Academic

One sentence under the title, the way each homepage section has a blurb.

Two or three paragraphs on what you work on and why. This is the thing the site currently has nowhere to say.

Teaching

How you teach and what you care about in a classroom.

Research PapersAll 13 papers →

A short lead-in to the papers — themes, threads, what connects them.

Showing the 4 most recent — add featured: true to pick these by hand.

  1. 2026
    Illustrating Hyperbolic Surfaces with Mesh Embeddings Bridges 2026 with Anna Wienhard, Diaaeldin Taha, Erik Loffelholtz, Fabian Lander
    Abstract

    Hyperbolic geometry exhibits geometric phenomena, such as fast area growth, that are difficult to visualize faithfully in Euclidean space, and which standard models like the Poincaré disk can obscure. To bring hyperbolic geometry to life, we embed hyperbolic surfaces in Euclidean space by discretizing the surfaces into meshes, and minimizing a distortion energy so that the edge lengths in the embeddings match those in the hyperbolic plane. The resulting surfaces buckle and ruffle to accommodate the extra area, making visible what flat models hide. We present exemplary illustrations, such as embedded disks, equidistant strips, diverging geodesics, and also artistic organic-like renders. We discuss our use of these models, as renders and 3D prints, in research talks, public engagement, outreach, and education.

  2. 2025
    Classical Optics for Charged Black Holes Bridges Math Art 2025 Conference Proceedings
    Abstract

    We describe a method for rendering multiple extremally charged black holes using an analogous system in classical optics, simplifying the prerequisite mathematics for generating accurate images. A primary goal of this work is to showcase a suite of techniques from geometry and relativity that may be of interest to illustrators and artists.

  3. 2025
    Elliptic Curves and the Hopf Fibration Bridges Math Art Conference Proceedings, 2025 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.

  4. 2024
    Degenerations of the product geometries in projective space that contain Nil Topology and its Applications with Tom Shifley
    Abstract

    This paper produces explicit conjugacy paths for the product geometries H^2 x R and S^2 x R whose limits contain the geometry of the Heisenberg group's action on itself. These are the first such conjugacy limits to any model of Nil, continuing the program of Daryl Cooper, Jeffrey Danciger, and Anna Wienhard to determine all possible degenerations between Thurston geometries in (PGL(4,R), RP^3).

TalksAll 25 talks →

The kinds of talks you like to give — research seminars, undergraduate colloquia, public lectures — and what you aim for in each.

Note: grouping these by kind needs a kind field on the talks schema. Right now the type is only inferable from venue-name strings ("Penn State — Student Colloquium", "UT Austin — Topology Seminar"), so this list is chronological.

Showing the 4 most recent — add featured: true to pick these by hand.

  1. 2026
    Mathematics 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.

  2. 2026
    Surfaces 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.

  3. 2026
    Visualizing 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.

  4. 2025
    A 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 →

How the art side connects to the mathematics, and what showing work in galleries and museums has meant for the research.
  1. 2026
    Creation: Between Art and Mathematics Maison Poincaré, Paris, France — April–July 2026 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.

Film

A few sentences on the documentary work.
  1. 2025
    Felix Klein: Insights from the Outside ↗ Co-Presenter & Director of Computer Graphics

    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

Optional lead-in — or delete this slot and let the list speak.
  1. 2024
    What's Happening in the Mathematical Sciences, Vol. 13 Published in AMS Books with Henry Segerman, Rémi Coulon, Sabetta Matsumoto

    Main, or 'Cover Chapter' of this year's book features prominently our work

  2. 2024
    On the Importance of Illustration for Mathematical Research Published in Notices of the AMS

    Article surveys recent work in Mathematical Illustration, features three of my research projects

  3. 2024
    Cover Art Published in Notices of the AMS with Henry Segerman, Rémi Coulon, Sabetta Matsumoto

    January 2024 edition, full cover illustration from Nil Geometry

  4. 2023
    Cover Art Published in MAA Focus

    Full cover illustration from my work on Knot Complements

  5. 2023
    Burning 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

  6. 2021
    Spectral Geometry in the Presence of Symmetry Published in Notices of the AMS

    Invited to do all illustrations for this survey article by Craig Sutton

  7. 2020
    Illustrating Mathematics Published in AMS Books

    Feature on my work with Projective Tilings in Laser Cut Wood