Nathan Kaplan

nckaplan@math.uci.edu


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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, cokernels of random p-adic and integer matrices, and quadratic forms and lattices.  I also have a strong interest in promoting undergraduate research.

Publications and Preprints

  1. J. De Loera, L. Escobar, N. Kaplan and C. Wang, Sums of weighted lattice points of polytopes. Submitted (2024) 24 pp.
  2. J. Cullinan, S. Dobson, L. Frey, A. Hamakiotes, R. Hernandez, N. Kaplan, J. Mello, and G. Scullard, The probability of non-isomorphic groups structures of isogenous elliptic cuves in finite field extensions, II. J. Num. Theory 266 (2025), 131-165.
  3. N. Kaplan and H. Polo, A Goldbach theorem for Laurent series semidomains. Submitted (2024) 12 pp.
  4. N. Kaplan and J.-L. Kim, Hulls of projective Reed-Muller codes. Des. Codes Cryptogr. (2024), 17 pp.
  5. J. De Loera, L. Escobar, N. Kaplan and C. Wang, Sums of weighted lattice points of polytopes. 35th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2023), Sém. Lothar. Combin. 89B (2023), Article #17, 12 pp.
  6. E. Cotterill, N. Kaplan, and R. Vieira Costa, Cusps in 3 with prescribed ramification. Submitted (2023) 23 pp.
  7. J. Cullinan and N. Kaplan, The probability of non-isomorphic group structures of isogenous elliptic curves in finite field extensions, I. Res. Number Theory 9 (2023), Paper No. 57, 31 pp.
  8. A. Chen, N. Kaplan, L. Lawson, C. O'Neill, and D. Singhal, Enumerating numerical sets associated to a numerical semigroup. Discrete Appl. Math. 341 (2023), 218-231.
  9. N. Kaplan and D. Singhal, The expected embedding dimension, type and weight of a numerical semigroup. Enumer. Comb. Appl. 3 (2023), no. 2, Paper No. S2R14, 28 pp.
  10. G. Cheong and N. Kaplan, Generalizations of results of Friedman and Washington on cokernels of random p-adic matrices. J. Algebra 604 (2022), 636-663.
  11. N. Kaplan and V. Matei, Counting plane cubic curves over finite fields with a prescribed number of rational intersection points. Eur. J. Math. 7 (2021), 1137-1181.
  12. N. Kaplan and C. O'Neill, Numerical semigroups, polyhedra, and posets I: the group cone. Combinatorial Theory 1 (2021), #19.
  13. D. Glass and N. Kaplan,  D. Glass and N. Kaplan, Chip-firing games and critical groups. A project-based guide to undergraduate research in mathematics, 107--152, Found. Undergrad. Res. Math., Birkhäuser/Springer, Cham, 2020.
  14. J. Fulman, N. Kaplan, Random partitions and Cohen-Lenstra heuristics. Ann. Comb. 23 (2019), no. 2, 295--315.
  15. N. Kaplan, Weight enumerators of Reed-Muller codes from cubic curves and their duals. Arithmetic Geometry: Computation and Applications, 59--77 Contemp. Math., 722, Amer. Math. Soc., Providence, RI, 2019.
  16. G. Chinta, N. Kaplan, and S. Koplewitz, The cotype zeta function of ℤd. Indag. Math. 34 (2023), 643-659.
  17. 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. Discrete Math. 341 (2018), no. 10, 2949--2963.
  18. S. Anderson, W. Halbawi, N. Kaplan, H. H. López, F. Manganiello, E. Soljanin, and J. Walker, Representations of the multicast network problem. Algebraic Geometry for Coding Theory and Cryptography-- IPAM, Los Angeles, CA February 2016, Association for Women in Mathematics Series, 9, Springer, (2017), 1-23.
  19. N. Kaplan, Counting numerical semigroups. Amer. Math. Monthly 124 (2017), no. 9, 862-875.
  20. N. Kaplan, Where should I open my restaurant? Math. Mag. 90 (2017), no. 4, 278-285.
  21. D. Short, N. Kaplan, and D. Narayan. Flanking numbers and arankings of cyclic groups. J. Combin. Math. Combin. Comput. 99 (2016), 131-150.
  22. 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, 351-370.
  23. N. Kaplan and I. Petrow, Elliptic curves over a finite field and the trace formula. Proc. London Math. Soc., 115 (2017), 1317-1372.
  24. 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.
  25. S. Colton and N. Kaplan, The realization problem for delta sets of numerical semigroups. J. Commut. Algebra 9 (2017), no. 3, 313-339.
  26. 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, 4297-4340.
  27. N. Kaplan,  J. Marcinek, and R. Takloo-Bighash, Distribution of orders in number fields. Res. Math. Sci. 2 (2015) Art. 6, 57 pp.
  28. 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.
  29. N. Kaplan, MacWilliams identities for m-tuple weight enumerators. SIAM J. Discrete Math. 28-1 (2014), 428-444.
  30. N. Kaplan, Rational Point Counts for del Pezzo Surfaces over Finite Fields and Coding Theory. Ph.D. Thesis, Harvard University, 2013. 203 pp.
  31. 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.
  32. N. Kaplan and L. Ye, The proportion of Weierstrass semigroups, J. Algebra 373 (2013), 377-391. 
  33. 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.

SUMRY Papers

  1. 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.
  2. S. Atanasov, N. Kaplan, B. Krakoff, and J. H. Menzel, Counting finite index subrings of ℤn.  Acta Arith. 197 (2021), no. 3, 221-246.
  3. H. Constantin, B. Houston-Edwards, and N. Kaplan, Numerical sets, core partitions, and integer points in polytopes. Combinatorial and Additive Number Theory. II, 99-127, Springer Proc. Math. Stat., 220, Springer, Cham, 2017.

Trinity REU Papers

  1. S. Chapman, N. Kaplan, T. Lemburg, A. Niles, and C. Zlogar, Shifts of generators and delta sets of numerical monoids. Internat. J. Algebra Comput. (2014) no. 5, 655-669.
  2. 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.
  3. S. Chapman, J. Daigle, R. Hoyer, and N. Kaplan. Delta sets of numerical monoids using non-minimal sets of generators. Comm. Algebra 38 (2010), no. 7, 2622-2634.
  4. S. Chapman, R. Hoyer, and N. Kaplan, Delta sets of numerical monoids are eventually periodic, Aequationes Math. 77 (2009), no. 3, 273-279.

    These 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 domain.

Papers from Undergraduate Research Projects

  1. N. Kaplan, Flat cyclotomic polynomials of order four and higher, Integers 10 (2010), 357-363.
  2. N. Kaplan, Bounds for the maximal height of divisors of xn-1,  J. Num. Theory 129 (2009), 2673-88.
  3. N. Kaplan, Flat cyclotomic polynomials of order three, J. Num. Theory 127 (2007), no. 1, 118-126.
  4. 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, 141-147.
  5. D. Bowles, S. Chapman, N. Kaplan, and D. Reiser, On delta sets of numerical monoids,  J. Algebra Appl. 5 (2006), no. 5, 695-718.

Online Talks

In January 2021 I gave an MAA Invited Address at the virtual Joint Mathematics Meetings, Codes from Polynomials over Finite Fields. For the slides that accompany this talk, click here.

In January 2021 I gave a talk in the New York Number Theory Seminar, Counting Subrings of Zn. For the slides that accompany this talk, click here.

In March 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 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.