Math Experimental Lab: Random Surfaces and Random Permutations

Spring 2026, UVA Math Experimental Lab

Mentored by Leonid Petrov

Project description

Imagine you have a 100x100x100 room, and you stack some number of 1x1x1 unit cubes in its corner. If you do this at random, what would be the shape of the pile? We will investigate the mathematical structure of such random 3D stepped surfaces, which leads to beautiful limit shapes. We will use basic combinatorics, coding, and we will potentially 3D print our results. We will also explore cutting edge research directions in this area, in particular, connections of random surfaces to random permutations. Most of the models we will explore are hands-on and computational, and we will be able to visualize and manipulate them.

Regular meeting times

• 9:00 AM on Thursdays, in Kerchof 209

MEL group meetings

• Meeting 1: Tuesday, February 17th, 5-6pm, in Kerchof 111: Introductory meeting. Groups will introduce themselves and say a little about their project (informal - no need for slides unless the students want to make them). Food will be provided.

• Meeting 2: Friday, March 20th, 9:30-11:30am, in Chemistry Building 202 (Maker Open Lab): Starr Hill Pathways visit. Each group should prepare an interactive activity suitable for high school students which will introduce them to your project. There will also be a short career panel at the end of the visit, so if you know of anyone interested in being on the panel, please let us know!

• Meeting 3: Thursday, April 23rd, 5-6pm, in Kerchof 111: Final presentations (15 minutes + time for questions). Food will be provided.

Notes

• (Feb 19) Dimers on bipartite graph (blackboard photos)

• (Feb 26) Dinic’s Algorithm and Lozenge Tilings: the Undo Mechanic (PDF)

• (Mar 11) Coupling From the Past: Why Doubling With Reuse is Essential (PDF)

• (Mar 11) Height Functions and Extremal Matchings on General Planar Bipartite Graphs (PDF)

Useful links

• UVA Rapid Prototyping Lab (for 3D printing our random surfaces)
• Markov chain video
• Random rhombus (lozenge) tilings video