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Series: Geometry Topology Seminar

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.

Series: Geometry Topology Seminar

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.

Series: Geometry Topology Seminar

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.

Series: Geometry Topology Seminar

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.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

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.

Series: Geometry Topology Seminar

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.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

The knot group has played a central role in classical knot theory
and has many nice properties, some of which fail in interesting ways for
knotted surfaces. In this talk we'll introduce an invariant of
knotted surfaces called ribbon genus, which measures the failure of a
knot group to 'look like' a classical knot group. We will show that
ribbon genus is equivalent to a property of the group called Wirtinger
deficiency. Then we will investigate some examples
and conclude by proving a connection with the second homology of the
knot group.