All talks will be in Skiles 006.

* This will be a colloquium talk.

Inanc Baykur
Title:Topological complexity of symplectic 4-manifolds and Stein fillings
Abstract:Following the ground-breaking works of Donaldson and Giroux, Lefschetz fibrations and open books have become central tools in the study of symplectic 4-manifolds and contact 3-manifolds. An open question at the heart of this relationship is whether or not there exists an a priori bound on the topological complexity of a symplectic 4-manifold, coming from the genus of a Lefschetz fibration with a maximal section on it. A similar question inquires if there is such a bound on any Stein filling of a fixed contact 3-manifold (possibly coming from the genus and the number of binding components of a compatible open book). We will present our solutions to both questions, while making heroic use of positive factorizations and commutators in surface mapping class groups of various flavors. This is joint work with M. Korkmaz and N. Monden, and independently with J. Van Horn-Morris.

Joan Birman
Title: The curve complex of a surface
Abstract: This will be a Colloquium talk, aimed at a general audience. The topic is the curve complex, introduced by Harvey in 1974. It's a simplicial complex, and was introduced as a tool to study mapping class groups of surfaces. I will discuss recent joint work with Bill Menasco about new local pathology in the curve complex, namely that its geodesics can have dead ends and even double dead ends.

Jeff Brock
Title: Fat, exhausted, integer homology spheres
Abstract: Since Perelman's groundbreaking proof of the geometrization conjecture for three-manifolds, the possibility of exploring tighter correspondences between geometric and algebraic invariants of three-manifolds has emerged. In this talk, we address the question of how homology interacts with hyperbolic geometry in 3-dimensions, providing examples of hyperbolic integer homology spheres that have large injectivity radius on most of their volume. (Indeed such examples can be produced that arise as (1,n)-Dehn filling on knots in the three-sphere). Such examples fit into a conjectural framework of Bergeron, Venkatesh and others and provide a counterweight to phenomena arising in asymptotic L^2 invariants of families of covers of hyperbolic manifolds.

Matt Hedden
Title: Splicing knot complements and bordered Floer homology
Abstract: In this talk I'll discuss recent work which aims to understand which 3-manifolds can have Heegaard Floer homology isomorphic to that of the 3-sphere. With an eye towards proving a non-triviality result for the Floer homology of toroidal manifolds, I'll discuss how to use bordered Floer homology to prove that the integer homology sphere obtained by splicing two nontrivial knot complements in the 3-sphere (or, more generally, in integer homology sphere L-spaces) has Heegaard Floer homology rank strictly greater than one. This is joint work with Adam Levine.

Sarah Koch
Title: Eigenvalues of the Thurston Pullback Map
Abstract: Given a critically finite rational map, one can define a holomorphic endomorphism of a Teichmueller space associated to it; this endomorphism is called the Thurston pullback map. With the exception of one class of examples, this endomorphism has a unique fixed point, and the eigenvalues of the derivative at this fixed point are all *algebraic*. What do these eigenvalues mean? What algebraic numbers arise this way? We establish some facts about these eigenvalues if the rational map is a quadratic polynomial (for example, we prove in this case that there are no "small eigenvalues"), but the situation is still mysterious.

Jeremy Van Horn-Morris
Title: Positive factorizations in the mapping class group
Abstract: The technology of Lefschetz pencils/fibrations and open book decompositions reduces many interesting questions in contact and symplectic topology to combinatorial questions in the mapping class group. At a fundamental level, we want to know: "What does the genus of the Lefschetz fibration or open book tell you about the symplectic or contact manifold?" One can translate a question about complexity bounds to the following question about mapping class groups: "For any mapping class element $\phi$ of a genus g surface with some number of boundary components, is there a bound on the length of a factorization of $\phi$ as a product of right-handed Dehn twists?" (That is, is there a bound on the number of Dehn twists in a positive factorization.) We will discuss an answer to this question and some of the applications. This is joint with I. Baykur.

Rebecca Winarski
Title: Mapping Class Groups and Covering Spaces
Abstract: A fundamental question of the study of mapping class groups is finding maps between mapping class groups of different surfaces. Given a covering space of surfaces, one may wish to relate the mapping class groups of the two surfaces. We say a cover of S over X has the Birman-Hilden property if there is a finite index subgroup of the mapping class group of X that injects into the mapping class group of S, modulo the deck transformations. In the 1970s, Birman and Hilden proved that regular covers have the Birman-Hilden property. We extend their work to certain irregular branched covers. As new applications, we prove: (1) Simple covers do not have the Birman-Hilden property and (2) All covers with at most one branch point do have the Birman-Hilden property.