I came to Bard in 2006 after teaching for a year at Colby College. Before that, 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. For more, see my CV.
In my non-mathematical life I race bikes, go hiking and do a lot of yoga.
Office: Albee 300. Email: cullinan at bard dot edu. Off campus until Fall 2018
My main mathematical interests are in arithmetic geometry, representations of finite groups, and the arithmetic of orthogonal polynomials. My current research focuses on the torsion structure of abelian varieties and the arithmetic of orthogonal polynomials. I also have side interests in mathematics and politics (voting theory).
Links to my publications are below.
Subgroups of GL(3,q) with applications to Galois representations.
Algebraic properties of Kaneko-Zagier lifts of supersingular polynomials
(w/ F. Hajir). Proc. Amer. Math. Soc. 145 (2017), no. 6, 2291-2304.
Real preimages of duplication on elliptic curves. Missouri J. Math. Sci. 29 (2017), no. 1, 19-26.
On the Newton polygons of Kaneko-Zagier lifts of supersingular polynomials
(w/ R. Gajek-Leonard). Res. Number Theory 2 (2016), Art. 34
Voting with partially ordered preferences
(w/ S. Hsiao). To appear in the Mathematics and Social Justice edited volume, MAA.
On the solution to a generalized Fermat equation: arithmetic and combinatorics
(w/ A. Barghi). J. Comb. Number Theory 7 (2015), no. 3, 157-169.
Arithmetic properties of generalized Rikuna polynomials (w/ Z. Chonoles et al). Publ. Math. Besancon Algebre Theorie Nr., 2014/1, 19-33.
11. On the ranks of Jacobians of curves defined by Jacobi polynomials (w/ A. Etropolski and E. Sell). Rocky Mountain J. Math. 44 (2014), no. 1, 23-33.
10. On the Galois groups of Legendre polynomials (w/ F. Hajir). Indag. Math. (N.S.) 25 (2014), no. 3, 534-552.
9. A Borda Count procedure for partially ordered ballots (w/ S. Hsiao and D. Polett). Soc. Choice Welf. 42 (2014), no. 4, 913-926.
8. Specializations of generalized Rikuna polynomials (w/ C. Cass, A. Rasmussen, and D. Trifunovski). Int. J. 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. Int. J. Number Theory 8 (2012), no. 2, 321-334.
5. Ramification in iterated towers for rational functions (w/ F. Hajir). Manuscripta Math. 137 (2012), no. 3-4, 273-286.
4. Points of small order on three-dimensional abelian varieties; With an appendix by Yuri Zarhin. J. Algebra 324 (2010), no. 3, 565-577.
3. Algebraic properties of a family of Jacobi polynomials (w/ F. Hajir and E. Sell). J. Theor. Nombres Bordeaux 21 (2009), no. 1, 97-108.
2. A computational approach to the 2-torsion structure of an abelian threefold. Math. Comp. 78 (2009), no. 267, 1825-1836.
1. Local-global properties of torsion points on three-dimensional abelian varieties. J. Algebra 311 (2007), no. 2, 736-774
Some of my favorites:
Serre's Minerva lectures 1, 2, and 3 and Serre on Writing Mathematics Badly
Katz's IHES colloquium
The FLT Conference.
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 student at Minnesota), Rylan Gajek-Leonard (Bard '16, Cambridge Part III, PhD student at UMass).
Most of my teaching material is on Moodle, but here are a few additional things I've written up for students: Steinberg Representation, Discriminant Lemma, Vector Calculus Handout.