Organizing Committee
- Matthew Hedden
Michigan State University - Matt Hogancamp
Northeastern University - Jonathan Johnson
Oklahoma State University - Miriam Kuzbary
Amherst College - Nancy Scherich
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.
![Image for "Braids Reunion Workshop"](http://app.icerm.brown.edu/img/489_image.jpg)
Confirmed Speakers & Participants
Talks will be presented virtually or in-person as indicated in the schedule below.
- Speaker
- Poster Presenter
- Attendee
- Virtual Attendee
-
Samantha Allen
Duquesne University
-
Byung Hee An
Kyungpook National University
-
Ishan Banerjee
Ohio State University
-
Inanc Baykur
University of Massachusetts Amherst
-
Orsola Capovilla-Searle
University of California, Davis
-
Rima Chatterjee
University of Cologne
-
Brandy Doleshal
Sam Houston State University
-
Megan Fairchild
Louisiana State University
-
Sergei Gukov
Caltech
-
Jacob Guynee
Georgia Institute of Technology
-
Touseef Haider
Rutgers University-Newark
-
Matthew Hedden
Michigan State University
-
Matt Hogancamp
Northeastern University
-
Diana Hubbard
Brooklyn College
-
James Hughes
Duke University
-
Jonathan Johnson
Oklahoma State University
-
Sungkyung Kang
University of Oxford
-
Keiko Kawamuro
University of Iowa
-
Marc Kegel
Humboldt-Universität zu Berlin
-
Siddhi Krishna
Columbia University
-
Miriam Kuzbary
Amherst College
-
Khanh Le
Rice University
-
Christine Ruey Shan Lee
Texas State University
-
Caitlin Leverson
Bard College
-
Jun Li
University of Dayton (Dayton, OH, US)
-
Abhishek Mallick
Rutgers University - New Brusnwick
-
Porter Morgan
University of Massachusetts Amherst
-
Seyed Ali Naseri Sadr
Boston College
-
Peter Patzt
University of Oklahoma
-
Nicolas Petit
Loyola University Chicago
-
Stanley Pritchard
Duke University
-
Nancy Scherich
Elon University
-
Jose Simental Rodriguez
Universidad Nacional Autónoma de México
-
Manpreet Singh
University of South Florida
-
Ian Sullivan
UC Davis
-
Joshua Sussan
CUNY
-
Mina Teicher
Bar-Ilan University
-
Joshua Turner
UC Davis
-
Hannah Turner
Stockton University
-
Rithwik Susheel Vidyarthi
Michigan State University
-
Biji Wong
Duke University
-
Angela Wu
Louisiana State University
-
Jiajun Yan
University of Virginia
-
Melissa Zhang
University of California, Davis
Workshop Schedule
Monday, July 15, 2024
-
9:20 - 9:30 am EDTWelcome11th Floor Lecture Hall
- Brendan Hassett, ICERM/Brown University
-
9:30 - 10:15 am EDTShortest word problem in braid theory11th 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 EDTCoffee Break11th Floor Collaborative Space
-
11:00 - 11:45 am EDTtba11th Floor Lecture Hall
- Speaker
- Joshua Sussan, CUNY
- Session Chair
- Nancy Scherich, Elon University
-
12:00 - 2:00 pm EDTLunch/Free Time
-
2:00 - 2:45 pm EDTBi-ordering link complements via braids11th 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 EDTCoffee Break11th Floor Collaborative Space
-
3:30 - 4:15 pm EDTUniform twisted homological stability of braid groups and moments of quadratic L-functions11th 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 EDTReception11th Floor Collaborative Space
Tuesday, July 16, 2024
-
9:30 - 10:15 am EDTSkein Lasagna Modules and Categorified Projectors11th 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. -
10:30 - 11:00 am EDTCoffee Break11th Floor Collaborative Space
-
11:00 - 11:45 am EDTCorrection terms of branched double covers and symmetries of immersed curves11th Floor Lecture Hall
- Speaker
- Biji Wong, Duke University
- Session Chair
- Miriam Kuzbary, Amherst College
Abstract
In this talk, we'll discuss recent work to use the immersed curves description of bordered Floer theory to study the d-invariants of branched double covers Sigma_2(L) of links L in the 3-sphere. We'll show that when L is a 2-component plumbing link and Sigma_2(L) is an L-space, then the spin d-invariants of Sigma_2(L) are determined by the signatures of L. This project is joint with J. Hanselman and M. Marengon.
-
12:00 - 2:00 pm EDTProblem Session in small groups over catered lunchLunch/Free Time - 11th Floor Lecture Hall
-
2:00 - 2:45 pm EDTCluster structures on braid varieties11th Floor Lecture Hall
- Speaker
- Jose Simental Rodriguez, Universidad Nacional Autónoma de México
- Session Chair
- Nicolas Petit, Loyola University Chicago
Abstract
In the previous 'Braids' meeting, I defined braid varieties, explained some of their properties and conjectured that they admit a cluster structure, that is, their coordinate algebra can be given the structure of a cluster algebra. I will recall the definition and examples of braid varieties, and explain how a cluster structure on them may indeed be constructed using the graphical calculus of weaves. If time permits, I will discuss how the cluster structure reflects properties of the braid and vice versa. This is based on joint works with various subsets of {Roger Casals, Marco Castronovo, Eugene Gorsky, Mikhail Gorsky, Ian Le, Linhui Shen, David Speyer}.
-
3:00 - 3:30 pm EDTCoffee Break11th Floor Collaborative Space
-
3:30 - 4:15 pm EDTA topological model for the HOMFLY-PT polynomial11th Floor Lecture Hall
- Speaker
- Christine Ruey Shan Lee, Texas State University
- Session Chair
- Nicolas Petit, Loyola University Chicago
Abstract
A topological model for a knot invariant is a realization of the invariant as graded intersection pairings on coverings of configuration spaces. In this talk I will describe a topological model for the HOMFLY-PT polynomial. I plan to discuss the motivation from previous work by Lawrence and Bigelow giving topological models for the Jones and SL_n polynomials, and the construction, joint with Cristina Anghel, which uses a state sum formulation of the HOMFLY-PT polynomial to construct an intersection pairing on the configuration space of a Heegaard surface of the link.
Wednesday, July 17, 2024
-
9:30 - 10:15 am EDTtba11th Floor Lecture Hall
- Speaker
- Inanc Baykur, University of Massachusetts Amherst
- Session Chair
- Orsola Capovilla-Searle, University of California, Davis
-
10:30 - 11:00 am EDTCoffee Break11th Floor Collaborative Space
-
11:00 - 11:45 am EDTTBA11th Floor Lecture Hall
- Speaker
- Siddhi Krishna, Columbia University
- Session Chair
- Orsola Capovilla-Searle, University of California, Davis
-
12:00 - 12:05 pm EDTGroup Photo (Immediately After Talk)11th Floor Lecture Hall
-
12:05 - 2:00 pm EDTLunch/Free Time
-
2:00 - 2:45 pm EDTNonorientable broken Lefschetz fibrations11th Floor Lecture Hall
- Speaker
- Porter Morgan, University of Massachusetts Amherst
- Session Chair
- Marc Kegel, Humboldt-Universität zu Berlin
Abstract
Lefschetz fibrations are a special type of map from a 4-manifold to a surface, usually either S^2 or D^2. Although they provide a lot of information about their source manifold, they’re only admitted by a limited collection of 4-manifolds. This motivates the study of broken Lefschetz fibrations; by relaxing the definition of a Lefschetz fibration, we get a family of maps that all closed, smooth 4-manifolds admit. In this talk, we’ll review the basic topology of broken Lefschetz fibrations, both in the more-established orientable case, and in the more-recent nonorientable case. We’ll examine the data that such a map provides, and see how we can obtain a broken Lefschetz fibration from a more general map onto S^2.
-
3:00 - 3:30 pm EDTCoffee Break11th Floor Collaborative Space
-
3:00 - 4:30 pm EDTDesignated collaboration/ work in groups timeGroup Work - 11th Floor Collaborative Space
Thursday, July 18, 2024
-
9:30 - 10:15 am EDTCharacterizing and non-characterizing knots by 3-manifolds11th Floor Lecture Hall
- Speaker
- Marc Kegel, Humboldt-Universität zu Berlin
- Session Chair
- Siddhi Krishna, Columbia University
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.
-
10:30 - 11:00 am EDTCoffee Break11th Floor Collaborative Space
-
11:00 - 11:45 am EDTContact surgery numbers11th Floor Lecture Hall
- Speaker
- Rima Chatterjee, University of Cologne
- Session Chair
- Marc Kegel, Humboldt-Universität zu Berlin
Abstract
A fundamental result in 3-dimensional contact topology due to Ding-Geiges tells us that any contact 3-manifold can be obtained via doing a surgery on a Legendrian link in the standard contact 3-sphere. So it's natural to ask how simple or complicated a surgery diagram could be for a particular contact manifold? Contact surgery number is a measure of this complexity. In this talk, I will define this notion of complexity and discuss some examples. This is joint work with Marc Kegel.
-
12:00 - 2:00 pm EDTLunch/Free Time
-
2:00 - 2:45 pm EDTTBA11th Floor Lecture Hall
- Speaker
- Orsola Capovilla-Searle, University of California, Davis
- Session Chair
- Biji Wong, Duke University
-
3:00 - 3:30 pm EDTCoffee Break11th Floor Collaborative Space
-
3:30 - 4:15 pm EDTTBA11th Floor Lecture Hall
- Speaker
- Angela Wu, Louisiana State University
- Session Chair
- Biji Wong, Duke University
Friday, July 19, 2024
-
9:30 - 10:15 am EDTWork in progress — a gauge-theoretic interpretation of the McKay correspondence11th Floor Lecture Hall
- Speaker
- Jiajun Yan, University of Virginia
- Session Chair
- Jonathan Johnson, Oklahoma State University
Abstract
Let Gamma be a finite subgroup of SU(2). The McKay correspondence states that the McKay quiver of Gamma is isomorphic to the graph of minimal resolution of C^2/Gamma. There are various proofs of the McKay correspondence coming from algebraic geometry and symplectic geometry which we will survey in the talk. Then, we will present an idea / work in progress of a new gauge-theoretic interpretation of the McKay correspondence. To do so, we will review a gauge-theoretic construction of the 4-dimensional hyperkähler ALE spaces for each of which the underlying topological space is the minimal resolution of C^2/Gamma. The main approach is to make use of an S^1-invariant Morse-Bott function arising from the gauge-theoretic construction by identifying its critical points with certain flat connections that induce representations of Gamma via holonomy representation. The idea is partially inspired by the Chern-Simons theory, and some tools from contact geometry also come into play.
-
10:30 - 11:00 am EDTCoffee Break11th Floor Collaborative Space
-
11:00 - 11:45 am EDTNew quantum invariants from braiding Verma modules11th Floor Lecture Hall
- Speaker
- Sergei Gukov, Caltech
- Session Chair
- Jonathan Johnson, Oklahoma State University
Abstract
In this talk, I will describe recent construction of new link and 3-manifold invariants associated with Verma modules of $U_q (sl_N)$ at generic $q$. The resulting invariants can be combined into a Spin$^c$-decorated TQFT and have a nice property that, for links in general 3-manifolds, they have integer coefficients. In particular, they are expected to admit a categorification and, if time permits, I will outline various ingredients that may go into a construction of 3-manifold homology categorifying $U_q (sl_N)$ invariants at generic $q$.
-
12:00 - 2:00 pm EDTLunch/Free Time
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