I am a
primarily a number theorist but am also very interested in algebraic
geometry and combinatorics. More specifically, I am interested in
problems about rational points on varieties over finite fields,
arithmetic statistics, coding theory, and quadratic forms and
I also have a strong interest in promoting undergraduate research.
- J. Fulman, N. Kaplan, Random Partitions and Cohen-Lenstra Heuristics. Submitted (2018), 17 pp.
- N. Kaplan, Weight enumerators of Reed-Muller codes from cubic curves and their duals. Accepted pending minor revisions, Proceedings of AGC^2T 2017, (2017), 17 pp.
- G. Chinta, N. Kaplan, and S.
Koplewitz, The cotype zeta
function of ℤd. Submitted (2017), 21 pp.
- B. Braun, H. Corrales, S. Corry, L. García Puente, D. Glass, N. Kaplan, J. Martin, G. Musiker, and C. Valencia, Counting arithmetical
structures on paths and cycles. Submitted (2017), 21 pp.
- S. Anderson, W. Halbawi, N.
Kaplan, H. H. López, F. Manganiello, E. Soljanin, and J. Walker, Representations of the multicast network
Algebraic Geometry for Coding Theory and Cryptography-- IPAM, Los
Angeles, CA February 2016, Association for Women in Mathematics Series,
9, Springer, (2017), 1-23.
- N. Kaplan, Counting numerical semigroups.
Amer. Math. Monthly 124 (2017), no. 9, 862-875.
- N. Kaplan, Where should I open my
restaurant? Math. Mag. 90 (2017), no. 4, 278-285.
Short, N. Kaplan, and D. Narayan. Flanking
numbers and arankings of cyclic groups. J. Combin.
Math. Combin. Comput. 99 (2016), 131-150.
- J. Balakrishnan, W. Ho, N.
Kaplan, S. Spicer, W. Stein, and J. Weigandt, Databases of elliptic curves ordered by
height and distributions of Selmer groups and ranks. LMS J. Comput. Math. 19 (2016), issue A,
- N. Kaplan and I. Petrow, Elliptic curves over a finite
field and the trace formula. Proc. London Math. Soc., 115 (2017), 1317-1372.
- N. Kaplan and I. Petrow, Traces
of Hecke operators and refined weight enumerators of Reed-Solomon codes. Trans. Amer. Math. Soc. 370 (2018), 2357-2561.
- S. Colton and N. Kaplan, The
realization problem for delta sets of numerical semigroups. J.
Commut. Algebra 9 (2017), no. 3, 313-339.
- A. Bucur, C. David, B. Feigon,
N. Kaplan, M. Lalín, E. Ozman, and M. Wood, The
distribution of 𝔽q points on cyclic l-covers of genus g. Int. Math. Res. Not. IMRN (2016), no. 14,
- N. Kaplan, J. Marcinek,
and R. Takloo-Bighash, Distribution of orders
in number fields. Res.
Math. Sci. 2 (2015) Art. 6, 57 pp.
- J. Clancy, N. Kaplan, T. Leake,
S. Payne, and M. Wood, On a Cohen-Lenstra
heuristic for Jacobians
of random graphs. J.
Algebraic Combin. 42 (2015), no. 3, 701-723.
- N. Kaplan, MacWilliams
identities for m-tuple weight enumerators. SIAM J. Discrete
Math. 28-1 (2014), 428-444.
- N. Kaplan, Rational Point
Counts for del Pezzo Surfaces over Finite Fields and Coding Theory.
Ph.D. Thesis, Harvard University, 2013. 203 pp.
- N. Elkies and N. Kaplan, Extended
Abstract: An application of weighted theta functions to $t$-core
partitions and numerical semigroups, in Optimal and Near Optimal
Configurations on Lattices and Manifolds, C. Bachoc, P. Grabner, E.
Saff, and A. Schürmann eds., Oberwolfach Reports (2013), 2453-2456.
- N. Kaplan and L. Ye, The
proportion of Weierstrass semigroups, J. Algebra 373 (2013),
- N. Kaplan, Counting numerical
semigroups by genus and some cases of a question of Wilf. J.
Pure Appl. Algebra 216 (2012), no. 5, 1016-1032.
I have experience as a mentor for undergraduate research
projects. In the summer of 2014 and 2015 I was a mentor for
SUMRY, a research program for Yale
undergraduates. I worked as a
graduate assistant at the University of Minnesota-Duluth REU program in
the summers of 2008 and 2009. In the summer of 2007 I was the
graduate assistant at the Trinity University REU program.
- N. Kaplan, S. Kimport, R.
Lawrence, L. Peilen, and M. Weinreich, Counting arcs in
the projective plane via Glynn's algorithm. J. Geom. 108 (2017), no. 3, 1013-1029.
- S. Atanasov, N. Kaplan, B.
Krakoff, and J. Menzel, Counting finite index subrings
of Z^n. Submitted
(2018), 18 pp.
Here are links to some data and some programs related to this project.
- H. Constantin, B.
Houston-Edwards, and N. Kaplan, Numerical sets, core partitions, and
integer points in polytopes.
Combinatorial and Additive Number Theory II-- CANT, New York, NY, USA,
2015 and 2016, Springer Proc. Math. Stat., 220, Springer, (2017),
Trinity REU Papers
- S. Chapman, N. Kaplan, T.
Lemburg, A. Niles, and C. Zlogar, Shifts of generators
and delta sets of
Internat. J. Algebra Comput. (2014) no. 5, 655-669.
- D. Anderson, S. Chapman, N.
Kaplan, and D. Torkornoo. An
algorithm to compute omega-primality in a numerical monoid.
Semigroup Forum 82 (2011), no. 1, 96-108.
- S. Chapman, J. Daigle, R. Hoyer,
and N. Kaplan. Delta
sets of numerical monoids using non-minimal sets of generators.
Algebra 38 (2010), no. 7, 2622-2634.
- S. Chapman, R. Hoyer, and N.
Kaplan, Delta sets of numerical
Aequationes Math. 77 (2009), no. 3, 273-279.
papers are the result of two undergraduate research groups working on
problems related to the factorization theory of numerical semigroups.
We studied the Delta sets and the Omega function, two related
measures of how far a semigroups is from being a unique factorization
Undergraduate Research Projects
- N. Kaplan, Flat cyclotomic
polynomials of order four and
higher, Integers 10 (2010),
- N. Kaplan, Bounds for the
maximal height of divisors of x^n-1, J. Num. Theory 129
- N. Kaplan, Flat cyclotomic
polynomials of order three,
J. Num. Theory 127 (2007), no. 1, 118-126.
- C. Erickson, N. Kaplan, N.
Mendoza, A. Pacelli, and T. Shayler, Parametrized families
of quadratic number
fields with 3-rank at least 2,
Acta Arith. 130 (2007), no. 2,
- D. Bowles, S. Chapman, N.
Kaplan, and D. Reiser, On delta sets of
J. Algebra Appl. 5 (2006), no. 5, 695-718.
In March of 2014 I gave two talks during the tutorial week of the
IPAM long program "Algebraic Techniques for Combinatorial and
Computational Geometry". You can see a video of my second talk here: Polynomials
Over Finite Fields and the Finite Field Kakeya Conjecture.
For the slides that accompany this talk, click here.
I also gave a talk for the fourth workshop of this long program,
"Finding Algebraic Structure in Extremal Combinatorial Configurations".
The video is here: Arcs
in the Projective Plane.
In December of 2016 I gave a general audience talk at the IPAM Reunion
Conference. Slides are available here: No-Three-in-Line,
Intransitive Dice, and Other Amusements in Mathematics.