Georgia Institute of Technology College of Sciences

The stable commutator length function measures the growth rate of the commutator length of powers of elements in the commutator subgroup of a group. In this talk, we will discuss the stable commutator length function on the mapping class groups of infinite-type surfaces which satisfy a certain topological characterization. In particular, we will show that stable commutator length is a continuous function on these big mapping class groups, as well as that the commutator subgroups of these big mapping class groups are both open and closed.

Let $M$ be the underlying smooth $4$-manifold of a degree $d$ del Pezzo surface. In this talk, we will discuss two related results about finite subgroups of the mapping class group $\text{Mod}(M) := \pi_0(\text{Homeo}^+(M))$. A motivating question for both results is the Nielsen realization problem for $M$: which finite subgroups $G$ of $\text{Mod}(M)$ have lifts to $\text{Diff}^+(M) \leq \text{Homeo}^+(M)$ under the quotient map $\pi: \text{Homeo}^+(M) \to \text{Mod}(M)$?

Let M be a complex-hyperbolic n-manifold, i.e. a quotient of the complex-hyperbolic n-space $\mathbb{H}^n_\mathbb{C}$ by a torsion-free discrete group of isometries, $\Gamma = \pi_1(M)$. Suppose that M is  convex-cocompact, i.e. the convex core of M is a nonempty compact subset. In this talk, we will discuss a sufficient condition on $\Gamma$ in terms of the growth-rate of its orbits in $\mathbb{H}^n_\mathbb{C}$ for which M is a Stein manifold. We will also talk about some interesting questions related to this result. This is a joint work with Misha Kapovich.

Given a knot $K$ in the 3-sphere, one can ask: what kinds of surfaces in the 3-sphere are bounded by $K$? One can also ask: what kinds of surfaces in the 4-ball (which is bounded by the 3-sphere) are bounded by $K$? In this talk we will discuss how to construct surfaces in both the 3-sphere and in the 4-ball bounded by a given knot $K$, how to obstruct the existence of such surfaces, and explore what is known and unknown about surfaces bounded by so-called torus knots.

In dimension three, Giroux developed the theory of convex surfaces in contact manifolds, and this theory has been used to prove many important results in contact geometry, as well as to establish deep connections with topology.  More recently, Honda and Huang have reformulated the work of Giroux in order to extend the theory to higher dimensions.  The purpose of this sequence of talks is to understand this reformulation and to see some of its applications.

In dimension three, Giroux developed the theory of convex surfaces in contact manifolds, and this theory has been used to prove many important results in contact geometry, as well as to establish deep connections with topology.  More recently, Honda and Huang have reformulated the work of Giroux in order to extend the theory to higher dimensions.  The purpose of this sequence of talks is to understand this reformulation and to see some of its applications.

We'll discuss various operations which can be applied to a knot to "simplify" or "unknot" it. Study of these "unknotting operations" began in the 1800s and continues to be an active area of research in low-dimensional topology. Many of these operations have applications more broadly in topology including to 3- and 4-manifolds and even to DNA topology. I will define some of these operations and highlight a few open problems.

A central problem in low-dimensional topology asks which homology 3-spheres bound contractible 4-manifolds and homology 4-balls. In this talk, we address this problem for plumbed 3-manifolds and we present the classical and new results together. Our approach is based on Mazur’s famous argument and its generalization which provides a unification of all results.

The mapping class group of a surface is well understood for surfaces of finite type. In contrast, the study of mapping class groups of infinite type surfaces is a new field with many opportunities to establish new results. In this talk, we will introduce infinite type surfaces and their mapping class groups.

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