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Organizing Committee
University of Massachusetts Amherst
Australian National University
University of Chicago
University of Cambridge
Australian National University
Abstract
Incarnations of braid groups, or generalizations thereof, naturally arise in a range of active research areas in symplectic and algebraic geometry. This is a rich and diverse ecosystem, and the workshop will aim to bring together speakers from all corners of it. A unifying theme is monodromy: on the one hand, generalized braid groups arise in symplectic and algebraic geometry as fundamental groups of moduli spaces, loosely construed -- for instance, of complements of discriminant loci of singularities or of hyperplane arrangements, or moduli spaces of deformations of complex or symplectic structures. On the other hand, monodromy ideas motivate representations of generalized braid groups as various flavors of geometric automorphisms -- for instance, as (framed) mapping class group elements, symplectic Dehn twists, spherical twists in derived categories, or flop functors for 3-folds. These perspectives lead in turn to a wide array of further geometric applications, from classifications of Stein fillings to the study of spaces of Bridgeland stability conditions.
From a community perspective, one aim of the workshop is to bring together mathematicians from adjacent research communities with a shared interest in braids as they arise in symplectic or algebraic geometry. We hope the conference will accelerate the cross-pollination of ideas, and help foster collaborations, at what is a very exciting time for the field. Much of this research also lies at an interface with other aspects of the thematic semester -- for instance, braids in representation theory (e.g. in connection with cluster algebras or Bridgeland stability conditions), or in low-dimensional topology (e.g. in connection with monodromies of open books) -- and many of the talks should be of interest to a broader set of semester participants.
From a community perspective, one aim of the workshop is to bring together mathematicians from adjacent research communities with a shared interest in braids as they arise in symplectic or algebraic geometry. We hope the conference will accelerate the cross-pollination of ideas, and help foster collaborations, at what is a very exciting time for the field. Much of this research also lies at an interface with other aspects of the thematic semester -- for instance, braids in representation theory (e.g. in connection with cluster algebras or Bridgeland stability conditions), or in low-dimensional topology (e.g. in connection with monodromies of open books) -- and many of the talks should be of interest to a broader set of semester participants.
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Workshop Schedule
Monday, March 21, 2022
8:30 - 8:50 AM EDT
Check In
11th Floor Collaborative Space
8:50 - 9:00 AM EDT
Welcome
11th Floor Lecture Hall
Brendan Hassett, ICERM/Brown University
9:00 - 9:45 AM EDT
Stable cohomology of braid groups with coefficients in symplectic representations
11th Floor Lecture Hall
Speaker
Craig Westerland, University of Minnesota
Session Chair
İnanç 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
İnanç 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
İnanç 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
İnanç 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
İnanç 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
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.
