko, Misha Kapovich, Peter Sarnak, and Rich Schwartz) the virtual ICERM workshop Computational Aspects of Discrete Subgroups of Lie Groups, Jun 14 - 18, 2021.

    I co-organized (with Hee Oh and Alireza Salehi Golsefidy) an Oberwolfach Arbeitsgemeinschaft ("Study Group") on Thin groups and super-approximation, Oct 22 - 26, 2020 Oct 11 - 15, 2021.

    I co-organized (with Alice Mark and Misha Belolipetsky) the virtual AIM workshop Arithmetic reflection groups and crystallographic packings, Dec 14-18, 2020.

    I co-organized (with Dubi Kelmer and Min Lee) the ICERM workshop Lattice Point Distribution and Homogeneous Dynamics, Jun 22 - 26, 2020.

    I co-organized (with Alice Mark and Daniel Allcock) the AIM workshop Arithmetic reflection groups and crystallographic packings, Mar 16 - 20 Dec 14-18, 2020.

    I co-organized (with Kate Stange, Ellen Eischen, and Joel Kamnitzer) the ICERM workshop Illustrating Number Theory and Algebra, Oct 21 - 25, 2019.

    As part of our NSF FRG, Darren Long, Alex Lubotzky, Alan Reid, and I organized a research conference at Rice, on Thin groups in Number Theory, Geometry and Topology, May 14 - 18, 2018. It is supported in part by the NSF, BSF, and Rice.

    As part of our NSF FRG, Darren Long, Alex Lubotzky, Alan Reid, and I organized an educational workshop at IAS (Princeton), on Thin Groups and Super Approximation in Geometry and Arithmetic, March 28 - April 1st, 2016. It is supported in part by the NSF, BSF, IAS, and Rutgers.

    Dubi Kelmer and I organized an AMS Eastern Sectional Meeting (Nov 14-15, 2015, Rutgers) Special Session on Number theory, spectral theory, and homogeneous dynamics.

    Jeff Lagarias, Will Abram, and I organized an AMS-MAA Invited Paper Session at the MAA Centennial (Aug 6, 2015, DC) on the Arithmetic of Spheres.

    Uri Shapira, Tammy Ziegler, and I organized an Special Session on Dynamics and Number Theory for the Joint International Meeting of the AMS and IMU (Israel Math Union), June 16-19, 2014.

    Dubi Kelmer and I organized an AMS Eastern Sectional Meeting (Apr 6-7th, 2013, Boston College) Special Session on Counting and Equidistribution on Symmetric Spaces.

    Outreach / Microblog

    Some more mathematically "significant" tweets recorded here:

    • Why aren't we introducing the main ideas of Calculus *much* earlier, in grade school, so students aren't bombarded with a million new ideas at once?
    • Thread on Triple-Product L-functions:
    • Thread on Multiplication Algorithms ("New Math"):
    • Followup: A simple algorithm for division:
    • Thread on Siegel Zeros, Prime Polynomials, and the Class Number One problem:
    • Thread on the Arithmeticity Conjecture for Polyhedra:
    • Thread on why the "evidence" for the Collatz Conjecture may not be so strong:
    • An attempt to search for more such evidence:
    • How could one easily discover Euler characteristic?
    • Thread on the origins of Fourier analysis:
    • Thread on the Prime Number Theorem, Part 1:
      and Part 2:
    • Thread on the origins of trigonometry (the Sun and Half-Moon):
      and Part 2: an explainer on the seasons:
    • Thread on new conjectures about Fibonacci numbers (based on joint work above with Jeff Lagarias):
    • Thread on decidability of Goldbach and Riemann hypothesis (see sub threads for more discussion...)
    • The Langlands program in a tweet:
    • Why is the Riemann hypothesis hard? (Spoiler: if we actually knew the answer, we might have a better chance of proving it...):
    • "Factor City" where the building blocks are... primes!
    • Be careful with convergence and tetration...
    • The local-global conjecture for Apollonian packings: "grade" the packing by height=curvature; every large enough, admissible height has some circles in it:
    • People seem to be surprised by how simple the notion of a "radian" is:
    • A fun use of the Green-Tao theorem:
    • People seem to be surprised by how simple the notion of Pi is:
    • RIP John Conway, Part 1:
      and Part 2:
    • Cardano, Tartaglia, and the Cubic Equation (also, why imaginary numbers were accepted *before* negative numbers!...):
    • Some weird decimal expansions:
    • And a fun game on digits:
    • How do you take a p-adic square root?
    • Chebotarev Density Theorem, "low tech version":
    • Moduli space of triangles up to homothety:
    • Taming the gods with your bare hands (aka the birth of trigonometry):
    • Some non-standard divisibility rules:
    • How you can teach calculus to very little kids:
    • The ancients were right, the planets *do* move on epicycles! (relative to us, of course...):
    • The "Music of the Primes" (from a MoMath lecture):
    • What is conditional expectation?:
    • How could you discover, all on your own, that the angles in a triangle add up to 180?
    • Subtlety in some elementary geometry constructions and the parallel postulate:
    • Linear algebra in a tweet:
    • Where did set theory come from? Analysis!
    • Cardinality for kids:
    • What might math and math publishing look like in 20-50 years?
    • On the evolution of Proof and Rigor in Mathematics. And on PhD education.
    • Perspective art and projective geometry (for kids). How easy it is to miss a beautiful and simple idea for millennia!
    • Interview on The Cartesian Cafe with Tim Nguyen


    • Interview on Steve Strogatz's "Joy of X" podcast (by Quanta Magazine).

    • Interview with 3blue1brown's Grant Sanderson