Mentorship
2023
I mentored the following projects at the Heidelberg Experimental Geometry Lab (HEGL).
Leniabreeder [CS Bachelor Thesis]
Reinforcement Learning for Counterexamples in Mathematics [REU]
Mentees: Hugo Hager Fernández, Karl Schamel, Chenyi Yang
Description: In a beautiful 2021 paper, Adam Wagner used reinforcement learning to find explicit constructions and counterexamples to several previously open conjectures in combinatorics. In this project, and following Wagner’s work, we will learn about reinforcement learning and how to set up some mathematical questions as problems that can be attacked with reinforcement learning.
Blog post: link
GitHub: link
Theorem Proving with Lean [REU]
Mentees: Nikita Grimm, Sebastian Meier, Frederick Vandermoeten
Description: Lean is an increasingly popular theorem prover used to assist with developing formal proofs through human-machine collaboration. In this project, we learned to use Lean and explore its applications in research mathematics, formal specification and verification, and mathematics education.
Blog post: link
Video presentation: link
Online app: link
Github: link
Simulating Broadway Boogie Woogie [REU]
Mentees: Bastian Hirschfeld, Heidi Rhodes James, Arjan Siddhpura, Ioana Todosi
Description: In this project, we used computer generative art techniques to illustrate works at the intersection of mathematics and art. In particular, we wrote software for generating recreations of the Broadway Boogie Woogie painting, Piet Mondrian’s last unfinished work.
Blog post: link
Video presentation: link
Online app: link
Github: link
Building Muqarnas [REU]
Mentees: Almoatasembellah Haggag
Description: The Mihrab (a concave niche situated in the wall of a mosque facing Mecca) of the 15th-century Badr al-Dın al-Aynı Madrasa in Cairo, renowned for its significance and uniqueness in the medieval city of Cairo, Egypt, was tragically lost in 1980 due to a structural collapse. This mihrab, an exemplar of artistic mastery, featured an exceptional instance of muqarnas – a distinctive form of stalactite vaulting with numerous smaller cells arranged in a complex geometric pattern to create a unified composition. In this report, I will provide a concise discussion of the process of reconstructing a 3D model of the muqarnas vaulting. This reconstruction is based on photographs taken of the mihrab in the 20th century, as well as a set of illustrations and sketches created by the French orientalist Jules Bourgoin in the 19th century.
Blog post: link
Video presentation: link
The Clebsch Surface with 27 Distinguished Lines [MathViz]
Mentee: Ricardo Waibel
Online app: clebsch-surf
GitHub: link
Rectangle to Torus [MathViz]
Mentee: Ricardo Waibel
Online app: rectangle-to-torus
GitHub: link
2022
I mentored the following projects at the Heidelberg Experimental Geometry Lab (HEGL).
Hyperbolic Billiards [Advanced REU]
Mentees: Christian Alber, Jannis Heising, Mara-Eliana Popescu
Description: In this project, we studied billiard orbits in hyperbolic polygons. Using symbolic dynamics, we visualized tiles in a parameter space corresponding to triangles with particular closed orbits. We are currently refining our method for describing these orbit tiles and exploring relations with billiards in Euclidean triangles.
Video presentation: link
Math and Data Science for Social Good [REU]
Co-Mentor: Valentina Disarlo
Mentees: Saifeldin Mandour, Samer Sheich Essa
Description: The goal of this project was to leverage mathematics, data analytics, machine learning, and visualization to create meaningful insights and solutions that address problems of a social nature. This project was a collaboration with DSSG-Berlin, the German Red Cross (DRK), and Harambee.
Report: available upon request
Video presentation: available upon request
GitHub: link
Political Geometry [REU]
Mentees: Ayşegül Peközsoy, Leonard Späth, Klaus Stier
Description: In this project, we analyzed the cross-party cooperation between US senators from 1973-2022 using co-sponsorship data. We were particularly interested in changes in cross-party cooperation over time.
Blog post: link
Video presentation: link
GitHub: link
Continuous Cellular Automata [REU]
Mentees: Adrian Becker
Description: Lenia is a family of cellular automata that generalize Conway’s game of life to a continuous domain. In this project, we implemented two Lenia systems and optimized our implementations with mathematics and GPU programming tricks.
Blog post: link
Root Systems and their Weyl Groups [REU]
Co-Mentors: Marius Leonhardt, Ricardo Waibel
Mentee: Amelie Strupp
Description: In this project, we created an app for exploring root systems and their transformations in 2 and 3 dimensions, and also in some higher dimensions through projections.
Blog post: link
Quasicrystals and the Cut-and-Project Method [REU]
Mentees: Noah Koopman, Ingmar Lowack
Description: In 1981, N. G. de Bruijn introduced a novel method for generating Penrose tilings by projecting particular points from a five-dimensional lattice onto the two-dimensional plane. Since then, it has been discovered that the de Bruijn method, now commonly known as the cut-and-project method, can generate many families of aperiodic tilings. In this project, we implemented and optimized the cut-and-project method for some families of aperiodic tilings.
Blog post: link
Poncelet’s Porism [REU]
Mentees: Adrian Becker, Miriam Compton, Lukas Schmidt
Description: Poncelet’s porism is a deceptively simple result in geometry that states that whenever a polygon is inscribed in one conic section and circumscribes another, the polygon must be part of an infinite family of polygons that are all inscribed in and circumscribe the same two conics. [Wikipedia] In this project, we learned the proof of this result and produced relevant visualizations.
Blog post: link
Ray Marching in Translation Surfaces [REU]
Mentees: Fabian Lander, Mara-Eliana Popescu
Description: A translation surface is a surface obtained by gluing together a finite collection of polygons in the Euclidean plane along parallel sides of the same length. In this project, we developed an immersive visualization of the geometry of translation surfaces (and mirror rooms, and surfaces of polyhedra) with raymarching.
2D Hyperbolic Game Engine [REU]
Mentees: Ines Bultmann, Filippa Piazolo
Description: In this project, we created a 2D hyperbolic geometry game engine that implemented basic features such as free motion, collision detection, and portals. Using our game engine, we built hyperbolic analogues of some classical arcade games like Combat (1977) and Asteroids (1979).
Blog post: link
Online app: link 1 (1vs1), link 2 (single player)
GitHub: link
Möbius Paint [REU]
Mentees: Mouna Dorothea Deubler, Yunus Sahin, Juliane Stehle
Description: In this project, we built a hyperbolic analogue of simple graphic editors such as MacPaint (1984) and Microsoft Paint (1985). Our editor implemented drawing features such as brushes and “straight lines” (i.e., geodesics) and canvas operations such as reflections, rotations, and translations.
Aperiodic Tilings of the Hyperbolic Plane [Highschool]
Co-Mentor: Anna Schilling
Mentee: Sebastian Ohlig
Description: In a beautiful 1998 paper, Margulis and Mozes constructed several aperiodic hyperbolic tiling systems consisting of a single convex tile. In this project, we studied and visualized some of these tilings.
Code: available upon request
Games of Chance and Skill [Highschool]
Co-Mentor: Anna Schilling
Mentees: Max Dörich, Lukas Kühlwein
Description: In this project, we explored the mathematics of games of chance and skill using computer simulations. We studied the game SET using algebra, geometry, and combinatorics. We also learned about the Sprague–Grundy theorem, and how to write programs that play popular impartial games perfectly.
GitHub: link
2021
I mentored the following project at the Heidelberg Experimental Geometry Lab (HEGL).
Laminations on Hyperbolic Surfaces in the Universal Cover [REU]
Co-mentor: Mareike Pfeil
Mentees: Christian Alber
Description: A lamination on a hyperbolic surface is a collection of simple disjoint geodesics whose union is closed. Laminations are very helpful when studying surfaces. For instance, they can be used to deform hyperbolic surfaces and to define coordinates on Teichmüller space. The goal of this project was to visualize lifts of laminations on hyperbolic surfaces in the Poincaré disk model for the hyperbolic plane.
GitHub: link
2019
I mentored the following reading course at Cairo University.
Sturmian Sequences [REU]
Mentees: Ahmed Badran, Amr Ezzat, Raghda Mohamed
Description: This reading course covered the basics of ergodic theory and symbolic dynamics by exploring circle rotation maps, linear flow on the torus, cutting sequences, and their different connections.
2017
I co-mentored the following project as a graduate student at the Washington Experimental Mathematics Lab (WXML).
Gravitational Billiards [REU]
Principal mentor: Jayadev Athreya
Mentees: Stephanie Anderson, Kush Gupta, N’Vida Yotcho
Description: This project explored gravitational billiards, a variant of classical billiards where a gravitational force affects the motion of the balls. We wrote software simulations to analyze the dynamics of gravitational billiards in frictionless domains in the shape of parabolas, circles, paraboloids, and spheres.
Report: link
Code: in report
2016
I co-mentored the following project as a graduate student at the Washington Experimental Mathematics Lab (WXML).
The Translation Surface of the Bothel Pentagon [REU]
Principal mentor: Jayadev Athreya
Mentees: Wenbo Gao, Maria Osborne, Matthew Staples
Description: In 2015, Casey Mann, Jennifer McLoud-Mann, and David Von Derau of the University of Washington Bothell announced that they had discovered a new convex pentagon capable of tiling the plane, namely the Bothell pentagon, and which was the 15th pentagon with this tiling property. In this project, we studied the translation surface of the Bothell pentagon, and in particular, we focused on analyzing some trajectories on and the cylindrical decomposition of the surface.
Report: link
AMS Notices mention: link
Code: available upon request
2015
I co-mentored the following project as a graduate student at the Mason Experimental Geometry Lab (MEGL).
Orbits on Arithmetic Moduli Spaces [REU]
Principal mentor: Sean Lawton
Mentees: Robert Argus, Patrick Brown, Jermain McDermott
Description: The project was to explore the nature of the periods resulting from certain dynamical systems. In particular, let 𝒳 = Hom(Fᵣ, SL(2, 𝔽ₚ))/SL(2, 𝔽ₚ) be the 𝔽ₚ-points of the character variety, where Fᵣ is the free group of r letters, and 𝔽ₚ is the finite field of order p. Then Out(Fᵣ) acts on 𝒳 by χ ↦ αχ for a given α ∈ Out(Fᵣ). Upon fixing α and considering the resulting dynamical system for a fixed χ, we were interested in the length of the orbits and how it changes as a function of p.
Slides: link
Poster: link