Braids

Institute for Computational and Experimental Research in Mathematics (ICERM)

https://icerm.brown.edu/programs/sp-s22/
February 1, 2022 - May 6, 2022
Tuesday, February 1, 2022
Braids
  • 9:00 am - 4:00 pm EST
    Check In
    11th Floor Collaborative Space
  • 10:30 - 11:50 am EST
    Signatures in topology, algebra, and dynamics
    Seminar - 10th Floor Classroom
    • Bena Tshishiku, Brown University
  • 4:00 - 4:30 pm EST
    Informal Welcome Tea
    Coffee Break - 11th Floor Collaborative Space
Wednesday, February 2, 2022
Braids
  • 9:30 - 10:00 am EST
    ICERM Director and Organizer Welcome
    Welcome - 11th Floor Lecture Hall
  • 3:00 - 3:30 pm EST
    Coffee Break
    11th Floor Collaborative Space
Thursday, February 3, 2022
Braids
  • 10:30 - 11:50 am EST
    Signatures in topology, algebra, and dynamics
    Seminar - 10th Floor Classroom
    • Bena Tshishiku, Brown University
  • 3:00 - 3:30 pm EST
    Coffee Break
    11th Floor Collaborative Space
Friday, February 4, 2022
Braids
  • 9:30 - 10:30 am EST
    Director and Organizer Meeting
    Meeting - 11th Floor Conference Room
  • 3:00 - 3:30 pm EST
    Coffee Break
    11th Floor Collaborative Space
Monday, February 7, 2022
Braids
  • 11:00 - 11:45 am EST
    Lightning Introductions
    Lightning Talks - 11th Floor Lecture Hall
    Abstract
    1 minute introductions for all "Braid" semester program attendees
  • 1:00 - 2:00 pm EST
    Director/Grad Student/Postdoc Meeting
    Meeting - 11th Floor Lecture Hall
  • 3:00 - 4:30 pm EST
    Welcome Reception
    Reception - 11th Floor Collaborative Space
Tuesday, February 8, 2022
Braids
  • 9:30 - 9:40 am EST
    Grad Student/Postdoc Intros
    Lightning Talks - 11th Floor Lecture Hall
    • Lei Chen, University of Maryland, College Park
  • 9:40 - 9:50 am EST
    Grad Student/Postdoc Intros
    Lightning Talks - 11th Floor Lecture Hall
    • María Cumplido Cabello, University of Seville
  • 9:50 - 10:00 am EST
    Grad Student/Postdoc Intros
    Lightning Talks - 11th Floor Lecture Hall
    • Hannah Turner, Georgia Institute of Technology
  • 10:00 - 10:10 am EST
    Grad Student/Postdoc Intros
    Lightning Talks - 11th Floor Lecture Hall
    • Jonathan Johnson, Oklahoma State University
  • 10:10 - 10:20 am EST
    Grad Student/Postdoc Intros
    Lightning Talks - 11th Floor Lecture Hall
    • Virtual Speaker
    • Angela Wu, Louisiana State University
  • 10:30 - 11:50 am EST
    Signatures in topology, algebra, and dynamics
    Seminar - 10th Floor Classroom
    • Bena Tshishiku, Brown University
  • 2:00 - 2:10 pm EST
    Grad Student/Postdoc Intros
    Lightning Talks - 11th Floor Lecture Hall
    • Virtual Speaker
    • Orsola Capovilla-Searle, University of California, Davis
  • 2:10 - 2:20 pm EST
    Grad Student/Postdoc Intros
    Lightning Talks - 11th Floor Lecture Hall
    • Sudipta Kolay, ICERM
  • 2:20 - 2:30 pm EST
    Grad Student/Postdoc Intros
    Lightning Talks - 11th Floor Lecture Hall
    • Miriam Kuzbary, Georgia Institute of Technology
  • 2:30 - 2:40 pm EST
    Grad Student/Postdoc Intros
    Lightning Talks - 11th Floor Lecture Hall
    • Virtual Speaker
    • Marc Kegel, Humboldt-Universität zu Berlin
  • 2:40 - 2:50 pm EST
    Grad Student/Postdoc Intros
    Lightning Talks - 11th Floor Lecture Hall
    • Siddhi Krishna, Columbia University
  • 2:50 - 3:00 pm EST
    Grad Student/Postdoc Intros
    Lightning Talks - 11th Floor Lecture Hall
    • Rima Chatterjee, University of Cologne
  • 3:00 - 3:10 pm EST
    Grad Student/Postdoc Intros
    Lightning Talks - 11th Floor Lecture Hall
    • Nancy Scherich, University of Toronto
  • 3:10 - 3:20 pm EST
    Grad Student/Postdoc Intros
    Lightning Talks - 11th Floor Lecture Hall
    • Yvon Verberne, Georgia Institute of Technology
  • 3:20 - 3:30 pm EST
    Grad Student/Postdoc Intros
    Lightning Talks - 11th Floor Lecture Hall
    • Biji Wong, Max Planck Institute for Mathematics
  • 3:30 - 3:35 pm EST
    Grad Student/Postdoc Intros
    Lightning Talks - 11th Floor Lecture Hall
    • Jiajun Yan, University of Virginia
  • 3:35 - 3:40 pm EST
    Grad Student/Postdoc Intros
    Lightning Talks - 11th Floor Lecture Hall
    • Willi Kepplinger, University of Vienna
  • 3:40 - 3:45 pm EST
    Grad Student/Postdoc Intros
    Lightning Talks - 11th Floor Lecture Hall
    • Eric Stenhede, University of Vienna
  • 3:45 - 3:50 pm EST
    Grad Student/Postdoc Intros
    Lightning Talks - 11th Floor Lecture Hall
    • Virtual Speaker
    • Edmund Heng, The Australian National University
  • 3:50 - 4:00 pm EST
    Grad Student/Postdoc Intros
    Lightning Talks - 11th Floor Lecture Hall
    • Marithania Silvero Casanova, Universidad de Sevilla
  • 4:00 - 4:30 pm EST
    Coffee Break
    11th Floor Collaborative Space
Wednesday, February 9, 2022
Braids
  • 2:00 - 3:00 pm EST
    Colloquium- Braids are everywhere (in low-dimensional topology)
    Seminar - 11th Floor Lecture Hall
    • Peter Feller, ETH Zurich
  • 3:00 - 3:30 pm EST
    Coffee Break
    11th Floor Collaborative Space
Thursday, February 10, 2022
Braids
  • 10:30 - 11:50 am EST
    Signatures in topology, algebra, and dynamics
    Seminar - 10th Floor Classroom
    • Bena Tshishiku, Brown University
  • 1:00 - 2:30 pm EST
    Prep talks for the workshop, "Braids in Representation Theory and Algebraic Combinatorics" - Artin Groups and normal forms
    Tutorial - 10th Floor Classroom
    • Juan González-Meneses, Universidad de Sevilla
  • 3:00 - 3:30 pm EST
    Coffee Break
    11th Floor Collaborative Space
Friday, February 11, 2022
Braids
  • 1:00 - 2:30 pm EST
    Prep talks for the workshop, "Braids in Representation Theory and Algebraic Combinatorics" - The braid group and categorification
    Tutorial - 10th Floor Classroom
    • Anthony Licata, Australian National University
  • 3:00 - 3:30 pm EST
    Coffee Break
    11th Floor Collaborative Space
Monday, February 14, 2022
Braids in Representation Theory and Algebraic Combinatorics
  • 8:50 - 9:00 am EST
    Welcome
    11th Floor Lecture Hall
    • Brendan Hassett, ICERM/Brown University
  • 9:00 - 9:45 am EST
    Higher structure and symmetry in Khovanov-Rozansky homology
    11th Floor Lecture Hall
    • Speaker
    • Matt Hogancamp, Northeastern University
    • Session Chair
    • Anthony Licata, Australian National University
    Abstract
    In this talk I will show how one constructs the action of a certain commutative dg algebra on the Khovanov-Rozansky complex of a link. The central application is a proof of the "mirror symmetry" property of triply graded Khovanov-Rozansky homology of a knot, originally conjectured in 2005 by Dunfield-Gukov-Rasmussen. This was proven first by Oblomov-Rozansky using their geometric link homology, but I will discuss an independent proof developed in joint work with Gorsky and Mellit.
  • 10:00 - 10:30 am EST
    Coffee Break
    11th Floor Collaborative Space
  • 10:30 - 11:15 am EST
    A skein theoretic Carlsson-Mellit algebra
    11th Floor Lecture Hall
    • Speaker
    • Nicolle Gonzalez, UCLA
    • Session Chair
    • Anthony Licata, Australian National University
    Abstract
    The shuffle theorem gives a combinatorial formula for the Frobenius character of the space of diagonal harmonics in terms of certain symmetric functions indexed by Dyck paths. In their proof, Carlsson and Mellit introduce a new interesting algebra denoted $A_{q,t}$. This algebra arises as an extension of the affine Hecke algebra by certain raising and lowering operators and acts on the space of symmetric functions via certain complicated plethystic operators. Afterwards Carlsson, Mellit, and Gorsky showed this algebra and its representation could be realized using parabolic flag Hilbert schemes and in addition to containing the generators of the elliptic Hall algebra. In this talk I will discuss joint work with Matt Hogancamp where we construct skein theoretic formulations of the representations of $A_{q,t}$ that arise in the proofs of the shuffle theorems and how this framework enables difficult computations to become simple diagrammatic manipulations as well as sheds light on potential applications to combinatorics and link homology.
  • 11:30 am - 12:15 pm EST
    Very positive braids are parity braids?
    11th Floor Lecture Hall
    • Speaker
    • Alexei Oblomkov, UMASS Amherst
    • Session Chair
    • Anthony Licata, Australian National University
    Abstract
    Based on joint work with Lev Rozansky. A braid is a parity braid if Khovanov-Rozansky homology of the closure of the braid has only odd or only even homological grading. It is expected that algebraic braids are parity, but probably there are more. It also seems to be natural to conjecture that after twisting by a very large power of the full twist any braid becomes parity. In our work we computed homology of the closure of composition of a quasi-Coxeter braid and a Jucys-Murphy braids. For these braids the answer to question in the title is yes.
  • 12:30 - 2:30 pm EST
    Lunch/Free Time
  • 2:30 - 3:15 pm EST
    A categorification of colored Jones polynomial at prime roots of unity
    11th Floor Lecture Hall
    • Speaker
    • You Qi, University of Virginia
    • Session Chair
    • María Cumplido Cabello, University of Seville
    Abstract
    We propose a categorification of the colored Jones polynomial evaluated at a 2pth root of unity by equipping a p-differential discovered by Cautis on the triply graded Khovanov-Rozansky homology.
  • 3:30 - 4:00 pm EST
    Coffee Break
    11th Floor Collaborative Space
  • 4:00 - 4:45 pm EST
    Braids: Classical, Virtual and Welded, Oh my!
    11th Floor Lecture Hall
    • Speaker
    • Nancy Scherich, University of Toronto
    • Session Chair
    • María Cumplido Cabello, University of Seville
    Abstract
    We will discuss the difference between the classical, virtual, and welded braid groups from an algebraic and topological perspective. We will discuss techniques to extend representations of classical braid groups to the virtual and welded settings.
  • 5:00 - 6:30 pm EST
    Reception
    11th Floor Collaborative Space
Tuesday, February 15, 2022
Braids
  • 10:30 - 11:50 am EST
    Signatures in topology, algebra, and dynamics
    Seminar - 10th Floor Classroom
    • Bena Tshishiku, Brown University
Braids in Representation Theory and Algebraic Combinatorics
  • 9:00 - 9:45 am EST
    Virtual Artin groups
    11th Floor Lecture Hall
    • Virtual Speaker
    • Luis Paris, University of Burgundy
    • Session Chair
    • You Qi, University of Virginia
    Abstract
    This talk concerns a joint work with Paolo Bellingeri and Anne-Laure Thiel. Starting from the observation that the standard presentation of a virtual braid group mixes the presentations of the corresponding braid group and the corresponding symmetric group together with the action of the symmetric group on its root system, we define a virtual Artin group ${\rm VA}[\Gamma]$ with a presentation that mixes the standard presentations of the Artin group $A[\Gamma]$ and of the Coxeter group $W[\Gamma]$ together with the action of $W[\Gamma]$ on its root system. By definition we have two epimorphisms $\pi_K:{\rm VA}[\Gamma]\to W[\Gamma]$ and $\pi_P:{\rm VA}[\Gamma]\to W[\Gamma]$ whose kernels are denoted by ${\rm KVA}[\Gamma]$ and ${\rm PVA}[\Gamma]$, respectively. In this talk we will focus on ${\rm KVA}[\Gamma]$. We will show that this group is an Artin group whose standard generating set is in one-to-one correspondence with the root system of $W[\Gamma]$. Afterwards, we use this presentation to show that the center of ${\rm VA}[\Gamma]$ is always trivial, and to show that ${\rm VA}[\Gamma]$ has a solvable word problem and finite virtual cohomological dimension when $\Gamma$ is of spherical type or of affine type.
  • 10:00 - 10:30 am EST
    Coffee Break
    11th Floor Collaborative Space
  • 10:30 - 11:15 am EST
    2-braid groups and positivity phenomenons in Hecke and Temperley-Lieb algebras
    11th Floor Lecture Hall
    • Speaker
    • Thomas Gobet, Université de Tours
    • Session Chair
    • You Qi, University of Virginia
    Abstract
    There is a well-known homomorphism from Artin's braid group to (the group of invertible elements of the) Iwahori-Hecke algebra of the symmetric group, or more generally from any Artin-Tits group to the corresponding Hecke algebra. Consider the positive lifts of the elements of the Coxeter group in the Artin-Tits group. Then their images in the Hecke algebra yield the so-called standard basis of the Hecke algebra. Elements of the standard basis have a positive expansion in one of Kazhdan and Lusztig's canonical bases, i.e., have coefficients which are Laurent polynomials with nonnegative coefficients.
    In the case where the Coxeter group is finite, the positive lifts of the elements of the Coxeter group in the Artin-Tits group are the so-called simple elements of the classical Garside structure. An alternative Garside structure, called dual Garside structure, was introduced for spherical type Artin-Tits groups. One can wonder if the images of these elements in the Hecke algebra still have a positive KL expansion or not. This is especially interesting in type A, as simple dual braids yield a basis of the Temperley-Lieb quotient of the Hecke algebra.
    We will explain how positivity of images of simple dual braids can be obtained in spherical type using a generalization of Kazhdan and Lusztig's inverse positivity, which predicts that certain elements of Artin-Tits groups, which we call ""Mikado braids"", have a positive Kazhdan-Lustig expansion, together with the fact that simple dual braids are Mikado braids. The positivity of the KL expansion of Mikado braids, shown for finite Weyl groups by Dyer and Lehrer, can be generalized to arbitrary Coxeter systems by adapting a result of Elias and Williamson on the perversity of minimal Rouquier complexes of positive simple braids to a ""twisted"" setting as introduced by Dyer, and asks the question of determining which braids have a minimal braid complex which is perverse.
  • 11:30 am - 12:15 pm EST
    How to know if a parabolic subgroup of an Artin group merges conjugacy classes
    11th Floor Lecture Hall
    • Speaker
    • María Cumplido Cabello, University of Seville
    • Session Chair
    • You Qi, University of Virginia
    Abstract
    Artin (or Artin-Tits) groups are generalizations of braid groups that are defined using a finite set of generators $S$ and relations $abab\cdots=baba\cdots$, where both words of the equality have the same length. Although this definition is quite simple, there are very few results known for Artin groups in general. Classic problems as the word problem or the conjugacy problem are still open. In this talk, we study a problem concerning a family of subgroups of Artin groups: parabolic subgroups. These subgroups have proven to be useful when studying Artin groups --for example, they are used to build interesting simplicial complexes--, but again, we do not know much about them in general. Our problem will be the following: Given two conjugate elements of a parabolic subgroup $P$ of an Artin group $A$, are they conjugate via an element of $P$? This is called the conjugacy stability problem. In 2014, González-Meneses proved that this is always true for braids, that is, geometric embedding of braids do not merge conjugacy classes. In an article with Calvez and Cisneros de la Cruz, we gave a classification for spherical Artin groups an proved that the answer to the question is not always affirmative. In this talk, we will explain how to give an algorithm to solve this problem for every Artin group satisfying three properties that are conjectured to be always true.
  • 12:30 - 2:30 pm EST
    Lunch/Free Time
  • 2:30 - 3:15 pm EST
    From Artin monoids to Artin groups
    11th Floor Lecture Hall
    • Speaker
    • Ruth Charney, Brandeis University
    • Session Chair
    • Matt Hogancamp, Northeastern University
    Abstract
    Braid groups belong to a broad class of groups known as Artin groups, which are defined by presentations of a particular form. These groups fall into two classes, finite-type and infinte-type Artin groups. The former come equipped with a powerful combinatorial structure, known as a Garside structure, while the latter are much less understood and present many challenges. However, if one restricts to the Artin monoid, a submonoid of the Artin group, then some aspects of Garside theory still apply in the infinite-type case. I will talk about joint work with Rachael Boyd and Rose Morris-Wright on geometric relations between Artin monoids and Artin groups.
  • 3:30 - 4:00 pm EST
    Coffee Break
    11th Floor Collaborative Space
  • 4:00 - 4:45 pm EST
    Dual Braids and the Braid Arrangement
    11th Floor Lecture Hall
    • Virtual Speaker
    • Jon McCammond, UC Santa Barbara
    • Session Chair
    • Matt Hogancamp, Northeastern University
    Abstract
    The braid groups have two well known Garside presentations. The elegant minimal standard presentation is closely related to the Salvetti complex, a cell complex derived from the complement of the complexification of the real braid arrangement. The dual presentation, introduced by Birman, Ko and Lee, leads to a second Garside structure and a second classifying space, but it has been less clear how the dual braid complex is related to the (quotient of the) complexified hyperplane complement, other than abstractly knowing that they are homotopy equivalent. In this talk, I will discuss recent progress on this issue. Following a suggestion by Daan Krammer, Michael Dougherty and I have been able to embed the dual braid complex into the complement of the complex braid arrangement. This leads in turn to a whole host of interesting complexes, combinatorics, and connections to other parts of the field. This is joint work with Michael Dougherty.
Wednesday, February 16, 2022
Braids in Representation Theory and Algebraic Combinatorics
  • 9:30 - 10:15 am EST
    Derived super equivalences from odd categorified quantum groups
    11th Floor Lecture Hall
    • Speaker
    • Aaron Lauda, University of Southern California
    • Session Chair
    • Hoel Queffelec, CNRS
    Abstract
    Since the pioneering work of Chuang and Rouquier, the construction of highly nontrivial derived equivalences has been one of the most powerful tools resulting from higher representation theory. Cautis-Kamnitzer-Licata showed these derived equivalences arising from categorified quantum groups gave rise to categorical actions of braid groups of the corresponding Lie type with Chuang-Rouquier's equivalences corresponding to the elementary braid generators. In 2011, motivated by the discovery of odd Khovanov homology, Ellis-Khovanov-Lauda proposed a new `odd' categorification of sl2. At the same time, this `odd sl2' was independently discovered by Kang-Kashiwara-Tsuchioka who were investigating super categorifications of Kac-Moody algebras. In this talk we will explain joint work with Mark Ebert and Laurent Vera giving new super analogs of the derived equivalences studied by Chuang and Rouquier coming from the odd categorification of sl2. Just as Chuang and Rouquier used their equivalences to achieve new results on the modular representation theory of the symmetric group, we will discuss how our new super equivalences can be applied to the spin symmetric group.
  • 10:00 - 10:30 am EST
    Coffee Break
    11th Floor Collaborative Space
  • 11:00 - 11:45 am EST
    The combinatorics and geometry of Harder-Narasimhan filtrations
    11th Floor Lecture Hall
    • Speaker
    • Anand Deopurkar, Australian National University
    • Session Chair
    • Hoel Queffelec, CNRS
    Abstract
    How does an object of a triangulated category evolve under repeated applications of an auto-equivalence? I will describe how this amorphous question can be made precise using a Bridgeland stability condition. For 2-CY categories associated to A_n quivers, I will describe how this investigation turns out to be a categorified version of well-studied notions in combinatorial geometry.
  • 12:00 - 12:10 pm EST
    Group Photo (Immediately After Talk)
    11th Floor Lecture Hall
  • 12:10 - 2:00 pm EST
    Lunch/Free Time
  • 2:00 - 3:00 pm EST
    Lightning Talks
    11th Floor Lecture Hall
    • Speakers
    • Edmund Heng, The Australian National University
    • Marc Kegel, Humboldt-Universität zu Berlin
    • Calder Morton-Ferguson, MIT
    • Marithania Silvero Casanova, Universidad de Sevilla
    • Session Chair
    • Ben Elias, University of Oregon
    Abstract
    Categorifying Burau Representations and Fusion Categories
    Edmund Heng, The Australian National University
    In this talk, we will look at a categorification of the Burau representations for the non-simply-laced type braid groups, generalising a construction given by Khovanov-Huefarno and Rouquier-Zimmermann. This will involve building certain algebra objects in the fusion categories associated to the quantum group sl2.

    Census L-space knots are braid positive, except for one that is not
    Marc Kegel, Humboldt-Universität zu Berlin
    I will explain and prove the statement in the title. This is based on joint work with Ken Baker.

    Kazhdan-Laumon Categories and the Symplectic Fourier Transform
    Calder Morton-Ferguson, MIT
    In 1988, Kazhdan and Laumon defined a “glued category” of perverse sheaves on the basic affine space. The key ingredient in their construction was the symplectic Fourier transform, which gives an action of the braid group on the category of perverse sheaves. They proposed a new construction of representations of Chevalley groups using this category, but this proposed construction depended on a conjecture which was later shown to be false. In this talk, we will discuss the action of the symplectic Fourier transform as a representation of the braid group. We will then discuss progress toward reworking Kazhdan-Laumon’s construction in the context of braids.

    A hooking conjecture on circle graphs motivated by Khovanov homology
    Marithania Silvero Casanova, Universidad de Sevilla
    We present a conjecture stating that the independence complex of any circle graph is homotopy equivalent to a wedge of spheres. This conjecture is motivated by the fact that extreme Khovanov homology of a link diagram $D$ coincides with the cohomology of the independence complex associated to its Lando graph (Lando graphs are bipartite circle graphs). We also give some advances on the proof of this conjecture; in particular, we prove it for permutation graphs, non-nested graphs, and graphs associated to closed braids with less than 5 strands.
  • 3:00 - 3:30 pm EST
    Coffee Break
    11th Floor Collaborative Space
  • 3:30 - 4:15 pm EST
    Braid groups and permutations of the Kazhdan-Lusztig basis
    11th Floor Lecture Hall
    • Virtual Speaker
    • Oded Yacobi, University of Sydney
    • Session Chair
    • Mee Seong Im, United States Naval Academy
    Abstract
    Let \lambda be a partition of n. We consider the Kazhdan-Lusztig basis of the corresponding Specht module, which is indexed by standard Young tableau of shape \lambda. One of the amazing features of this basis is that it can be used to relate representation theoretic properties of Specht modules to combinatorial properties of tableau. For example, in the 90s Berestein-Zelevinsky and Stembridge showed that the long element of the symmetric group acts on the Kazhdan-Lusztig basis by the Schutzenberger involution on tableau. Similarly, in 2010 Rhoades showed that the long cycle (1,2,...,n) acts by the jeu de taquin promotion operator when \lambda is rectangular. In this talk we will explain how to use braid groups acting on triangulated categories to generalize Rhoades' result in three directions: we lift the condition on the shape of the partition, we greatly enlarge the class of permutations for which the result holds, and we prove analogs in other Lie types. This is based on joint work with Martin Gossow.
Thursday, February 17, 2022
Braids
  • 10:30 - 11:50 am EST
    Signatures in topology, algebra, and dynamics
    Seminar - 10th Floor Classroom
    • Bena Tshishiku, Brown University
Braids in Representation Theory and Algebraic Combinatorics
  • 9:00 - 9:45 am EST
    Non-semisimple Hermitian TQFTs
    11th Floor Lecture Hall
    • Speaker
    • Joshua Sussan, CUNY
    • Session Chair
    • Mikhail Khovanov, Columbia University
    Abstract
    Topological quantum field theories coming from semisimple categories build upon interesting structures in representation theory and have important applications in low dimensional topology and physics. The construction of non-semisimple TQFTs is more recent and they shed new light on questions that seem to be inaccessible using their semisimple relatives. In order to have potential applications to physics, these non-semisimple categories and TQFTs should possess Hermitian structures. We will define these structures and give some applications.
  • 10:00 - 10:30 am EST
    Coffee Break
    11th Floor Collaborative Space
  • 10:30 - 11:15 am EST
    Braid varieties and positroid varieties
    11th Floor Lecture Hall
    • Virtual Speaker
    • Jose Simental Rodriguez, Max-Planck Institute for Mathematics
    • Session Chair
    • Mikhail Khovanov, Columbia University
    Abstract
    Associated to a positive braid, we define an affine algebraic variety via an explicit set of polynomial equations. I will give properties of these varieties, including their dimension, smoothness properties and a realization as a moduli space of chains of flags. I will also explain how some classical varieties in Lie theory, such as positroid and more generally Richardson varieties, appear in this way, as well as a connection to the computation of the Khovanov-Rozansky homology of the link obtained by closing the braid. This is joint work with Roger Casals, Eugene Gorsky and Mikhail Gorsky.
  • 11:30 am - 12:15 pm EST
    Braid varieties
    11th Floor Lecture Hall
    • Speaker
    • Eugene Gorsky, UC Davis
    • Session Chair
    • Mikhail Khovanov, Columbia University
    Abstract
    In the talk I will define braid varieties, a class of affine algebraic varieties associated to positive braids. I will discuss their relation to Richardson and positroid varieties, HOMFLY polynomial and HOMFLY homology, and Legendrian link invariants. This is a joint work with Roger Casals, Mikhail Gorsky and Jose Simental Rodriguez.
  • 12:30 - 2:30 pm EST
    Lunch/Free Time
  • 2:30 - 3:15 pm EST
    Braid groups and representation stability
    11th Floor Lecture Hall
    • Virtual Speaker
    • Jennifer Wilson, University of Michigan
    • Session Chair
    • Thomas Gobet, Université de Tours
    Abstract
    In 1970, Arnold proved that the homology groups of the braid groups on n strands stabilizes as n tends to infinity, a phenomenon called "homological stability". The pure braid groups, in contrast, are not homologically stable. In this (partly expository) talk I will describe a sense in which (co)homology groups of the pure braid groups do stabilize when we take into account the natural symmetric group actions. We will use tools from "representation stability" to shed light on the structure of the (co)homology of the pure braid groups, and many of their generalizations. This talk will survey work of Church, Ellenberg, and Farb, and joint work with Miller. 
  • 3:30 - 4:00 pm EST
    Coffee Break
Friday, February 18, 2022
Braids in Representation Theory and Algebraic Combinatorics
  • 9:00 - 9:45 am EST
    Khovanov-Seidel braid representation and geometric group theory
    11th Floor Lecture Hall
    • Speaker
    • Hoel Queffelec, CNRS
    • Session Chair
    • Juan González-Meneses, Universidad de Sevilla
    Abstract
    Khovanov and Seidel defined an action of the braid group by autoequivalences of a certain category of projective modules over the so-called zigzag algebra. Taking the Grothendieck group, one recovers the famous Burau representation, but unlike the latter, Khovanov-Seidel representation is faithful. In work with Licata, I showed how to use Khovanov-Seidel representation to extract metric data on braids. Building upon this idea, I'll try to convince the audience that such categorical tools should play in the larger context of geometric group theory.
  • 10:00 - 10:30 am EST
    Coffee Break
    11th Floor Collaborative Space
  • 10:30 - 11:15 am EST
    Categorical $q$-deformed rational numbers and compactifications of stability space
    11th Floor Lecture Hall
    • Speaker
    • Asilata Bapat, The Australian National University
    • Session Chair
    • Juan González-Meneses, Universidad de Sevilla
    Abstract
    We will discuss new categorical interpretations of two distinct $q$-deformations of the rational numbers. The first one was introduced in a different context by Morier-Genoud and Ovsienko, and enjoys fascinating combinatorial, topological, and algebraic properties. The second one is a natural partner to the first, and is new. We obtain these deformations via boundary points of a compactification of the space of Bridgeland stability conditions on the 2-Calabi--Yau category of the $A_{2}$ quiver. The talk is based on joint work with Louis Becker, Anand Deopurkar, and Anthony Licata.
  • 11:30 am - 12:15 pm EST
    From configurations on graphs to cohomology of M_{2,n}
    11th Floor Lecture Hall
    • Speaker
    • Nir Gadish, The University of Michigan
    • Session Chair
    • Juan González-Meneses, Universidad de Sevilla
    Abstract
    The configuration space of particles on a graph is a classifying space for the graph's braid group and thus computes the group cohomology. If instead one considers compactly supported cohomology the resulting groups depend only on the genus of the graph, or "loop order", and admit a particularly interesting action by Out(F_g). In this talk I will explain how tropical geometry relates these latter representations to the cohomology of the moduli spaces M_{g,n} and discuss computational approaches.
  • 3:00 - 3:30 pm EST
    Coffee Break
    11th Floor Collaborative Space
Monday, February 21, 2022
Braids
  • 3:00 - 3:30 pm EST
    Coffee Break
    11th Floor Collaborative Space
  • 4:00 - 5:00 pm EST
    Dehn Surgery: Why and How
    Post Doc/Graduate Student Seminar - 11th Floor Lecture Hall
    • Siddhi Krishna, Columbia University
    Abstract
    In this survey talk, I'll introduce Dehn surgery, a prominent technique within low-dimensional topology for building 3-manifolds. Dehn surgery can be studied using a variety of tools, including hyperbolic geometry, representation theory, and Floer homology. I'll provide an overview of major themes, questions, and results, as well as types of tools developed along the way. No background in 3- or 4-manifold topology will be assumed -- I will do my best to be as accessible as possible for grad students and postdocs across fields.
Tuesday, February 22, 2022
Braids
  • 9:00 - 10:00 am EST
    Professional Development: Ethics I
    Professional Development - 11th Floor Lecture Hall
  • 10:30 - 11:50 am EST
    Signatures in topology, algebra, and dynamics
    Seminar - 10th Floor Classroom
    • Bena Tshishiku, Brown University
  • 3:00 - 3:30 pm EST
    Coffee Break
    11th Floor Collaborative Space
Wednesday, February 23, 2022
Braids
  • 1:30 - 2:30 pm EST
    Computational Agenda Kick Off
    Problem Session - 11th Floor Lecture Hall
    Abstract
    Please join us with your ideas, questions, and ambitions for all things algorithmic, whether you're an expert programmer or a novice looking learn more about computational approaches to braids.
  • 3:00 - 3:30 pm EST
    Coffee Break
    11th Floor Collaborative Space
Thursday, February 24, 2022
Braids
  • 3:00 - 3:30 pm EST
    Coffee Break
    11th Floor Collaborative Space
Monday, February 28, 2022
Braids
  • 3:00 - 3:30 pm EST
    Coffee Break
    11th Floor Collaborative Space
  • 4:00 - 5:00 pm EST
    Surfaces, Triangulated Categories and Dynamics
    Post Doc/Graduate Student Seminar - 11th Floor Lecture Hall
    • Edmund Heng, The Australian National University
    Abstract
    Recent developments in the theory of Bridgeland’s stability conditions have established astounding analogues of dynamics and Teichmuller theory in triangulated categories. In this talk I will aim to introduce the study of dynamical systems in triangulated categories. In particular, I will introduce the notion of categorical entropy, which aims to measure the complexity of endofunctors of triangulated categories. If time allows, I will briefly explain a categorical Nielsen-Thurston classification for the rank two Artin groups, coming from a notion of HN-automaton that serves as a train-track automaton in the categorical world.
Tuesday, March 1, 2022
Braids
  • 10:30 - 11:50 am EST
    Signatures in topology, algebra, and dynamics
    Seminar - 10th Floor Classroom
    • Bena Tshishiku, Brown University
  • 3:00 - 3:30 pm EST
    Coffee Break
    11th Floor Collaborative Space
Wednesday, March 2, 2022
Braids
  • 9:00 - 10:00 am EST
    Professional Development: Ethics II
    Professional Development - 11th Floor Lecture Hall
  • 1:30 - 2:30 pm EST
    Colloquium - Taut foliations of 3-manifolds with Heegaard genus two
    Seminar - 11th Floor Lecture Hall
    • Virtual Speaker
    • Tao Li, Boston College
    Abstract
    Let M be a closed, orientable, and irreducible 3-manifold with Heegaard genus two. We prove that if the fundamental group of M is left-orderable then M admits a co-orientable taut foliation.
  • 3:00 - 3:30 pm EST
    Coffee Break
    11th Floor Collaborative Space
Thursday, March 3, 2022
Braids
  • 10:30 - 11:50 am EST
    Signatures in topology, algebra, and dynamics
    Seminar - 10th Floor Classroom
    • Bena Tshishiku, Brown University
  • 3:00 - 3:30 pm EST
    Coffee Break
    11th Floor Collaborative Space
Friday, March 4, 2022
Braids
  • 11:30 am - 12:30 pm EST
    Computational Seminar - Conjugacy problem and Nielsen-Thurston classification of braids.
    Seminar - 11th Floor Lecture Hall
    • Juan González-Meneses, Universidad de Sevilla
    Abstract
    We will explain some algorithms to solve the conjugacy problem in braid groups, and how they can be used to determine the Nielsen-Thurston classification of a braid into periodic, reducible, or pseudo-Anosov. Most of these algorithms are included in the C++ library "cbraid" and the program "braiding", available under GPL (https://github.com/jeanluct/cbraid).
  • 3:00 - 3:30 pm EST
    Coffee Break
    11th Floor Collaborative Space
Monday, March 7, 2022
Braids
  • 3:00 - 3:30 pm EST
    Coffee Break
    11th Floor Collaborative Space
  • 4:00 - 5:00 pm EST
    Braids and the fractional Dehn twist coefficient
    Post Doc/Graduate Student Seminar - 11th Floor Lecture Hall
    • Hannah Turner, Georgia Institute of Technology
    Abstract
    In this talk I'll discuss an invariant for braids (and other objects - but I'll focus on braids) called the fractional Dehn twist coefficient. This invariant heuristically measures how "twisted" a braid is. I will survey results that relate the fractional Dehn twist coefficient a braid to topological/geometric properties of its closure.
Tuesday, March 8, 2022
Braids
  • 10:30 - 11:50 am EST
    Signatures in topology, algebra, and dynamics
    Seminar - 10th Floor Classroom
    • Bena Tshishiku, Brown University
  • 3:00 - 3:30 pm EST
    Coffee Break
    11th Floor Collaborative Space
Wednesday, March 9, 2022
Braids
  • 9:00 - 10:00 am EST
    Professional Development: Job Applications in Academia
    Professional Development - 11th Floor Lecture Hall
  • 1:30 - 2:30 pm EST
    Colloquium - Alternating links and rational homology 4-balls
    Seminar - 11th Floor Lecture Hall
    • Brendan Owens, University of Glasgow
    Abstract
    I will discuss the related problems of determining which alternating knots are smoothly slice, and which have double branched covers which bound rational homology balls, as well as generalisations of these questions to links. I will describe some progress on these problems, including the extremal determinant case of a conjectured answer to the rational ball question. The latter is joint work with Josh Greene. The talk will be example-focused.
  • 3:00 - 3:30 pm EST
    Coffee Break
    11th Floor Collaborative Space
Thursday, March 10, 2022
Braids
  • 10:30 - 11:50 am EST
    Signatures in topology, algebra, and dynamics
    Seminar - 10th Floor Classroom
    • Bena Tshishiku, Brown University
  • 3:00 - 3:30 pm EST
    Coffee Break
    11th Floor Collaborative Space
Friday, March 11, 2022
Braids
  • 11:30 am - 12:30 pm EST
    Computational Seminar - Introduction to SnapPy
    Seminar - 11th Floor Lecture Hall
    • Virtual Speaker
    • Marc Culler, University of Illinois at Chicago
  • 3:00 - 3:30 pm EST
    Coffee Break
    11th Floor Collaborative Space
Monday, March 14, 2022
Braids
  • 3:00 - 3:30 pm EDT
    Pi Day Coffee Break
    Coffee Break - 11th Floor Collaborative Space
  • 4:00 - 4:30 pm EDT
    Peg problem
    Post Doc/Graduate Student Seminar - 11th Floor Lecture Hall
    • Lei Chen, University of Maryland, College Park
  • 4:30 - 5:00 pm EDT
    Order Preserving Braids
    Post Doc/Graduate Student Seminar - 11th Floor Lecture Hall
    • Jonathan Johnson, Oklahoma State University
    Abstract
    When does a braid preserve a bi-ordering of the free group? The answer to this question has interesting applications to the bi-orderability of link groups. Let’s explore this question together.
Tuesday, March 15, 2022
Braids
  • 10:30 - 11:50 am EDT
    Signatures in topology, algebra, and dynamics
    Seminar - 10th Floor Classroom
    • Bena Tshishiku, Brown University
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Wednesday, March 16, 2022
Braids
  • 1:30 - 2:30 pm EDT
    Colloquium
    Seminar - 11th Floor Lecture Hall
    • Anthony Licata, Australian National University
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Thursday, March 17, 2022
Braids
  • 10:30 - 11:50 am EDT
    Signatures in topology, algebra, and dynamics
    Seminar - 10th Floor Classroom
    • Bena Tshishiku, Brown University
  • 1:30 - 2:30 pm EDT
    Prep talks for the workshop "Braids in Symplectic and Algebraic Geometry"
    Tutorial - 11th Floor Lecture Hall
    • John Etnyre, Georgia Institute of Technology
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Friday, March 18, 2022
Braids
  • 1:30 - 2:30 pm EDT
    Prep talks for the workshop "Braids in Symplectic and Algebraic Geometry"
    Tutorial - 11th Floor Lecture Hall
    • Anand Deopurkar, Australian National University
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Monday, March 21, 2022
Braids in Symplectic and Algebraic Geometry
  • 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
    Stable cohomology of braid groups with coefficients in symplectic representations
    11th Floor Lecture Hall
    • Speaker
    • Craig Westerland, University of Minnesota
    • Session Chair
    • Inanc Baykur, University of Massachusetts Amherst
    Abstract
    The braid groups are equipped with symplectic representations via their connection with hyperelliptic mapping class groups. In this talk I'll describe joint work with Bergström, Diaconu, and Petersen in which we compute the stable cohomology of these representations. Time permitting, I will discuss connections to conjectures on moments of quadratic Dirichlet L-functions.
  • 10:00 - 10:30 am EDT
    Coffee Break
    11th Floor Collaborative Space
  • 10:30 - 11:15 am EDT
    Laudenbach’s sequence for mapping class groups of connect sums of S2 x S1.
    11th Floor Lecture Hall
    • Speaker
    • Tara Brendle, University of Glasgow
    • Session Chair
    • Inanc Baykur, University of Massachusetts Amherst
    Abstract
    Let Mn denote the connect sum of n copies of S2 x S1, and let Mod(Mn) denote its mapping class group. A theorem of Laudenbach from 1973 gives a short exact sequence realizing Mod(Mn) as an extension of Out(Fn) by (Z/2)n. In this talk we will show that Laudenbach’s sequence splits, with Out(Fn) embedded in Mod(Mn) as the stabilizer of a trivialization of TMn. This is joint work with Nathan Broaddus and Andrew Putman.
  • 11:30 am - 12:15 pm EDT
    Holomorphic maps between Configurations and Moduli spaces
    11th Floor Lecture Hall
    • Speaker
    • Lei Chen, University of Maryland, College Park
    • Session Chair
    • Inanc Baykur, University of Massachusetts Amherst
    Abstract
    In this talk, I will discuss holomorphic maps between Configuration spaces of complex plane and Moduli space. This is a joint work with Nick Salter
  • 12:30 - 2:30 pm EDT
    Lunch/Free Time
  • 2:30 - 3:15 pm EDT
    Unexpected fillings and braided curve arrangements: Part I
    11th Floor Lecture Hall
    • Virtual Speaker
    • Laura Starkston, UC Davis
    • Session Chair
    • Inanc Baykur, University of Massachusetts Amherst
    Abstract
    We examine Stein fillings of contact 3-manifolds that arise as links of certain isolated complex surface singularities. For a particular class of rational singularities, the contact structure is supported by a planar open book. This allows us to give a correspondence between Stein fillings and certain decorated plane curve arrangements. We can encode such a curve arrangement via a braided wiring diagram that captures the corresponding monodromy factorization of the Stein filling. Our correspondence gives a symplectic analog of a result by de Jong-van Straten on the smoothings for these singularities, which they encode by certain deformations of a reducible singular algebraic curve associated to the singularity.
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 4:00 - 4:45 pm EDT
    Unexpected fillings and braided curve arrangements - Part 2
    11th Floor Lecture Hall
    • Speaker
    • Olga Plamenevskaya, Stony Brook University
    • Session Chair
    • Inanc Baykur, University of Massachusetts Amherst
    Abstract
    Using the constructions described in Part I, we compare Stein fillings to Milnor fibers of smoothings for certain rational complex surface singularities. This addresses an important question on the interplay of symplectic and algebraic geometry: every Milnor fiber gives a Stein filling, but the converse is only known to be true in some very special cases. We will explain how to construct "unexpected" Stein fillings via "unexpected" pseudoline arrangements, and show that the topology of these fillings is different from that of any Milnor fiber.
  • 5:00 - 6:00 pm EDT
    Reception
    11th Floor Collaborative Space
Tuesday, March 22, 2022
Braids in Symplectic and Algebraic Geometry
  • 9:00 - 9:45 am EDT
    Pure braids in birational geometry
    11th Floor Lecture Hall
    • Speaker
    • Michael Wemyss, University of Glasgow
    • Session Chair
    • Anand Deopurkar, Australian National University
    Abstract
    I will give an overview of some joint work with Will Donovan, and with Yuki Hirano, where we show that certain surgeries in birational geometry (flopping contractions) admit actions of pure-braid type groups, and we prove various results (such as faithfulness) in that direction. These groups include pure braid groups of Type ADE, but they also contain fundamental groups of non-Coxeter hyperplane arrangements.
  • 10:00 - 10:30 am EDT
    Coffee Break
    11th Floor Collaborative Space
  • 10:30 - 11:15 am EDT
    Homological stability for Temperley-Lieb algebras
    11th Floor Lecture Hall
    • Speaker
    • Rachael Boyd, University of Cambridge
    • Session Chair
    • Anand Deopurkar, Australian National University
    Abstract
    Many sequences of groups and spaces satisfy a phenomenon called 'homological stability'. I will present joint work with Hepworth, in which we abstract this notion to sequences of algebras, and prove homological stability for the sequence of Temperley-Lieb algebras. The proof uses a new technique of 'inductive resolutions', to overcome the lack of flatness of the Temperley-Lieb algebras. I will also introduce the 'complex of planar injective words' which plays a key role in our work. Time permitting, I will explore some connections to representation theory and combinatorics that arose from our work.
  • 11:30 am - 12:15 pm EDT
    Braid monodromy and fundamental groups
    11th Floor Lecture Hall
    • Speaker
    • Michael Lönne, University Bayreuth
    • Session Chair
    • Anand Deopurkar, Australian National University
    Abstract
    While the braid monodromy of a divisor determines a finite presentation of the fundamental group of its complement, there lacks a general method for its computation. We want to give a review on ideas how to overcome these shortcomings and to exploit additional information in case of discriminants of isolated hypersurface singularities and linear systems to get presentations anyway.
  • 12:30 - 2:30 pm EDT
    Lunch/Free Time
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Wednesday, March 23, 2022
Braids in Symplectic and Algebraic Geometry
  • 9:30 - 10:15 am EDT
    Characterizing mapping classes of a K3 manifold
    11th Floor Lecture Hall
    • Speaker
    • Eduard Looijenga, Mathematisch Instituut Universiteit Utrecht
    • Session Chair
    • Benson Farb, University of Chicago
    Abstract
    This reports on joint work (in progress) with Benson Farb. Thurston characterized mapping classes of compact oriented 2-manifolds by singling out in each class a subclass enjoying special geometric properties (such as the preservation of a foliation on a subsurface). This is quite helpful in understanding such a mapping class. The ultimate goal is to develop something similar for K3 manifolds, where mapping classes often appear as monodromies of complex-analytic families. As we shall explain, in such a program a special role is played by genus one fibrations (in the differentiable category) and the associated representation of the spherical braid group.
  • 10:30 - 11:00 am EDT
    Coffee Break
    11th Floor Collaborative Space
  • 11:00 - 11:45 am EDT
    Plumbings and flops
    11th Floor Lecture Hall
    • Virtual Speaker
    • Ivan Smith, University of Cambridge
    • Session Chair
    • Benson Farb, University of Chicago
    Abstract
    We will discuss the symplectic topology of certain simple plumbings of 3-spheres which are related, at a derived level and depending interestingly on the characteristic of the ground field, to local threefolds containing a pair of floppable curves. This talk reports on joint work with Michael Wemyss.
  • 12:00 - 12:10 pm EDT
    Group Photo (Immediately After Talk)
    11th Floor Lecture Hall
  • 12:10 - 2:00 pm EDT
    Lunch/Free Time
  • 2:00 - 3:00 pm EDT
    Lightning Talks
    11th Floor Lecture Hall
    • Speakers
    • Ishan Banerjee, University of Chicago
    • Orsola Capovilla-Searle, University of California, Davis
    • Marc Kegel, Humboldt-Universität zu Berlin
    • Francesco Morabito, École Polytechnique
    • Virtual Speaker
    • Minh-Tam Trinh, MIT
    • Session Chair
    • Benson Farb, University of Chicago
    Abstract
    Extending homological stability for spaces of nonsingular hyper surfaces
    Ishan Banerjee, The University Of Chicago
    Spaces of nonsingular hypersurfaces in P^n (and more generally in smooth projective varieties) are known to exhibit homological stability. I will survey some of the work that has been done in this area before discussing my work on extending the ranges of stabiltiy and proving analogous results in similar contexts

    Artin Braids from Infinitesimal Loops
    Minh-Tâm Trinh, Massachusetts Institute of Technology
    In algebraic geometry, Spec of the field of formal Laurent series is an infinitesimal loop. Any map from this scheme into a quotient M / W, where M is the complex hyperplane-arrangement complement corresponding to a reflection group W, determines an element in the (profinite) braid group of W up to conjugacy. We classify the elements thus obtained, which we call algebraic braids. Our result roughly generalizes the classification of the links of plane curves. The form of the answer is related to Geck–Michel's notion of the "good" elements of W. We deduce that all algebraic braids have Burau spectral radius 1; in type A, we conjecture that more strongly, they all have topological entropy zero.

    A winding number filtration on the Morse complex
    Francesco Morabito, École Polytechnique
    In this talk I will present some results I have obtained during my PhD project; it is very much still a work in progress. Starting from some works by Patrice Le Calvez developed in the nineties, I have constructed a filtration on the power Morse complex of a generating function of a compactly supported Hamiltonian diffeomorphism on the plane. I will give an idea of the framework I used, and if time permits, of how to extend winding numbers to the diagonal in a consistent way. <b>
    Stein traces
    Marc Kegel, Humboldt University Berlin
    Every Legendrian knot L leaves a Stein trace in the 4-dimensional symplectic world by attaching a Weinstein 2-handle along L to the 4-ball. In this talk we will investigate whether a 4-dimensional tracker (with the necessary mathematical education) can determine the 3-dimensional creature that left the trace. This is based on joint work with Roger Casals and John Etnyre.

    Infinitely many planar exact Lagrangian fillings and symplectic Milnor fibers
    Orsola Capovilla-Searle, UC Davis
    We provide a new family of Legendrian links with infinitely many distinct exact orientable Lagrangian fillings up to Hamiltonian isotopy. This family of links includes the first examples of Legendrian links with infinitely many distinct planar exact Lagrangian fillings, which can be viewed as the smallest Legendrian links currently known to have infinitely many distinct exact Lagrangian fillings. As an application we find new examples of infinitely many exact Lagrangian spheres and tori 4-dimensional Milnor fibers of isolated hypersurface singularities with positive modality.
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 3:30 - 4:15 pm EDT
    Maps of braid groups
    11th Floor Lecture Hall
    • Speaker
    • Dan Margalit, Georgia Institute of Technology
    • Session Chair
    • Benson Farb, University of Chicago
    Abstract
    In joint work with Chen and Kordek, we recently classified all homomorphisms from the braid group on n strands to a braid group on at most 2n strands. I will present two proofs of this theorem, some related results, and a conjectural classification of all homomorphisms between braid groups.
Thursday, March 24, 2022
Braids in Symplectic and Algebraic Geometry
  • 9:00 - 9:45 am EDT
    Products of positive Dehn twists and their iterates
    11th Floor Lecture Hall
    • Speaker
    • Paul Seidel, MIT
    • Session Chair
    • Ailsa Keating, University of Cambridge
    Abstract
    (This is joint work in progress with S. Bao) Symplectic Floer homology provides a natural algebraic count of the fixed points and periodic points of a symplectic diffeomorphism. In the case of a product of Dehn twists, it is related to the intersections of the Lagrangian spheres involved. I will explain some simple applications of this idea, in particular to exponential growth rates, and then discuss the underlying Floer theory.
  • 10:00 - 10:30 am EDT
    Coffee Break
    11th Floor Collaborative Space
  • 10:30 - 11:15 am EDT
    Arrangements, duality, and local systems
    11th Floor Lecture Hall
    • Virtual Speaker
    • Alexandru Suciu, Northeastern University
    • Session Chair
    • Ailsa Keating, University of Cambridge
    Abstract
    We consider smooth, complex quasi-projective varieties that admit a compactification with a boundary which is an arrangement of smooth algebraic hypersurfaces. If the hypersurfaces intersect locally like hyperplanes, and the relative interiors of the hypersurfaces are Stein manifolds, we prove that the cohomology of certain local systems on the variety vanishes. As an application, we show that complements of linear, toric, and elliptic arrangements, as well as some orbit configuration spaces of Riemann surfaces are both duality and abelian duality spaces. This is joint work with Graham Denham.
  • 11:30 am - 12:15 pm EDT
    Thurston theory: a tale of two theorems
    11th Floor Lecture Hall
    • Speaker
    • Becca Winarski, MSRI/College of the Holy Cross
    • Session Chair
    • Ailsa Keating, University of Cambridge
    Abstract
    The Nielsen–Thurston classification of mapping classes and Thurston's theorem for the characterization of rational maps are central theorems in surface topology and complex dynamics, respectively. We give a single proof that unifies the two theorems. Moreover, we adapt mapping class group techniques to develop an algorithm that identifies when a branched self-cover of the plane is equivalent to a polynomial. This is joint work with Jim Belk, Justin Lanier, and Dan Margalit.
  • 12:30 - 2:30 pm EDT
    Lunch/Free Time
  • 2:30 - 3:15 pm EDT
    Surface Braids and Galois Cohomology
    11th Floor Lecture Hall
    • Speaker
    • Jesse Wolfson, University of California, Irvine
    • Session Chair
    • Ailsa Keating, University of Cambridge
    Abstract
    By a theorem of Artin and Griffiths, every sufficiently small Zariski open of a complex variety admits the structure of an iterated punctured curve fibration. Said another way, the absolute Galois group of a complex function field admits the structure of an (inverse limit of) iterated free by free groups with monodromy given by mapping classes. In the present talk, we describe joint work in progress with Benson Farb and Mark Kisin to use the theory of surface braids to construct Galois cohomology classes and control their behavior under finite extensions with specified ramification. Time permitting, we will sketch applications and limitations of this method for understanding how hard it is to solve a general degree n polynomial.
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 4:00 - 5:00 pm EDT
    Open Problem Session
    Problem Session - 11th Floor Lecture Hall
    • Session Chair
    • Ailsa Keating, University of Cambridge
Friday, March 25, 2022
Braids in Symplectic and Algebraic Geometry
  • 9:00 - 9:45 am EDT
    Geometric monodromy of families of framed Riemann surfaces
    11th Floor Lecture Hall
    • Speaker
    • Nick Salter, University of Notre Dame
    • Session Chair
    • Anthony Licata, Australian National University
    Abstract
    A family of Riemann surfaces gives rise to a geometric monodromy group valued in the mapping class group of the fiber. In a surprising diversity of examples in algebraic geometry (e.g. linear systems on algebraic surfaces, Milnor fibers of an isolated plane curve singularity, strata of abelian differentials), the fibers come endowed with a canonical framing (or some close cousin known as an "r-spin structure"). This forces the monodromy group to stabilize this framing up to isotopy, and one would like to know if this gives a complete description - is the monodromy group *equal* to the associated “framed mapping class group”? I will give an account of this story, with the ultimate aim of explaining how the methods of geometric group theory can be used to give a positive answer in each of the situations mentioned above, and the consequences this has for the study of vanishing cycles and of injectivity properties of monodromy groups. This incorporates joint work with Calderon and with Portilla Cuadrado.
  • 10:00 - 10:30 am EDT
    Coffee Break
    11th Floor Collaborative Space
  • 10:30 - 11:15 am EDT
    Rewrite systems in 3-free Artin groups
    11th Floor Lecture Hall
    • Speaker
    • Rose Morris-Wright, UCLA
    • Session Chair
    • Anthony Licata, Australian National University
    Abstract
    (Joint work with Maria Cumplido and Ruben Blasco) Artin groups are a generalization of braid groups, first defined by Tits in the 1960s. While specific types of Artin groups have many of the same properties as braid groups, other examples of Artin groups are still very mysterious. In particular, it is unknown whether the word problem is solvable for all Artin groups. I will discuss a new algorithm for solving the word problem in 3-free Artin groups. This is based on work by Holt and Rees for large type and sufficiently large type groups (2012 and 2013). Our work significantly broadens the class of Artin groups to which this result applies because it allows for groups that have commuting generators. This algorithm gives an explicit way to reduce a word to a geodesic word without ever increasing the length of the word.
  • 11:30 am - 12:15 pm EDT
    Braid factorizations and exotic complex curves
    11th Floor Lecture Hall
    • Speaker
    • Kyle Hayden, Columbia University
    • Session Chair
    • Anthony Licata, Australian National University
    Abstract
    Braid factorizations provide a link between the braid group and the study of embedded surfaces and complex curves in 4-manifolds. After reviewing a bit of this story, I will explain how quasipositive braid factorizations can help bridge the gap between the rigid complex realm and the exotic smooth setting, building the first examples of complex curves that are isotopic through homeomorphisms but not diffeomorphisms of complex 2-space. Time permitting, I will explain how this relates to a speculative connection between braid factorizations and Khovanov and Floer homologies.
  • 12:30 - 2:30 pm EDT
    Lunch/Free Time
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Monday, March 28, 2022
Braids
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 4:00 - 4:30 pm EDT
    Algorithms with fibered links and closures of positive braids
    Post Doc/Graduate Student Seminar - 11th Floor Lecture Hall
    • Marc Kegel, Humboldt-Universität zu Berlin
    Abstract
    We will follow beautiful work of D. Gabai explaining how to understand if a link is fibered and relate this to closures of positive braids. If time permits we will discuss an algorithm that decides in finite time if a knot is the closure of a positive braid and discuss its running time. Everyone is welcome to attend.
  • 4:30 - 5:00 pm EDT
    Legendrian knots and contact geometry
    Post Doc/Graduate Student Seminar - 11th Floor Lecture Hall
    • Rima Chatterjee, University of Cologne
    Abstract
    Legendrian knots in contact geometry has been extensively studied in the last couple of decades. In this talk, I'll introduce Legendrian knots and its properties. Specifically, I'll discuss some applications which show how these knots have become so important in the study of (contact) 3 manifolds. No prior knowledge of contact geometry will be assumed.
Tuesday, March 29, 2022
Braids
  • 10:30 - 11:50 am EDT
    Signatures in topology, algebra, and dynamics
    Seminar - 10th Floor Classroom
    • Bena Tshishiku, Brown University
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Wednesday, March 30, 2022
Braids
  • 9:00 - 10:00 am EDT
    Professional Development: Hiring
    Professional Development - 11th Floor Lecture Hall
  • 10:15 - 10:30 am EDT
    Graduate Student/Postdoc Group Photo
    Group Photo (Immediately After Talk) - 11th Floor Collaborative Space
  • 1:30 - 2:30 pm EDT
    Colloquium - Illustrating geometry (and topology)
    Seminar - 11th Floor Lecture Hall
    • Virtual Speaker
    • Saul Schleimer, University of Warwick
    Abstract
    According to Poincaré “geometry is the art of reasoning well from badly drawn figures” [1895]. In this talk I will give an informal discussion of some famous attempts to draw mathematical figures: some more, and some less, badly drawn. I will finish by discussing some of my own work in this direction, attempting to use computer graphics, interactive web apps, and 3D printing to illustrate mathematics.
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Thursday, March 31, 2022
Braids
  • 10:30 - 11:50 am EDT
    Signatures in topology, algebra, and dynamics
    Seminar - 10th Floor Classroom
    • Bena Tshishiku, Brown University
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Friday, April 1, 2022
Braids
  • 11:30 am - 12:30 pm EDT
    Computational Seminar - The search for alternating and quasi-alternating surgeries
    Seminar - 11th Floor Lecture Hall
    • Marc Kegel, Humboldt-Universität zu Berlin
    Abstract
    A slope r is called alternating slope for a knot K in the 3-sphere if the r-surgery along K is diffeomorphic to the double branched cover branched over an alternating link in the 3-sphere. We will quickly discuss the relevance of this notion and then algorithmically classify for concrete example knots all their alternating slopes. For that we will use mainly SnapPy with help from sage, regina, and KLO (Knot Like Objects). I believe that the methods we are using are so general that they might be useful in other situations as well. The audience is encouraged to bring their own laptops and follow the algorithms interactively. This talk is based on joint work with Ken Baker and Duncan McCoy.
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Monday, April 4, 2022
Braids
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Tuesday, April 5, 2022
Braids
  • 10:30 - 11:50 am EDT
    Signatures in topology, algebra, and dynamics
    Seminar - 10th Floor Classroom
    • Bena Tshishiku, Brown University
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Wednesday, April 6, 2022
Braids
  • 9:00 - 10:00 am EDT
    Professional Development: Papers and Journals
    Professional Development - 11th Floor Lecture Hall
  • 1:30 - 2:30 pm EDT
    Colloquium - Counting geometric objects arithmetically with inspiration from topology
    Seminar - 11th Floor Lecture Hall
    • Isabel Vogt, Brown University
    Abstract
    The answers to many classical counting problems in complex algebraic geometry can be computed by taking the number of zeros of a section of an appropriate vector bundle. In the same way that a degree d polynomial over a non-algebraically closed field need not have d roots defined over the field, results about constant counts in enumerative geometry can fail over arithmetically interesting ground fields. Building on ideas of Morel and Levine coming from topology, Kass-Wickelgren gave an explicit "enrichment" of the number of zeros of a section of a relatively-orientable vector bundle that takes extra arithmetic data into account to yield constant enriched counts. I'll discuss an application of these ideas to the classical problem of counting the bitangents of a smooth plane quartic curve. Subtleties arise because the relevant vector bundle is not "relatively-orientable"! This is joint work with Hannah Larson.
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Thursday, April 7, 2022
Braids
  • 10:30 - 11:50 am EDT
    Signatures in topology, algebra, and dynamics
    Seminar - 10th Floor Classroom
    • Bena Tshishiku, Brown University
  • 3:00 - 3:30 pm EDT
    Donut Break!
    Coffee Break - 11th Floor Collaborative Space
Friday, April 8, 2022
Braids
  • 11:30 am - 12:30 pm EDT
    Computational Seminar - Introduction to KLO
    Seminar - 11th Floor Lecture Hall
    • Virtual Speaker
    • Frank Swenton, Middlebury College
    Abstract
    In this talk, we'll present an introduction to KLO, both as a tool for manipulating diagrams for Knot-Like Objects and as a computational framework for exploring questions and conjectures concerning them. After an overview of KLO's basic functionality with knot and knotted surface diagrams, Kirby diagrams, and surgery diagrams, we'll give an overview of the implementation of Brendan Owens's algorithm for finding ribbon disks for alternating knots, including a look at the computational results obtained thus far. Accretion of additional features of KLO has been largely user-directed, so ample time will be built in for discussion and suggestions; attendees are invited to bring their laptops and install KLO via the download links at http://klo-software.net for the talk.
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Monday, April 11, 2022
Braids
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 4:00 - 5:00 pm EDT
    Link homology in a nutshell
    Post Doc/Graduate Student Seminar - 11th Floor Lecture Hall
    • Marithania Silvero Casanova, Universidad de Sevilla
Tuesday, April 12, 2022
Braids
  • 10:30 - 11:50 am EDT
    Signatures in topology, algebra, and dynamics
    Seminar - 10th Floor Classroom
    • Bena Tshishiku, Brown University
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Wednesday, April 13, 2022
Braids
  • 9:00 - 10:00 am EDT
    Professional Development: Grant Proposals
    Professional Development - 11th Floor Lecture Hall
  • 1:30 - 2:30 pm EDT
    Colloquium - Nielsen realization problems
    Seminar - 11th Floor Lecture Hall
    • Bena Tshishiku, Brown University
    Abstract
    The Nielsen realization problem is about group actions on manifolds. For a manifold M, it asks when a subgroup of the mapping class group Mod(M) can be lifted to a group of diffeomorphisms under the natural projection Diff(M) → Mod(M). We will discuss some aspects of this problem and its connections to geometry.
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 3:00 - 4:00 pm EDT
    Double branched covers of links, signatures, and invariants from Heegaard Floer theory
    Kassar 105 - Brown University Department of Math
    • Biji Wong, Max Planck Institute for Mathematics
    Abstract
    Signatures are useful integer invariants that appear in many settings. They can be defined for manifolds, links, etc. In this talk we'll focus on 3-manifolds that can be obtained as double branched covers of (multi-component) links in the 3-sphere. We'll relate the signature of the branched link to an invariant of the branched cover that comes from Heegaard Floer theory, a powerful set of tools for studying 3-manifolds. This generalizes work of Owens-Manolescu, Owens-Lisca and others, and is work in progress with Marco Marengon.
Thursday, April 14, 2022
Braids
  • 10:30 - 11:50 am EDT
    Signatures in topology, algebra, and dynamics
    Seminar - 10th Floor Classroom
    • Bena Tshishiku, Brown University
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Friday, April 15, 2022
Braids
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Monday, April 18, 2022
Braids
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 4:00 - 4:30 pm EDT
    A Quick Trip through String Link Concordance
    Post Doc/Graduate Student Seminar - 11th Floor Lecture Hall
    • Miriam Kuzbary, Georgia Institute of Technology
    Abstract
    In this brief survey talk we will learn about a subclass of tangles called string links which can be contextualized as a generalization of pure braids. These objects are fundamental to the classification of links up to link homotopy by Habegger and Lin in the early nineties, and are interesting in their own right as a way to define a link concordance group for links with a fixed number of components. We will discuss Habegger and Lin’s results on both link homotopy and concordance and see how the pure braid group fits into the story.
  • 4:30 - 5:00 pm EDT
    Branched covers
    Post Doc/Graduate Student Seminar - 11th Floor Lecture Hall
    • Biji Wong, Max Planck Institute for Mathematics
    Abstract
    We'll introduce the theory of branched covers and use it to construct some interesting 3-manifolds.
Tuesday, April 19, 2022
Braids
  • 10:30 - 11:50 am EDT
    Signatures in topology, algebra, and dynamics
    Seminar - 10th Floor Classroom
    • Bena Tshishiku, Brown University
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Wednesday, April 20, 2022
Braids
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 4:15 - 5:05 pm EDT
    Intro to Machine Learning
    11th Floor Lecture Hall
    • Maurizio Parton, University of Chieti-Pescara
Thursday, April 21, 2022
Braids
  • 10:00 - 10:50 am EDT
    Knot theory and machine learning
    11th Floor Lecture Hall
    • Virtual Speaker
    • András Juhász, University of Oxford
    Abstract
    The signature of a knot K in the 3-sphere is a classical invariant that gives a lower bound on the genera of compact oriented surfaces in the 4-ball with boundary K. We say that K is hyperbolic if its complement admits a complete, finite volume hyperbolic metric. I will explain how we have used methods from machine learning to find an unexpected relationship between the signature and the cusp shape of a hyperbolic knot. This is joint work with Alex Davies, Marc Lackenby, and Nenad Tomasev.
  • 10:30 - 11:50 am EDT
    Signatures in topology, algebra, and dynamics
    Seminar - 10th Floor Classroom
    • Bena Tshishiku, Brown University
  • 11:00 - 11:50 am EDT
    Machine learning and the combinatorial invariance conjecture
    11th Floor Lecture Hall
    • Virtual Speaker
    • Alex Davies, DeepMind
    Abstract
    In this talk I will give an overview of using machine learning together with mathematicians to aid in conjecture discovery and describe work that we have done in collaboration with Geordie Williamson, using machine learning to help understand surprising new structure in representation theory.
  • 1:00 - 1:50 pm EDT
    Applying Machine Learning to mathematics brainstorming/problem session
    Problem Session - 11th Floor Lecture Hall
    • Moderators
    • Julia Grigsby, Boston College
    • Mark Hughes, Brigham Young University
    • Maurizio Parton, University of Chieti-Pescara
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Friday, April 22, 2022
Braids
  • 11:30 am - 12:30 pm EDT
    Cars, Interchanges, Traffic Counters, and a Pretty Darned Good Knot Invariant
    11th Floor Lecture Hall
    • Dror Bar-Natan, University of Toronto
    Abstract
    Reporting on joint work with Roland van der Veen, I'll tell you some stories about ρ1, an easy to define, strong, fast to compute, homomorphic, and well-connected knot invariant. ρ1 was first studied by Rozansky and Overbay, it has far-reaching generalizations, and I wish I understood it. http://drorbn.net/waco22
  • 2:00 - 3:00 pm EDT
    Introductory talk on (mostly Heegaard) Floer homology
    11th Floor Lecture Hall
    • Matthew Hedden, Michigan State University
    Abstract
    I'll give a brief overview/introduction to Heegaard Floer homology in preparation for some of next week's conference talks.
  • 3:00 - 3:30 pm EDT
    End of Semester Program Coffee Break
    Coffee Break - 11th Floor Collaborative Space
Monday, April 25, 2022
Braids in Low-Dimensional Topology
  • 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
Braids in Low-Dimensional Topology
  • 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.
  • 10:00 - 10:30 am EDT
    Coffee Break
    11th Floor Collaborative Space
  • 10:30 - 11:15 am EDT
    Tight surgeries on torus knots
    11th Floor Lecture Hall
    • Speaker
    • Bulent Tosun, University of Alabama
    • Session Chair
    • Joan Licata, Australian National University
    Abstract
    In 3-dimensional contact geometry, tight contact structures constitute a geometrically interesting class of contact structures because they form natural boundary conditions for symplectic and certain complex 4-manifolds, and moreover they are deeply related to the topology of 3-manifolds, and Heegaard and Monopole Floer theories. An outstanding open problem in 3-dimensional contact geometry concerns the classification of tight contact structures: When a closed oriented 3-manifold admits a tight contact structure, can one classify all tight contact structures on the manifold? A great deal of important work in the last 25 years has been put towards the resolution of this fundamental question. But at the moment it is fair to say a thorough understanding is far from complete. For example, if one considers the classification question on 3-manifolds obtained by Dehn filling of knots in three sphere, then currently the only such complete classification result for all surgeries available is for the unknot. In this talk, I will report on ongoing joint work with J. Etnyre and H. Min that gives the first complete classification result for all surgeries on an infinite family of torus knots. I will provide further context and motivations for the result, and give some details of the proof.
  • 11:30 am - 12:15 pm EDT
    Braids, homogenization, and the slice-Bennequin inequality
    11th Floor Lecture Hall
    • Speaker
    • Peter Feller, ETH Zurich
    • Session Chair
    • Joan Licata, Australian National University
    Abstract
    We investigate braid invariants, such as the writhe and the fractional Dehn twist coefficient (FDTC), that arise as the homogenization of a concordance homomorphism, such as tau and Upsilon from the Heegaard Floer tool box. After providing a new characterization of the FDTC, we turn to connections with low-dimensional topology via the concept of homogenization of knot invariants. Concretely, we view the slice Bennequin inequality---a celebrated  inequality due to Kronheimer-Mrowka and Rudolph that relates a knot invariant (the smooth 4-genus) and a braid invariant (the writhe)---as a special case of relating knot concordance homomorphisms and their homogenizations. As an application we find that the slice-Bennequin inequality holds with the FDTC in place of the writhe. Teaser: As a motivation for the concept of homogenization, this talk features a neat construction of the field of real numbers you probably dont know about.
  • 12:30 - 2:00 pm EDT
    Lunch/Free Time
  • 2:00 - 2:45 pm EDT
    Pure Braids, Legendrian Knots, and open book decompositions
    11th Floor Lecture Hall
    • Speaker
    • Sinem Onaran, Hacettepe University
    • Session Chair
    • John Etnyre, Georgia Institute of Technology
    Abstract
    In this talk, I will discuss the pure braided plat presentation for knots and links in lens spaces. Using such presentations, I will present an algorithm to put the knots on a planar page of an open book decomposition whose monodromy is a product of positive Dehn twists and discuss its applications.
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 3:30 - 4:15 pm EDT
    Knotted handlebodies
    11th Floor Lecture Hall
    • Speaker
    • Maggie Miller, Stanford University
    • Session Chair
    • John Etnyre, Georgia Institute of Technology
    Abstract
    We construct 3-dimensional genus-g handlebodies H and H' in the 4-sphere so that H and H' have the same boundary and are homeomorphic rel boundary, but are not smoothly isotopic rel boundary (for all g ≥ 2). In fact, H and H' are not even topologically isotopic rel boundary, even when their interiors are pushed into the 5-ball. This proves a conjecture of Budney and Gabai for g ≥ 2 in a very strong sense, and is a surprising answer to a 1-dimension up version of an open question about Seifert surfaces in S^3. In this talk, I'll give 3- and 4-dimensional motivation and discuss some interesting theorems about knotted surfaces that go into the construction. (This is joint with Mark Hughes and Seungwon Kim.)
Wednesday, April 27, 2022
Braids in Low-Dimensional Topology
  • 9:00 - 9:45 am EDT
    An algorithm to distinguish Legendrian knots
    11th Floor Lecture Hall
    • Virtual Speaker
    • Ivan Dynnikov, Steklov Mathematical Institute
    • Session Chair
    • Hannah Turner, Georgia Institute of Technology
    Abstract
    In recent joint works with Maxim Prasolov and Vladimir Shastin we developed a method to decide algorithmically whether or not two given Legendrian knots are equivalent. The method is based on the formalism of rectangular diagrams, which we have extended to Giroux' convex surfaces and Legendrian graphs. In many cases, the approach allows to distinguish Legendrian knots that are not distinguished by other known means or have so large complexity that other methods are not feasible to apply to them. We provide an example of two non-equivalent Legendrian knots that cobound an annulus embedded in the three-sphere and tangent to the standard contact structure along the entire boundary. Such examples have not been known earlier.
  • 10:00 - 10:30 am EDT
    Coffee Break
    11th Floor Collaborative Space
  • 10:30 - 11:15 am EDT
    Computations of ECH capacities and infinite staircases of 4D symplectic embeddings
    11th Floor Lecture Hall
    • Speaker
    • Morgan Weiler, Cornell University
    • Session Chair
    • Hannah Turner, Georgia Institute of Technology
    Abstract
    Since McDuff proved that embedded contact homology can be used to characterize 4D ellipsoid embeddings in 2011 and McDuff and Schlenk discovered the first "infinite staircase" of ellipsoid embeddings in 2012, a growing body of work has analyzed which toric domains in R^4 (regions symmetric under the natural torus action from C^2) admit infinite staircases of ellipsoid embeddings. From the ECH (embedded contact homology) perspective, symplectic embeddings into a toric domain are determined by a certain set of torus knots on its boundary. We will discuss an algorithm used to identify these torus knots and find a fundamentally new type of infinite staircase in recent work with Magill and McDuff, as well as its possible generalizations and limitations.
  • 11:30 am - 12:15 pm EDT
    Computers, complex curves, and Khovanov homology
    11th Floor Lecture Hall
    • Speaker
    • Kyle Hayden, Columbia University
    • Session Chair
    • Hannah Turner, Georgia Institute of Technology
    Abstract
    Khovanov homology provides a powerful tool for studying knots and links in 3-space and surfaces in 4-space. I will discuss recent developments that use Khovanov homology to distinguish non-isotopic surfaces in the 4-ball. We will see how braids relate two seemingly disparate strengths of these tools from Khovanov homology: their amenability to calculation (including recent software), and their sensitivity to complex curves.
    Based on joint work with Isaac Sundberg and with Alan Du.
  • 12:30 - 12:40 pm EDT
    Group Photo -Immediately following talk
    Group Photo (Immediately After Talk) - 11th Floor Lecture Hall
  • 12:40 - 2:30 pm EDT
    Lunch/Free Time
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Thursday, April 28, 2022
Braids in Low-Dimensional Topology
  • 9:00 - 9:45 am EDT
    Birman-Menasco finiteness theorem revisited
    11th Floor Lecture Hall
    • Virtual Speaker
    • Tetsuya Ito, Kyoto University
    • Session Chair
    • Antonio Alfieri, Université du Québec à Montréal (CRM)
    Abstract
    Birman-Menasco proved the remarkable finiteness theorem: modulo exchange move, the number of the closed n-braid representatives of genus g knots/links is finite. In this talk, I would like to explain various topics related or inspired by Birman-Menasco finiteness theorem. Among them, I will explain its quantitative refinement and applications.
  • 10:00 - 10:30 am EDT
    Coffee Break
    11th Floor Collaborative Space
  • 10:30 - 11:15 am EDT
    The fractional Dehn twist coefficient: from braids to mapping class groups
    11th Floor Lecture Hall
    • Speaker
    • Diana Hubbard, Brooklyn College
    • Session Chair
    • Antonio Alfieri, Université du Québec à Montréal (CRM)
    Abstract
    The braid group is a particular example of the mapping class group of a surface with boundary. An invariant that roughly measures how much a mapping class “twists” about the boundary of the surface is the fractional Dehn twist coefficient. In this talk I will discuss some reasons why we might care about this invariant, several results about the fractional Dehn twist coefficient for braids, and explore to what extent these results can be extended to more general mapping class groups. This talk will include joint work with Peter Feller and Hannah Turner.
  • 11:30 am - 12:15 pm EDT
    VIrtual Artin groups II: pure subgroups and applications
    11th Floor Lecture Hall
    • Speaker
    • Paolo Bellingeri, University of Caen Normandy
    • Session Chair
    • Antonio Alfieri, Université du Québec à Montréal (CRM)
    Abstract
    This talk can be seen as the second part of the one given by Luis Paris last February. To make the exposition self-contained, I will start with the definition (and the motivation) of this new family of groups and therefore I will present some results (and questions) on pure subgroups and crystallographic quotients of virtual Artin groups. At the end, if time permits, I will present other families of groups that admit similar notions of virtual extension. Joint work with Luis Paris and Anne Laure Thiel.
  • 12:30 - 2:30 pm EDT
    Lunch/Free Time
  • 2:30 - 3:15 pm EDT
    An Unknotting Number for Transverse Knots
    11th Floor Lecture Hall
    • Speaker
    • Lisa Traynor, Bryn Mawr College
    • Session Chair
    • Keiko Kawamuro, University of Iowa
    Abstract
    I will review some classic results about transverse knots in contact manifolds and then introduce the definition of the transverse unknotting number. For smooth knots, an “ancestor-descendant” relation has been defined by Cantarella-Henrich-Magness-O’Keefe-Perez-Rawdon-Zimmer; I will describe a transverse analog of this relation and then define and calculate some “transverse family trees.” This is joint work with Blossom Jeong.
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Friday, April 29, 2022
Braids in Low-Dimensional Topology
  • 9:00 - 9:45 am EDT
    Non-positive open books of Stein fillable contact 3-manifolds
    11th Floor Lecture Hall
    • Speaker
    • Andy Wand, University of Glasgow
    • Session Chair
    • Vera Vertesi, University of Vienna
    Abstract
    We will discuss motivation for and approaches to the question of when the monoid in the mapping class group of a surface with boundary corresponding to monodromies of open book decompositions of Stein fillable contact 3-manifolds differs from the monoid of mapping classes which admit factorizations into positive Dehn twists. In particlar, combining new results with previous work of several people, we give a complete solution to this problem for all but the case of the genus 1 surface with 1 boundary component. This is joint work with Vitalijs Brejevs.
  • 10:00 - 10:30 am EDT
    Coffee Break
    11th Floor Collaborative Space
  • 10:30 - 11:15 am EDT
    Taut Foliations and Braid Positivity
    11th Floor Lecture Hall
    • Speaker
    • Siddhi Krishna, Columbia University
    • Session Chair
    • Vera Vertesi, University of Vienna
    Abstract
    The L-space conjecture has been in the news a lot lately: this conjecture predicts that three seemingly different ways to measure the "size" of a 3-manifold are equivalent. In particular, it predicts that a manifold with the "extra" geometric structure of a taut foliation also has "extra" Heegaard Floer homology. In this talk, I'll discuss the motivation for this conjecture, and describe some new results which produce taut foliations by leveraging special properties of positive braid knots. Along the way, we will produce some novel obstructions to braid positivity. I will not assume any background knowledge in Floer or foliation theories; all are welcome!
  • 11:30 am - 12:15 pm EDT
    Symmetric knots and Heegaard Floer homology
    11th Floor Lecture Hall
    • Speaker
    • Antonio Alfieri, Université du Québec à Montréal (CRM)
    • Session Chair
    • Vera Vertesi, University of Vienna
    Abstract
    I will discuss some open problems regarding symmetric knots, and group actions on 3- and 4-manifolds. In the second part of the talk I will discuss how techniques from Floer theory can be employed to approach some of these problems.
  • 12:30 - 2:30 pm EDT
    Lunch/Free Time
  • 2:30 - 3:15 pm EDT
    Khovanov homology and knot detection
    11th Floor Lecture Hall
    • Speaker
    • Steven Sivek, Imperial College London
    • Session Chair
    • Marc Kegel, Humboldt-Universität zu Berlin
    Abstract
    In this talk I’ll outline a proof that Khovanov homology detects the T(2,5) torus knot. In particular, I’ll explain why a knot with the same Khovanov homology as T(2,5) must be fibered, and then use some deep results in Floer homology to see what we can deduce about the monodromy of that fibration. This is based on joint work with John Baldwin and Ying Hu.
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 4:00 - 4:45 pm EDT
    Floer homology and right-veering monodromy
    11th Floor Lecture Hall
    • Speaker
    • John Baldwin, Boston College
    • Session Chair
    • Marc Kegel, Humboldt-Universität zu Berlin
    Abstract
    In this talk I'll explain how knot Floer homology detects whether the monodromy of a fibered knot is right-veering. This gives a purely Floer-theoretic characterization of tight contact structures, and has applications to Dehn surgery and taut foliations. Our proof uses a relationship between Heegaard Floer homology and the dynamics of surface diffeomorphisms. This is based on joint work with Yi Ni and Steven Sivek.
Monday, May 2, 2022
Braids
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 4:00 - 4:30 pm EDT
    In Sync—a course design on math, art and aesthetics
    Post Doc/Graduate Student Seminar - 11th Floor Lecture Hall
    • Jiajun Yan, University of Virginia
    Abstract
    In this talk, I will speak about the design of an engagement course for freshman undergraduates I created which I will be teaching at UVA in Fall 2023. The course will seek traces of mathematical ideas in the realms of literature, visual arts and philosophy in the first half and will involve student projects in the later half. By presenting the design of this course, I want to encourage outreaching questions relating math and other subjects, introduce more daring approaches to make math accessible to the general audience and convey an alternative way to appreciate math.
  • 4:30 - 5:00 pm EDT
    Some observations on isotopy classes of simple closed curves on closed non orientable surfaces
    Post Doc/Graduate Student Seminar - 11th Floor Lecture Hall
    • Willi Kepplinger, University of Vienna
    Abstract
    I will discuss some curiosities of isotopy classes of simple closed curves on closed non orientable surfaces
Tuesday, May 3, 2022
Braids
  • 10:30 - 11:50 am EDT
    Signatures in topology, algebra, and dynamics
    Seminar - 10th Floor Classroom
    • Bena Tshishiku, Brown University
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Wednesday, May 4, 2022
Braids
  • 1:30 - 2:30 pm EDT
    Meeting for Braids participants
    Meeting - 11th Floor Lecture Hall
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Thursday, May 5, 2022
Braids
  • 10:30 - 11:50 am EDT
    Signatures in topology, algebra, and dynamics
    Seminar - 10th Floor Classroom
    • Bena Tshishiku, Brown University
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Friday, May 6, 2022
Braids
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space

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