Helene Tyler

My primary area of research is in the representation theory of artin algebras. Broadly speaking, representation theory allows one to understand an abstract algebraic structure, such as a group or a Lie algebra, by representing its elements as linear transformations of vector spaces. Thus, the abstract notion becomes a concrete one. I study modules over artin algebras by realizing them as representations of quivers (directed graphs). My earlier work in the field concerns the classification of quiver representations via Auslander-Reiten theory, a well-developed theory that emerged over the last fifty years. More recent work concerns the relationship between the Auslander-Reiten theory and an alternative classification mechanism, called Gabriel-Roiter theory, which was introduced in the mid-2000's. These publications employed a variety of techniques from homological algebra, category theory, and algebraic combinatorics.

Publications

• Helene R. Tyler. Karate and the Art of Mathematical Maintenance, Notices of the American Mathematical Society, Vol. 71, No. 7, August 2024. https://doi.org/10.1090/noti2972
• Helene R. Tyler. A Friendly Introduction to Path Algebras and Quiver Representations, The American Mathematical Monthly (2022)  https://doi.org/10.1080/00029890.2022.2115778.
• Andre P. Oliveira & Helene R. Tyler, Measurement and comparison of passing networks in collegiate soccer, Minnesota Journal of Undergraduate Mathematics, Vol 1, No 1 (2015), 16 pp. https://mjum.math.umn.edu/index.php/mjum/article/view/005/11.
• Markus Schmidmeier & Helene R. Tyler (2014) The Auslander–Reiten Components in the Rhombic Picture, Communications in Algebra, 42:3, 1312-1336, http://dx.doi.org/10.1080/00927872.2012.738341
• M. Kleiner, H.R. Tyler, Sequences of reflection functors and the preprojective component of a valued quiver, Journal of Pure and Applied Algebra, 212 (2008) 718-726, http://dx.doi.org/10.1016/j.jpaa.2007.06.014
• M. Kleiner, H.R. Tyler, Admissible sequences and the preprojective component of a quiver, Advances in Mathematics, 192 (2005) 376-402, http://dx.doi.org/10.1016/j.aim.2004.04.006
• D.Gagliardi, H.Tyler, M.Lewinter, Three-colorable maximal planar graphs and self-complementary graphs, Graph Theory Notes of NY XLVII (40-41) 2004.

Selected Presentations

• "Reflections on Mathematical Development in Southeast Asia," Joint Mathematics Meetings, Virtual, January 8, 2021.
• “Classification of Module Categories via the Gabriel-Roiter Measure”, Seminario Padova - Verona MALGA, Università di Verona, May 31, 2017. 
• “Classification of Module Categories via the Gabriel-Roiter Measure”, Universiteit Antwerpen, June 16, 2017. 
• “Measurement and Comparison of Passing Networks in Collegiate Soccer”, Ninth International Conference on Science and Mathematics Education in Developing Countries, Mandalay, Myanmar, November 2016.
• “Problem Seminar: a capstone experience for undergraduates”, Ninth International Conference on Science and Mathematics Education in Developing Countries, Mandalay, Myanmar, November 2016. 
• "Gabriel-Roiter Measures for Struwwelpeter Algebras", XVII International Conference on Representations of Algebras, Syracuse University, August 2016.
• "Stepping onto the Pitch: A Pure Mathematician's Foray into Applied Research with Undergraduates", Pi Mu Epsilon Induction Ceremony, Fordham University, May 2015.
• "String algebras with one cycle - the rhombic picture", Canadian Mathematical Society Winter Meeting, Ottawa, ON, December 2013.
• "The Auslander-Reiten Components in the Rhombic Picture", International Conference on Representations of Algebras, Bielefeld, Germany, August 2012, http://www.math.uni-bielefeld.de/icra2012/presentations/icra2012_tyler.pdf
• "The Auslander-Reiten Components in the Rhombic Picture", Maurice Auslander International Conference, Woods Hole, MA, April, 2012.
• "Auslander-Reiten Layers in the Rhombic Picture of an Algebra," Canadian Mathematical Society Winter Meeting, Toronto, Canada, December, 2011.
• “Auslander-Reiten Layers in the Rhombic Picture,” BIREP Seminar, Bielefeld University, Germany, July, 2011.
• “Gabriel-Roiter Families Occurring in Tubes,”American Mathematical Society Fall Eastern Section Meeting, Syracuse University, October, 2010.
• “A Comparison of Orderings for Preprojective Modules,”Canadian Mathematical Society Summer Meeting, University of New Brunswick at Fredericton, June, 2010.
• “Reflections on Graphs and a Measure for the Position of a Module,”Algebra Seminar, Florida Atlantic University, April, 2010.
• “Slices, Tilting Sets, and Annihilating Sequences of BGP-Reflection Functors,”American Mathematical Society Fall Western Section Meeting, University of California at Riverside, November, 2009.
• “Shortest Annihilating Sequences for Preprojective Quiver Representations,” International Conference on Representations of Algebras and Related Topics, Woods Hole, Massachusetts, April, 2009.
• “Reflection Functors Acting on Systems of Vector Spaces,” Algebra/Cryptology Seminar, Florida Atlantic University, October 2007.
• “Sequences of Reflection Functors and the Preprojective Component of a Valued Quiver,” XII International Conference on Representations of Algebras, Nicolaus Copernicus University, Torun, Poland, August 2007.
• “(+)-Admissible Sequences and the Preprojective Component,” American Mathematical Society Eastern Section Meeting, Northeastern University, October, 2003.

I have served four times as a Volunteer Lecturer at the Royal University of Phnom Penh (RUPP) in Cambodia, most recently in June 2014. During each of my first two visits, I taught a course in Ordinary Differential Equations. I taught a course in Group Theory during my third and fourth visits. The Volunteer Lecturer Program is supported by the National Academy of Sciences and focuses on scientific capacity-building in Cambodia and other developing countries. Volunteer lecturers from the USA, France, Japan, and other countries collaborate with RUPP to offer a masters program in mathematics. More information about the program can be found on their website: http://sites.nationalacademies.org/pga/biso/IMU/PGA_050583

I am a member of the American Mathematical Society, the Mathematical Association of America, and the Association for Women in Mathematics.

I have previously served as a member of the Carus Monograph Series Editorial Board.

• Award for Distinguished Teaching, MAA Metro NY Section 

• Distinguished Lasallian Faculty Award, Manhattan College, 2015-2016.

• 2003-2004 Exxon-Mobil Project NexT Fellow