Monday, November 19, 2018 - 14:00 , Location: Skiles 006 , Livio Liechti , Paris-Jussieu , Organizer: Balazs Strenner
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.
Friday, November 16, 2018 - 14:00 , Location: Skiles 006 , Stavros Garoufalidis , Georgia Tech and MPI , Organizer: John Etnyre
I will explain some connections between the counting of incompressible surfaces in hyperbolic 3-manifolds with boundary and the 3Dindex of Dimofte-Gaiotto-Gukov. Joint work with N. Dunfield, C. Hodgson and H. Rubinstein, and, as usual, with lots of examples and patterns.
Monday, November 12, 2018 - 14:00 , Location: Skiles 006 , Tom Bachmann , MIT , Organizer: Kirsten Wickelgren
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.
Monday, November 5, 2018 - 14:00 , Location: Skiles 006 , Min Hoon Kim , Korea Institute for Advanced Study , Organizer: Jennifer Hom
The still open topological 4-dimensional surgery conjecture is equivalent to the statement that all good boundary links are freely slice. In this talk, I will show that every good boundary link with a pair of derivative links on a Seifert surface satisfying a homotopically trivial plus assumption is freely slice. This subsumes all previously known methods for freely slicing good boundary links with two or more components, and provides new freely slice links. This is joint work with Jae Choon Cha and Mark Powell.
Monday, October 22, 2018 - 14:00 , Location: Skiles 006 , Luis Alexandre Pereira , Georgia Tech , Organizer: Kirsten Wickelgren
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.
Monday, October 15, 2018 - 14:00 , Location: Skile 006 , Lev Tovstopyat-Nelip , Boston College , Organizer: John Etnyre
Let K be a link braided about an open book (B,p) supporting a contact manifold (Y,x). K and B are naturally transverse links. We prove that the hat version of the transverse link invariant defined by Baldwin, Vela-Vick and Vertesi is non-zero for the union of K with B. As an application, we prove that the transverse invariant of any braid having fractional Dehn twist coefficient greater than one is non-zero. This generalizes a theorem of Plamenevskaya for classical braid closures.