The main goal of this project was to further research in the area of algebraic combinatorics. Roughly speaking, algebraic combinatorics is the study of discrete objects using techniques from algebra or geometry, or conversely, it is the study of modeling phenomena in algebra and geometry using discrete objects. For instance, one classical problem in geometry is the following: If I give you 4 random, straight lines in three-dimensional space, how many lines can you find that intersect all 4? The answer is 2, which is also the answer to the following seemingly unrelated question: How many ways are there to place the numbers 1 through 4 into a 2x2 grid such that the numbers in each row and column increase from left to right and top to bottom? In fact, there is a deep and nonobvious connection explaining why these two questions have the same answer, a connection that has been developed through the study of algebraic combinatorics. This project sought to relate similar geometric questions to other types of combinatorial objects. Such questions relate to many other areas of math, including algebraic geometry and representation theory, as well as other disciplines such as particle physics. As a result of this award, I was able to work with other combinatorialists at the University of Minnesota and the University of Michigan towards solving such problems. For example, I was able to show how some questions about certain algebraic objects called symmetric group representations can be reduced to solving certain simpler combinatorial problems. In the end, at least six papers will result from this work and be published in peer-reviewed mathematical journals. This award has also given me the opportunity to teach a number of mathematics classes at the University of Michigan, ranging from introductory to more advanced. It has also allowed me to develop other professional skills such as creating course curricula, advising undergraduate students, giving research seminars, and developing a research program. Such skills will allow me to continue my educational and research career in academia.
