Alex McDonough


Graduate Student
Brown University

About Me

I'm a fifth year graduate student at Brown University working under the supervision of Caroline Klivans.

Contact Info

Department of Mathematics
Box 1917
151 Thayer Street
Providence, RI 02912

amcd [at] math [dot] brown [dot] edu

Curriculum Vitae

Research

I am interested in algebraic, geometric, and topological combinatorics, particularly chip-firing and matroids.

One of the classic results of chip-firing is that the number of elements of the sandpile group of a graph is the same as the number of spanning trees. I've spent some time exploring "sandpile torsors" which give families of bijections between these two sets. I recently introduced an analogue of a sandpile torsor for higher-dimensional chip-firing, using many-to-one maps instead of bijections. These maps are constructed using a periodic tiling of n-space out of non-convex polyhedra.

I've also had an opportunity to explore some topics outside of chip-firing. In particular, I have collaborated on a paper on matching complexes and a paper on the Neighborhood Grid data structure. I love that fact that a combinatorial perspective provides an approachable entry point to a wide variety of subjects, and I look forward to exploring more in the future!


Here is an example tile from my higher-dimensional chip-firing map.
This polyhedron periodically tiles the plane. The different colors
correspond to different matroid bases. See here for details.

Papers

A Combinatorial Mapping for the Higher-Dimensional Matrix-Tree Theorem.

Matching Complexes of Trees and Applications of the Matching Tree Algorithm. With Marija Jelić Milutinović, Helen Jenne, and, Julianne Vega.

Determining Genus From Sandpile Torsor Algorithms. Extended Abstract in Séminaire Lotharingien de Combinatoire. Issue 80B.82 (2018) Poster.

Combinatorial and Asymptotic Aspects of the Neighborhood Grid Data Structure. With Martin Skrodzki and Ulrich Reitebuch.

Videos and Slides

Chip-firing on Cell Complexes and Matroids: video slides
A Higher-Dimensional Sandpile Map: video slides(extended)

Teaching

Spring 2020: TA for Math 0180: Intermediate Calculus.
Fall 2019: Instructor for Math 0180: Intermediate Calculus.
Summer 2019: Instructor for Probability and Its Applications (Pre-college for high school students).
Spring 2019: Instructor for Math 0100: Introductory Calculus II.
Fall 2018: Instructor for Math 0090: Introductory Calculus I.
Summer 2018: TA for Spy Training 101 (Stem 1 for middle school students).
Summer 2018: TA for Million Dollar Math (Stem 2 for 9th and 10th grade students).
Spring 2018: TA for Math 0100: Introductory Calculus II .
Fall 2017: TA for Math 0090: Introductory Calculus I .
Summer 2017: TA for Spy Training 101 (Stem 1 for middle school students).

Seminars

I created the Graduate Online Combinatorics Colloquium which I organize with Galen Dorpalen-Barry and Andrés R. Vindas Meléndez. This is a low-pressure weekly online seminar for combinatorics graduate students from all around the world. If you'd like to join the mailing list or are interested in speaking, send an email to GOCCcombinatorics [at] gmail [dot] com.

I am also coordinator for the Math Resource Center along with fellow grad student Olivia Vasquez. The MRC is a walk-in tutoring center where undergraduate students can get help on math homework from graduate and advanced undergraduate tutors. It meets two hours a day every Monday-Thursday while school is in session. During Covid, the MRC will be meeting virtually in a custom-made room on gather.town.

Finally, I'm involved in the horizons seminar. The goals of this seminar are (quoting from the seminar webpage) "to discuss issues of gender, racial, and sexual inclusivity in STEM fields; to provide career advancement and job placement advice to graduate students; and to promote the research and work of traditionally under-represented mathematicians by hosting several colloquium-style mathematics talks." Feel free to contact me if you are interested in presenting a topic that is in line with these goals.

Personal

One of my favorite hobbies is playing, collecting, and reading about board games. A well written board game instruction manual has many of the same characteristics as a well written research paper (such as clear logic, judicious structure, and plenty of examples). I feel that learning new games has been great practice for learning new mathematics, and teaching rules to others has been great practice for lecturing! If you are curious about my collection, here is my BoardGameGeek Profile.

I also love playing all kinds of sports, my favorite being basketball. In 2017, I started two intramural soccer teams for my department, and I've also played for several teams in the applied math and computer science departments. Click here for information about the current soccer season a sad shrine for what could have been.