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Organizing Committee
Georgia Institute of Technology
Michigan State University
University of Iowa
Australian National University
University of Vienna
Abstract
Braids are deeply entwined with low-dimensional topology. Closed braids are knots and links, while viewing braid groups as surface mapping class groups connects the topic to fundamental constructions of three- and four-manifolds. The question of how properties of braids or mapping classes reflect the associated manifolds arises in Dehn surgery, link invariants, and contact and symplectic geometry. The workshop will highlight recent advances in these and other areas of low-dimensional topology where braids and mapping classes play a significant role. The workshop will also explore related algorithms, with an eye towards their (efficient) implementation.
Please note that this program is nearing capacity for in-person participation. Consider applying for this program virtually.
Please note that this program is nearing capacity for in-person participation. Consider applying for this program virtually.
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Workshop Schedule
Monday, April 25, 2022
8:30 - 8:50 AM EDT
Check In
11th Floor Collaborative Space
8:50 - 9:00 AM EDT
Welcome
11th Floor Lecture Hall
Brendan Hassett, ICERM/Brown University
9:00 - 9:45 AM EDT
New algebraic structures for Legendrian links
11th Floor Lecture Hall
Speaker
Lenny Ng, Duke University
Session Chair
Matthew Hedden, Michigan State University
Abstract
I'll discuss a number of (arguably) new holomorphic-curve invariants of Legendrian knots and links. These come from an L-infinity structure on commutative Legendrian contact homology, derived from rational symplectic field theory. The new invariants include a symplectic structure on the augmentation variety of a Legendrian link, as well as a Poisson bracket (on a polynomial ring) associated to any positive braid. Parts of this are joint work in progress with Roger Casals, Honghao Gao, Linhui Shen, and Daping Weng.
10:00 - 10:30 AM EDT
Coffee Break
11th Floor Collaborative Space
10:30 - 11:15 AM EDT
Fractional Dehn twists and left-orders on mapping class groups
11th Floor Lecture Hall
Speaker
Hannah Turner, Georgia Institute of Technology
Session Chair
Matthew Hedden, Michigan State University
Abstract
Three-manifolds (and closed braids inside them) admit descriptions called open book decompositions; in this setting a surface with boundary and a mapping class describe the 3-manifold (and braid). One invariant of an open book is the fractional Dehn twist coefficient (FDTC). The FDTC is a real number invariant of a mapping class of a surface with boundary, which has connections to contact topology and foliation theory. I'll show that the FDTC of a given mapping class can be computed using a multitude of geometrically defined left-orders on the mapping class group. This is joint work with Diana Hubbard.
11:30 AM - 12:15 PM EDT
Computational bounds on the band rank
11th Floor Lecture Hall
Speaker
Mark Hughes, Brigham Young University
Session Chair
Matthew Hedden, Michigan State University
Abstract
The band rank of a braid is the length of its shortest decomposition into a product of conjugates of Artin generators. Using braided surfaces the band rank of a braid can be used to express the ribbon genus of its closure, and thus it is very difficult to compute in general. In this talk I will outline computational approaches to producing upper and lower bounds on the band rank of a braid. This is joint work with Justin Meiners.
12:30 - 2:30 PM EDT
Lunch/Free Time
2:30 - 3:15 PM EDT
Braided Embeddings
11th Floor Lecture Hall
Speaker
Sudipta Kolay, ICERM
Session Chair
Siddhi Krishna, Columbia University
Abstract
Braided embeddings are a natural generalization of closed braids in three dimensions, which gives a way to construct many higher dimensional embeddings. We will discuss the lifting and isotopy problem for braided embeddings. Time permitting, we will also mention some applications to the existence of nice branched coverings, contact geometry and homomorphisms between braid groups.
3:30 - 4:00 PM EDT
Coffee Break
11th Floor Collaborative Space
4:00 - 4:45 PM EDT
Algebraic singular fibers from the symplectic perspective
11th Floor Lecture Hall
Speaker
Jeremy Van Horn-Morris, University of Arkansas
Session Chair
Siddhi Krishna, Columbia University
Abstract
Kodaira classified all singular fibers that can arise in an algebraic fibration with genus 1 fiber. Sakali and I have been looking into a similar classification for genus 2 fibrations by Ogg, Iitaka and later Namikawa and Ueno and we determine how these singular fibrations deform into Lefschetz fibrations. The gamut of tools runs from deformation theory, to Lefschetz fibrations and open books, to braids and braided surfaces, and the branched covers that relate all of them. This is joint work with S. Sakalli.
5:00 - 6:30 PM EDT
Reception
11th Floor Collaborative Space
Tuesday, April 26, 2022
9:00 - 9:45 AM EDT
Genericity of pseudo-Anosov mapping classes
11th Floor Lecture Hall
Speaker
Yvon Verberne, Georgia Institute of Technology
Session Chair
Joan Licata, Australian National University
Abstract
The Nielsen-Thurston classification theorem states that mapping classes fall into three types: periodic, reducible, and pseudo-Anosov. One of the famous open problems in the study of mapping class groups is to show that pseudo-Anosov mapping classes are generic. This problem remains open, but it has been proven in one sense. In this talk, I will introduce this open problem and share the progress which has been made in solving the problem.
