Organizing Committee
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"
Image credit: Joan Licata

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
  • İnanç Baykur
    University of Massachusetts Amherst
  • Michele Capovilla-Searle
    University of Iowa
  • Orsola Capovilla-Searle
    University of California, Davis
  • Rima Chatterjee
    University of Cologne
  • Baris Coskunuzer
    UT Dallas
  • Brandy Doleshal
    Sam Houston State University
  • Megan Fairchild
    Louisiana State University
  • Sam Freedman
    University of Chicago
  • Jacob Garcia
    University of California, Riverside
  • 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)
  • Zijun Li
    Duke University
  • Trent Lucas
    Brown University
  • 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
  • Felix Roz
    Columbia University
  • Takuto Sato
    University of Massachusetts Amherst
  • Nancy Scherich
    Elon University
  • Ben Shapiro
    Dartmouth College
  • Manpreet Singh
    University of South Florida
  • Dean Spyropoulos
    Michigan State University
  • Ian Sullivan
    UC Davis
  • Joshua Sussan
    CUNY
  • Mina Teicher
    Bar-Ilan University
  • Max Throm
    Michigan State 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
  • Yuxiang Yao
    University of California, Irvine
  • Melissa Zhang
    University of California, Davis

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.
  • 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.
  • 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.
  • 12:00 - 2:00 pm EDT
    Lunch/Free Time
  • 2:00 - 2:45 pm EDT
    Detecting corks
    11th Floor Lecture Hall
    • Speaker
    • Abhishek Mallick, Rutgers University - New Brusnwick
    • Session Chair
    • Nicolas Petit, Loyola University Chicago
    Abstract
    Corks are fundamental to the study of exotic smooth structures on 4-manifolds. In this talk, I will describe how to detect corks and their usefulness. This is joint work with many collaborators over several projects.
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 3:30 - 4:15 pm EDT
    A topological model for the HOMFLY-PT polynomial
    11th 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 EDT
    tba
    11th Floor Lecture Hall
    • Speaker
    • İnanç Baykur, University of Massachusetts Amherst
    • Session Chair
    • Orsola Capovilla-Searle, University of California, Davis
  • 10:30 - 11:00 am EDT
    Coffee Break
    11th Floor Collaborative Space
  • 11:00 - 11:45 am EDT
    TBA
    11th Floor Lecture Hall
    • Speaker
    • Siddhi Krishna, Columbia University
    • Session Chair
    • Orsola Capovilla-Searle, University of California, Davis
  • 12:00 - 12:05 pm EDT
    Group Photo (Immediately After Talk)
    11th Floor Lecture Hall
  • 12:00 - 2:00 pm EDT
    Problem Session in small groups over catered lunch
    Lunch/Free Time - 11th Floor Lecture Hall
  • 2:00 - 2:45 pm EDT
    Nonorientable broken Lefschetz fibrations
    11th Floor Lecture Hall
    • Speaker
    • Porter Morgan, University of Massachusetts Amherst
    • Session Chair
    • Marc Kegel, Humboldt-Universität zu Berlin
    Abstract
    Broken Lefschetz Fibrations (BLFs) are surjections from a 4-manifold to a sphere with only Lefschetz and indefinite fold singularities. Unlike Lefschetz fibrations, any generic map M^4\to S^2 is homotopic to a BLF. Among BLFs, a particularly nice subset is Simplified Broken Lefschetz Fibrations, which give an explicit handlebody decomposition of their source manifold. In this talk, we’ll review the basic topology of simplified BLFs for closed, smooth, nonorientable 4-manifolds. Then we’ll use simplified BLFs to construct an explicit embedding of nonorientable 4-manifolds into certain 6-manifolds.
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 3:00 - 4:30 pm EDT
    Designated collaboration/ work in groups time
    Group Work - 11th Floor Collaborative Space
Thursday, July 18, 2024
  • 9:30 - 10:15 am EDT
    Correction terms of branched double covers and symmetries of immersed curves​
    11th Floor Lecture Hall
    • Speaker
    • Biji Wong, Duke University
    • Session Chair
    • Siddhi Krishna, Columbia University
    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.
  • 10:30 - 11:00 am EDT
    Coffee Break
    11th Floor Collaborative Space
  • 11:00 - 11:45 am EDT
    Contact surgery numbers
    11th Floor Lecture Hall
    • Speaker
    • Rima Chatterjee, University of Cologne
    • Session Chair
    • Siddhi Krishna, Columbia University
    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 EDT
    Lunch/Free Time
  • 2:00 - 2:45 pm EDT
    Exact Lagragian fillings of Legendrian links and Newton polytopes.
    11th Floor Lecture Hall
    • Speaker
    • Orsola Capovilla-Searle, University of California, Davis
    • Session Chair
    • Biji Wong, Duke University
    Abstract
    An important problem in contact topology is to understand Legendrian submanifolds; these submanifolds are always tangent to the plane field given by the contact structure. Legendrian links can also arise as the boundary of exact Lagrangian surfaces in the standard symplectic 4-ball. Such surfaces are called fillings of the link. In the last decade, our understanding of the moduli space of fillings for various families of Legendrians has greatly improved thanks to tools from sheaf theory, Floer theory and cluster algebras. I will talk about connections between fillings and Newton polytopes.
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 3:30 - 4:15 pm EDT
    High-dimensional anti-surgery for Weinstein manifolds
    11th Floor Lecture Hall
    • Speaker
    • Angela Wu, Louisiana State University
    • Session Chair
    • Biji Wong, Duke University
    Abstract
    A Legendrian knot in the boundary of a Weinstein domain of dimension at least 6 which bounds a Lagrangian disk can be considered the boundary of the co-core of a handle. A Weinstein anti-surgery amounts to carving out this handle from the Weinstein domain. In this talk, I’ll explain an algorithm which constructs explicit handle decompositions of many of these high-dimensional Weinstein anti-surgery manifolds using a new high-dimensional Legendrian isotopy. I’ll give a specific application of this algorithm to Lazarev and Sylvan’s class of Weinstein manifolds which they called P-flexible, formed from handle attachment along P-loose Legendrians. This talk is based on work in progress with Ipsita Datta, Oleg Lazarev, and Chindu Mohanakumar.
Friday, July 19, 2024
  • 9:30 - 10:15 am EDT
    Work in progress — a gauge-theoretic interpretation of the McKay correspondence
    11th 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 EDT
    Coffee Break
    11th Floor Collaborative Space
  • 11:00 - 11:45 am EDT
    New quantum invariants from braiding Verma modules
    11th 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 EDT
    Lunch/Free Time

All event times are listed in ICERM local time in Providence, RI (Eastern Daylight Time / UTC-4).

All event times are listed in .

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