In this talk, we will first discuss the topology of the Frobenius complex, the order complex of a poset motivated by the classical Frobenius problem. Specifically, we will determine the homotopy type of the Frobenius complex in certain cases using discrete Morse theory. In the second half of the talk, we will extend the classical excedancestatistic of the symmetric group to the affine symmetric group and determine the generating function of its distribution. The proof involves enumerating lattice points in a skew version of the root polytope of type $A$.

There will be a reception for Eric at 4:00 p.m. in POT 745.

A cornerstone theorem of free groups is the Nielsen-Schreier theorem - that every subgroup of a free group is itself free. In this talk we'll explore this result via Nielsen's proof and then discuss the statement of the Andrew-Curtis conjecture. This is an open problem that is a natural extension of the tools developed by Nielsen to prove the Nielsen-Schreier theorem, yet a proof has eluded mathematicians for nearly 50 years (and continues to do so!) The talk should be accessible to everyone with an interest in algebra.

I will give an elementary introduction to wall crossing phenomena from the point of view of string theory. In particular I will explain how simple kinematics of molecule-like bound states in this context leads to remarkably universal wall crossing formulae that have been uncovered in recent years.

Equivalence relations show up at all levels in mathematics from kindergarten to graduate school: regrouping in addition and subtraction; equivalent fractions; equivalent algebraic expressions and equations; vectors; modular arithmetic; row reduction in matrices; cardinality; etc. I hypothesize that one reason students have difficulties with these topics is the subtle difference between "equivalent" and "equal" in these settings. In spite of the centrality of equivalence relations to understanding so many math topics, we don't explicitly talk about this to students, even to math majors and prospective math teachers, until late in their education, if at all.

I will talk about the mathematics that underlies the various uses of equivalence relations in these diverse settings. There will be something here for everyone who teaches math, or who prepares math teachers, at all grade levels.