Research

Download Research Statement


Research Interests

  • Hopf Algebras
  • Actions of Hopf Algebras on Path Algebras
  • Quantum Groups
  • Invariant Theory
  • Combinatorics

Current Work

A Hopf algebra is an algebraic structure that can be thought of as a collection of elements, which interact in a very specific way and have a lot of structure. We can multiply, add, and divide elements in a Hopf algebra, as well as perform dual operations to the aforementioned. This type of structure is referred to as a bialgebra, and it appears naturally in algebraic topology, scheme theory, group theory and physics, which makes it a very popular and studied bialgebra.

A quiver is a directed graph, so a collection of vertices and directed arrows, and the path algebra of a quiver is a vector space where all the paths of the quiver form a basis, and multiplication is given by concatenation of paths whenever possible, and zero otherwise. In other words, we may fomally add paths, and we can physically multiply the paths together by concatinating them: as long as the paths meet arrow head to arrow tail, the multiplication results in a longer path.

The focus of my research is twofold. First, I look at a specific Hopf algebra and classify the possible ways it can act on a path algebra, with the fewest number of restrictions possible. Second, once I have a good description of the action, I can describe the elements that are unaffected by this action. In other words, I study the invariant ring of the path algebra under the action from a Hopf algebra.

Teaching

Download Teaching Philosophy

ePortfolio

Courses Taught

  • Graduate Number Theory
  • Abstract Algebra
  • Topology
  • Discrete Mathematics
  • Introduction to Proofs
  • Geometries
  • Explorations in Mathmematics
  • Computational Linear Algebra
  • Calculus I and II
  • Precalculus
  • College Algebra
  • Introduction to Mathematics

Curriculum Vitae

Download CV

Selected Teaching Credentials

  • Assistant Professor of Mathematics at Colorado Mesa University, 2018-2024
  • Mathematics PhD from The University of Iowa, 2018
  • The University of Iowa's Graduate Certificate in College Teaching, 2018
  • The University of Iowa's Outstanding Teaching Assistant Award, 2017
  • Graduate Mentor for the Mathematical Science Research Institute Undergraduate Program, 2012
  • Master's in Mathematics with a Minor in Education, University of Texas, Austin 2012
  • Teaching Assistant, University of Iowa, 2015-2018
  • Teaching Assistant, University of Texas Austin, 2011-2012

Publications:

  • "Escape the Math Room." Berrizbeitia, A. (2024). PRIMUS, v34 n8 p830-841.
    Escape-The-Math-Room.pdf
  • "Parking functions, Shi arrangements, and mixed graphs." Berrizbeitia, A., et al. (2015). American Mathematical Monthly, 122(7), 660-673.
    Shi-Arrangements.pdf
  • "The p-adic valuation of Stirling numbers." Berrizbeitia, Ana, et al. Journal for Algebra and Number Theory Academia 1 (2010): 1-30.
    Stirling-Numbers.pdf
  • PhD Thesis (unpublished)
    Thesis.pdf

About Me

What About Me?

Hi, I'm Ana Berrizbeitia, a Math PhD and former professor at Colorado Mesa University, now transitioning from academia to industry as an instructional designer. My strength lies in making learning fun, engaging, and interactive. I have a talent for transforming complex concepts into exciting experiences through interactive activities, games (like escape rooms), and technology-driven learning. I believe that education should be dynamic and enjoyable, which is why I always incorporate active learning and creativity into my courses.

At my core, I value honesty and authenticity, always striving to lead by example. My background in education has equipped me with strong skills in curriculum development, instructional technology, and e-learning design—skills I'm eager to apply in industry settings to create impactful learning experiences.

Outside of work, I love painting, singing, and playing board games with my family. I'm also a dedicated TV fan, with plenty of guilty pleasures I fully embrace (ask me about my latest binge!).

I'm always open to new opportunities and connections—whether it's collaborating on an exciting project, discussing instructional design, or just exchanging TV show recommendations. Let's connect!