Papers

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.

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.

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).

Recent research has shown that alignment between the structure of graph data and the geometry of an embedding space is crucial for learning high-quality representations of the data. The uniform geometry of Euclidean and hyperbolic spaces allows for representing graphs with uniform geometric and topological features, such as grids and hierarchies, with minimal distortion. However, real-world graph data is characterized by multiple types of geometric and topological features, necessitating more sophisticated geometric embedding spaces. In this work, we utilize the Riemannian symmetric space of symmetric positive definite matrices (SPD) to construct graph neural networks that can robustly handle complex graphs. To do this, we develop an innovative library that leverages the SPD gyrocalculus tools to implement the building blocks of five popular graph neural networks in SPD. Experimental results demonstrate that our graph neural networks in SPD substantially outperform their counterparts in Euclidean and hyperbolic spaces, as well as the Cartesian product thereof, on complex graphs for node and graph classification tasks.

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.

We propose the use of the vector-valued distance to compute distances and extract geometric information from the manifold of symmetric positive definite matrices (SPD), and develop gyrovector calculus, constructing analogs of vector space operations in this curved space. We implement these operations and showcase their versatility in the tasks of knowledge graph completion, item recommendation, and question answering. In experiments, the SPD models outperform their equivalents in Euclidean and hyperbolic space. The vector-valued distance allows us to visualize embeddings, showing that the models learn to disentangle representations of positive samples from negative ones.

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.

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.

This article presents virtual reality software designed to explore the Sol geometry. The simulation is available on 3-dimensional.space/sol.html

We describe a method of rendering real-time scenes in Nil geometry, and use this to give an expository account of some interesting geometric phenomena. You can play around with the simulation at www.3-dimensional.space/nil.html.

The geometry of the Heisenberg group acting on the plane arises naturally in geometric topology as a degeneration of the familiar spaces S^2, H^2 and E^2 via conjugacy limit as defined by Cooper, Danciger, and Wienhard. This paper considers the deformation and regeneration of Heisenberg structures on orbifolds, adding a carefully worked low-dimensional example to the existing literature on geometric transitions. In particular, the closed orbifolds admitting Heisenberg structures are classified, and their deformation spaces are computed. Considering the regeneration problem, which Heisenberg tori arise as rescaled limits of collapsing paths of constant curvature cone tori is completely determined in the case of a single cone point.