I developed these small interactive demos of calculus concepts for a class I was teaching. They are written in Mathematica 8 and are downloadable as Mathematica notebooks or as CDF (Computable Document Format) documents which can be played using the free CDF Player software from Wolfram Research.
- limit.cdf - a cute interactive illustration of the continuity of functions and the epsilon-delta definition of a limit. Mathematica source code.
- limit2.cdf - another widget similar to limit.cdf but with a discontinuous function. Mathematica source code.
- slopes.cdf - illustrates the definition of the average slope of a function on an interval. Mathematica source code.
- demand-curve.cdf - an example of function maximization from economics, involving the demand curve for a product with fixed per-unit production costs. Mathematica source code.
My coauthors and I have written several software packages as an aid to studying various interesting mathematical objects.
- SofaBounds: A software package for computer-assisted proofs in the moving sofa problem. A Unix command line tool, written in C++, that is a companion package to the paper "Improved upper bounds in the moving sofa problem" written jointly with Yoav Kallus.
- A Mathematica package MovingSofas, or a PDF version. This is a companion package to the paper "Differential equations and exact solutions in the moving sofa problem."
- A Mathematica notebook SU3-ASYM.nb, or a PDF version. This is a companion package to the paper "On the number of n-dimensional representations of SU(3), the Witten zeta function, and the Bernoulli numbers."
- A Mathematica-based slide presentation titled "The combinatorics of fully packed loops and Razumov-Stroganov conjectures", which I presented on Aug. 8, 2014 at MAA MathFest 2014. Download the Mathematica notebook of the presentation, written in Mathematica 8, or a (non-interactive) PDF version. The notebook contains as embedded code two Mathematica function libraries I wrote, linkpatterns.m and loops.m, that implement many operations and several interactive simulations related to fully packed loops, noncrossing matchings and related mathematical objects.
- A Maple package RefinedASM1234.txt. This is a companion package to my paper "New enumeration formulas for alternating sign matrices and square ice partition functions" written jointly with Arvind Ayyer.
- A Mathematica notebook GammaDelta.nb. Another companion package to the same paper with Arvind Ayyer. This notebook verifies a claim added in the revised version of the paper.
- A Mathematica notebook RefinedASM.nb. This is a companion package to the paper "More refined enumerations of alternating sign matrices" written jointly with Ilse Fischer.
- A SageMath package qKZ, written by Joshua Sumpter. This package implements operations on wheel polynomials, a family of polynomials described in my paper "Connectivity patterns in loop percolation I: the rationality phenomenon and constant term identities".
Mac simulation software
Here is some software I’ve written to simulate some interesting mathematical objects I’m thinking about. It runs on any Mac running Mac OS version 10.5 and higher.
A program for experimenting with random alternating sign matrices and domino tilings of the Aztec diamond. It lets you visualize alternating sign matrices in several ways and witness the remarkable Arctic Circle Theorem for random domino tilings. The mathematics behind this is described in several papers. You can find a bibliography and some more information in the application itself by clicking the button marked “?”. The download package includes the source code, written in Objective-C.
A program for experimenting with optimal and fair allocation of areas to points in a square domain. Starting from the Voronoi diagram, the areas of the cells allocated to the points are adjusted until all areas are equal. The theoretical idea behind this construction is described in the paper Minkowski-type theorems and least-squares clustering, by F. Aurenhammer, F. Hoffmann and B. Aronov, [Algorithmica 20 (1998), 61-76], and in the context of analyzing the behavior for random points was suggested to me by Yuliy Baryshnikov.
A program for experimenting with random Young diagrams and random Standard Young Tableaux (SYT’s). Simulate Plancherel measure on the set of Young diagrams, Rost’s corner growth model for random Young diagrams, or uniformly random SYT of given shape (including the cases of square, rectangular, and staircase shapes which my coauthors and I have studied). Also implemented are the Robinson-Schensted algorithm, the jeu de taquin sliding operations, and the non-crossing path interpretation of SYT’s.
A program for experimenting with random sorting networks, a.k.a. reduced word representations of the maximal permutation. The two main random models are the Uniform Sorting Networks (USN), which were studied in the paper Random Sorting Networks, by O. Angel, A. E. Holroyd, B. Virag and myself [Adv. Appl. Math. 215 (2007), 839-868], and the Oriented Swap Process, discussed in the paper The Oriented Swap Process, by O. Angel, A. E. Holroyd and myself [Ann. Probab. 37 (2009), 1970-1998].