Institute for Computational and Experimental Research in Mathematics (ICERM)

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.

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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.

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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

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.

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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.

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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.

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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.

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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.

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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.

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.

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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.

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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.

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(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.

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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.

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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.

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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.

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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.

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(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.

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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.

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