Workshop

A Harshad number is a positive integer that is divisible by the sum of its digits.  The word “Harshad” comes from the Sanskrit harsa (joy) + da (give), meaning joy-giver, which was defined by the Indian mathematician D.R. Kaprekar.  All one digit numbers are Harshad numbers and it is fairly simple to determine which two digit numbers are Harshad.  In 1994, H. Grundman generalized the concept to b-Harshad (or b-Niven) numbers.  Simply put, for b > 1, a b-Harshad number is a positive integer that is divisible by the sum of the digits of its base b expansion.  The mini-course will provide undergraduate students with an opportunity to learn about Harshad numbers and how to compute some of their properties using the freely available mathematical program Sage. No prior programing experience is required.

Pre-requisite: An undergraduate algebra or number theory course

The sandpile model developed by Bak, Tang, and Wiesenfeld in 1987 is a mathematical model first used to exemplify the concept of self-organized criticality (SOC).  SOC is a property of certain dynamical systems that naturally evolve toward critical states and it is considered to be one of the mechanisms by which complexity arises in nature.  The abelian sandpile model introduced by Dhar in 1990 is a special class of sandpile model defined on a combinatorial graph whose dynamic structure is encoded in a finite abelian group known as the sandpile group.  This algebraic structure has played a central role in the study of diverse properties of the abelian sandpile model. Moreover, the sandpile group has also been an important object of study in several distinct areas of mathematics, including algebraic combinatorics, algebraic, tropical and arithmetic geometry, the theory of computation, and the study of pattern formation.

In this mini-course, we will give an introduction to the theory of sandpiles.  In particular, we will study the interactions between the combinatorics of the graph, the algebraic information of the sandpile group and the dynamics of the abelian sandpile model.

Prerequisites: One undergraduate course in linear algebra.