Fellowship of the Ring, National Seminar: A truly mutually beneficial friendship: how Stanley-Reisner theory enhanced both combinatorics and algebra
Hailong Dao (University of Kansas)
Given a simplicial complex on n vertices, one can associate to it a quotient of the polynomial ring in n variables, called the Stanley-Reisner ring. Starting with the proof of the Upper Bound Conjecture for spheres, this approach has been spectacularly useful in bringing tools from commutative algebra to the study of simplicial complexes. In the first part of the talk I will sketch some relevant parts of this story. In the second, I will describe how modern tools, including cohomological vanishing results and characteristic p methods, have inspired new developments. At the same time, results obtained on the combinatorics side now can be brought back to induce interesting new questions and theorems on the algebra side. One thing I really like about this topic is that it can be used to generate good problems at all levels, including for high school students.