A picture of me
Mark Gillespie

I'm a sixth year Computer Science PhD student at Carnegie Mellon University, advised by Keenan Crane. I work on algorithms for reliably and efficiently processing geometric data, drawing inspiration from classical topology and differential geometry. Previously I was an undergraduate at Caltech, where I worked on plasma simulation with Peter Schröder, chromosomal shape embeddings with Mathieu Desbrun, and interval analysis root finding techniques with Alan Barr.

In my spare time, I like to fold origami and knit. You can find more here.


Research
Winding Numbers on Discrete Surfaces
Nicole Feng, Mark Gillespie, Keenan Crane
ACM TOG (SIGGRAPH 2023)
Winding numbers are a basic component of geometric algorithms such as point-in-polygon tests, and their generalization to data with noise or topological errors has proven invaluable in robust geometry processing. However, standard definitions do not immediately apply on surfaces, where not all curves bound regions. We develop a meaningful generalization, starting with the well-known relationship between winding numbers and harmonic functions. Ultimately, our algorithm yields (i) a closed completion the input curves, (ii) integer labels for regions that are meaningfully bounded by these curves, and (iii) the complementary curves that do not bound any region.
Surface Simplification using Intrinsic Error Metrics
ACM TOG (SIGGRAPH 2023)
This paper describes a method for fast simplification of surface meshes. Rather than approximate the extrinsic geometry, we construct a coarse intrinsic triangulation of the input domain. In the spirit of the quadric error metric (QEM), we perform greedy decimation while agglomerating global information about approximation error. In lieu of extrinsic quadrics, however, we store intrinsic tangent vectors that track how far curvature “drifts” during simplification. The overall payoff is a “black box” approach to geometry processing, which decouples mesh resolution from the size of matrices used to solve equations.
Integer Coordinates for Intrinsic Geometry Processing
Mark Gillespie, Nicholas Sharp, Keenan Crane
ACM TOG (SIGGRAPH Asia 2021)
This paper describes a numerically robust data structure for encoding intrinsic triangulations of polyhedral surfaces. Our starting point is the framework of normal coordinates from geometric topology, which we extend to a broader set of operations needed for mesh processing. As a stress test, we successfully compute an intrinsic Delaunay refinement and associated subdivision for all manifold meshes in the Thingi10k dataset.
Geometry Processing with Intrinsic Triangulations
Nicholas Sharp, Mark Gillespie, Keenan Crane
ACM SIGGRAPH Courses (2021), SIAM IMR 2021 Courses
Intrinsic triangulations de-couple the mesh used to encode geometry from the one used for computation. The basic shift in perspective is to encode the geometry of a mesh not in terms of ordinary vertex positions, but instead only in terms of edge lengths. This course provides a first introduction to intrinsic triangulations and their use in mesh processing algorithms, covering the theory, practical details, and some cutting-edge research in the area.
Discrete Conformal Equivalence of Polyhedral Surfaces
Mark Gillespie, Boris Springborn, Keenan Crane
ACM TOG (SIGGRAPH 2021)
We present a numerical method for surface parameterization, leveraging hyperbolic geometry to yield maps that are locally injective and discretely conformal in an exact sense. Stress tests involving difficult cone configurations and near-degenerate triangulations indicate that the method is extremely robust in practice, and provides high-quality interpolation even on meshes with poor elements.
Magnetohydrodynamics Simulation via Discrete Exterior Calculus
Mark Gillespie
Senior Thesis (Caltech)
Magnetohydrodyamics (MHD) models the behavior of current-carrying fluids such as plasma on the surface of the sun, or the earth's molten core. In this thesis, I gave an integrator for ideal MHD in two-dimensional domains with boundary and showed that the integrator preserves total energy and cross helicity.

Awards
2019

NSF Graduate Research Fellowship

2017

SIGGRAPH ACM Turing Award Celebration Grant

I was one of 10 students sponsored by SIGGRAPH to attend the ACM Turing Award Celebration.
2017, 2016

Arthur R. Adams SURF Fellowship

Fellowship to fund my summer research.
2016

William Lowell Putnam Mathematics Competition

31 points (rank: 365/3214)

Education

Schools

Carnegie Mellon University
2018-present
  • PhD student in the Computer Science Department
California Institute of Technology
2014-2018
  • Majors: Computer Science, Mathematics
  • GPA: 4.1

Talks
Dec. 2023

Dynamic Intrinsic Geometry Processing

CMU
Thesis proposal talk. My slides are available here and my proposal document is available here.
Sept. 2023

Intrinsic Triangulations in Geometry Processing

ISTA
90 minute talk about my work on intrinsic triangulations.
Aug. 2023

Intrinsic Triangulations in Geometry Processing

Geometry Workshop in Obergurgl
30 minute talk about my work on intrinsic triangulations.
Jul. 2023

Intrinsic Triangulations in Geometry Processing

TU Berlin SFB TRR 109 Colloquium
45 minute talk about some of my assorted work on intrinsic triangulations. The slides are available here.
Apr. 2022

Discrete Conformal Equivalence of Polyhedral Surfaces

UCSD Pixel Cafe
45 minute talk about my paper of the same name. The slides are available here.
Mar. 2022

Discrete Conformal Equivalence of Polyhedral Surfaces

Toronto Geometry Colloquium
10 minute talk about my paper of the same name. The slides are available here, and the beautiful poster designed by Rachel Joan Wallis is available here.
Dec. 2021

Integer Coordinates for Intrinsic Geometry Processing

Siggraph Asia Technical Papers
20 minute talk about my paper of the same name. The full talk is available on YouTube here, and a 5-minute version is available here.
Aug. 2021

Discrete Conformal Equivalence of Polyhedral Surfaces

Siggraph Technical Papers
20 minute talk about my paper of the same name. The full talk is available on YouTube here, and a 5-minute version is available here.
Oct. 2019

Origami and Geometry

CMU Graphics Lab
1 hour talk overview of assorted algorithms for origami design, and how they relate to geometetry. The slides are available here.
Jun. 2019

Hyperbolic Geometry and Discrete Conformal Maps

CMU Graphics Lab
1 hour talk about discrete conformal maps, length cross ratios, and hyperbolic geometry. The slides are available here.
Jan. 2019

Magnetohydrodynamics and the Geometry of Conservation Laws

CMU Graphics Lab
1 hour talk on my previous summer research. Focused on general background about geometric mechanics and MHD. The slides are available here.
Oct. 2017

2D Plasma Simulation via Discrete Exterior Calculus

Caltech Summer Research Seminar Day
15 minute presentation on the results of my summer research. The slides are available here.
Sept. 2017

Combinatorics and the Probabilistic Method

Westfield High School Seminar in College Mathematics
30 minute presentation to a high school math class. Gave an introduction to elementary combinatorics and presented some simple applications of the probabilistic method. My notes are available here.
Mar. 2017

Continuous and Discrete Mechanics for Variational Integrators

Caltech CS 177b
1.5 hour final presentation for a computer graphics class. Gave an overview of Hamiltonian and Lagrangian mechanics, and discussed how to discretize them to produce variational time integrators. Extended notes are available here.

Miscellanea
Web demo which traces out geodesics on a mesh. The code is available here.
Notes on hyperbolic geometry and discrete conformal maps.
A shader to make meshes tartan.
GLSL implementation of BPM: Blended Piecewise Möbius Maps by Rorberg, Vaxman & Ben-Chen, incororated into polyscope.
Takes any triangle mesh and turn it into an orientable manifold mesh by constructing the orientable double cover.
This is a rough implementation of the Delaunay edge split algorithm presented in Efficient construction and simplification of Delaunay meshes by Yong-Jin Liu, Chunxu Xu, Dian Fan, and Ying He. It takes in a triangle mesh and then performs edge splits to make the mesh Delaunay.
Web demo computing the Karcher mean of points on a sphere.
Visualization of the phase space of a pendulum as a cylinder.
Comparison of explicit, implicit, and symplectic Euler integrators for a pendulum.
Demo of an implementation of the combinatorial map data structure for n-dimensional simplicial complexes that I experimented with in geometry-central. It has not yet made its way into the library itself, but the implementation can be found here.
Online tool to check if a word is in the dictionary.