Geometry and Topology Seminar

Abstract: The fundamental group of a closed hyperbolic 3-manifold is known to act geometrically on a CAT(0) cube complex. We ask whether the same is true for the fundamental group of negatively curved 3-pseudomanifolds, i.e., 3-manifolds with isolated singularities. While many 3-pseudomanifolds are cubulated, such as those arising from RACGs and strict hyperbolization, in this talk we give the first examples of closed 3-pseudomanifolds that are locally CAT(-1) but whose fundamental group cannot be cubulated. These examples are obtained from certain compact hyperbolic 3-manifolds with totally geodesic boundary by coning off the boundary components. This is joint work with J. Manning.

Abstract: Let $M$ be an orientable 3-manifold.  A celebrated theorem of Kneser-Milnor states that $M$ admits a unique connected sum decomposition, up to permutations of the prime factors.  We prove a space-level version of this theorem by introducing a poset of decompositions of $M$ along collections of essential 2-spheres (called separating systems) and showing that the geometric realization of this poset is contractible.  As an application, we prove that for any $M$ the classifying space $BDiff(M)$ is homotopy equivalent to a CW complex with finite $k$-skeleton for every $k$. When M is a connected sum of $g$ copies of $S^1 \times S^2$, we also show that every topological $M$-bundle fiberwise extends to a bundle of 4-dimensional handlebodies, generalizing another classical result due to Laudenbach-Poenaru. This is joint work with Rachael Boyd and Jan Steinebrunner. 

Abstract: How can we tell if a group element can be written as a $k$-fold nested commutator? One approach is to find computable invariants of words in groups, that vanish on all $(k-1)$-fold commutators but not on $k$-fold ones. We introduce the theory of letter-braiding invariants - these are "polynomial" functions on words, inspired by the homotopy theory of loop-spaces and Koszul duality, and carrying deep geometric content. They extend the influential Magnus expansion of free groups, which already had countless applications in low dimensional topology, into a functorial invariant defined on arbitrary groups. As a consequence we get new combinatorial formulas for braid and link invariants, and a way to linearize automorphisms of general groups which specializes to the Johnson homomorphism of mapping class groups.

PATCH Seminar (joint with Bryn Mawr, Haverford, Penn, and Swarthmore)

Abstract: Given a relation between sets A and X, one can build two simplicial complexes that each have A and X as the potential vertex set according to the relation. Dowker's Theorem states that the two simplicial complexes are homotopy equivalent. I will present short, new proofs of Dowker's Theorem using modifications of joins and products of simplicial complexes, called relational join and relational product complexes. Using the relational product complex, I will then discuss generalizations of Dowker's Theorem to settings of relations among three (or more) sets. 

In the morning background talk (at 11am, in room KINSC E309), I will provide a brief overview of how topology has been used in applied settings. The talk will provide some landscape of the field, but I will focus mostly on the topics relevant to my research. 

PATCH Seminar (joint with Bryn Mawr Haverford, Penn, and Swarthmore)

Abstract: The Burau representation of the braid group and its cousin, the Gassner representation of the pure braid group, have been studied for more than 80 years. Even so, questions remain about their faithfulness. Separately, these braid representations share a close connection with knot Floer homology through the Fox calculus. We explore the relationship between the Burau representation and knot Floer homology, and show that a large number of matrices related to the Burau representation can be categorified by appropriate Heegaard Floer theories. As an application, we demonstrate a simple correspondence between knot Floer homology and the $gl(1|1)$ quantum tangle invariant. This contributes to a larger program of understanding knot Floer theory through the lens of quantum groups.

In the morning background talk (at 9:30am in room KINSC E309), I will discuss Fox Calculus and Heegaard diagrams. Knot Floer homology, and related Heegaard Floer theories, are powerful tools with a variety of applications in low-dimensional topology. In his paper defining knot Floer homology, Rasmussen remarks that the theory is “essentially just a geometric realization of the Fox calculus.” We’ll explore the meaning of this observation, with the goal of understanding how knot Floer theory fits into the context of classical three-dimensional topology and knot theory. No background knowledge of the theory will be needed.

Abstract: The fractional Dehn twist coefficient (FDTC) is an invariant of a self-map of a surface which is some measure of how the map twists near a boundary component of the surface. It has mostly been studied for compact surfaces; in this setting the invariant is always a fraction. I will discuss work to give a new definition of the invariant which has a natural extension to infinite-type surfaces and show that it has surprising properties in this setting. In particular, the invariant no longer needs to be a fraction - any real number amount of twisting can be achieved! I will also discuss a new set of examples of (tame) big mapping classes called wagon wheel maps which exhibit irrational twisting behavior. This is joint work with Diana Hubbard and Peter Feller.

PATCH Seminar (joint with Bryn Mawr, Haverford, Penn, and Swarthmore)

Abstract: One of the crowning achievements of modern mathematics is the Geometrization Theorem of Thurston and Perelman, which provides a structural description of all 3-dimensional manifolds. In the morning session, we will have discussed an operation called Dehn filling, and the results presented there will rely heavily on Geometrization.

The goal of this talk is to give an overview of Geometrization and its interaction with Dehn fillings. As motivation, we will discuss a recent algorithm that decides whether a given 3-manifold can be obtained from another given manifold via Dehn filling. We will see how Geometrization allows us to reduce this topological question to a classical result about Diophantine quadratic equations.

In the morning background talk (10am, in DRL A6), I will give an introduction to Dehn surgery and discuss the question of (non-) uniqueness of this construction.

PATCH Seminar (join with Bryn Mawr, Haverford, Penn, and Swarthmore)

Abstract: This is a report on work in progress with David Gabai and Daniel Hartman. Our goal is to show that there exists an orientation preserving diffeomorphism of $S^4$ that is not isotopic to the identity. I will outline the proposed proof and then focus on the key underlying combinatorial data used to construct an invariant: systems of Whitney disks for intersecting spheres in connected sums of $S^2 \times S^2$'s. I will explain what these are, how they relate to diffeomorphisms of $S^4$, and what the moves are that relate systems of disks describing isotopic diffeomorphisms, which are the moves that any invariant constructed this way need to be invariant under.

In the morning background talk (11am, in DRL A6), I will give an introduction to pseudoisotopies and parameterized Morse theory (a.k.a. Cerf theory).

Abstract: Randomness is a way to discuss generic or typical behavior in a (class of) group(s).  In this talk, I will discuss random quotients of certain classes of groups.  Quotients of hyperbolic groups (groups that act geometrically on a hyperbolic space) and their generalizations have long been a powerful tool for proving strong algebraic results.  I will focus on random quotients of acylindrical and hierarchically hyperbolic groups (HHGs), two generalizations of hyperbolic groups that include mapping class groups, most CAT(0) cubical groups including right-angled Artin and Coxeter groups, many 3-manifold groups, and various combinations of such groups. In this context, I will explain why a random quotient of an HHG that does not split as a direct product is again an HHG, definitively showing that the class of HHGs is quite broad. I will also describe how the result can also be applied to understand the geometry of random quotients of hyperbolic and relatively hyperbolic groups. This is joint work with Dan Berlyne, Giorgio Mangioni, Thomas Ng, and Alexander Rasmussen.

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