Lecture Notes

Hyperbolic Trigonometry and Special Relativity

A master's course for high school teachers I designed and taught in Summer 2025 as part of Hollins' M.A.L.S. program. The premise is beautiful in its simplicity — by replacing Euclidean distances with Minkowski distance, the locus of points a fixed distance from some center point becomes a hyperbola rather than a circle. We can define angles and trigonometric functions exactly as before, but with all distances interpreted as Minkowski distances. The result is hyperbolic trigonometry, and it is the mathematical framework of Special Relativity.

This course was inspired by Austin Gleeson's Introduction to Modern Physics class at UT Austin. I was a TA for the course for a few years as a grad student.

• Hyperbolic Trigonometry and Special Relativity (PDF)

Toric Varieties

An LTCC graduate course I designed and taught on toric varieties in Winter 2023.

• Toric Varieties- Lecture 1 (PDF) Why study toric varieties, algebraic geometry background, algebraic tori
• Toric Varieties- Lecture 2 (PDF) Affine toric varieties, normality
• Toric Varieties- Lecture 3 (PDF) Fans, smooth toric varieties
• Toric Varieties- Lecture 4 (PDF) Separated toric varieties
• Toric Varieties- Lecture 5 (PDF) Orbit cone correspondence, morphisms of toric varieties, projective toric varieties

Polyhedral Geometry

I team-taught a Math Lab course with Molly Weselcouch at Hollins in Spring 2025. My portion of the course was a survey of polyhedral geometry, culminating with Lev Borisov's dual nef partitions.

• Polyhedral Geometry (PDF) Survey of polyhedral geometry building to dual nef partitions

Several Variable Calculus

The first time I taught this course, I had the impression that I was exerting more effort avoiding the topic of wedge products than it would take to simply teach it. So, this semester (Fall 2025) I redesigned the first half of the course. I give a more careful treatment of real vector spaces and real affine spaces, discuss dot products as inner products, then treat \(k\)-volumes of \(k\)-parallelotopes as a stepping stone to wedge products. I do explain how to recover cross products from wedge products in case the topic comes up in other classes.

• Several Variable Calculus (PDF) Notes for the first half of Several Variable Calculus: Vector Spaces, Affine Spaces, Dot Product, Wedge Product, Planes and Quadric Surfaces

From Toric Varieties to Cluster Varieties Minicourse

I gave a minicourse at Centro de Ciencias Matemáticas UNAM Campus Morelia in Fall 2022 on the path from the theory of toric varieties to an analogous theory for minimal models of cluster varieties.

• De Variedades Tóricas a Variedades de Conglomerado (PDF) (In Spanish)

Cluster Varieties Lecture

Slides from a talk on cluster varieties in the Symplectic Cut seminar.

• Cluster Varieties (PDF)

Convexity in Tropical Spaces and Compactifications of Cluster Varieties talk

Slides from a talk given in the Bath Algebra, Geometry and Number Theory Seminar, based on joint work with Mandy Cheung and Alfredo Nájera Chávez.

• Convexity in Tropical Spaces and Compactifications of Cluster Varieties (PDF)

Littlewood-Richardson Coefficients from Mirror Symmetry talk

Slides from a talk given in the Clusters and Geometry Seminar at Yale, based on my PhD thesis.

• Littlewood-Richardson Coefficients from Mirror Symmetry (PDF)

Grassmannians, Plabic Graphs, and Mirror Symmetry for Cluster Varieties talk

Slides from a talk given in the Fanosearch Group Seminar at Imperial College London, based on ongoing joint work with Lara Bossinger, Mandy Cheung, and Alfredo Nájera Chávez.

• Grassmannians, Plabic Graphs, and Mirror Symmetry for Cluster Varieties (PDF)

Toric Degenerations of Cluster Varieties talk

Slides from a talk given at the Tropical Geometry Meets Representation Theory II Workshop at the University of Leicester, based on joint work with Lara Bossinger, Juan Bosco Frías Medina, and Alfredo Nájera Chávez.

• Toric Degenerations of Cluster Varieties (PDF)