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publications

talks

Relationship of Mutation Dimer Graphs at 4-Faces and Quiver Mutations at 4-Valent Vertices

Published:

Abstract (written for general audience)

We introduce quivers, a mathematical abstraction of a network with a sense of direction. These arose naturally from triangulations of polygons of \(n\)-sides with special interest in mathematical problems involved in counting (combinatorics). Today, they are popular as areas of research in combinatorics, representation theory, and theoretical physics.

Part of our main focus is on the concept of restricted mutations of a quiver. In the case of triangulated surfaces this means we are interested in how many distinct triangulations we can find by modifying only one diagonal of a triangulation at a time. In the more general case, this corresponds to applying the mutation transformation on a vertex with 2 arrows incoming and 2 arrows outgoing.

The conjecture this project focused on was to show that infinitely many restricted mutation transformations lead to only finitely many unique quivers. While unable to give a proof/disproof of this conjecture, we did compute many examples which lead finitely many unique quivers. To achieve this, we modified a pre-existing program by Dr. Bernhard Keller and ran it on many different example quivers.

The other portion this project focused on was to relate quiver mutations to another construction known as dimer graphs. Dimer graphs can be thought of as undirected networks where all of the vertices are either black or white with the additional restriction that one can only connect black vertices to white vertices and vice versa. To each dimer graph, there is also an associated dimer quiver. Provided certain conditions met, we were able to define and prove a new transformation on the dimer graph that corresponds exactly to a modified version of quiver mutation at vertices with 2 arrows incoming and 2 arrows outgoing. We refer to this transformation as a 4-face mutation on the dimer graph.

Mutations of Quivers with Potential and Dimer Models

Published:

Abstract

In this work, we aim to give a brief exposition to a relationship between mutations of quivers with potential and dimer models. Quiver mutation arose naturally from studying the flips of triangulations of regular \(n\)-gons and are combinatorial in nature. Dimer models can be interpreted as finite bipartite tilings of a compact oriented Riemann surface and arose from statistical physics. The work of Derksen, Weymann, and Zelevinsky provide an approach to lift quiver mutation to an algebraic setting by introducing the notion of a quiver with potential. Every dimer model has a naturally associated quiver with potential known as its dimer quiver. We show that a specific kind of transformation, known as urban renewal, of a dimer model induces mutation of the dimer quiver in a natural way with respect to quiver potential.

Restricted Quiver Mutation and n-Face Urban Renewal in Dimer Models

Published:

Abstract

Quivers that arise from triangulations of surfaces have been extensively studied by Fomin, Shapiro, and Thurston in the context of flips of triangulations. In this talk, we introduce restricted quiver mutation as a mild generalization of flips of triangulations and how this generalization can be connected to a variant of urban renewal in dimer models via Derksen, Weyman, and Zelevinsky’s mutation of quivers with potential.

Cluster-Like Algebras from Triangulations of Non-Orientable Surfaces

Published:

Abstract

In 2006, Fomin, Shapiro, and Thurston studied various properties of cluster algebras that arise from triangulations of orientable bordered surfaces with marked points. In this talk, we will give an expository introduction to quasi-cluster algebras, defined by Dupont and Palesi in 2011, which generalize the surface-type cluster algebras by removing the assumption of orientability of the surface being triangulated. If time permits, we will present various recent developments of quasi-cluster algebras such as partitioned quivers and quasi-cluster complexes.

teaching

Art of Problem Solving

Teaching Assistant, Art of Problem Solving Online School, June 2020 - Present

My role as a teaching assistant at the Art of Problem Solving’s Online School entails:

  • Answering questions, engaging, and encouraging students in live text-based class sessions ranging from 30 to 75 students.
  • Grading student submissions and provides detailed written feedback.

To date, I have assisted the following courses in some form or another:

MATH 093 (UOP)

Teaching assistant, University of the Pacific, Fall 2022

Answered student questions on in-class worksheets for a precalculus companion course.

Math Hub Tutor

Tutor, University of the Pacific, Fall 2023 - Spring 2024

Stationed as a CRLA-certified drop-in tutor at University of the Pacific’s library to answer student questions from the college algebra, precalculus, and calculus classes. I also was the primary drop-in tutor for a variety of other higher-level math content such as linear algebra, differential equations, proof-writing, and abstract algebra.

Pacific Math Club Lecture Series: Complex Analysis

Lecturer, Pacific Math Club, Fall 2023

Ran a weekly lecture series through University of the Pacific’s mathematics club to lecture on topics from a typical complex analysis course which is not offered at University of the Pacific. Topics were presented in a way friendly to those with background in calculus at the levels of MATH 051, MATH 053, and MATH 055 at the University of the Pacific.

Resources used

  1. Handouts written by myself that are inspired by or reference the following books:
    • Complex Variables and Applications (9th edition) by Brown and Churchill
    • Visual Complex Analysis by Needham
    • Complex Analysis by Gamelin
    • Visual Complex Functions by Wegert
    • Complex Analysis by Stein and Shakarchi
  2. Regular usage of Complex Function Explorer and cplot for plotting functions
  3. Regular usage of MATLAB and Numpy/Matplotlib for numerical computations

Topics Covered

  1. Naive definition of complex numbers
  2. Euler’s formula \(e^{it} = \cos ⁡t+i\sin ⁡t\)
  3. Complex number algebra
  4. Complex transcendental functions
  5. Branch cuts and multivalued functions
  6. Graphing Techniques
  7. Stereographic projection
  8. Limits and continuity of complex functions
  9. Derivatives of complex functions
  10. Cauchy-Riemann equations
  11. Line integrals and Green’s theorem
  12. Line integrals of complex functions
  13. Cauchy’s theorem and Cauchy’s integral formulae
  14. Liouville’s theorem and the fundamental theorem of algebra
  15. Sequences and series of complex numbers
  16. Taylor and Laurent decompositions
  17. Classifications of singularities
  18. Calculus of residues

Pacific Math Club Lecture Series: Topology

Lecturer, Pacific Math Club, Spring 2024

Ran a weekly lecture series through University of the Pacific’s mathematics club to lecture on topics from point-set topology which is not offered as a course at the University of the Pacific. Topics were presented in a way friendly to those with background in calculus at the levels of MATH 051 and MATH 053.

Resources used

Handouts written by myself that are inspired by or reference the following books:

  • Real Mathematical Analysis by Pugh
  • Introduction to Topological Manifolds by Lee

Topics Covered

  1. Real analysis in \(\mathbb R\) and \(\mathbb R^n\)
    1. Sequences, subsequences, and convergence
    2. \(\varepsilon\)-\(\delta\) condition
    3. Continuity
  2. Metric spaces
    1. Sequences, subsequences, and convergence
    2. \(\varepsilon\)-\(\delta\) condition
    3. Open subsets and closed subsets
    4. Continuity and homeomorpisms
    5. Metric subspaces and product spaces
  3. Topological spaces
    1. Topology and open subsets
    2. Sequences, subsequences, and convergence
    3. Continuity and homeomorphisms
    4. Hausdorff Property
    5. Bases and countability properties
    6. Manifolds
    7. Subspaces
    8. Product spaces
    9. Quotient spaces
    10. Compact spaces
    11. Connected spaces

The University of Kansas

Instructor, University of Kansas, Fall 2024 - Present

As a Graduate Teaching Assistant, I receive teaching assignments at the University of Kansas. To date, I have received the following assignments:

UGA Trio Upward Bound/Upward Bound Math & Science

Instructor, University of Georgia, Summer 2025

In the summer of 2025, I served as one of the mathematics instructors for the University of Georgia’s Trio UB/UBMS program where I taught geometry, algebra, and precalculus to traditionally underrepresented students in collegiate eduation.

Lecture Series on Cluster Algebras and Quiver Representations

Lecturer, University of Kansas Math Club, Fall 2025

Ran a weekly lecture series through KU’s math club on surface type cluster algebras and quiver representations. We covered the Laurent Phenonmenon, coefficient positivity, cluster complexes, analogous developments in quasi-cluster algebras, quiver representations, path algebras, and Bernstein-Gelfand-Ponomarev reflection functors and briefly sketched out the connection to quiver mutations via cluster categories.