Braids, Links, and Applications

Abstract: We present braiding algorithms and derive braid equivalences for knots and links in various topological settings (classical, in 3-manifolds, virtual, singular, long knots), focusing in the end in the recent theory of knotoids and braidoids.

Abstract: We introduce to the proof of the classical $K(\pi,1)$ conjecture, recently given in the case of affine Artin groups (G. Paolini, M. Salvetti, "Proof of the $K(\pi, 1)$ conjecture for affine Artin groups", Inven. Math, {\bf 224}, 2 (2021)").

Abstract: We describe a refinement of the bounds on the number of reducible fibers in a pencil of curves on a smooth projective surface assuming that irreducible components of reducible members belong to a fixed subset of Neron-Severi group. The results give a substantial generalization of results by the speaker on free quotients of the fundamental groups of the complements to arrangements of lines. Joint work with J.I. Cogolludo.

Abstract: Braid groups, configuration spaces, and hyperplane arrangements have been tightly intertwined for at least 60 years. I will discuss some recent advances in our understanding of fundamental groups of complements of complex line arrangements, with emphasis on several of the Lie algebras associated to them.

Abstract: I overview some of the earlier work with collaborators, Diaconscu and Yau, on reduction of virtual fundamental classes on moduli spaces of sheaves in Calabi Yau 4 folds to  virtual cycles of moduli spaces of sheaves on threefolds which embed in the 4 fold as suitable divisors.  In particular I discuss some examples of such virtual reductions which turn out to be useful in the study of Donaldson-Thomas (DT) invariants of torsion sheaves with support on a smooth projective surface in an ambient non-compact Calabi Yau fourfold given by the total space of a rank 2 bundle on the surface.

We prove that in certain cases, when the rank 2 bundle is chosen appropriately, the universal truncated Atiyah class of these codimension 2 sheaves reduces to one, defined over the moduli space of such sheaves realized as torsion codimension 1 sheaves in a noncompact divisor (threefold) embedded in the ambient fourfold. Such reduction property of universal Atiyah class enables us to relate our fourfold DT theory to a reduced DT theory of a threefold and subsequently then to the moduli spaces of sheaves on the base surface using results in arXiv:1701.08899 and arXiv:1701.08902. We then make predictions about modularity of such fourfold invariants when the base surface is an elliptic K3.

If time permits, I will elaborate on extensions of this project to the case where we study particular framed sheaves on cotangent bundles of surfaces and show how a version of degeneration technique and our results, can help relating invariants on the 4 fold to Vafa-Witten invariants of the surface. Finally, if time permits, I will provide hints about how to generalize these results to more general case of sheaves with support on general 4 manifolds (non-algebraic surfaces) and hidden structures which one can discover in their associated partition functions.

Abstract: Elliptic surfaces form an important class of surfaces both from the theoretical perspective (appearing in the classification of surfaces) and the practical perspective (they are fascinating to study, individually and as a class, and are amenable to many particular computations).  Elliptic surfaces that are also rational are a special sub-class. 

The first example is to take a general pencil of plane cubics (with 9 base points) and blow up the base points to obtain an elliptic fibration; these are so-called Jacobian surfaces, since they have a section (the final exceptional curve of the sequence of blowups).  Moduli spaces for rational elliptic surfaces with a section were constructed by the speaker, and further studied by Heckman and Looijenga.  In general, there may not be a section, but a similar description is possible: all rational elliptic surfaces are obtained by taking a pencil of curves of degree 3k with 9 base points, each of multiplicity k. 

There will always be the k-fold cubic curve through the 9 points as a member, and the resulting blowup produces a rational elliptic surface with a multiple fiber of multiplicity m (called the index of the fibration).  A. Zanardini has recently computed the GIT stability of such pencils for m=2; in joint work with her we have constructed a moduli space for them via toric constructions.  I will try to tell this story in this lecture.

Abstract: These lectures will present a board survey of recent work on new q-series invariants of 3-manifolds labeled by Spin-C structures. While the original motivation for studying these invariants is rooted in topology, they exhibit a number of unexpected properties and connections to other areas of mathematics, e.g. turn out to be characters of logarithmic vertex algebras. The integer coefficients of these q-series invariants can be understood as the answer to a certain enumerative problem, and when q tends to special values these invariants relate to other invariants of 3-manifolds labeled by Spin and Spin-C structures.

Abstract: We introduce to the proof of the classical $K(\pi,1)$ conjecture, recently given in the case of affine Artin groups (G. Paolini, M. Salvetti, "Proof of the $K(\pi, 1)$ conjecture for affine Artin groups", Inven. Math, {\bf 224}, 2 (2021)").

Abstract: Divergence of a metric space is an interesting quasi-isometry invariant of the space which measures how geodesic rays diverge outside of a ball of radius r, as a function of r. Divergence of a finitely generated group is defined as the divergence of its Cayley graph. For symmetric spaces of non-compact type the divergence is either linear or exponential, and Gromov suggested that the same dichotomy should hold in a much larger class of non-positively curved CAT(0) spaces. 

However, this turned out not to be the case and we now know that the spectrum of possible divergence functions on groups is very rich. In a joint project with Pallavi Dani, Yusra Naqvi, and Anne Thomas, we initiate the study of the divergence in the general Coxeter groups. We introduce a combinatorial invariant called the `hypergraph index', which is computable from the Coxeter graph of the group and use it to characterize when a Coxeter group has linear, quadratic or exponential divergence, and also when its divergence is bounded by a polynomial.

Abstract: These lectures will present a board survey of recent work on new q-series invariants of 3-manifolds labeled by Spin-C structures. While the original motivation for studying these invariants is rooted in topology, they exhibit a number of unexpected properties and connections to other areas of mathematics, e.g. turn out to be characters of logarithmic vertex algebras. The integer coefficients of these q-series invariants can be understood as the answer to a certain enumerative problem, and when q tends to special values these invariants relate to other invariants of 3-manifolds labeled by Spin and Spin-C structures.

Abstract: The configuration space F2(M) of ordered pairs of distinct points in a manifold M, also known as the deleted square of M, is not a homotopy invariant of M: Longoni and Salvatore produced examples of homotopy equivalent lens spaces M and N of dimension three for which F2(M) and F2(N) are not homotopy equivalent. We study the natural question whether two arbitrary 3-dimensional lens spaces M and N must be homeomorphic in order for F2(M) and F2(N) to be homotopy equivalent. Among our tools are the Cheeger–Simons differential characters of deleted squares and the Massey products of their universal covers. This is a joint work with Kyle Evans-Lee.

Abstract: Welded or w-tangles are a higher dimensional analogue of classical tangles, which admit a yet further generalization to welded foams, or w-trivalent graphs, a class of knotted tubes in 4-dimensional space. Welded foams can be presented algebraically as a circuit algebra.

Together with Dancso and Robertson we show that their automorphisms can be realized in Lie theory as the Kashiwara-Vergne group, which plays a key role in the setting of the Baker-Campbell-Hausdorff series. In the process, we use a result of Bar-Natan and Dancso which identifies homomorphic expansions for welded foams, a class of powerful knot invariants, with solutions to the Kashiwara-Vergne equations.

Abstract: The virtual and welded braid groups are generalizations of the classical braid group by adding new kinds of crossings and relations. We will discuss constructions of classical braid group representations that extend to representations of the virtual and welded braid groups.

Abstract: To an isolated complex surface singularity, we can assign a 3-manifold - the link of the singularity.  There has been a lot of recent work on modularity properties of GPPV invariants $\hat(Z)_b$ of such 3-manifolds. I will present a new relation of these modular forms with the spectrum of the corresponding singularity in the case of Brieskorn homology spheres and links of ADE singularities. Joint work with L. Katzarkov and Kyoung-Seog Lee.

Abstract: Stallings folds are a tool that has had a tremendous impact in the study of subgroups of free groups. For instance, they can be deployed to easily determine whether a subgroup is normal, to find a basis, or to determine its index. After a brief introduction, I will talk about joint work with Ivan Levcovitz, in which we develop an analogue of such folds for the setting of right-angled Coxeter groups.

I will describe several applications, including a recent new construction of non-quasiconvex subgroups of hyperbolic groups.

Abstract: Plane curve singularities have provided bridges between algebraic geometry and low dimensional topology. For example, the HOMFLY-PT polynomial of an algebraic link can be expressed in terms of Hilbert schemes of the plane curve singularity thanks to the works of Oblomkov-Shende and Maulik. On the other hand, there have been lots of interests in mirror symmetry of hypersurface singularities these days and plane curve singularities again have provided natural testing grounds for mirror symmetry conjecture. In this talk, we will discuss the relation between Hilbert schemes of plane curve singularities, certain topological data of some algebraic links, and matrix factorizations, stability conditions on them.