Neranga Fernando

Research Interests

Algebra and Number Theory

  • Permutation polynomials over finite fields
  • Self-reciprocal polynomials and their applications in coding theory.
  • Complete mappings over finite fields
  • Irreducible polynomials over finite fields
  • Dickson polynomials, reversed Dickson polynomials, and polynomials defined by functional equations over finite fields.

    Quandle Theory

  • Quandles
  • Racks
  • Shelves
  • Ring theoretic aspects of Quandles
  • Dihedral quandle rings

    Commutative Algebra

  • Gröbner bases and their applications in Quandle rings

    HERE is my Research Statement.

    Talks

  • Reversed Dickson polynomials over finite fields, California Polytechnic, San Luis Obispo, CA, AMS Meeting, May 3 - 4, 2025. (abstract submitted)
  • Idempotents and tripotents in quandle rings, Knots in Washington, George Washington University, December 6 - 8, 2024. (abstract submitted)
  • (invited) Reversed Dickson permutation polynomials over finite rings, and their cycle types, Canadian Mathematical Society Winter Meeting 2024, Vancouver, Canada, November 29 - December 2, 2024.
  • Idempotents in quandle rings, AMS Meeting, University of Albany, Albany, New York, NY, October 19 - 20, 2024.
  • Quandles, racks and shelves, Clavius Group Meeting, College of the Holy Cross, Worcester, MA, June 22, 2023.
  • Reversed Dickson polynomials of the (k+1)-th kind over fintie fields, Clavius Group Meeting, College of the Holy Cross, Worcester, MA, June 20, 2023.
  • Reversed Dickson polynomials of the (k+1)-th kind over fintie fields, Canadian Mathematical Society Summer Meeting 2023, Ottawa, Canada, June 4, 2023.
  • Classification of connected shelves and their applications to positive knots, Claremont Topology Seminar, Claremont Center for the Mathematical Sciences, Claremont, CA, March 7, 2023.
  • (invited) Knots, quandles and shelves, Worcester Polytechnic University, Worcester, MA, February 20, 2022.
  • (invited) Quandles, racks and shelves and their applications to positive knots, Purdue University, West Lafayette, IN, February 7, 2023.
  • Quandles, racks and shelves, School of Mathematics and Statistics, Carleton University, Ottawa, Canada, October 14, 2022. Abstract slides
  • (invited) Self-reciprocal polynomials and reversed Dickson polynomials, Mathematics Colloquium, University of South Florida, Tampa, FL, March 11, 2022.
  • Self-reciprocal polynomials arising from reversed Dickson polynomials, College of the Holy Cross, Worcester, MA, February 9, 2022.
  • Permutation polynomials over finite fields, College of the Holy Cross, Worcester, MA, November 3, 2021.
  • Sel-reciprocal polynomials and reversed Dickson polynomials, Rose-Hulman Institute of Technology, Terre Haute, IN, May 6, 2021. slides
  • Self-reciprocal polynomials and Zumkeller numbers, Amherst College, Amherst, MA, March 4, 2019.
  • Fibonacci permutation polynomials, Math Club, California State University, Northridge, CA, March, March 9, 2018. Abstract (Chalkboard presentation)
  • Reversed Dickson permutation polynomials, Algebra, Number Theory, and Discrete Mathematics Seminar, California State University, Northridge, CA, March 7, 2018. Abstract slides
  • Reversed Dickson permutation polynomials over finite fields, Algebra, Number Theory, and Combinatorics Seminar, Claremont Center for the Mathematical Sciences, Claremont, CA, March 6, 2018. Abstract slides
  • Self-reciprocal polynomials arising from reversed Dickson polynomials, AMS Meeting, State University of New York at Buffalo, Buffalo, NY, September 16 - 17, 2017. slides
  • Reversed Dickson polynomials of the (k+1)-th kind over finite fields, II, AMS Meeting, University of North Texas, Denton, TX, September 9 - 10, 2017. (Click HERE for further details) slides
  • (invited) Reversed Dickson polynomials of the (k+1)-th kind over finite fields, AMS Meeting, Hunter College, CUNY, New York, NY, May 6 - 7, 2017. slides
  • A generalization of the reversed Dickson polynomials over finite fields, Discrete Mathematics Seminar, University of South Florida, March 6, 2017. slides
  • (invited) Permutation polynomials, self-reciprocal polynomials, and their role in coding theory, Northeastern University, December 1, 2016. (Chalkboard presentation)
  • (invited) Reversed Dickson polynomials of the third kind, 47th Southeastern International Conference on Combinatorics, Graph Theory & Computing, March 7-11, 2016. slides
  • (invited) From r-linearized polynomial equations to rm-linearized polynomial equations, Fq12: The 12th International Conference on Finite Fields and Their Applications, Saratoga Springs, NY, July 13-17, 2015. slides
  • New classes of permutation polynomials over finite fields defined by functional equations, Joint Mathematics Meeting , San Diego, Jan 9 - 12, 2013. slides
  • A New Approach to Permutation Polynomials over Finite Fields, Discrete Mathematics Seminar, University of South Florida, November 19, 2012. slides
  • (invited) Permutation Polynomials over Finite Fields defined by functional equations, AMS Meeting, University of South Florida , March 10-11, 2012. slides
  • A Matrix form of the Jacobi-Trudi Identity, Discrete Mathematics Seminar, University of South Florida, February 8, 2010. slides

    Research Problems for Ph.D. students

    Here are some research problems for graduate students who are interested in working on permutation polynomials and self-reciprocal polynomials. I will add more soon.
    1. Here are some open problems on the permutation behaviour of reversed Dickson polynomials.
    2. There are many unexplained desirable triples of polynomial gn,q, especially in odd characteristic. Take a look at the tables in "A new approach to permutation polynomials over finite fields, II" listed under publications.
    3. Orthomorphism polynomials over finite fields arising from polynomial gn,q would be an interesting project. I already have numerical results obtained from computer. If you'd like to see them, please send me an email with your Ph.D. thesis advisor's name. I will be happy to share them with you.



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