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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

FORMALIZATIONS:

"Prime Number Theorem +" project (with Terry Tao), 2024--.

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

55. (with Swee Hong Chan and Igor Pak) Effective resistance in planar graphs and continued fractions, submitted 2025.
54. (with Swee Hong Chan and Igor Pak) Spanning trees and continued fractions, submitted 2024.
53. (with Dubi Kelmer and Chris Lutsko) Sign changes along geodesics of modular forms, submitted 2024.
52. (with Xin Zhang) On the Local-Global Conjecture for Combinatorial Period Lengths of Closed Billiards on the Regular Pentagon, submitted 2024.
51. (with Chris Lutsko) Counting in Lattice Orbits, submitted 2024.
50. (with Will Chen and Shehryar Sikander) Remarks on the Z/p Dijkgraaf-Witten invariants of 3D mapping tori, to appear, Comm Math Phys, 2025.
49. (with Dubi Kelmer and Chris Lutsko) Norm Bounds on Eisenstein series, IJNT, Vol. 20, No. 8, pp. 2083--2098 (2024).
48. (with Alla Detinko and Misha Kapovich) Problems on discrete subgroups of Lie groups and their computational aspects. Contemp. Math., 783, pp. 113--126 (2023).
47. (with Dubi Kelmer and Chris Lutsko) Sarnak's spectral gap question, Journal d'Analyse Mathematique, Vol. 151, pp. 171--179, 2023.
46. (with Chris Lutsko) Effective Counting in Sphere Packings, Journal of the Association for Mathematical Research, Vol 2, No 1, 15--52 (2024).
45. (with Kolya Bogachev and Sasha Kolpakov) Kleinian sphere packings, reflection groups, and arithmeticity, Math. of Comp. 93, No 345 (2024), 505--521.
44. (with Xin Zhang) On Length Sets of Subarithmetic Hyperbolic Manifolds, Math Annalen, Vol 389 (2024), 2783--2855.
43. (with Misha Kapovich) On Superintegral Kleinian Sphere Packings, Bugs, and Arithmetic Groups, J. Reine Angew. Math. 798 (2023), 105--142.
42. On Trace Sets of Restricted Continued Fraction Semigroups, In: Avila, A., Rassias, M.T., Sinai, Y. (eds) Analysis at Large: Dedicated to the Life and Work of Jean Bourgain. Springer (2022), 253--272.
41. (with Darren Long, Alex Lubotzky, and Alan Reid) What is... A Thin Group?, Notices AMS Vol 66 No 6, pp. 905--910, 2019.
40. (with Jeff Lagarias) On Toric Orbits in the Affine Sieve, Exper. Math, 30 (4), pp. 575--586 (2021).
39. (with Dubi Kelmer) Exponents for the Equidistribution of Shears and Applications, J. Number Theory 208 (2020), 1--46.
38. (with Kei Nakamura) Geometry and Arithmetic of Crystallographic Sphere Packings, Proc. Natl. Acad. Sci., 116 (2) 436-441, 2019.
37. The Local-Global Principle for Integral Soddy Sphere Packings, J. Mod. Dynamics Vol 15, pp. 209--236 (2019).
36. (with Arthur Baragar) Efficiently constructing tangent circles, Math Magazine Vol. 93, No. 1, 2020, 27--32.
35. (with Jean Bourgain) Beyond Expansion IV: Traces of Thin SemiGroups, Discrete Anal. (2018), Paper No. 6, 27 pp.
34. (with Jean Bourgain) Beyond Expansion III: Reciprocal Geodesics, Duke Math. J. 168 (2019), no. 18, 3413--3435.
33. Applications of Thin Orbits, Dynamics and analytic number theory, 289--317, London Math. Soc. Lecture Note Ser., 437, Cambridge Univ. Press, Cambridge, 2016.
32. (with Jean Bourgain and Michael Magee) Thermodynamic expansion to arbitrary moduli (appendix), Crelles Journal 753 (2019), 89--135
31. (with Dubi Kelmer) Effective equidistribution of shears and applications, Math. Ann. 370 (2018), no. 1-2, 381--421.
30. (with Peter McNamara and Geordie Williamson) Appendix to Schubert calculus and torsion explosion by Geordie Williamson, J. Amer. Math. Soc. 30 (2017), no. 4, 1023--1046.
29. (with Jiuzu Hong) Almost Prime Coordinates for Anisotropic and Thin Pythagorean Orbits, Israel J. Math, 209 no. 1, (2015) 397--420.
Corrigendum 2024.
28. (with Jean Bourgain) Beyond Expansion II: Low-Lying Fundamental Geodesics, JEMS, 19 (2017), 1331--1359.
27. Levels of Distribution and the Affine Sieve, Ann. Fac. Sci. Toulouse Math 23, (2014), no. 5, 933--966.
26. (with Dubi Kelmer) On the Pair Correlation Density for Hyperbolic Angles, Duke Math J, (2015), no. 3, 473--509.
25. (with Jean Bourgain) The Affine Sieve Beyond Expansion I: Thin Hypotenuses, IMRN (19), 9175--9205 (2015).
24. (with Valentin Blomer, Farrell Brumley, and Nicolas Templier) Bounding hyperbolic and spherical coefficients of Maass forms, J. Theor. Nombres Bordeaux 26, (2014), no. 3, 559--579.
23. From Apollonius To Zaremba: Local-Global Phenomena in Thin Orbits, Bull. AMS 50, (2013), No. 2, 187--228.
22. The Orbital Circle Method, in "Thin Groups and Super Approximation", MSRI Publications Vol 61, Cambridge University Press (2014), 93--106.
21. (with Jean Bourgain) On the Local-Global Conjecture for Integral Apollonian Gaskets, Inventiones Math, 196 (2014), no. 3, 589--650.
20. (with Dorian Goldfeld) On the GL(3) Kuznetsov formula with applications to symmetry types of families of L-functions, in Automorphic Representations and L-Functions, (Ed. by D. Prasad et al), Tata Institute, (2013), 263-310.
19. (with Jean Bourgain) On Zaremba's Conjecture, Annals Math 180 Issue 1, (2014), 137--196.
18. (with Dorian Goldfeld and Eric Stade), On the Kontorovich-Lebedev Transform, Commentarii Mathematici 60, (2011), no. 1-2, (Fujii volume) 37-46.
17. (with Jean Bourgain) On Zaremba's Conjecture, research announcement, CR Math Paris 349 (2011) no. 9, 493-495.
16. (with Dorian Goldfeld) On the Determination of the Plancherel Measure for Lebedev-Whittaker transforms on GL(n), Acta Arith 155 (Schinzel volume), 2012, pp. 15-26.
15. (with Jean Bourgain) On a Theorem of Friedlander and Iwaniec, Comptes Rendus Math. Acad. Sci. Paris, Vol. 348, No. 17-18, Sept 2010, pp. 947-950.

14. (with Jean Bourgain) Erratum to [13], GAFA Vol. 20, No.6, 2010, pp. 1548-1549.

13. (with Jean Bourgain) On Representations of Integers in Thin Subgroups of SL(2,Z), GAFA Vol. 20, No.5, 2010, pp. 1144-1174.

12. (with Jean Bourgain and Peter Sarnak) Sector Estimates for Hyperbolic Isometries, GAFA Vol. 20, No.5, 2010, pp. 1175-1200.
11. (with Jeff Lagarias) Stochastic Models for the 3x + 1 and 5x + 1 Problems, in "The Ultimate Challenge: The 3x+1 Problem," AMS 2010.
10. (with Hee Oh) Apollonian Circle Packings and Closed Horospheres on Hyperbolic 3-Manifolds, JAMS 24, (2011), no. 3, 603-648.

9. (with Hee Oh) Almost Prime Pythagorean Triples in Thin Orbits, J. Reine Angew. Math., Vol 667 (2012), 89--131.

8. (with Jeff Hoffstein) The First Non-Vanishing Quadratic Twist of an Automorphic L-series, preprint, 2009.

7. [PhD thesis] The Hyperbolic Lattice Point Count in Infinite Volume with Applications to Sieves, Duke Math J., Vol. 149, No. 1, 2009, pp. 1-36.

6. The Dirichlet Series for the Exterior Square L-function on GL(n), Ramanujan J. Vol 21, Issue 3 (2010).

5. A Pseudo-Twin Primes Theorem, in "Multiple Dirichlet Series, L-functions and Automorphic Forms", Birkhauser Progress in Math Series, Vol. 300, (2012), 287--298.

4. [undergrad "junior paper"] (with Aviezri Fraenkel) The Sierpinski Sieve of Nim-varieties and Binomial Coefficients, Combinatorial number theory, 209--227, de Gruyter, Berlin, 2007.

3. (with Mel Nathanson) Quadratic addition rules for quantum integers, Journal of Number Theory, Vol 117, Issue 1, March 2006, pp. 1-13.

2. (with Steven J. Miller) Benford's Law, values of L-functions and the 3x+1 Problem, Acta Arithmetica 120 (2005), 269-297.

1. [undergrad thesis] (with Yakov Sinai) Structure Theorem for (d,g,h)-Maps, Bulletin of the Brazilian Mathematical Society, New Series 33(2), 2002, pp. 213-224.

LETTERS / EXPOSITORY:

(with Glen Whitney) Polyplane: Art, Visualization, and Outreach, Math Horizons, 31(2), p. 34, 2023.
Notes on a Path to AI Assistance in Mathematical Reasoning, 2023. From National Academies panel discussion, video available here.
(with Joe Buhler and Steve Miller) Jerry Tunnell (1950--2022), Notices AMS Vol 70 No 6, pp. 945--952, 2023.
Foreword to: the Enjoyment of Math (by Rademacher and Toeplitz), Jan 2023, Princeton Press.
Foreword to Special Issue on Interactive Theorem Provers, June 2022, Experimental Mathematics.
Quanta magazine column: Glimpse at Langlands, June 2022. Accompanying video here .
Quanta magazine column: Remembering Eli Stein, Jan 2021. Accompanying video on RH here .
(with Brandon Gomes) Formalization of the Riemann Hypothesis in the Lean Theorem Prover, Oct 2020.
Book Review of Ghys's "A Singular Mathematical Promenade", The American Mathematical Monthly, Vol 127 (4), pp 381--384, 2020.
(with Amanda Folsom) Advice for the Campus Interview, Notices AMS Vol 66 No 10, pp. 1651--1655, 2019.
Remembering Jean Bourgain, Notices AMS Vol 68 No 6, pp. 948--949, 2021.
If you're going to celebrate Pi Day, you should be celebrating Archimedes! (Mar 2019).
Letter to Stephen Wolfram on The Audioactive Decay Mod 2. See accompanying mathematica file (Nov 2018).
Letter to Arthur Baragar on Crystallographic Sphere Packings. See accompanying mathematica file (Dec 2017).
Letter to Bill Duke on Inversive Coordinates (Nov 2017).
Letter to Kei Nakamura about Apollonian equilateral triangles (June 2017).
Letter to Jayadev Athreya about Apollonian radial densities (Oct 2014).

Brun's theorem via Selberg's upper bound sieve (Sept 2011).
A brutally explicit example of a compact arithmetic surface (Sept 2011). This has a corresponding Mathematica file. See Brooke Ogrodnik's version for a compact arithmetic 3-fold.
Notes on: ``Agencies Method'' in Coalitions and Games (posted Oct 2021, written ~2003/2004). This was undergraduate research work assisting John Nash.

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-organized (with Antoine Chambert-Loir and Heather Macbeth) the Simons Foundation workshoop Lean for Mathematicians, June 16 - 27, 2025.

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.