Georgia Institute of Technology College of Sciences

Mapping classes are the natural topological symmetries of surfaces. Their study is often restricted to the orientation-preserving ones, which form a normal subgroup of index two in the group of all mapping classes. In this talk, we discuss orientation-reversing mapping classes. In particular, we show that Lehmer's question from 1933 on Mahler measures of integer polynomials can be reformulated purely in terms of a comparison between orientation-preserving and orientation-reversing mapping classes.

It is a classical theorem in algebraic topology that the loop space of a suitable Lie group can be modeled by an infinite dimensional variety, called the loop Grassmannian. It is also well known that there is an algebraic analog of loop Grassmannians, known as the affine Grassmannian of an algebraic groop (this is an ind-variety). I will explain how in motivic homotopy theory, the topological result has the "expected" analog: the Gm-loop space of a suitable algebraic group is A^1-equivalent to the affine Grassmannian.

A fundamental result in equivariant homotopy theory due to Elmendorf states that the homotopy theory of G-spaces, with w.e.s measured on all fixed points, is equivalent to the homotopy theory of G-coefficient systems in spaces, with w.e.s measured at each level of the system. Furthermore, Elmendorf’s result is rather robust: analogue results can be shown to hold for, among others, the categories of (topological) categories and operads. However, it has been known for some time that in the G-operad case such a result does not capture the ”correct” notion of weak equivalence, a fact made particularly clear in work of Blumberg and Hill discussing a whole lattice of ”commutative operads with only some norms” that are not distinguished at all by the notion of w.e. suggested above. In this talk I will talk about part of a joint project which aims at providing a more diagrammatic understanding of Blumberg and Hill’s work using a notion of G-trees, which are a generalization of the trees of Cisinski-Moerdijk-Weiss. More specifically, I will describe a new algebraic structure, which we dub a ”genuine equivariant operad”, which naturally arises from the study of G-trees and which allows us to state the ”correct” analogue of Elmendorf’s theorem for G-operads.