Greg Knapp

Number Theory

Most of my research attention is devoted to number theory from an analytic perspective (which is different from analytic number theory), where my interests include Diophantine approximation, Thue equations, polynomial root separation, and other topics.

You can find my full research statement here (intended for all mathematicians, but please reach out if you have questions about something)!

Papers

(with Chi Hoi Yip.) Upper Bounds on Polynomial Root Separation. Accepted for publication in the Bulletin of the Australian Mathematical Society. arXiv preprint. 2024.
(with Seda Albayrak, Samprit Ghosh, and Khoa D. Nguyen.) On Certain Polytopes Associated to Products of Algebraic Integer Conjugates. Accepted for publication in the Proceedings of the American Mathematical Society. arXiv preprint. 2024.
The Number of Solutions to the Trinomial Thue Equation. arXiv preprint. Functiones et Approximatio Commentarii Mathematici, 69 (2): 247 - 270, 2023.

Here is the associated Jupyter notebook: trinomial_computations.ipynb
Here are the .csv files containing data about the number of solutions to trinomial Thue equations of fixed degree and height: trinomial_solution_data.zip

Polynomial Root Distribution and its Impact on Solutions to Thue Equations. dissertation.pdf. PhD dissertation. June, 2023.

Talks

I have given several versions of the same talk on my research into Thue equations. All have been titled something like "Bounds on the Number of Solutions to Thue's Inequality." The slides for the best version of the talk (which was given in the PIMS Emergent Research Seminar on October 11, 2023) are here: thue_equation_talk.pdf. Moreover, the talk was recorded, and you can view the talk here: https://mathtube.org/lecture/video/bounds-number-solutions-thue-equations.

While I have given talks about my work on polynomial root separation, none of them contain the substantial progress I made with Chi Hoi Yip in the paper above. As such, I won't link to any slides or recordings here.

I spoke about my research on products of powers of algebraic integer conjugates (joint work with Seda Albayrak, Samprit Ghosh, and Khoa Nguyen) at the most recent Alberta Number Theory Days conference at the Banff International Research Station on March 23, 2024. You can find the slides here: exp_inequalities_talk.pdf. Moreover, the talk was recorded, and you can view the talk here: https://www.birs.ca/events/2024/2-day-workshops/24w2020/videos/watch/202403231735-Knapp.html

Logic

I did my master's thesis under Colin McLarty at Case Western Reserve University in 2017.

"Minkowski's Linear Forms Theorem in Elementary Function Arithmetic." Electronic Thesis or Dissertation. Case Western Reserve University, 2017. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=case1495545998803274