I am interested in combinatorics and geometry. I particularly like incidence geometry and Euclidean Ramsey theory.
József Solymosi & Joshua Zahl
Email: kjmoore@math.ubc.ca
Office: AUDX128
Mailing Address: 1984 Mathematics Road. Vancouver, BC, Canada V6T 1Z2
I am good at an engineering game called "Fantastic Contraption". You can find my machines on YouTube.
I am planning to record some works of Scott Joplin on piano. Those will appear here eventually.
We provide a general framework to construct colorings avoiding short monochromatic arithmetic progressions in Euclidean Ramsey theory.
Arxiv Preprint. Our resources page is here.
A conjecture of Erdos et. al. states that, with the exception of equilateral triangles, any two-coloring of the plane will have a monochromatic congruent copy of every three-point configuration. This conjecture is known in some special cases. In this manuscript, we confirm one of the most natural open cases (stated in the title).
Arxiv Preprint. Our resources page is here.
We prove new upper and lower bounds on the minimal axiality (reflective symmetry) of planar convex sets, as well as the minimal folding symmetry. We also obtain the first (to our knowledge) exact upper bounds on hyperplane symmetry in any dimension.
We obtain new upper and lower bounds on the number of unit perimeter triangles spanned by points in the plane. We also establish improved bounds in the special case where the point set is a section of the integer grid.
Published in Discrete & Computational Geometry, or see the original Arxiv Preprint.
We prove a Bakry-Émery generalization of a theorem of Petersen and Wilhelm, that closed minimal hypersurfaces in a complete manifold with a suitable curvature bound must intersect. We then prove splitting theorems of Croke-Kleiner type for manifolds bounded by hypersurfaces obeying Bakry-Émery curvature bounds.
Published in the Journal of Mathematical Physics, or see the original Arxiv Preprint.
We give a rigorous computer-assisted proof that a triangle has a periodic billiard path when all its angles are at most 112.3 degrees. This nears the apparent boundary for computerized methods occuring at 112.5 degrees.
Reviewed Version, or see the original Arxiv Preprint. Featured in Quanta Magazine.
This page contains a list of point sets that form many unit distance pairs. To my knowledge, these are the best lower bounds for most small numbers.
One can use this program to easily check the axial symmetry of convex polygons, as well as run our searching algorithm to try to lower the current bound.
A surprising way of approximating conformal maps is with corresponding circle packings. This program implements an algorithm for finding such packings.
Periodic path unfoldings are very cumbersom to draw, so this program was designed to help make figures. The best takeaway from this project is the display of the amazing stable behaviour of certain periodic paths.
I wanted to quickly make a couple diagrams for a graph theory course I was taking. I plan to put in some functions to compute properties of graphs as well if I work on anything that would benefit from that.
These notes were transcribed in Latex during some classes I have taken. They are quickly typed live, so there may be some typos and odd formatting here and there. If you would like the .tex files and figures, you may request them over email.
Taken at the 2023 Summer School for Algbraic Methods in Combinatorics at UIUC
'Math 503' at UBC, taught by József Solymosi
'Math 541' at UBC, taught by Izabella Łaba
'Math 532' at UBC, taught by Jim Bryan
'Math 201' at the University of Alberta, taught by me. The textbook mentioned is Differential Equations with Boundary Value Problems, Boyce (11th Edition).