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Nicolas Allen Smoot

 

Nicolas Allen Smoot

 

Contact


Office: RISC, Hagenberg, 0.2-5
           Science Park 2, 0079

Phone: ++43(0)732/2468-9981

EMail: ,
          nicolas.smoot@risc.jku.at,
          ns02570@georgiasouthern.edu,
          n.smoot@hotmail.com

Address:
Doctoral Program Computational Mathematics
Johannes Kepler University
Altenberger Straße 69
A-4040 Linz
Austria

Publications

 

Publications (Complete List)

Published

  • "A Family of Congruences for Rogers--Ramanujan Subpartitions," Journal of Number Theory 196, pp. 35-60, 2018. (arXiv)
  • "A Partition Function Connected With The Göllnitz-Gordon Identities," Harmonic Analysis, Partial Differential Equations, Banach Spaces, and Operator Theory (Volume 2), Springer International, 2017, pp. 373-400. (arXiv)
  • (With Silviu Radu) "A Method of Verifying Partition Congruences by Symbolic Computation," Journal of Symbolic Computation 104, pp. 105-133, 2021. (arXiv)
  • "On the Computation of Identities Relating Partition Numbers in Arithmetic Progressions with Eta Quotients: An Implementation of Radu's Algorithm," Journal of Symbolic Computation 104, pp. 276-311, 2021. (arXiv)

Submitted

  • "A Single-Variable Proof of the Omega SPT Congruence Family Over Powers of 5," 2020. (arXiv)

Miscellaneous

  • Solution 11908 II: "A Generalized Bijection for Partitions," The American Mathematical Monthly, Volume 125, Issue 3, 2018, pp. 276-283.

 

Refereed for the Following Journals

  • Journal of Number Theory
  • Ramanujan Journal
  • Annals of Combinatorics
  • Journal of Integer Sequences

 

Grants

  • October, 2020: Principal investigator for FWF grant (ca. €145,000), Stand-alone Project P33933, "Partition Congruences by the Localization Method."

 

Awards

  • Awarded in 2020: JKU Young Researchers' Award -- Awarded for excellent scientific achievement in the course of doctoral studies

 

Short Curriculum Vitae

  • 2021 - Present: FWF Project Leader, "Partition Congruences by the Localization Method."
  • 2016 - 2020: PhD Student in "Computational Mathematics'' Program, Johannes Kepler University (Linz,
 Austria).
  • 2014 -2016: M.S. in Mathematics, Georgia Southern University (Statesboro, GA USA)
  • 2014: B.S. Cum Laude in Mathematics, Armstrong State University (Savannah, GA USA)

 

Doctoral Dissertation

  • "Computer Algebra with the Fifth Operation: Applications of Modular Functions to Partition Congruences."  Thesis advisor: Prof. Peter Paule.

 

Research interests

  • Number theory
  • Integer partitions
  • Modular forms
  • q-Series
  • Complex analysis
  • Riemann surfaces
  • Symbolic computation

  

Software

RaduRK: Ramanujan-Kolberg Program

 

RaduRK is a Mathematica implementation of an algorithm developed by Cristian-Silviu Radu.  The algorithm takes as input an arithmetic sequence a(n) generated from a large class of q-Pochhammer quotients, together with a given arithmetic progression mn+j, and the level of a given congruence subgroup.  The algorithm produces expressions for the generating function of a(mn+j) in terms of Q-linear combinations of Dedekind eta quotients which are modular over the subgroup.  Identities of this form include famous results by Ramanujan which demonstrate the divisibility properties of p(5n+4) and p(7n+5).

The algorithm relies on certain powerful finiteness conditions imposed by the study of modular functions, and illustrates the utility of the subject to computational number theory.

For using the package, download the file RaduRK.m, which can be found at

Place this file into a directory where Mathematica will find it.

For a demonstration of how to use the package see 

For more ambitious examples, see:

The package requires the packages 4ti2 and math4ti2.m.

Instructions for the installation for these packages and RaduRK, together with a review of the basic theory, can be found here.

For details concerning the design of the algorithm, consult the following:

  • S. Radu, "An Algorithmic Approach to Ramanujan's Congruences," Ramanujan Journal, 20, pp. 215-251 (2009).
  • S. Radu, "An Algorithmic Approach to Ramanujan-Kolberg Identities," Journal of Symbolic Computation, 68, pp. 225-253 (2015).

Please report any bugs or other suggestions to nicolas.smoot@risc.jku.at.

 

Honors

  • 2013 - 2014: President, Pi Mu Epsilon, Armstrong State University Chapter

 

Teaching

  • 2015 - 2016: Instructor of Record for College Algebra, Georgia Southern University
  • 2014 - 2015: Teaching Assistant for Differential and Integral Calculus, Georgia Southern University
  • 2012 - 2014: Tutor, Savannah State University
  • 2011 - 2014: Tutor, Armstrong State University 

 

Activities

Participation at conferences and workshops

  • March, 2012: Southeastern Regional Meeting On Numbers, Western Carolina University
  • "Consecutive Pairs of Octic Residues," PANTS XVIII, Wake Forest University, Winston-Salem, NC, 16 September, 2012.
  • "The Distribution of Consecutive $2^t$ Power Residues and Implications," Kennesaw Mountain Undergraduate Mathematics Conference, 20 October, 2012.
  • "The Structure of Consecutive Octic Residues," Joint Meetings of the AMS and MAA, San Diego, CA, 10 January, 2013.
  • "Commutative Rings and the Invariant Basis Property," Hudson Colloquium, Armstrong State University, Savannah, GA, 30 October, 2013.

  • "On the Stability of Ring Structures in Direct Limits," Joint Meetings of the AMS and MAA, Baltimore, MD, 17 January, 2014.

  • "On the Stability of Ring Structures in Direct Limits," Hudson Colloquium, Armstrong State University, Savannah, GA, 19 February, 2014.

  • "Partitions Connected with Modulus 8: An Application of the Circle Method I," Georgia Southern Analysis Seminar, Statesboro, GA, 2015.

  • "Partitions Connected with Modulus 8: An Application of the Circle Method II," Georgia Southern Analysis Seminar, Statesboro, GA, 2015.

  • "Enumerating the Partitions of the Göllnitz--Gordon Identities," Joint Meetings of the AMS and MAA, Baltimore, MD, 6 January, 2016.

  • "Partitions Connected with Modulus 8: An Application of the Circle Method I," Algorithmic Combinatorics Seminar, RISC, Hagenberg, 12 October, 2016.
  • "Partitions Connected with Modulus 8: An Application of the Circle Method II," Algorithmic Combinatorics Seminar, RISC, Hagenberg, 16 November, 2016.
  • "Modular Forms and Associated Congruences: Applying Sturm's Theorem to Eta Quotients," Algorithmic Combinatorics Seminar, RISC, Hagenberg, 10 May, 2017.
  • "Some Arithmetic Properties of Integer Partitions," Computational Mathematics Workshop, JKU Linz, 11 July, 2017.
  • "The Computation of Ramanujan--Kolberg Identities," Algorithmic Combinatorics Seminar, RISC, Hagenberg, 29 November, 2017.

  • "The Computation of Ramanujan--Kolberg Identities," Joint Meetings of the AMS and MAA, San Diego, CA, 12 January, 2018.

  • "The Ramanujan--Kolberg Algorithm and Some New Partition Identities," Computational Mathematics Workshop, JKU Linz, 5 February, 2018.
  • "Towards a New Family of Partition Congruences," Algorithmic Combinatorics Seminar, RISC, Hagenberg, 9 May, 2018.
  • "A Proof of a New Family of Partition Congruences," Algorithmic Combinatorics Seminar, RISC, Hagenberg, 6 June, 2018.

  • "A Family of Congruences for Rogers--Ramanujan Subpartitions," Contributed Talk, Combinatory Analysis 2018: A Conference in Honor of George Andrews' 80th Birthday, Pennsylvania State University, University Park, PA, 21 June, 2018.

  • "A New Family of Congruences for Rogers--Ramanujan Subpartitions," Computational Mathematics Workshop, JKU Linz, 9 July, 2018.
  • "A Method of Verifying Partition Congruences by Symbolic Computation," Algorithmic Combinatorics Seminar, RISC, Hagenberg, 31 October, 2018.

  • "A Family of Congruences for Rogers--Ramanujan Subpartitions," Joint Meetings of the AMS and MAA, Baltimore, MD, 17 January, 2019.

  • "Eta Generalized: An Overview of its Wonderful Properties," Algorithmic Combinatorics Seminar, RISC, Hagenberg, 10 April, 2019.
  • "Some New Partition Identities, Discovered on the Balance of Theory and Computation," SFB F50 Algorithmic and Enumerative Combinatorics Colloquium, University of Vienna, 12 April, 2019.
  • "Some New Identities From the Ramanujan--Kolberg Algorithm,'' Contributed Talk, Analytic and Combinatorial Number Theory: The

    Legacy of Ramanujan, University of Illionois at Urbana--Champaign, 7 June, 2019.

  • "A Connection Between Arithmetic, Analysis, and Topology," Computational Mathematics Workshop, JKU Linz, 8 July, 2019.
  • "Partitions, Genus, and a Beautiful New Identity,'' Algorithmic Combinatorics Seminar, RISC, Hagenberg, 13 November, 2019.
  • "Verifying Partition Congruences with Symbolic Computation," AMS Special Session on Interactions Among Partitions, Basic Hypergeometric Series, and Modular Forms, Joint Meetings of the AMS and MAA, Denver, CO, 15--18 January, 2020.
  • "A Single-Variable Proof of the Omega SPT Congruence Family Over Powers of 5," Algorithmic and Enumerative Combinatorics Converence (AEC 2020), Vienna, 29 June - 3 July, 2020.
  • "A Connection Between Arithmetic, Analysis, and Topology II: 5-adic Growth on Localized Rings," Computational Mathematics Workshop, JKU Linz, 20 February, 2020.
  • "A Connection Between Arithmetic, Algebra, Analysis, and Topology: Congruences by the Localization Method," Algorithmic Combinatorics Seminar, RISC, Hagenberg (via Zoom), 29 April, 2020.
  • "A Connection Between Arithmetic, Algebra, Analysis, and Topology: 5-adic Growth on Localized Rings," Algorithmic Combinatorics Seminar, RISC, Hagenberg (via Zoom), 6 May, 2020.
  • "Partition Congruences and the Localization Method," Vanderbilt University Number Theory Seminar (via Zoom), 8 September, 2020.