Refreshments at 3:30 pm, outside of Davis 216
Pamela Harris, Williams
Consider the following combinatorial problem: In how many ways can the positive integer n be written as a sum of positive integers (ignoring the order)? For example, five can be written as a sum of positive integers in the following seven ways:
5; 4 + 1; 3 + 2; 3 + 1 + 1; 2 + 2 + 1; 2 + 1 + 1 + 1; and 1 + 1 + 1 + 1 + 1:
Although this process is simple, determining a formula for the value of the partition function, which counts the number of integer partitions of n, alluded generations of mathematicians and was only recently solved by Ken Ono, Jan Bruinier, Amanda Folsom, and Zach Kent in 2011. Their formula relied on the new and surprising discovery that partitions of integers are fractal in nature. In this talk, we discuss the long history of this problem along with many related mathematical tools used in its study. Lastly, we consider vector partition functions, a family of functions generalizing the integer partition function, which count the number of ways to express a vector as a nonnegative integer linear combination of a finite set of vectors. These generalizations fuel many new avenues for research and provide ample mathematical problems for further investigation.