Research
My work uses methods of applied mathematics, including spectral theory and other analytic tools, to address problems of number theory and particle physics. I use modern techniques of functional analysis and applications of partial differential equations to identify the causal mechanisms underlying the location of zeros of certain automorphic L-functions. These causal mechanisms go beyond recognizing correlation but serve to provide an understanding for how certain functions behave.
The following are questions central to my work:
What characteristics influence the location of zeros of certain functions? How can we use these qualities to provide explanatory results and proofs of this behavior?
How can tools not traditionally used in number theory (i.e. modern functional analysis and PDEs) be used to study number theoretic objects.
What are the connections between particle physics and number theory? How does the study of each field influence the other?
While these questions may appear unrelated, they are in fact deeply intertwined, as we will show below. My work thus far has applied functional analysis to two distinct areas: zeros of L-functions and graviton scattering.
This work is currently supported by NSF grant DMS-2302309.
Papers
On the size of families of unimodular roots with Henry Azubuike and Carrington Thom (submitted)
Convolution sums and Modular forms with Ksenia Fedosova and Danylo Radchenko (submitted: arXiv 2312.00722)
The D^6R^4 interaction as a Poincaré series, and a related shifted convolution sum with Stephen D. Miller and Danylo Radchenko (submitted: arXiv 2210.00047)
A Dedkind-Rademacher cocycle for Bianchi groups with Kalani Thalagoda and Tian An Wong (to appear in JTNB)
Shifted convolution sums from string theory with Ksenia Fedosova in Journal of Number Theory 2024 (arXiv 2307.03144 )
Equidistribution of elliptic Dedekind sums and generalized Selberg-Kloosterman sums with Tian An Wong in Research in Number Theory 2024
Whittaker Fourier type solutions to differential equations arising in string theory with Ksenia Fedosova in Communications in Number Theory and Physics 2023 (arXiv 2209.09319)
Pair correlation of Gamma_1(q) L-functions with Vorrapan Chandee and Xiannan Li in Mathematische Zeitschrift 2022
Linear Operators, the Hurwitz Zeta Function and Dirichlet L-Functions with Bernardo Bianco Prado in Journal of Number Theory 2020 (arXiv 1910.01192)
Differential equations in automorphic forms in Communications in Number Theory and Physics 2018 (arXiv 1801.00838)
In preparation...
A spectral interpretation of zeros of certain functions (arXiv 1706.08552)
Education and Equity...
Planning to Include chapter to appear in Advocating for Students of Color (edited by Pamela E. Harris and Aris Winger)
A Workshop to Build Community and Broaden Participation in Mathematics: Reflections on the Mathematics Project at Minnesota, with E. Banaian, S. Brauner, H. Chandramouli, A. Nadeau, and M. Philbin in PRIMUS
Other...
This is what success feels like: What I learned from applying to he NSF Postdoc twice, AMS Notices (August 2022)
Talks
Upcoming...
Arithmetic QFT Conference at Harvard's Center for Mathematical Sciences and Applications, March 25, 2024
Past...
U of Oklahoma, February 2024
JMM 2023, January 4-7, 2023
AMS Midwest Sectional, Omaha, NE, October 7-8, 2023
Pollica Physics Centre, May 2023
LSU Number Theory Seminar October 18, 2022
PANTS at UNCC, September 24-25, 2022
Crossing the Bridge, Newton Institute Cambridge, August 22-26, 202
AWM Symposium, Minneapolis, June 16-19
AMS Sectional Denver, May 14-15
TORA XI, OK State, April 1-3
AMS Sectional Purdue, March 26
Western Ontario Number Theory Seminar, Dec 10
Southern Illinois University Colloquium, Nov 1
OleMiss Number Theory Seminar, Oct 13
Maine/Quebec Number Theory Conference, Oct 2-3
IPMU conference on Number Theory, Strings and Quantum Physics, 31 May 2021
"Graviton scattering and differential equations in automorphic forms"
Graduate Research
If you are a graduate student potentially interested in working with me, please send me an email (kklingerlogan (at) ksu (dot) edu). I am very happy to make time to chat.
Foundational books in order of difficulty-ish:
Bruinier, Geer, Harder, Zagier’s The 1-2-3 of Modular Forms (Part 1)
Iwaniec’s Spectral Methods of Automorphic Forms
Goldfeld’s Automorphic forms and L-functions on GL(n,R)
Fleig, Gustafsson, Kleinschmidt, Persson Eisenstein series and automorphic representations (Chapters 1 and 2)
Some key papers:
Zagier 1982 The Rankin-Selberg method for automorphic functions which are not of rapid decay
Green, Miller, Vanhove 2015 SL(2,Z) invariance and D-instanton Contributions to the D^6R^4 Interaction
Garrett and Bombieri 2020 Designed Pseudo-Laplacians
Undergraduate Research
Aside from working with KSU's Math REU SUMaR, I have mentored undergraduate research for three students. They have received funding through the University of Minnesota Undergraduate Research Opportunities Program (UROP) and North Star STEM Alliance (The Minnesota Louis Stokes Alliance for Minority Participation). I have also helped organize undergraduate research in mathematics at the UMN.
I'm happy to serve as a mentor for undergraduate research projects or directed readings in number theory and analysis. Students who are interested in working with me should have taken Calculus I and II. If you're interested in working on a project with me, please send me a short email with the following items:
the reason you are interested in doing a math research project,
what interests you about math, and
a few times you are available to meet in the coming week for a 30 minute discussion about possible projects.