Helene Tyler

Department Chairperson

Associate Professor, Mathematics

Since joining the Manahttan College Faculty in 2002, Helene Tyler has taught two dozen different Mathematics courses, ranging in breadth and depth, and serving students across the college. Her main research interests lie in the representation theory of finite-dimensional algebras. In addition, she has guided several undergraduate students through projects that concern network theory applications to team sports. Dr. Tyler has delivered mathematics lectures in eight countries, including Cambodia and Myanmar. She has served four times as a Volunteer Visiting Lecturer at the Royal University of Phnom Penh. 

Education

  • PHD, Syracuse University
  • MS, Syracuse University
  • BA, SUNY Purchase College

Courses Taught

MATH 153      Linear Math Analysis 

MATH 154      Calculus for Business Decisions

MATH 185      Calculus I

MATH 187      Calculus I (Honors)
MATH 186      Calculus II

MATH 188      Calculus II (Honors)
MATH 243      Foundations for Higher Mathematics

MATH 272      Linear Algebra I

MATH 285      Calculus III

MATH 286      Differential Equations

MATH 287      Honors Calculus III

MATH 377      Algebra I

MATH 471      Linear Algebra II
MATH 478      Algebra II

MATH 489      Problem Seminar

MATH 490      Complex Analysis

MATH 492      Topics in Applied Network Theory

MATH 492      Elementary Number Theory

MATH 499      Independent Study: Complex Analysis II

MATH 499      Independent Study: Linear Algebra III
MATH 499      Independent Study: Intro to Representation Theory of Algebras
MATG 571      Advanced Linear Algebra with Applications

  • Research

    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.

    A secondary line of research, which I developed in order to work with undergraduate students, concerns applications of network theory to team sports. Specifically, we used various centrality measures, including Google PageRank, to analyze the play style of the Manhattan College Women's Soccer and Lacrosse teams. 
  • Publications and Scholarly Activities

    Publications

    • Helene R. Tyler. A Friendly Introduction to Path Algebras and Quiver Representations, The American Mathematical Monthly (to appear).
    • 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.
  • Professional Experience and Memberships

    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.

  • Honors, Awards, and Grants
     
    • Award for Distinguished Teaching, MAA Metro NY Section¬†
    • Distinguished Lasallian Faculty Award, Manhattan College, 2015-2016.

    • 2003-2004 Exxon-Mobil Project NexT Fellow