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Organizing Committee
Michigan State University
Northeastern University
Oklahoma State University
Amherst College
Elon University
Abstract
This conference is intended to celebrate and amplify the mathematics of the Braids Semester Program at ICERM in 2022. The aim is to bring together mathematicians who participated in the program, or whose research interacts with its themes, for an event that will rekindle the interactions between fields that the subject of braid groups naturally stimulated during the semester. A central goal is to showcase work that resulted from the semester's activities, and a further goal is to incorporate new participants whose research has fruitful connections with researchers who were a part of the semester.
The workshop will have a variety of activities, with research talks, problem sessions, and dedicated work time for collaboration. Special emphasis will be placed on highlighting the work of early-career mathematicians and providing space to develop new collaborations.
The workshop will have a variety of activities, with research talks, problem sessions, and dedicated work time for collaboration. Special emphasis will be placed on highlighting the work of early-career mathematicians and providing space to develop new collaborations.
Image credit: Joan Licata
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Workshop Schedule
Monday, July 15, 2024
9:20 - 9:30 AM EDT
Welcome
11th Floor Lecture Hall
Brendan Hassett, ICERM/Brown University
9:30 - 10:15 AM EDT
Shortest word problem in braid theory
11th Floor Lecture Hall
Speaker
Keiko Kawamuro, University of Iowa
Session Chair
Nancy Scherich, Elon University
Abstract
Given a braid element in B_n, searching for a shortest braid word representative (using the band-generators) is called the Shortest Braid Problem. Up to braid index n = 4, this problem has been solved by Kang, Ko, and Lee in 1997. In this talk I will discuss recent development of this problem for braid index 5 or higher. I will also show diagrammatic computational technique of the Left Canonical Form of a given braid, that is a key to the three fundamental problems in braid theory; the Word Problem, the Conjugacy Problem and the Shortest Word Problem. This is joint work with Rebecca Sorsen and Michele Capovilla-Searle.
10:30 - 11:00 AM EDT
Coffee Break
11th Floor Collaborative Space
11:00 - 11:45 AM EDT
Symmetries of link homology
11th Floor Lecture Hall
Speaker
Joshua Sussan, CUNY
Session Chair
Nancy Scherich, Elon University
Abstract
We construct an action of sl(2) on equivariant Khovanov-Rozansky link homology.
This is joint with You Qi, Louis-Hadrien Robert, and Emmanuel Wagner.
This is joint with You Qi, Louis-Hadrien Robert, and Emmanuel Wagner.
12:00 - 2:00 PM EDT
Lunch/Free Time
2:00 - 2:45 PM EDT
Bi-ordering link complements via braids
11th Floor Lecture Hall
Speaker
Hannah Turner, Stockton University
Session Chair
Caitlin Leverson, Bard College
Abstract
Any link (or knot) group – the fundamental group of a link complement – is left-orderable. However, not many link groups are bi-orderable – that is, admit an order invariant under both left and right multiplication. It is not well understood which link groups are bi-orderable, nor is there is a conjectured topological characterization of links with bi-orderable link groups. I will discuss joint work in progress with Jonathan Johnson and Nancy Scherich to study this problem for braided links – braid closures together with their braid axis. Inspired by Kin-Rolfsen, we focus on braided link groups because algebraic properties of the braid group can be employed in this setting. In particular, I will discuss our implementation of an algorithm which, given a braided link group which is not bi-orderable, will return a definitive "no" and a proof in finite time. Using our program, we give a new infinite family of non-bi-orderable braided links.
3:00 - 3:30 PM EDT
Coffee Break
11th Floor Collaborative Space
3:30 - 4:15 PM EDT
Uniform twisted homological stability of braid groups and moments of quadratic L-functions
11th Floor Lecture Hall
Speaker
Peter Patzt, University of Oklahoma
Session Chair
Caitlin Leverson, Bard College
Abstract
A conjecture of Conrey-Farmer-Keating-Rubinstein-Snaith aims to describe the asymptotics of moments of quadratic L-functions. In joint work with Miller, Petersen, and Randal-Williams and in combination with a paper by Bergström–Diaconu–Petersen–Westerland, we proved a version of this conjecture for function fields. Using the Grothendieck-Lefschetz trace formula, Bergström–Diaconu–Petersen–Westerland showed a connection between the conjecture and the twisted homology of the braid groups. In our paper, we showed what was needed to make this connection. Homological stability says that the k-dimensional homology groups are all isomorphic for a large enough number of strands of the braid groups. This is even known for twisted coefficients pulled back from polynomial representations of the symplectic groups. We proved that the starting point of stability is independent of which irreducible polynomial representation of the symplectic groups one uses.
In the talk, I will explain the connections between number theory, the braid groups, the symplectic groups, and homological stability.
4:30 - 6:00 PM EDT
Reception
11th Floor Collaborative Space
Tuesday, July 16, 2024
9:30 - 10:15 AM EDT
Skein Lasagna Modules and Categorified Projectors
11th Floor Lecture Hall
Speaker
Melissa Zhang, University of California, Davis
Session Chair
Miriam Kuzbary, Amherst College
Abstract
In 2018, Morrison, Walker, and Wedrich’s skein lasagna modules are 4-manifold invariants defined using Khovanov-Rozansky homology similarly to how skein modules for 3-manifolds are defined. In 2020, Manolescu and Neithalath developed a formula for computing this invariant for 2-handlebodies by defining an isomorphic object called cabled Khovanov-Rozansky homology; this is computed as a colimit of cables of the attaching link in the Kirby diagram of the 4-manifold.
In joint work with Ian Sullivan, we lift the Manolescu-Neithalath construction to the level of Bar-Natan's tangles and cobordisms, and trade colimits of vector spaces for a homotopy colimit in Bar-Natan's category. As an application, we give a proof that the skein lasagna module of S2xS2 is trivial, confirming a conjecture of Manolescu. Our local techniques also allow for computations of the skein lasagna invariant for other 4-manifolds whose Kirby diagram contains a 0-framed unknot component. Our methods also allow us to relate the Rozansky-Willis invariant of links in S2xS1 to skein lasagna modules.
In joint work with Ian Sullivan, we lift the Manolescu-Neithalath construction to the level of Bar-Natan's tangles and cobordisms, and trade colimits of vector spaces for a homotopy colimit in Bar-Natan's category. As an application, we give a proof that the skein lasagna module of S2xS2 is trivial, confirming a conjecture of Manolescu. Our local techniques also allow for computations of the skein lasagna invariant for other 4-manifolds whose Kirby diagram contains a 0-framed unknot component. Our methods also allow us to relate the Rozansky-Willis invariant of links in S2xS1 to skein lasagna modules.
10:30 - 11:00 AM EDT
Coffee Break
11th Floor Collaborative Space
11:00 - 11:45 AM EDT
Characterizing and non-characterizing knots by 3-manifolds
11th Floor Lecture Hall
Speaker
Marc Kegel, Humboldt-Universität zu Berlin
Session Chair
Miriam Kuzbary, Amherst College
Abstract
From a knot K, we can build 3-manifolds by performing Dehn surgery on that knot. We will discuss some new results explaining in which sense the diffeomorphism types of these 3-manifolds characterize the isotopy class of the knot K. This talk is based on joint work with Abe-Weiss, Baker, Baker-McCoy, Casals-Etnyre, and Piccirillo.