GSCAGT Abstracts

Plenary Talks

Julie Bergner (UC Riverside)
Title: Spaces, Categories, and Homotopical Structures
Abstract: Simplicial sets were developed as a combinatorial alternative to topological spaces. Yet, they also have a close relationship to categories. We'll look at the interplay between these topological and algebraic perspectives and show how they can be brought together to define homotopical algebraic structures.

Jessica Purcell (BYU)
Title: Low-Dimensional Interactions Between Geometry and Topology
Abstract: In dimension three, geometry and topology of a manifold are known to be intricately related. For example, within the last ten years, the Geometrization conjecture has been proved, which states that every (topological) 3-manifold breaks into geometric pieces. However, it seems to be very difficult to determine geometric information from a topological or combinatorial description of a 3-manifold. It also seems difficult to relate classical topological properties to classical geometric ones. In other words, even though we know we are working in the same field now, it is still difficult for geometers and topologists to use each others' results! Many open problems remain. In this talk, we will discuss what it means to give a manifold a geometric structure, and we will look at several examples of geometric structures. We will also discuss examples of open problems concerning these structures.

Chelsea Walton (MIT)
Title: Quantum symmetry from an algebraic point-of-view
Abstract: Symmetry has long been a crucial notion in mathematics and physics. Groups arose to axiomatize the notion of symmetry; namely, groups are comprised of a set of invertible transformations of an object of interest. But it is common practice to replace the object of study \(X\) with an algebra \(A\) of functions on \(X\). Symmetries of \(X\) are then realized as the set of group actions on \(A\). (Here, the group is a set of automorphisms of \(A\).)
So let's kick this up a notch- let's study symmetries of quantum objects. Indeed such objects are impossible to visualize, yet there are natural noncommutative algebras \(B\) that arise as 'quantum function algebras' on these objects. We can certainly still consider group actions on \(B\) in this setting. But the aim of this talk is to convince you that studying actions of "Hopf algebras" (or of "quantum groups") on \(B\) is more appropriate.
Classification results and lots of examples will be included.

Dani Wise (McGill)
Title: Counting cycles in graphs: A rank-1 version of the Hanna Neumann Conjecture
Abstract: A "\(W\)-cycle" in a labelled digraph \(\Gamma\) is a closed path whose label is the word \(W\). I will describe a simple result about counting the number of \(W\)-cycles in a deterministically labelled connected digraph. Namely: the number of \(W\)-cycles in \(\Gamma\) is bounded by \(|E(\Gamma)| - |V(\Gamma)|+1\). I will outline the proof which uses left orderable groups.

Student Talks

Alyson Bittner (SUNY Buffalo)
Title: Constructing Infinite Loopspaces from Categories
Abstract: An abelian monoid is a familiar algebraic object whose definition can be stated in a way which lends itself to a generalization. I will introduce the concept of an abelian monoid object in an arbitrary category and discuss how such objects in the category of topological spaces give rise to infinite loop spaces. We will then explore Segal's construction of an infinite loop space from a category.

Kathryn Bryant (Bryn Mawr)
Title: Obstructing Sliceness in 5-Stranded Pretzel Knots
Abstract: Within any infinite family of knots, a compelling question to ask is which members of the family hold certain knot theoretic properties and which do not. One such property of interest is that of being {\it smoothly slice}, which means that the knot bounds a smoothly embedded disk in the 4-ball. This talk will focus on the problem of determining slice knots within the family of 5-stranded pretzel knots. A brief overview of the history of this and related problems will be given, citing recent results of Lisca, Greene-Jabuka, Lecuona, and Herald-Kirk-Livingston. One of the main aspects of this problem that makes it intriguing are the seemingly disparate tools used in solving it so far; one of the sliceness obstructions comes from Donaldson's Theorem in gauge theory, another comes from the \(d\)-invariant in Heegaard-Floer theory, and still another comes from twisted Alexander polynomials. Each obstruction has a very different flavor. The general idea behind each of these tools as slice obstructions will be explained, but highly technical details will be omitted and only basic knowledge of knots, covering spaces, and surgery diagrams will be assumed. This is joint work with Paul Melvin.

Edward Chien (Rutgers)
Title: Rectangular Tiling of Closed Surfaces
Abstract: In a paper from the 1940s, Brooks, Smith, Stone, and Tutte proved a classical theorem which produces a tiling of a rectangle by squares associated to any connected planar graph. We generalize this result to high genus surfaces. Given a closed surface \(S\), a non-zero first homology class and a graph \(G\) on \(S\) so that each component of \(S \backslash G\) is simply connected, we show that exists a singular flat metric and a square tiling on S associated to the graph and the homology class. The proof uses analogues of Kirchoff's circuit laws and discrete harmonic forms.

Michelle Chu (UT Austin)
Title: Small generators of arithmetic Fuchsian groups
Abstract: Arithmetic Fuchsian groups form a subclass of Fuchsian groups with a special connection to number theory and the theory of automorphic forms. These groups are necessarily of finite covolume and thus finitely generated. In this talk, we consider the problem of finding a set of small generators. For any cocompact Fuchsian group, we determine a set of generators which are a bounded distance from the identity in \(SL(2,\mathbb{R})\). This bound depends only on the degree of the invariant trace field, the covolume, and the spectral gap. This work is joint with Han Li.

Thomas Crawford (Boston College)
Title: Just How Exceptional Can Manifolds Be?
Abstract: Given a compact irreducible hyperbolic 3-manifold with torus boundary, all but a finite number of dehn fillings yield a hyperbolic 3-manifold. Cameron Gordon conjectured that no such manifold could have more than 10 of these exceptional (non-hyperbolic) fillings, and further that the figure-8 knot complement is the only one with 10. Marc Lackenby and Robert Meyerhoff proved the former; we will discuss an attempt to prove the later.

Vinh Dang (BYU)
Title: The geometry of compression bodies
Abstract: I will discuss some aspects of the space of marked hyperbolic structures on compression bodies and examples where an understanding of geometric structures on compression bodies is used to construct tunnel number one manifolds with interesting cusp properties.

Matt Evans (Binghamton)
Title: Finite Planar Groups
Abstract: A group is called planar if its subgroup lattice is a planar graph. In 2004, Starr and Turner classified all planar Abelian groups. In 2006, Bohanon and Reid classified all finite planar groups. In this talk I will state the general classification theorem, but focus my attention on the finite Abelian case by sketching the necessary arguments.

Mustafa Hajij (LSU)
Title: Quantum Invariants and \(q\)-Series
Abstract: Quantum invariants connect Hopf algebras, Lie Algebras and 3-manifolds. In this talk we give a discuss the relationship between a particular quantum invariant called the colored Jones polynomial and Ramanujan type identities.

Robert Haraway (Boston College)
Title: Is It \(T^2 \times I\)? and Other Questions about 3-Manifolds
Abstract: I'll explain part of an algorithm to determine the hyperbolicity of compact 3-manifolds with nonempty boundary, namely the test of whether a 3-manifold is a torus cross an interval.

Jesse Hicks (Utah State)
Title: Equivalence Of Lie Algebra-Subalgebra Pairs And Why It Matters Geometrically
Abstract: The equivalence problem in general relativity is the problem of determining if two solutions to the Einstein equations are related by a coordinate transformation. While there is a theoretic solution to the problem, the implementation of this solution in a computer algebra system is very problematic. At the heart of this is an interplay between algebra and geometry, and in the case of homogeneous space-times, is the need to know whether or not certain Lie algebra-subalgebra pairs are related by isomorphism. We will discuss this need and the approaches to its solution.

Ying Hu (LSU)
Title: Left-orderability and cyclic branched covers
Abstract: A group is called left-orderable if one can put a total order \(\lt\) on the set of group elements so that inequalities are preserved by group multiplication on the left. The left-orderability of 3-manifold groups is closely related to the concepts of L-spaces and taut foliations, as conjectured by Boyer-Gordon-Watson. In this talk, we will discuss the left-orderability of fundamental groups of cyclic branched covers of the three sphere.

Diana Hubbard (Boston College)
Title: Sutured Khovanov Homology and the Burau Representation Abstract: I will discuss a connection between the Euler characteristic of the Sutured Khovanov Homology (\(SKH\)) of braids and the classical Burau Representation. This yields a straightforward method for distinguishing, in many cases, the \(SKH\) of two braids. As a corollary I will explain why \(SKH\) is not necessarily invariant under braid axis preserving mutation.

Kyle Istvan (LSU)
Title: \(N\)-Equivalences of \(6\)-\(J\) Symbols
Abstract: The tail of the Colored Jones Polynomial of links, introduced by Armond and independently by Garoufalidis, is a formal power series associated to certain classes of links that arises as a set of stabilized coefficients of the sequence of Colored Jones Polynomials. This talk will begin with a discussion of the high computational complexity of the CJP. Then, using special skein elements, I will show how the tail can be more efficiently calculated by simplifying the computationally intensive \(6\)-\(J\) symbols. I will be using diagrammatic representations of the algebras involved, and will not assume any background in knot theory.

Kelli Karcher (Virginia Tech)
Title: The Space of Biorders on Some Solvable Groups
Abstract: A group is said to be biorderable if it has a total order invariant under left and right multiplication. These orders can be given a topology and is called the space of biorders on this group. There has been intensive study on the space of left-orders recently, but less on the space of biorders. We will focus on solvable groups to show under certain conditions the space of biorders is either finite or homeomorphic to the Cantor set.

Seungwon Kim (CUNY Graduate Center)
Title: Link diagrams with Turaev genus one
Abstract: The Turaev genus of a link is a topological measure of how far a given link is from being alternating. We show that every prime Turaev genus one link is a cycle of alternating 2-tangles. The proof provides a new way to obtain a meridian longitude basis for the Turaev surface from the link diagram.

Caitlin Leverson (Duke)
Title: Augmentations and rulings of Legendrian knots
Abstract: A Legendrian knot in \(\mathbb{R}^3\) with the standard contact structure is a knot for which \(dz-ydx=0\). Given a Legendrian knot, one can associate the Chekanov- Eliashberg differential graded algebra (DGA) over \(\mathbb{Z}/2\). Fuchs and Sabloff showed there is a correspondence between augmentations to \(\mathbb{Z}/2\) of the DGA and rulings of the knot diagram. Etnyre, Ng, and Sabloff showed that one can define a lift of the Chekanov-Eliashberg DGA over \(\mathbb{Z}/2\) to a DGA over \(\mathbb{Z}[t,t^{-1}]\). This talk will give an extension of the relationship between rulings and augmentations to \(\mathbb{Z}/2\) for the DGA over \(\mathbb{Z}/2\), to a relationship between rulings and augmentations to a field of the DGA over \(\mathbb{Z}[t,t^{-1}]\). No knowledge of the Chekanov-Eliashberg DGA will be assumed.

Jesse Levitt (LSU)
Title: Connected Hopf algebras as deformations of universal enveloping algebras
Abstract: The classification problem for connected Hopf algebras of finite GK dimension has attracted much attention in recent years. Recently, a classification of all such Hopf algebras of GK dimension 4 was devised in part using Coassociative Lie Algebras to construct deformed universal enveloping algebras. Motivated by this, we invoke a Drinfeld construction to produce non-cocommutative Hopf algebras on completions of universal enveloping algebras. In this joint work with Milen Yakimov, we describe necessary and sufficient conditions for deformed universal enveloping algebras to be connected Hopf Algebras and show how previous classifications follow from the Drinfeld construction.

Ash Lightfoot (Indiana)
Title: Link maps in the 4-sphere
Abstract: To a link map \(f:S^2_+\cup S^2_-\to S^4\), a map of two 2-spheres in the 4-sphere with disjoint images, Kirk associated a pair of integer polynomials \(\sigma(f)\) that obstructs homotoping \(f\) through link maps to an embedding. The obstruction measures whether (up to link homotopy) the self-intersections of each component \(f|S^2_{\pm}:S^2_\pm\to S^4-f(S^2_\mp)\) occur as pairs of double points equipped with Whitney discs --- the devices for performing the ``Whitney trick'' to eliminate self-intersections. A priori (since \(2+2=4\)) these discs may have interior intersections with \(f(S^2_\pm)\). In this talk I will introduce these concepts and sketch a proof that if a link map \(f\) has \(\sigma(f)=(0,0)\), then after a link homotopy the double points of \(f|S^2_{\pm}\) can be equipped with Whitney discs whose interiors are, in fact, disjoint from \(f(S^2_\pm)\). This is a necessary step towards addressing the open question of whether \(\sigma(f)\( is a complete obstruction.

Veny Liu (U Alabama)
Title: Free inverse semigroupoids and their inverse subsemigroups
Abstract: Semigroupoids are generalizations of semigroups and of small categories. However, the quotient of a semigroupoid in general is not a semigroupoid and the homomorphisms of semigroupoids can also behave badly. We define such congruence and homomorphism to form the first isomorphism theorem for semigroupoid homomorphisms for this special type. Hence, we can investigate inverse semigroupoid which is a semigroupoid in which each element has a unique inverse. Using Stalling's folding and Munn tree, this will lead us to free inverse semigroupoids and their inverse subsemigroup.

Jonathan Paprocki (Georgia Tech)
Title: Overview of Quantum Topology
Abstract: We will try to answer the question "What is quantum topology?" In brief, it is a collection of methods and ideas stemming from quantum mechanics and quantum field theory that have been used to study topological problems. Despite being somehow about topology, many methods are highly algebraic or combinatorial in nature, making it a fascinating crossroads of many areas of mathematics. In particular, quantum topology has been central to revolutionizing knot theory over the last 25 years.

Joe Quinn (CUNY Graduate Center)
Title: Arithmetic Kleinian groups and Macfarlane spaces
Abstract: I will briefly review the role of arithmetic Kleinian groups in studying hyperbolic 3-manifolds, using the upper half space model. I will then review some methods from relativity theory that can be applied to hyperbolic 3-space, using the hyperboloid model. Then I will show some original work connecting these two approaches in the context of quaternion algebras, using a generalization of a construction by Macfarlane dating back to 1900. Lastly I will show some applications, deriving topological information from the number theory invited by arithmetic invariants.

Ahmad Rafiqi (Cornell)
Title: Constructing pseudo-Anosov maps with given dilatations.
Abstract: We give sufficient conditions for a Perron number, given as the leading eigenvalue of an aperiodic matrix, to be a pseudo-Anosov dilatation of a compact surface. We give an explicit construction of the surface and the map when the sufficient condition is met.

Clinton Reece (Utah State)
Title: Modified Reshetikhin-Turaev and Turaev-Viro Invariants
Abstract: For certain weak conditions on a category, Turaev and Virelizier and independently Balsam and Kirillov, recently extended the well-known Turaev-Viro (TV) invariants to prove that they are equal to the Reshetikhin-Turaev (RT) invariants defined on the center of the category. The proof is through the construction of extended TQFTs. Several years back, Geer, Patureau, and Turaev renormalized quantum dimensions to produce modified T-V invariants that are non-zero in cases when the standard invariants are zero. Constantino, Geer, and Patureau have recently constructed a modified R-T invariants which they have shown distinguish Lens spaces. In this talk I will discuss Turaev and Virelizier's construction of extended TQFTs and if time allows explain how I am generalizing this to the setting of the modified T-V invariants. It is a long term goal of Geer and mine to show this leads to a version of the T-V-B-K theorem in the modified setting.

Adan Saltz (Boston College)
Title: Branched Diagrams and the Ozsváth-Szabó Spectral Sequence
Abstract: Khovanov homology is a combinatorially defined link invariant with roots in representation theory, while Heegaard Floer homology is an invariant of three-manifolds inspired by symplectic topology. Despite these differences in character and origin, the two theories are connected by a spectral sequence first constructed by Ozsváth-Szabó. I will present a fresh set of Heegaard multidiagrams which clarify the correspondence with Khovanov homology. These diagrams may have applications to Szabó geometric spectral sequence, contact topology and transverse knots, and the construction of a spectral sequence over \(\mathbb{Z}\).

Matthew Sartwell (SUNY Buffalo)
Title: A P-local Delooping Machine
Abstract: The question of whether a given space is an n-fold loop space comes down to whether or not it can be endowed with certain algebraic structures. It is interesting to ask whether other mapping spaces can be characterized by such algebraic data. In this talk we discuss an algebraic structure which detects mapping spaces out of a P-local sphere, where P is a set of primes.

Dan Scofield (NC State)
Title: Torsion in chromatic graph cohomology
Abstract: Khovanov homology is an invariant of knots and links that categorifies the Jones polynomial. Helme-Guizon and Rong (2005) introduced a Khovanov-type, bigraded homology for graphs that categorifies the chromatic polynomial. We will present results about torsion in homology for graphs and use it to describe torsion in Khovanov homology. The occurrence of torsion is of particular interest, as it may lead towards a combinatorial proof that Khovanov homology detects the unknot.

Michael Vaiana (SUNY Buffalo)
Title: Modular Reductions for Symmetric Group Representation
Abstract: In characteristic zero, the irreducible representations of the symmetric group are in fact defined over the integers. This allows us to reduce coefficients mod p to obtain a modular representation. We will give a survey of this reduction process including some recent results on the simplicity of these reductions.

Nicholas Vlamis (Boston College)
Title: Basmajian's identity for real projective surfaces
Abstract: Basmajian's identity for a bordered hyperbolic surfaces gives a formula for calculating the length of the boundary in terms of a particular set of length data, the orthospectrum. In this talk, we will introduce real projective structures on surfaces and see how this identity generalizes to the setting of convex real projective surfaces.

Shida Wang (Indiana)
Title: Minimal Cobordisms Between Algebraic Knots
Abstract: This talk will start from an introduction to Heegaard-Floer theory and then give some interesting applications to the cobordism between knots. The knot Floer complex has shown its power on detecting 4-dimensional properties of knots. The Upsilon invariant, defined by Ozsvath, Stipsicz and Szabo recently, is a piecewise linear function on \([0,2]\) derived from the knot Floer complex. It gives lower bounds for the slice genus of knots. We will briefly introduce the properties of the Upsilon invariant and show how it obstructs the genus of cobordisms between algebraic knots. In particular, we will show the \((p,q)\)-torus knot cannot appear in the minimal unknotting sequence of the \((r,s)\)-torus knot if \(p \gt r\).

Biji Wong (Brandeis)
Title: Orbifold Turaev Torsion
Abstract: The Seiberg-Witten invariant of an admissible 3-manifold has a combinatorial interpretation as Turaev torsion. This remarkable result is due to Turaev in the late 90s. Since then, Seiberg-Witten invariants of admissible 3-orbifolds have been introduced and studied. In this talk, I will generalize Turaev torsion to admissible 3-orbifolds, and show it satisfies properties similar to the ones on the Seiberg-Witten side.

Andrew Yarmola (Boston College)
Title: An Improved Bound for Sullivan's Convex Hull Theorem
Abstract: A quasi-Fuchsian 3-manifold (a class of hyperbolic manifolds homeomorphic to a surface cross R) carries a natural complex structure on the boundary at infinity and a hyperbolic structure on the boundary of the convex core. A remarkable theorem of Sullivan states that these two structures are a universally bounded distance apart in Teichmuller space. Using pleated planes and extending techniques of Epstein, Marden, and Markovic, we provide a new upper bound of 7.1695 for this universal constant. This is joint work with Martin Bridgeman and Richard D. Canary.

Andrew Zemke (Cornell)
Title: The simple loop conjecture for torus bundles
Abstract: The simple loop conjecture for 3-manifolds states that every 2-sided immersion of a closed surface into a 3-manifold is either injective on fundamental groups or admits a compression. This can be viewed as a generalization of the Loop Theorem to immersed surfaces. We will give a brief history of this problem and outline a solution when the target 3-manifold is a torus bundle over a circle.