In this presentation we consider aggregation procedures (consensus functions) over median algebras (ternary algebras that subsume several ordered structures such as distributive lattices as well as several combinatorial structures such as median graphs). Our starting point is a recent Arrow type impossibility result that states that any median preserving consensus function over linearly ordered sets is trivial in the sense that it only depends on a single argument. In view of this result, a natural problem is then to identify those median algebras that lead to such impossibility results. In particular, we will show that such impossibility results are inevitable when the codomain contains no cycle, i.e., it is a "tree", and we will provide a surprisingly simple condition that completely describes the latter as median algebras. To broaden the talk, we will also present some recent results that answer the parametrized version of this problem in which dependence is restricted to k arguments. We will conclude by observing that the underlying property to proving such results is that of congruence distributivity, which naturally raises the question whether these results extend to other varieties of algebras, e.g., congruence modular varieties.
Impossibility theorems over median algebras and beyond
Fully commutative elements in Coxeter groups and are completely characterized and counted by Stembridge. Feinberg-Kim-Lee-Oh have extended the study of fully commutative elements from Coxeter groups to the complex setting, giving an enumeration of such elements in complex reflection groups . In this talk, we will present a combinatorial connection between fully commutative elements in and , which allows us to characterize fully commutative elements in by pattern avoidance and enumerate them. Further, we will show that fully commutative elements in do not have the pattern avoidance property and we will explore full commutativity in with different generating sets.
Fully commutative elements in complex reflection groups Sponsored by the Meyer Fund
In a precise sense, the prime ideals of the stable homotopy category are given by a class of extraordinary co/homology theories called the Morava K-theories K(n). Much like we study rings by localizing at prime ideals, we can study the stable homotopy category by localizing at K(n). Associated to each K(n) is a certain formal group law whose automorphism group G appears on the E_2 page of a spectral sequence that computes the K(n)-local homotopy groups of spheres. In this talk, I hope to provide the necessary background on categorical localization, cohomology theories, and formal group laws to define G (called the Morava stabilizer group) and to describe some related original results.
A computation of the action of the Morava stabilizer group on the Lubin-Tate deformation ring