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Luis Ragognette, Federal University of São Carlos, Brazil
The theory of hyperfunctions deals with generalized functions that are even more general than distributions. Our goal in this talk is to discuss techniques that allowed us to study microlocal regularity of a hyperfunction with respect to different subspaces of the space of hyperfunctions. In other to do that we used a subclass of the FBI transforms introduced by S. Berhanu and J. Hounie. This is a joint work with Gustavo Hoepfner.
Lee Mosher, Rutgers Newark
In the course of our theorem on the \(H^2_b\)-alternative for \(Out(F_n)\) — every finitely generated subgroup of \(Out(F_n)\) is either virtually abelian or has second bounded cohomology of uncountable dimension — the case of subgroups of natural embeddings of \(Aut(F_k)\) into \(Out(F_n)\) led us to subgroups of \(Aut(F_k)\) which have interesting new hyperbolic actions arising from “suspension” constructions, generalizing a thread of hyperbolic suspension constructions which goes back to a theorem of W. Thurston. In this talk we will describe these suspension constructions, and we will speculate on what may unify them.
This is joint work with Michael Handel.
Emre Mengi, Koc University, Istanbul
Christine Lee, University of Texas
PATCH Seminar, joint with Bryn Mawr, Haverford, and Penn
Abstract: Quantum link invariants lie at the intersection of hyperbolic geometry, 3-dimensional manifolds, quantum physics, and representation theory, where a central goal is to understand its connection to other invariants of links and 3-manifolds. In this talk, we will introduce the colored Jones polynomial, an important example of quantum link invariants. We will discuss how studying properly embedded surfaces in a 3-manifold provides insight into the topological and geometric content of the polynomial. In particular, we will describe how relating the definition of the polynomial to surfaces in the complement of a link shows that it determines boundary slopes and bounds the hyperbolic volume of many links, and we will explore the implication of this approach on these classical invariants.
In the background talk (9:30, AM) I'll introduce the colored Jones polynomial and discuss the many conjectures/open problems surrounding the polynomial, to give the research talk more context.
Olga Plamenevskaya, SUNY Stony Brook
PATCH Seminar, joint with Bryn Mawr, Haverford, and Penn
Abstract: Due to work of Giroux, contact structures on 3-manifolds can be topologically described by their open books decompositions (which in turn can be encoded via fibered links). A contact structure is called planar if it admits an open book with fibers of genus 0. Symplectic fillings of such contact structures can be understood, by a theorem of Wendl, via Lefschetz fibrations with the same planar fiber. Using this together with topological considerations, we prove a new obstruction to planarity (in terms of intersection form of fillings) and obtain a few corollaries. In particular, we consider contact structures that arise in a canonical way on links of surface singularities, and show that the canonical contact structure on the link is planar only if the singularity is rational. (Joint work with P. Ghiggini and M. Golla.)
In the background talk (11:00 AM), I will discuss topological properties of Lefschetz fibrations over a disk, focusing on the case where fiber is a surface of genus 0. The boundary of the 4-manifold given by Lefschetz fibration has an induced open book and a contact structure. This will be the setting for my second talk.
There are no conferences this week.