About Me
I came to Bard in 2006 after teaching for a year at Colby College. After two years as a visitor, I became tenure-track in 2008, received tenure in 2014, and was promoted to full professor in 2021. I chaired the Mathematics Department from 2014 -- 2023. I also teach regularly for the Bard Prison Initiative at all levels of the curriculum, including supervising senior theses.Before Bard, I received my PhD in mathematics from the University of Massachusetts in 2005 under the direction of Siman Wong. I graduated from Bates College in 1999 with a double-major in mathematics and physics. I am originally from Yonkers, NY and went to high school at ECFS. In my non-mathematical life I race bikes, go hiking, and do a lot of yoga.
Contact: cullinan at bard dot edu; Albee 300
Research
My research lies broadly within number theory with specific interests in arithmetic geometry (variation of arithmetic in isogeny classes) as well as the arithmetic of orthogonal polynomials with an emphasis on Galois theory. I also study nonlinear voting. Links to my publications are below.
32. The probability of non-isomorphic group structures of isogenous elliptic curves in finite field extensions, II (w/ S. Dobson, L. Frey, A. Hamakiotes, R. Hernandez, N. Kaplan, J. Mello, G. Scullard). Submitted 2024.
31. On the modular Plesken Lie algebra
Submitted, 2024.
30. On the discriminant of the truncated logarithmic polynomials (w/ R. Gajek-Leonard)
Submitted, 2023.
29. On the Irreducibility of the Krawtchouck Polynomials
Submitted, 2023.
28. Towards a Universal Gibbs Constant (w/ Antu Santanu)
Submitted, 2023.
27. Realizations of unisingular representations by hyperelliptic Jacobians.
Communications in Algebra 52 (12) 5297-5312 (2023)
26. The probability of non-isomorphic group structures of isogenous elliptic curves in finite field extensions, I (w/ N. Kaplan).
Research in Number Theory 9, 57 (2023)
25. On the arithmetic of Pade approximants to the exponential function (w/ V. Scheel).
Journal of the Ramanujan Mathematical Society 37 (3) 1-13 (2022)
24. On a probabilistic local-global principal for torsion on elliptic curves (w/ J. Voight, M. Kenney).
Journal de Théorie des Nombres de Bordeaux 34 (1) 41-90 (2022)
23. Divisibility of torsion subgroups of abelian surfaces over number fields (w/ J. Yelton).
Canadian Journal of Matheamatics 74 (1) 266-298 (2022)
22. Unisingular representations in arithmetic and Lie theory (w/ A. Zalesski).
European Journal of Mathematics (2021) 7 1645-1667
21. A remark on the group structure of 2-isogenous elliptic curves in towers of finite fields.
New York Journal of Mathematics 26 207 - 217 (2020)
20. Fixed-Point Subgroups of GL(3,q).
Journal of Group Theory 22 (5) 893 - 914 (2019)
19. On the Jacobians of curves defined by the Generalized Laguerre Polynomials.
Experimental Mathematics 28 (2019), no. 2, 223-232.
18. Voting with partially ordered preferences (w/ S. Hsiao).
Mathematics for Social Justice: Resources for the College Classroom
AMS/MAA Classroom Resource Materials 60 123-138 (2019)
17. A remark on the group structure of elliptic curves in towers of finite fields.
New York Journal of Mathematics. 24 857-865 (2018)
16. Algebraic properties of Kaneko-Zagier lifts of supersingular polynomials (w/ F. Hajir).
Proceedings of the American Mathematical Society 145 (2017), no. 6, 2291-2304.
15. Real preimages of duplication on elliptic curves.
Missouri Journal of Mathematical Sciences 29 (2017), no. 1, 19-26.
14. On the Newton polygons of Kaneko-Zagier lifts of supersingular polynomials (w/ R. Gajek-Leonard).
Research in Number Theory 2 (2016), Art. 34
13. On the solution to a generalized Fermat equation: arithmetic and combinatorics (w/ A. Barghi).
Journal of Combinatorics and Number Theory 7 (2015), no. 3, 157-169.
12. Arithmetic properties of generalized Rikuna polynomials (w/ Z. Chonoles et al).
Publications Mathematiques de Besancon: Algebre et Theorie des Nombres, 2014/1, 19-33.
11. On the ranks of Jacobians of curves defined by Jacobi polynomials (w/ A. Etropolski and E. Sell).
Rocky Mountain Journal of Mathematics 44 (2014), no. 1, 23-33.
10. On the Galois groups of Legendre polynomials (w/ F. Hajir).
Indagationes Mathematicae (N.S.) 25 (2014), no. 3, 534-552.
9. A Borda Count procedure for partially ordered ballots (w/ S. Hsiao and D. Polett).
Social Choice Welfare 42 (2014), no. 4, 913-926.
8. Specializations of generalized Rikuna polynomials (w/ C. Cass, A. Rasmussen, and D. Trifunovski).
International Journal of Number Theory 10 (2014), no. 3, 585-600.
7. Primes of prescribed congruence class in short intervals (w/ F. Hajir).
Integers 12 (2012) #A56.
6. Symplectic stabilizers with applications to abelian varieties.
International Journal of Number Theory 8 (2012), no. 2, 321-334.
5. Ramification in iterated towers for rational functions (w/ F. Hajir).
Manuscripta Mathematica 137 (2012), no. 3-4, 273-286.
4. Points of small order on three-dimensional abelian varieties; With an appendix by Yuri Zarhin.
Journal of Algebra 324 (2010), no. 3, 565-577.
3. Algebraic properties of a family of Jacobi polynomials (w/ F. Hajir and E. Sell).
Journal de Theorie des Nombres de Bordeaux 21 (2009), no. 1, 97-108.
2. A computational approach to the 2-torsion structure of an abelian threefold.
Mathematics of Computation 78 (2009), no. 267, 1825-1836.
1. Local-global properties of torsion points on three-dimensional abelian varieties.
Journal of Algebra 311 (2007), no. 2, 736-774
Miscellaneous
Some of my senior thesis students have gone on to pursue graduate degrees in mathematics: Andrija Perunicic (Bard '08, PhD Brandeis), Mona Merling (Bard '09, PhD Chicago), Anastassia Etropolski (Bard '11, PhD Emory), Galen Dorpalen-Barry (Bard '15, PhD Minnesota), Rylan Gajek-Leonard (Bard '16, Cambridge Part III, PhD UMass), Meagan Kenney (Bard '19, PhD student at Minnesota), Riti Bahl (Bard '21, PhD student at Emory).My research is primarily in arithmetic geometry and representation theory. The online databases that I use most in my research are the LMFDB, the Atlas of finite groups, and Tim Dokchitser's GroupNames project.
Here are a few expository things I've written up for students: Steinberg Representation, Discriminant Lemma.
I enjoy writing for Math Reviews. To read any of the ~100 reviews I've written, or to access the database in general, you'll need a site license.
Here are some of my favorite mathematical videos:
Serre's Minerva lectures 1, 2, and 3 and Serre on Writing Mathematics Badly
Katz's IHES colloquium
The FLT Conference.