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Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

Smooth simply connected 4-manifolds can admit second homology classes
not representable by smoothly embedded spheres; knot traces are the
prototypical example of 4-manifolds with such classes. I will show that
there are knot traces where
the minimal genus smooth surface generating second homology is not of
the canonical type, resolving question 1.41 on the Kirby problem list. I
will also use the same tools to show that Conway knot does not bound a
smooth disk in the four ball,
which completes the classification of
slice knots under 13 crossings and gives the first example of a
non-slice knot which is both topologically slice and a positive mutant
of a slice knot.

Series: Geometry Topology Seminar

We show that the three-dimensional homology cobordism group admits an infinite-rank summand. It was previously known that the homology cobordism group contains an infinite-rank subgroup and a Z-summand. Our proof relies on the involutive Heegaard Floer package of Hendricks-Manolescu and Hendricks-Manolescu-Zemke. This is joint work with I. Dai, M. Stoffregen, and L. Truong.

Series: Geometry Topology Seminar

We will present an h-principle for the simplification of singularities of Lagrangian and Legendrian fronts. The h-principle says that if there is no homotopy theoretic obstruction to simplifying the singularities of tangency of a Lagrangian or Legendrian submanifold with respect to an ambient foliation by Lagrangian or Legendrian leaves, then the simplification can be achieved by means of a Hamiltonian isotopy. We will also discuss applications of the h-principle to symplectic and contact topology.

Series: Geometry Topology Seminar

The notion of an acylindrically hyperbolic group was introduced by Osin as a
generalization of non-elementary hyperbolic and relative hyperbolic groups. Ex-
amples of acylindrically hyperbolic groups can be found in mapping class groups,
outer automorphism groups of free groups, 3-manifold groups, etc. Interesting
properties of acylindrically hyperbolic groups can be proved by applying techniques such as Monod-Shalom rigidity theory, group theoretic Dehn filling, and
small cancellation theory. We have recently shown that non-elementary convergence groups are acylindrically hyperbolic. This result opens the door for
applications of the theory of acylindrically hyperbolic groups to non-elementary
convergence groups. In addition, we recovered a result of Yang which says a
finitely generated group whose Floyd boundary has at least 3 points is acylindrically hyperbolic.

Series: Geometry Topology Seminar

In this talk, we describe some applications of link Floer homology to the topology of surfaces in 4-space. If K is a knot in S^3, we will consider the set of surfaces in B^4 which bound K. This space is naturally endowed with a plethora of non-Euclidean metrics and pseudo-metrics. The simplest such metric is the stabilization distance, which is the minimum k such that there is a stabilization sequence connecting two surfaces such that no surface in the sequence has genus greater than k. We will talk about how link Floer homology can be used to give lower bounds, as well as some techniques for computing non-trivial examples. This is joint work with Andras Juhasz.

Series: Geometry Topology Seminar

I will discuss knot concordances in 3-manifolds. In particular I will talk about knot concordances of knots in the free homotopy class of S^1 x {pt} in S^1 x S^2. It turns out, we can use some of these concordances to construct Akbulut-Ruberman type exotic 4-manifolds. As a consequence, at the end of the talk we will see absolutely exotic Stein pair of 4-manifolds. This is joint work with Selman Akbulut.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

Weinstein cobordisms give a natural relationship on the set of Weinstein domains. Flexible Weinstein domains are minimal with respect to this relationship. In this talk, I will use these minimal domains to construct maximal Weinstein domains: any two high-dimensional Weinstein domains with the same topology are Weinstein subdomains of a maximal Weinstein domain also with the same topology. Using this construction, a wide range of new Weinstein domains can be produced, for example exotic cotangent bundles of spheres containing many different closed exact Lagrangians. On the other hand, I will explain how the same line of ideas can be used to prove restrictions on which categories can arise as the Fukaya categories of certain Weinstein domains.

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.