Academic
Curriculum Vitae →Research
- Paper Tori
Flat polyhedral tori in three-space with eight vertices, the fewest possible, and which shapes of flat torus they can take.
- Surfaces of Revolution in Homogeneous 3-Manifolds
Which rotationally symmetric surfaces can be realized as surfaces of revolution in which homogeneous three-dimensional geometries, Riemannian and Lorentzian.
- Certifying Arithmeticity for Degree-Six Symplectic Hypergeometric Monodromy Groups
The last three open degree-six symplectic hypergeometric monodromy groups, settled with machine-found certificates verified in exact arithmetic, and a program to do the same for every hypergeometric group in degree six and below.
- Discrete Isometric Embeddings
Finding isometric embeddings of discretized surfaces and manifolds into curved ambient spaces by minimizing an energy, beginning with hyperbolic surfaces drawn in three-dimensional space.
- Visualizing Elliptic Curves
Faithful pictures of elliptic curves over the complex numbers and over finite fields, and the equivalence of categories that makes the finite field pictures work.
- Optics of Black Holes
Rendering spacetimes with many black holes by turning general relativity into classical optics.
- Visualizing Knot Complements
Drawing the space around a knot from the inside, by thickening knots in the 3-sphere and projecting from a point on the knot.
- Symmetric Spaces for Machine Learning
Embedding graphs and knowledge into Riemannian symmetric spaces, where Euclidean and hyperbolic pieces coexist.
- Algebraic Number Starscapes
Pictures of every complex algebraic number of low degree, and the hyperbolic geometry and Diophantine approximation they reveal.
- Ray-Marching the Thurston Geometries
Real-time, geometrically exact views from inside all eight Thurston geometries
- Geometric Transitions
How one geometry degenerates into another inside projective space, from the Heisenberg plane to the first limits that reach Nil.
PapersAll 13 papers →
- 2025Elliptic Curves and the Hopf Fibration with Nadir Hajouji
Abstract
By combining tools from different areas of mathematics, we obtain 3D visualizations of elliptic curves over different fields that faithfully capture the underlying algebra and geometry.
- 2022Algebraic Number Starscapes with Edmund Harriss, Kate Stange
Abstract
We study the geometry of algebraic numbers in the complex plane, and their Diophantine approximation, aided by extensive computer visualization. Motivated by these images, called algebraic starscapes, we describe the geometry of the map from the coefficient space of polynomials to the root space, focussing on the quadratic and cubic cases. The geometry describes and explains notable features of the illustrations, and motivates a geometric-minded recasting of fundamental results in the Diophantine approximation of the complex plane. The images provide a case-study in the symbiosis of illustration and research, and an entry-point to geometry and number theory for a wider audience. The paper is written to provide an accessible introduction to the study of homogeneous geometry and Diophantine approximation. We investigate the homogeneous geometry of root and coefficient spaces under the natural PSL(2;C) action, especially in degrees 2 and 3. We rediscover the quadratic and cubic root formulas as isometries, and determine when the map sending certain families of polynomials to their complex roots (our starscape images) are embeddings. We consider complex Diophantine approximation by quadratic irrationals, in terms of hyperbolic distance and the discriminant as a measure of arithmetic height.
- 2022Ray-marching Thurston Geometries with Henry Segerman, Rémi Coulon, Sabetta Matsumoto
Abstract
We describe algorithms that produce accurate real-time interactive in-space views of the eight Thurston geometries using ray-marching. We give a theoretical framework for our algorithms, independent of the geometry involved. In addition to scenes within a geometry X, we also consider scenes within quotient manifolds and orbifolds X/Gamma. We adapt the Phong lighting model to non-euclidean geometries. The most difficult part of this is the calculation of light intensity, which relates to the area density of geodesic spheres. We also give extensive practical details for each geometry.
- 2021Symmetric Spaces for Graph Embeddings: A Finsler-Riemannian Approach with Anna Wienhard, Beatrice Pozzetti, Federico Lopez, Michael Strube
Abstract
Learning faithful graph representations as sets of vertex embeddings has become a fundamental intermediary step in a wide range of machine learning applications. We propose the systematic use of symmetric spaces in representation learning, a class encompassing many of the previously used embedding targets. This enables us to introduce a new method, the use of Finsler metrics integrated in a Riemannian optimization scheme, that better adapts to dissimilar structures in the graph. We develop a tool to analyze the embeddings and infer structural properties of the data sets. For implementation, we choose Siegel spaces, a versatile family of symmetric spaces. Our approach outperforms competitive baselines for graph reconstruction tasks on various synthetic and real-world datasets. We further demonstrate its applicability on two downstream tasks, recommender systems and node classification.
Teaching
- Differential Equations ↗
A first course in differential equations, told through the geometry of flows: what equations mean, when they can be solved, and how to see the behavior of solutions even when they can't.
- Modern Geometry ↗
Notes / textbook for a 1 semester undergraduate course in geometry, taking calculus as the foundations. Covers Euclidean, Spherical and a bit of Hyperbolic geometry.
- Multivariable Calculus ↗
Supplementary Course Notes for the multivariable calculus course I teach at USF, with live animations.
- Reaching for Infinity ↗
A year-long advanced undergraduate real analysis textbook, with an emphasis on historically important problems.
- Shader Programming ↗
Course notes for GPU shader programming as a tool for mathematical visualization, covering fractals, implicit surfaces, tilings, and physical simulations in GLSL.
- Fall 2026 Calculus I (MATH 109) — 2 sections, Differential Equations (MATH 340)
- Fall 2025 Linear Algebra (MATH 230), Multivariable Calculus (MATH 211)
- Spring 2025 Multivariable Calculus (MATH 211), Real Analysis (MATH 453)
- Fall 2024 Linear Algebra (MATH 229) — 2 sections, Multivariable Calculus (MATH 211)
- Spring 2024 Calculus I (MATH 109), Real Analysis (MATH 453)
- Fall 2023 Multivariable Calculus (MATH 211), Foundations of Geometry (MATH 380)
- Spring 2023 Real Analysis (MATH 453), Multivariable Calculus (MATH 211)
- Fall 2022 Calculus I (MATH 109), Calculus II (MATH 110)
TalksAll 25 talks →
- 2026Mathematics of Beautiful Graphics
- MAA MathFest, Boston — Invited Address 2026
Abstract
Photorealistic computer-generated images are everywhere — from blockbuster films to video games to architectural visualization. Modern software tends to hide the mathematics behind them, but it's beautiful stuff: differential geometry, measure theory, and statistics all play essential roles. In this talk, we'll see how simple questions about light and color naturally lead to an infinite-dimensional problem with a curious recursive structure — to compute anything, you seemingly need to have already computed everything. The way out is to recast the problem as integration over a space of light paths. Different visual phenomena — mirrors, glass, fog — correspond to different measures on this space. Understanding this structure puts realistic rendering within reach of anyone willing to do the analysis. It also offers mathematicians a source of compelling high-dimensional problems and a new way of seeing the world — from the translucence of a jade sculpture to the glow of a sunset, the geometry of path space is all around us.
- 2026Surfaces of Revolution in Homogeneous Spaces
- Casa Matemática Oaxaca (BIRS) 2026
Abstract
The pseudosphere realizes the hyperbolic plane as a surface of revolution in Euclidean space, but only up to a circular rim, beyond which no isometric extension exists. This failure reflects a limitation of the ambient geometry rather than of the surface itself: in hyperbolic space, the same surface extends without obstruction. Such behavior is common — rotationally symmetric surfaces arise throughout geometry, and whether a given one can be concretely realized inside an ambient 3-manifold depends sharply on the interplay between their geometries. In this talk we seek a classification: which surfaces embed in which ambient spaces, as the surfaces range over all rotationally symmetric (Riemannian and Lorentzian) surfaces, and the ambient spaces over all homogeneous 3-manifolds of either signature. This leads to a web of implications between geometries: if a surface embeds in geometry X, what does this force about its embeddability in geometry Y? Joint work with Fabian Lander.
- 2026Visualizing Elliptic Curves over Finite Fields
- Institut Henri Poincare 2026
- TU Eindhoven 2025
Abstract
Elliptic curves over finite fields are central to modern number theory and cryptography, yet they are rather difficult to visualize — their points form finite sets without obvious geometric structure, and the group law that makes them so useful is obscured in most pictures. This is quite different from the more familiar story over the complex numbers, where geometry runs the show: every elliptic curve is a torus, and the group law is simply addition. In this talk, we explore a way to bridge the gap between these worlds. Using ideas from lattices with complex multiplication, we construct a way to lift any elliptic curve over a finite field to a (subset of a) complex torus, in a way that makes the group structure, the action of Frobenius, and the points over all field extensions simultaneously visible in a single picture. We will begin with an introduction to elliptic curves aimed at a general mathematical audience, building from curves over the reals and complex numbers to the finite field setting, before describing the lifting construction and the pictures it produces. This is joint work with Nadir Hajouji.
- 2025A Gravitational Photograph
- ICERM — Brown University 2025
Abstract
Massive bodies focus light by curving spacetime so that initially parallel null geodesics converge, producing the well‑known phenomenon of gravitational lensing. In this talk we explore the extent to which this name can be taken literally, imagining a camera where the optical lenses system has been replaced by a black hole. Using some tricks from Lorentzian geometry, path tracing, and gpu computing, we simulate such a camera, and the images it produces.
Art ExhibitionsAll 19 exhibitions →
- 2026Elliptic Curves with Nadir Hajouji
Elliptic Curves over Finite Fields, shown in the art collections in the exhibition hall at the International Congress of Mathematicians — joint work with Nadir Hajouji.
- 2026Creation: Between Art and Mathematics with Claudio Gomez-Gonzales, Gabriel Dorfsman-Hopkins
My piece "Lines on a Cubic Surface" was selected for the Creation exhibition at the Maison Poincaré in Paris.
- 2025Warped Realities: The Art of Differential Geometry with Edmund Harriss, Henry Segerman, Robert Fathauer, Stepan Paul
Lightfall, The Geometry of Spacetime, Geodesic Woodcuts, and Many Paths to the Beach, shown in MoMath's "Warped Realities" exhibition.
- 2025West Coast Biennial
Singularity, selected for the juried West Coast Biennial.
- 2024Chaos
Julia and Many Paths to the Beach, selected for the juried show "Chaos".
Film
- 2025Felix Klein: Insights from the Outside ↗
Feature documentary; I co-presented the mathematical segments with Anna Wienhard and, as Director of Computer Graphics, produced all of its mathematical illustration.
Featured Work
- 2024What's Happening in the Mathematical Sciences, Vol. 13 with Henry Segerman, Rémi Coulon, Sabetta Matsumoto
Main, or 'Cover Chapter' of this year's book features prominently our work
- 2024On the Importance of Illustration for Mathematical Research
Article surveys recent work in Mathematical Illustration, features three of my research projects
- 2024Cover Art with Henry Segerman, Rémi Coulon, Sabetta Matsumoto
January 2024 edition, full cover illustration from Nil Geometry
- 2023Cover Art
Full cover illustration from my work on Knot Complements
- 2023Burning Man with Henry Segerman, Rémi Coulon, Sabetta Matsumoto
Commissioned by MIT Mathematician Daniel Álvarez-Gavela to make 3D Geometry videos for large immersive 180-degree dome screen
- 2021Spectral Geometry in the Presence of Symmetry
Invited to do all illustrations for this survey article by Craig Sutton
- 2020Illustrating Mathematics
Feature on my work with Projective Tilings in Laser Cut Wood