Hi, I'm Mark. I'm a senior undergrad in math and computer science at Caltech. My research interests are in computer graphics and physical simulation. Right now, I am interested in variational integrators and the use of Discrete Exterior Calculus to discretize dynamical systems.
Here is a blog that I write with some friends. And here is my CV.
Email: mgillesp@caltech.edu
Over the summer of 2017, I worked on simulating MHD using discrete exterior calculus. Following this paper by Kraus and Maj, I implemented a 2D MHD simulation in periodic domains. I then proved that the method conserves energy and crosshelicity, and I extended the method to domains with boundaries.
Below are some results of the simulation. The magnetic field strength is visualized by a glowing effect. You can see the fluid carrying the magnetic field upwards, where it collides with the container's boundary and starts to flow sideways.
I implemented the algorithm in HoudiniFX, and used Houdini to render these images.
In the winter of 2016, I implemented a variational integrator for multiple pendula as a final project for a graphics class. I used the method of Marsden and West described here (also available here) to simulate a physical system with constraints.
Here are some interesting assignments that I implemented for a discrete differential geometry class in the fall of 2016.
I computed the normal modes of the discrete Laplacian on the Stanford bunny. Here are the first 4.
Using these normal modes, I numerically solved a Poisson equation to find the electric field induced by charges on the bunny's surface.
Recently, I've been working with Professor Alan Barr on using Interval Analysis to solve rootfinding and minimization problems. Interval analysis is the technique of doing computations with intervals representing uncertainty instead of regular real numbers. This provided guaranteed bounds on the accuracy of your results. Our code is available on GitHub.
Fall 2017  Teaching Assistant for CS 171, Introduction to Computer Graphics (Caltech) 
Under Professor Alan Barr, graded problem sets, held weekly office hours, delivered recitation lectures 

Summer 2017  Arthur R. Adams Undergraduate Researcher (Caltech) 
Under Professor Peter Schroeder, implemented an energypreserving integrator for 2D MHD on grids and proved its conservation properties


Summer 2016  Arthur R. Adams Undergraduate Researcher (Caltech) 
Under Professor Mathieu Desbrun, developed a new algorithm for computing polymer conformation using dimensionality reduction techniques.


Jan. 2016  2017  Undergraduate Researcher (Caltech) 
Under Professor Alan Barr, explored applications of interval analysis to rootfinding and solving differential equations


Spring 2016,2017  Teaching Assistant for CS 38: Algorithms (Caltech) 
I graded problem sets every few weeks and held weekly office hours. 

Summer 2015  Software Engineering Intern (Google) 
I prototyped a new credit card information input interface for an Android library. I wrote code in Java. 
CS 177ab  Discrete Differential Geometry 
discrete study of: differential forms, deRham cohomology, Poisson problems, variational mechanics  
CS 176  Introduction to Computer Graphics Research 
geometry processing, data visualization, vector fields and flows  
CS 171  Introduction to Computer Graphics Laboratory 
shaders, geometry processing, physical simulation, ray tracing  
Ma 109bc  Introduction to Geometry and Topology 
manifolds, vector fields, GaussBonnet theorem, geodesics, differential forms  
Ma 151a  Algebraic Topology 
fundamental groups, covering spaces, homology  
CS 150  Probability and Algorithms 
analysis of probabilistic algorithms, the probabilistic method  
CS 139  Analysis and Design of Algorithms 
streaming algorithms, experts algorithm, SDPs, spectral graph theory  
CS 151  Complexity Theory 
nondeterminism, nonuniform circuits, randomized algorithms, polynomial hierarchy, interactive proofs, approximation  
Ma 120a  Abstract Algebra 
graduate course in commutative algebra 
Oct. 2017  2D Plasma Simulation via Discrete Exterior Calculus 
Caltech Summer Research Seminar Day  
15 minute presentation on the results of my summer research. Slides 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. Notes 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/Lagrangian mechanics and how to discretize them to produce variational time integrators. Extended notes available here. 