mixedmath

I'm a senior research scientist at ICERM and Brown University. I'm supported by the Simon's Collaboration on arithmetic geometry, number theory, and computation. I mostly research analytic number theory, arithmetic geometry, computational number theory, cryptography, and computation more generally.

Previously, I was a postdoctoral researcher at the University of Warwick. I received my PhD at Brown University with Jeff Hoffstein as my PhD advisor.

See this page for a broad overview of my research.

Themes of my Research

Most of my research falls along certain major themes (and sometimes falls across multiple themes), described here. The names are clickable and should point your browser to the specific reference from my complete list of publications and preprints below.^{1 1}If for some reason this doesn't work, each title string is unique and can be searched. Note the years in the titles are from the availability of the first preprint, which is different from both submission order and publication order.

1. Murmuration Phenomena and Machine Learning: This is very new, but already interesting. See my work with weight $k$ modular forms [BBLLD23] and with Maass forms [BLLDSHZ24].
   More recently, I've been looking at other applications of machine learning to problems in number theory and geometry. See [LD24Technical] and [LD24General] for preliminary reports.
2. Arithmetic Geometry: Classifying points on modular curves [BHKKLDMNS23].
   Counting polynomials with small Galois group [AGLOLDSZ22], or (very closely related) counting number fields [AGHLOLDTWZ22].
   Studying the distribution of right triangle triples [LDH20], or (very closely related again) congruent numbers [HKLDW19].
   Studying the distribution of low-lying zeros of families of hyperelliptic curves [BCDGLD16].
3. Arithmetic Statistics: My very first (eventually) published research project on one-level density [BCDGLD16]; combining harmonic analysis with algebraic number theory [AGHLOLDTWZ22] and [AGLOLDSZ22]; congruent numbers [HKLDW19],
4. Computational Number Theory: Classifying isolated points on modular curves [BHKKLDMNS23].
   Computing modular forms rigorously [BBBCCDLDLRSV20], and plotting modular forms [LD20b], [LDS21]. Also, computing and making visualizations for modular curves [LD22].
   Testing conjectures related to the distribution of right triangles [LDH20].
   Interoperability theories between different computer algebra systems [CKLDMPRTW18]. (This latter project was not ultimately successful).
5. Visualization: Visualizing modular forms [LD20b] and [LDS21]. Note that [LD20b] also describes how to do complex plotting catering to different types of vision, and is now implemented in sagemath.
   Visualizing modular curves [LD22].
6. Lattice Point Estimates: Uniform lattice point estimates with explicit dependence on the lattice: [LDTT17].
   Counting lattice points in hyperboloids [LD17a], $d$-dimensional spheres [HKLDW17d], and circles [HKLDW21].
7. Automorphic Forms: With my longest-running collaboration group, we investigate problems stemming from shifted convolutions for cusp forms in [HKLDW17a], [HKLDW17b], [HKLDW17c]; then adapt this to the Gauss circle and sphere problems in [HKLDW17d], [HKLDW21]; and then to triple correlations, first for cusp forms in [HKLDW20a], then for theta functions in [HKLDW20b]; we then further modify our arguments to apply to restricted sums along quadratic squares, first for cusp forms in [KLDWS23], then for divisor functions in [KLDW23].
   A technical set of pure harmonic analysis sharpening our previous results: [LD17a].
   Along the way, we'd thought we had an interesting approach for the congruent number problem [HKLDW19] (relating to triple correlations, but ultimately too hard to make work).
   Computing modular forms rigorously: [BBBCCDLDLRSV20].
   And visualizing modular forms: [LD21], [LDS21].
8. Explicit Number Theory: By Explicit Number Theory, I mean classical analytic number theory, either improving highly studied famous problems (such as the Gauss circle problem) or producing stronger estimates using nothing more than classical techniques (such as pure harmonic analysis based on nothing more than functional equations).
   $\Omega_{\pm}$ type limiting behavior using pure harmonic analysis [LD20a].
   Work in the Gauss circle and sphere problems [HKLDW17d] and [HKLDW21]. Also consider the sphere problem with uniform bounds for lattices, using nothing more than functional equations and harmonic analysis [LDTT17].
   Lattice points in hyperboloids, divisor sums, and the circle problem all together in [LD17a].
   Counting right triangles [LDH20]. Equivalently (nonobviously), counting arithmetic progressions of squares or certain points on elliptic curves HKLDW20b], [HKLDW19].
   Restricted divisor problem [KLDW23],
9. Research with Young Scientists: I like to guide young scientists (high schoolers and undergraduate mathematicians) through research projects. This includes PROMYS projects on Konigsberg primes [BCLMELD22] and prime sums [DEFLDX21]; and the report [LD15].
10. Expository: Work on the mean value theorem (yes, really) [LDW19] and [CLD16].
    Classroom capsule on quadratic reciprocity [LD15].
    A brief expository note on prime gaps [LD17b]. And a different expository note on sieves [LD13].

All Publications and Preprints

This is mostly in reverse chronological order. Please note that I upload preprints of all of my papers to the arxiv. If you have trouble accessing one of my papers, email me and I'll send you a copy.

1. [BLLDSHZ24]: Andrew R. Booker, Min Lee, David Lowry-Duda, Andrei Seymour-Howell, and Nina Zubrilina. Murmurations of Maass forms.
   arxiv preprint
   Talks discussing this work: July 2023.

2. [BBLLD23]: Jonathan Bober, Andrew R. Booker, Min Lee, and David Lowry-Duda. Murmurations of Modular Forms in the Weight Aspect.
   arxiv preprint
   Paper discussion page
   Talks discussing this work: July 2023.
   Quanta article discussing this work: link to article

3. [KLDW23]: Chan Ieong Kuan, David Lowry-Duda, and Alexander Walker. Counting divisors in the outputs of a binary quadratic form.
   arxiv preprint

4. [BHKKLDMNS23]: Abbey Bourdon, Sachi Hashimoto, Timo Keller, Zev Klagsbrun, David Lowry-Duda, Travis Morrison, Filip Najman, and Himanshu Shukla. Towards a classification of isolated $j$-invariants.
   To appear in Mathematics of Computation.
   arxiv preprint
   code repository.
   Paper discussion page

5. [KLDWS23]: Chan Ieong Kuan, David Lowry-Duda, Alexander Walker, and Raphael S. Steiner. Sums of cusp form coefficients along quadratic sequences.
   arxiv preprint.

6. [BCLMELD22]: Smitali Bhandari, Twyla Colburn, Patrick Lu, Haran Mouli, Helena Ellis Do Amaral, and David Lowry-Duda. Königsberg pseudoprimes and Fermat's little theorem for matrices.
   (This comes from a PROMYS project for high schoolers). Preprint available by request.

7. [AGHLOLDTWZ22]: Theresa C. Anderson, Ayla Gafni, Kevin Hughes, Robert J. Lemke Oliver, David Lowry-Duda, Frank Thorne, Jiuya Wang, and Ruixiang Zhang. Improved bounds on number fields of small degree.
   To appear in Discrete Analysis.
   arxiv preprint
   Paper discussion page.
   Other pages discussing this work: simplified proofs.
   Talks discussing this work: May 2022, October 2022.

8. [AGLOLDSZ22]: Theresa C. Anderson, Ayla Gafni, Robert J. Lemke Oliver, David Lowry-Duda, George Shakan, and Ruixiang Zhang. Quantitative Hilbert irreducibility and almost prime values of polynomial discriminants. To appear in International Mathematics Research Notices.
   arxiv preprint.
   Paper discussion page.

9. [LD21]: David Lowry-Duda. Sign changes of cusp form coefficients on indices that are sums of two squares.
   arxiv preprint.

10. [DEFLDX21]: Anupam Datta, Nir Elber, Raymond Feng, David Lowry-Duda, and Henry Xie. Prime Sums.
    arxiv preprint.
    (This comes from a PROMYS research project for high schoolers).

11. [HKLDW20b]: Thomas Hulse, Chan Ieong Kuan, David Lowry-Duda, and Alexander Walker. Arithmetic progressions of squares and multiple Dirichlet series.
    To appear in Mathematische Zeitschrift.
    arxiv preprint.

12. [LDTT17]: David Lowry-Duda, Takashi Taniguchi, and Frank Thorne. Uniform bounds for lattice point counting and partial sums of zeta functions. To appear in Mathematische Zeitschrift.
    arxiv preprint.
    Other pages discussing this work: technical point, bounds from FE.

13. [LD20b]: David Lowry-Duda. Visualizing modular forms. To appear in Arithmetic Geometry, Number Theory, and Computation volume in Simons Symposia series.
    arxiv preprint.
    Springer SharedIt link to published article
    Talks discussing this work: November 2019 (ICERM), November 2019 (Bowdoin), May 2021, August 2021

14. [LDH20]: David Lowry-Duda, with an appendix by Brendan Hassett. Congruent numbers with the same hypotenuse. To appear in Arithmetic Geometry, Number Theory, and Computation volume in Simons Symposia series.
    arxiv preprint.
    Springer SharedIt link to published article
    Paper discussion page

15. [BBBCCDLDLRSV20]: Alex J. Best, Jonathan Bober, Andy R. Booker, Edgar Costa, John Cremona, Martin Derickx, David Lowry-Duda, Min Lee, David Roe, Andrew V. Sutherland, and John Voight. Computing classical modular forms. To appear in Arithmetic Geometry, Number Theory, and Computation volume in Simons Symposia.
    arxiv preprint.
    Springer SharedIt link to published article

16. [HKLDW20a]: Thomas Hulse, Chan Ieong Kuan, David Lowry-Duda, and Alexander Walker. Triple correlation sums of coefficients of cusp forms. Journal of Number Theory 2020 (2021): 1-18.
    arxiv preprint.

17. [HKLDW21]: Thomas Hulse, Chan Ieong Kuan, David Lowry-Duda, and Alexander Walker. The Laplace transform of the second moment in the Gauss circle problem. Algebra and Number Theory 15 No. 1 (2021): 1-27.
    arxiv preprint.

18. [LDS21]: David Lowry-Duda and Adam Sakareassen. Towards flying through modular forms. Proceedings of Bridges (2021).
    arxiv preprint.
    Talks discussing this work: August 2021 at Bridges

19. [LD20a]: David Lowry-Duda. Non-real poles and irregularity of distribution. Journal of Number Theory, 217 (2020): 23-35.
    arxiv preprint.

20. [LDW19]: David Lowry-Duda and Miles Wheeler. Perturbing the mean value theorem: implicit functions, the Morse lemma, and beyond. The American Mathematical Monthly, 128 No. 1 (2020): 50-61.
    This paper received the Halmos - Ford award for expository excellence.
    arxiv preprint.
    Paper discussion page
    A separate note on choosing functions and making figures for this paper

21. [HKLDW19]: Thomas Hulse, Chan Ieong Kuan, David Lowry-Duda, and Alexander Walker. A shifted problem for the congruent number problem. Ramanujan Journal (2019): 1-8.
    arxiv preprint.
    Paper discussion page.

22. [HKLDW17d]: Thomas Hulse, Chan Ieong Kuan, David Lowry-Duda and Alexander Walker. Second moments in the generalized Gauss circle problem. Forum of Mathematics, Sigma, 6 (2018): e24.
    arxiv preprint.
    Paper discussion page.

23. [BCDGLD16]: Alina Bucur, Edgar Costa, Chantal David, João Guerreiro, and David Lowry-Duda. Traces, high powers and one level density for families of curves over finite fields. Mathematical Proceedings of the Cambridge Philsophical Society 165 No. 2 (2018): 225-248.
    arxiv preprint.

24. [CKLDMPRTW18]: John Cremona, Michael Kohlase, David Lowry-Duda, Dennis Müller, Markus Pfeiffer, Florian Rabe, Nicolas M. Thiéry, and Tom Wiesing. GAP/SAGE/LMFDB Interface Theories and Alignment in OMDoc/MMT for System Interoperability, part of the OpenDreamKit reports 2018.
    Available on github.

25. [HKLDW17c]: Thomas Hulse, Chan Ieong Kuan, David Lowry-Duda, and Alexander Walker. Short-Interval Averages of Sums of Fourier Coefficients of Cusp Forms. Journal of Number Theory, 173 (2017): 394-415.
    arxiv preprint.
    Paper discussion page.

26. [HKLDW17b]: Thomas Hulse, Chan Ieong Kuan, David Lowry-Duda, and Alexander Walker. Sign-Changes of Sums of Fourier Coefficients of Cusp Forms. Journal of Number Theory, 177 (2017): 112-135.
    arxiv preprint.
    Paper discussion page.

27. [HKLDW17a]: Thomas Hulse, Chan Ieong Kuan, David Lowry-Duda, and Alexander Walker. The Second Moment of Sums of Coefficients of Cusp Forms. Journal of Number Theory,  173 (2017): 304-331.
    arxiv preprint.
    Paper discussion page.

28. [CLD16]: Paul Carter and David Lowry-Duda. On Functions whose Mean-Value Abscissas are Midpoints. The American Mathematical Monthly, 124 No. 6 (2017): 535-542. arxiv preprint.
    Paper discussion page.

29. [LD17a]: David Lowry-Duda. On Some Variants of the Gauss Circle Problem. PhD Thesis at Brown University, 2017.
    arxiv preprint.

30. [LD15]: David Lowry-Duda. Unexpected Conjectures about -5 modulo primes, Classroom Capsule in College Math Journal Vol 46 No. 1, January 2015.

Technical Notes

The following are technical notes, typically describing implementation details of an algorithm or program that I've written.

1. [LD24General] David Lowry-Duda. General report on machine learning experiments for the Möbius function. A preliminary report (2024).
   preprint
   Discussion page

2. [LD24Technical] David Lowry-Duda. Technical report on machine learning experiments for the Möbius function. A preliminary technical report, adding detail for the general report (2024).
   preprint
   Discussion page

3. [LD22] David Lowry-Duda. Visualizing Modular Curves. A short technical note on implementation details for modular curve visualizations (2022).
   preprint
   Discussion page

Unpublished research notes

In addition to the unpublished notes on this site, I've written the following notes and do not intend to publish them.

1. [LD17b] David Lowry-Duda. A Short Note on Gaps between Powers of Consecutive Primes. Unpublished note (2017).
   arxiv preprint.
   Discussion page.

2. [LD13] David Lowry-Duda. A Friendly Intro to Sieves with a Look Towards Recent Progress on the Twin Primes Conjecture. Unpublished note (2013).
   arxiv preprint.
   Discussion page.

Finally, I have informal notes (i.e. written just for me or my collaborators) on my notes page.