My current research interests are in dynamical systems, specifically symbolic dynamics and automorphism groups of shifts of finite type. I am also interested in the application of symbolic dynamics to hyperbolic dynamics.
Current Problems:
- ¨ How large of a subgroup of the automorphism group of the full 2-shift commutes with the flip map?
- I am also trying to generalize several classes of examples of mixing shifts of finite type with surjective dimension representations. These examples are the only known examples of MSFTs with surjective dimension representations that come necessarily from non-elementary strong shift equivalences. The first class is presented in my thesis (Chapter 5) linked below.
- ¨ Given a dynamical system with a hyperbolic set, under what conditions is the set hyperbolic for other maps?
I am also preparing papers addressing:
- Conditions for injective and surjective maps of mixing SFTs to preserve finite group actions.
- The quotient spaces of 1-sided SFT by strictly order n automorphisms.
I have an interest into several other problems dealing with various aspects of matrix theory and the classification problem of mixing SFTs.
IBL Based Research:
- Does an IBL based course in the precalculus sequence have better outcomes in calculus 1?
- The effectiveness of virtual reality in increasing geometric understanding of multivariable calculus
- On the Culture of Making Things, AMS Blog: on Teaching and Learning Mathematics Nov 2017
- Inquiry Based Learning Notes and Problems for Plane Analytic Geometry, Submitted to the Journal for Inquiry Based Learning in Mathematics (JIBLM) Fall 2018.
- Inquiry Based Learning Notes and Problems for Linear Algebra with Applications, Submitted to the Journal for Inquiry Based Learning in Mathematics (JIBLM) Summer 2018.
- Many Voices Are Greater Than One, MAA Math Ed Matters Blog, October 2015
Student Research Projects:
- Randal Robin – Marching Cubes
- Josh Harris – Surfaces of Revolution in Virtual Reality
- Dylan Jager-Kujawa –
- Tyler Velvin – Maximum increments of the Perron-Frobenius Eigenvalue
- Jeremy Bigger-Examples of Shifts of Finite Type with Surjective Dimension Representations
- Chris Brown- Voting Theory
- Kim Fruge- Geometry of Adding Issues to an Election
- Kelsie Howard- Bad Assumptions in Pension Management
- Cassandra Lindsey- Epidemiology of a College Campus
- Tyler Velvin- Urban Sprawl and Cellular Automata
Preprints, Papers, and CV
- Curriculum Vitae
- Mixing Shifts of Finite Type with Non-Elementary Surjective Dimension Representations, Acta Appl. Math. 126 (2013), 277–295.
- Fixed Point Shifts of Inert Involutions DCDS Volume: 25, Number: 4, December 2009
- Strictly Order n Automorphisms of 1-Sided Shifts of Finite Type
- Embedding Finite Order Automorphisms
- Flip Commuting Maps of the 2-shift
- Involutions of Shifts of Finite Type: Fixed Point Shifts, Orbit Quotients, and the Dimension Representation (PhD Thesis)
- Dynamical Systems Inset in SciDac Review Spring 2007 Issue 3 p. 48
Grants:
- Texas Undergraduate Mathematics Conference 2018-2019 Collaborative Proposal with D. Milan at UT-Tyler National Science Foundation: (Funded)
DMS-1834888 $14,553 - Calculus and Virtual Reality (CalcVR) with J. Becnel et. al National Science Foundation: (Funded) DUE-1820724 $294,523
- Multivariable Calculus in Virtual Reality April 2017 SFASU Provost’s Office: (Funded)
- Inquiry-Based Learning in Linear Algebra and Trigonometry August 2015
Academy For Inquiry Based Learning (Funded) - REU Site: Dynamics and Iteration REU with Becnel and Beauregard August 2015
National Science Foundation: (Not Funded) - Implementing IBL in Plane Analytic Geometry Faculty Research Engagement Grant –Internal SFA (Funded) May 2015
- REU Site: REU in Mathematics at SFA with J. Long and J. Becnel August 2014
National Science Foundation (Not Funded) - Framework for Surjective Dimension Representations Faculty Research Grant –Internal SFA Summer 2014 (Funded)
Talks:
Dr. Nick Long's SFA Webpage 