Towards a Unified Theory of Partition Congruences

The subject of my research is in what mathematicians usually call number theory. This is the study of the whole numbers and their properties. Most of the motivating questions in this subject are extremely easy to understand; if you know what whole numbers are, and how to add, subtract, multiply, and divide, then you can probably understand the really interesting questions. These questions very often appear pretty and whimsical; however, they are very often very difficult to prove, they call upon various advanced mathematical methods, and they provide astonishing and extremely important applications. Most electronic security systems rely on encryption and cryptography, which rely heavily on understanding prime number theory. More surprising is that the mathematical foundation of some of our most advanced subjects in physics like quantum field theory and string theory rely on a very deep knowledge of number theory. My own work in number theory involves what are called partitions. A partition is an expression of a whole number as a sum of other whole numbers. For example, the number 4 has 5 partitions: we can write 4 by itself, 3+1, 2+2, 2+1+1, and 1+1+1+1. Notice that we do not care about the ordering; 3+1 and 1+3 are the same thing for us. Notice that the number 1 has just 1 partition: 1. The number 2 has 2 partitions: 2, and 1+1. Next, 3 has 3 partitions (3, 2+1, 1+1+1), 4 has 5 partitions, 5 has 7 partitions (5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1). If we just list the number of partitions of each number 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,20, then we have the list 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, If you do not see an easy pattern in this sequence, do not worry. It looks really random, and some really talented mathematicians believed that it was random for many centuries. Only about a hundred years ago, in 1918, did an Indian mathematician named Ramanujan discover that there are some beautiful and simple properties in this sequence. To see one of them, just look at the number of partitions for each number with a 4 or 9 as its last digit. So this sequence begins 4, 9, 14, 19,, and the partition numbers look like 5, 30, 135, 490, Yes! Each one is divisible by 5. This is called a congruence property. Ramanujan was able to find and prove a lot of similar but considerably deeper congruence properties for the numbers 5, 7, 11, and other prime numbers. We have since adapted his methods and studied similar congruence properties of a huge variety of other numbers that are closely related to partitions. Sometimes Ramanujans methods work really well, but sometimes they dont; after a hundred years, we have found a lot of similar congruence properties that we still cannot properly understand. The goal of my project is to develop new methods of understanding congruence properties. These methods have applications to difficult problems in physics, especially string theory.