Advertisement

Lateral Movement in Undergraduate Research: Case Studies in Number Theory

Chapter
  • 435 Downloads
Part of the Foundations for Undergraduate Research in Mathematics book series (FURM)

Abstract

We explore the thought processes, strategies, and pitfalls involved in entering new territory, developing novel projects, and seeing them through to publication. We propose twenty-one general principles for developing a sustainable undergraduate research pipeline and we illustrate those ideas in three case studies.

References

  1. 1.
    Soren Aletheia Zomlefer, Stephan Ramon Garcia, and Lenny Fukshansky. One conjecture to rule them all: Bateman–Horn. Expo. Math. (in press). https://arxiv.org/abs/1807.08899.
  2. 2.
    Carlos A. M. André. Basic characters of the unitriangular group. J. Algebra, 175(1):287–319, 1995.Google Scholar
  3. 3.
    Carlos A. M. André. The basic character table of the unitriangular group. J. Algebra, 241(1):437–471, 2001.Google Scholar
  4. 4.
    Carlos A. M. André. Basic characters of the unitriangular group (for arbitrary primes). Proc. Amer. Math. Soc., 130(7):1943–1954 (electronic), 2002.Google Scholar
  5. 5.
    Levon Balayan and Stephan Ramon Garcia. Unitary equivalence to a complex symmetric matrix: geometric criteria. Oper. Matrices, 4(1):53–76, 2010.Google Scholar
  6. 6.
    Paul T. Bateman and Roger A. Horn. A heuristic asymptotic formula concerning the distribution of prime numbers. Math. Comp., 16:363–367, 1962.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Paul T. Bateman and Roger A. Horn. Primes represented by irreducible polynomials in one variable. In Proc. Sympos. Pure Math., Vol. VIII, pages 119–132. Amer. Math. Soc., Providence, R.I., 1965.Google Scholar
  8. 8.
    Samuel G. Benidt, William R. S. Hall, and Anders O. F. Hendrickson. Upper and lower semimodularity of the supercharacter theory lattices of cyclic groups. Comm. Algebra, 42(3):1123–1135, 2014.Google Scholar
  9. 9.
    Bruce C. Berndt, R.J. Evans, and K.S. Williams. Gauss and Jacobi sums. Canadian Mathematical Society series of monographs and advanced texts. Wiley, 1998.Google Scholar
  10. 10.
    Dario Bini and Milvio Capovani. Spectral and computational properties of band symmetric Toeplitz matrices. Linear Algebra Appl., 52/53:99–126, 1983.Google Scholar
  11. 11.
    Bryan Brown, Michael Dairyko, Stephan Ramon Garcia, Bob Lutz, and Michael Someck. Four quotient set gems. Amer. Math. Monthly, 121(7):590–599, 2014.Google Scholar
  12. 12.
    J. L. Brumbaugh, Madeleine Bulkow, Patrick S. Fleming, Luis Alberto Garcia German, Stephan Ramon Garcia, Gizem Karaali, Matt Michal, Andrew P. Turner, and Hong Suh. Supercharacters, exponential sums, and the uncertainty principle. J. Number Theory, 144:151–175, 2014.Google Scholar
  13. 13.
    J. L. Brumbaugh, Madeleine Bulkow, Luis Alberto Garcia German, Stephan Ramon Garcia, Matt Michal, and Andrew P. Turner. The graphic nature of the symmetric group. Exp. Math., 22(4):421–442, 2013.Google Scholar
  14. 14.
    J. Bukor, P. Erdős, T. Šalát, and J. T. Tóth. Remarks on the (R)-density of sets of numbers. II. Math. Slovaca, 47(5):517–526, 1997.Google Scholar
  15. 15.
    Jozef Bukor, Tibor Šalát, and János T. Tóth. Remarks on R-density of sets of numbers. Tatra Mt. Math. Publ., 11:159–165, 1997. Number theory (Liptovský Ján, 1995).Google Scholar
  16. 16.
    Paula Burkhardt, Alice Zhuo-Yu Chan, Gabriel Currier, Stephan Ramon Garcia, Florian Luca, and Hong Suh. Visual properties of generalized Kloosterman sums. J. Number Theory, 160:237–253, 2016.Google Scholar
  17. 17.
    Paula Burkhardt, Gabriel Currier, Mathieu de Langis, Stephan Ramon Garcia, Bob Lutz, and Hong Suh. An exhibition of exponential sums: visualizing supercharacters. Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture, pages 475–478, 2015.Google Scholar
  18. 18.
    Alice Zhuo-Yu Chan, Luis Alberto Garcia German, Stephan Ramon Garcia, and Amy L. Shoemaker. On the matrix equation XA + AX T = 0, II: Type 0-I interactions. Linear Algebra Appl., 439(12):3934–3944, 2013.Google Scholar
  19. 19.
    Jeffrey Danciger, Stephan Ramon Garcia, and Mihai Putinar. Variational principles for symmetric bilinear forms. Math. Nachr., 281(6):786–802, 2008.Google Scholar
  20. 20.
    Jean-Marie De Koninck and Armel Mercier. 1001 Problems in Classical Number Theory. American Mathematical Society, Providence, RI, 2007.zbMATHGoogle Scholar
  21. 21.
    Persi Diaconis and I. M. Isaacs. Supercharacters and superclasses for algebra groups. Trans. Amer. Math. Soc., 360(5):2359–2392, 2008.Google Scholar
  22. 22.
    Christopher Donnay, Stephan Ramon Garcia, and Jeremy Rouse. p-adic quotient sets II: Quadratic forms. J. Number Theory, 201:23–39, 2019.Google Scholar
  23. 23.
    William Duke, Stephan Ramon Garcia, and Bob Lutz. The graphic nature of Gaussian periods. Proc. Amer. Math. Soc., 143(5):1849–1863, 2015.Google Scholar
  24. 24.
    Ephraim Feig and Michael Ben-Or. On algebras related to the discrete cosine transform. Linear Algebra Appl., 266:81–106, 1997.MathSciNetCrossRefGoogle Scholar
  25. 25.
    Benjamin Fine and Gerhard Rosenberger. Number Theory: An Introduction via the Distribution of Primes. Birkhäuser, Boston, 2007.zbMATHGoogle Scholar
  26. 26.
    Patrick S. Fleming, Stephan Ramon Garcia, and Gizem Karaali. Classical Kloosterman sums: representation theory, magic squares, and Ramanujan multigraphs. J. Number Theory, 131(4):661–680, 2011.Google Scholar
  27. 27.
    Christopher F. Fowler, Stephan Ramon Garcia, and Gizem Karaali. Ramanujan sums as supercharacters. Ramanujan J., 35(2):205–241, 2014.Google Scholar
  28. 28.
    Stephan Ramon Garcia. Quotients of Gaussian Primes. Amer. Math. Monthly, 120(9):851–853, 2013.Google Scholar
  29. 29.
    Stephan Ramon Garcia, Yu Xuan Hong, Florian Luca, Elena Pinsker, Carlo Sanna, Evan Schechter, and Adam Starr. p-adic quotient sets. Acta Arith., 179(2):163–184, 2017.Google Scholar
  30. 30.
    Stephan Ramon Garcia, Trevor Hyde, and Bob Lutz. Gauss’s hidden menagerie: from cyclotomy to supercharacters. Notices Amer. Math. Soc., 62(8):878–888, 2015.Google Scholar
  31. 31.
    Stephan Ramon Garcia, Elvis Kahoro, and Florian Luca. Primitive root bias for twin primes. Exp. Math., 28(2):151–160, 2019.Google Scholar
  32. 32.
    Stephan Ramon Garcia, Gizem Karaali, and Daniel J. Katz. On Chebotarëv’s nonvanishing minors theorem and the Biró–Meshulam–Tao discrete uncertainty principle. (submitted). https://arxiv.org/abs/1807.07648.
  33. 33.
    Stephan Ramon Garcia and Florian Luca. Quotients of Fibonacci numbers. Amer. Math. Monthly, 123(10):1039–1044, 2016.Google Scholar
  34. 34.
    Stephan Ramon Garcia and Florian Luca. On the difference in values of the Euler totient function near prime arguments. In Irregularities in the distribution of prime numbers, pages 69–96. Springer, Cham, 2018.Google Scholar
  35. 35.
    Stephan Ramon Garcia, Florian Luca, and Timothy Schaaff. Primitive root biases for prime pairs I: Existence and non-totality of biases. J. Number Theory, 185:93–120, 2018.Google Scholar
  36. 36.
    Stephan Ramon Garcia, Florian Luca, Kye Shi, and Gabe Udell. Primitive root bias for twin primes II: Schinzel-type theorems for totient quotients and the sum-of-divisors function. J. Number Theory, 208:400–417, 2020.Google Scholar
  37. 37.
    Stephan Ramon Garcia and Bob Lutz. A supercharacter approach to Heilbronn sums. J. Number Theory, 186:1–15, 2018.Google Scholar
  38. 38.
    Stephan Ramon Garcia, Bob Lutz, and Dan Timotin. Two remarks about nilpotent operators of order two. Proc. Amer. Math. Soc., 142(5):1749–1756, 2014.Google Scholar
  39. 39.
    Stephan Ramon Garcia and Daniel E. Poore. On the closure of the complex symmetric operators: compact operators and weighted shifts. J. Funct. Anal., 264(3):691–712, 2013.Google Scholar
  40. 40.
    Stephan Ramon Garcia and Daniel E. Poore. On the norm closure problem for complex symmetric operators. Proc. Amer. Math. Soc., 141(2):549, 2013.Google Scholar
  41. 41.
    Stephan Ramon Garcia, Daniel E. Poore, and William T. Ross. Unitary equivalence to a truncated Toeplitz operator: analytic symbols. Proc. Amer. Math. Soc., 140(4):1281–1295, 2012.Google Scholar
  42. 42.
    Stephan Ramon Garcia, Daniel E. Poore, and James E. Tener. Unitary equivalence to a complex symmetric matrix: low dimensions. Linear Algebra Appl., 437(1):271–284, 2012.Google Scholar
  43. 43.
    Stephan Ramon Garcia, Daniel E. Poore, and Madeline K. Wyse. Unitary equivalence to a complex symmetric matrix: a modulus criterion. Oper. Matrices, 5(2):273–287, 2011.Google Scholar
  44. 44.
    Stephan Ramon Garcia, Vincent Selhorst-Jones, Daniel E. Poore, and Noah Simon. Quotient sets and Diophantine equations. Amer. Math. Monthly, 118(8):704–711, 2011.Google Scholar
  45. 45.
    Stephan Ramon Garcia and Amy L. Shoemaker. On the matrix equation XA + AX T = 0. Linear Algebra Appl., 438(6):2740–2746, 2013.Google Scholar
  46. 46.
    Stephan Ramon Garcia and Amy L. Shoemaker. Wetzel’s problem, Paul Erdős, and the continuum hypothesis: a mathematical mystery. Notices Amer. Math. Soc, 62(3):243–247, 2015. (part of Erdős retrospective).Google Scholar
  47. 47.
    Stephan Ramon Garcia and George Todd. Supercharacters, elliptic curves, and the sixth moment of Kloosterman sums. J. Number Theory, 202:316–331, 2019.Google Scholar
  48. 48.
    Stephan Ramon Garcia and Samuel Yih. Supercharacters and the discrete Fourier, cosine, and sine transforms. Comm. Algebra, 46(9):3745–3765, 2018.Google Scholar
  49. 49.
    Rafael C. Gonzalez and Richard E. Woods. Digital image processing. Pearson, 2017. Fourth Edition.Google Scholar
  50. 50.
    Fernando Q. Gouvêa. p-adic numbers. Universitext. Springer-Verlag, Berlin, second edition, 1997. An introduction.Google Scholar
  51. 51.
    Ben Green and Terence Tao. The primes contain arbitrarily long arithmetic progressions. Ann. of Math. (2), 167(2):481–547, 2008.MathSciNetCrossRefGoogle Scholar
  52. 52.
    G. H. Hardy and J. E. Littlewood. Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes. Acta Math., 114(3):215–273, 1923.MathSciNetzbMATHGoogle Scholar
  53. 53.
    G. H. Hardy and E. M. Wright. An introduction to the theory of numbers. Oxford University Press, Oxford, sixth edition, 2008. Revised by D. R. Heath-Brown and J. H. Silverman, With a foreword by Andrew Wiles.Google Scholar
  54. 54.
    D. R. Heath-Brown. An estimate for Heilbronn’s exponential sum. In Analytic number theory, Vol. 2 (Allerton Park, IL, 1995), volume 139 of Progr. Math., pages 451–463. Birkhäuser Boston, Boston, MA, 1996.Google Scholar
  55. 55.
    D. R. Heath-Brown. Heilbronn’s exponential sum and transcendence theory. In A panorama of number theory or the view from Baker’s garden (Zürich, 1999), pages 353–356. Cambridge Univ. Press, Cambridge, 2002.Google Scholar
  56. 56.
    Shawn Hedman and David Rose. Light subsets of ℕ with dense quotient sets. Amer. Math. Monthly, 116(7):635–641, 2009.MathSciNetzbMATHGoogle Scholar
  57. 57.
    Anders Olaf Flasch Hendrickson. Supercharacter theories of cyclic p-groups. ProQuest LLC, Ann Arbor, MI, 2008. Thesis (Ph.D.)–The University of Wisconsin - Madison.Google Scholar
  58. 58.
    David Hobby and D. M. Silberger. Quotients of primes. Amer. Math. Monthly, 100(1):50–52, 1993.Google Scholar
  59. 59.
    Kenneth Ireland and Michael Rosen. A classical introduction to modern number theory, volume 84 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1990.Google Scholar
  60. 60.
    Henryk Iwaniec and Emmanuel Kowalski. Analytic number theory, volume 53 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2004.Google Scholar
  61. 61.
    Nicholas M. Katz. Gauss sums, Kloosterman sums, and monodromy groups, volume 116 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1988.CrossRefGoogle Scholar
  62. 62.
    H. D. Kloosterman. On the representation of numbers in the form ax 2 + by 2 + cz 2 + dt 2. Acta Math., 49(3-4):407–464, 1927.Google Scholar
  63. 63.
    Neal Koblitz. A course in number theory and cryptography, volume 114 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1994.Google Scholar
  64. 64.
    Philip C. Kutzko. The cyclotomy of finite commutative P.I.R.’s. Illinois J. Math., 19:1–17, 1975.Google Scholar
  65. 65.
    Serge Lang. Math talks for undergraduates. Springer-Verlag, New York, 1999.CrossRefGoogle Scholar
  66. 66.
    Piotr Miska, Nadir Murru, and Carlo Sanna. On the p-adic denseness of the quotient set of a polynomial image. J. Number Theory, 197:218–227, 2019.MathSciNetCrossRefGoogle Scholar
  67. 67.
    Piotr Miska and Carlo Sanna. p-adic denseness of members of partitions of \(\mathbb {N}\) and their ratio sets. Bulletin of the Malaysian Mathematical Sciences Society. (in press) https://arxiv.org/abs/1808.00374.
  68. 68.
    Andrzej Nowicki. Editor’s endnotes. Amer. Math. Monthly, 117(8):755–756, 2010.MathSciNetGoogle Scholar
  69. 69.
    Paul Pollack. Not Always Buried Deep: A Second Course in Elementary Number Theory. American Mathematical Society, Providence, RI, 2009.CrossRefGoogle Scholar
  70. 70.
    Paulo Ribenboim. The Book of Prime Number Records. Springer-Verlag, New York, 2nd edition, 1989.Google Scholar
  71. 71.
    T. Šalát. On ratio sets of sets of natural numbers. Acta Arith., 15:273–278, 1968/1969.Google Scholar
  72. 72.
    T. Šalát. Corrigendum to the paper “On ratio sets of sets of natural numbers”. Acta Arith., 16:103, 1969/1970.Google Scholar
  73. 73.
    V. Sanchez, P. Garcia, A. Peinado, J. Segura, and Rubio A. Diagonalizing properties of the discrete cosine transforms. IEEE Transactions on Signal Processing, 43(11):2631–2641, 1995.Google Scholar
  74. 74.
    Victoria Sanchez, Antonio M. Peinado, Jose C. Segura, Pedro Garcia, and Antonio J. Rubio. Generating matrices for the discrete sine transforms. IEEE Transactions on Signal Processing, 44(10):2644–2646, 1996.CrossRefGoogle Scholar
  75. 75.
    Carlo Sanna. The quotient set of k-generalised Fibonacci numbers is dense in ℚp. Bull. Aust. Math. Soc., 96(1):24–29, 2017.MathSciNetCrossRefGoogle Scholar
  76. 76.
    J.-P. Serre. A course in arithmetic. Springer-Verlag, New York-Heidelberg, 1973. Translated from the French, Graduate Texts in Mathematics, No. 7.Google Scholar
  77. 77.
    I. D. Shkredov. On Heilbronn’s exponential sum. Q. J. Math., 64(4):1221–1230, 2013.MathSciNetCrossRefGoogle Scholar
  78. 78.
    Brian D. Sittinger. Quotients of primes in an algebraic number ring. Notes Number Theory Disc. Math., 24(2):55–62, 2018.MathSciNetCrossRefGoogle Scholar
  79. 79.
    Paolo Starni. Answers to two questions concerning quotients of primes. Amer. Math. Monthly, 102(4):347–349, 1995.MathSciNetCrossRefGoogle Scholar
  80. 80.
    S. A. Stepanov. The number of points of a hyperelliptic curve over a finite prime field. Izv. Akad. Nauk SSSR Ser. Mat., 33:1171–1181, 1969.MathSciNetzbMATHGoogle Scholar
  81. 81.
    S. A. Stepanov. Estimation of Kloosterman sums. Izv. Akad. Nauk SSSR Ser. Mat., 35:308–323, 1971.MathSciNetGoogle Scholar
  82. 82.
    P. Stevenhagen and H. W. Lenstra, Jr. Chebotarëv and his density theorem. Math. Intelligencer, 18(2):26–37, 1996.Google Scholar
  83. 83.
    Oto Strauch and János T. Tóth. Asymptotic density of A ⊂N and density of the ratio set R(A). Acta Arith., 87(1):67–78, 1998.Google Scholar
  84. 84.
    Oto Strauch and János T. Tóth. Corrigendum to Theorem 5 of the paper: “Asymptotic density of A ⊂ℕ and density of the ratio set R(A)” [Acta Arith. 87 (1998), no. 1, 67–78; MR1659159 (99k:11020)]. Acta Arith., 103(2):191–200, 2002.Google Scholar
  85. 85.
    Terence Tao. An uncertainty principle for cyclic groups of prime order. Math. Res. Lett., 12(1):121–127, 2005.MathSciNetCrossRefGoogle Scholar
  86. 86.
    S. Vajda. Fibonacci & Lucas numbers, and the golden section. Ellis Horwood Series: Mathematics and its Applications. Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley & Sons, Inc.], New York, 1989. Theory and applications, With chapter XII by B. W. Conolly.Google Scholar
  87. 87.
    André Weil. On some exponential sums. Proc. Nat. Acad. Sci. U. S. A., 34:204–207, 1948.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Pomona CollegeClaremontUSA

Personalised recommendations