Topology Students Workshop











TSW 2012

TSW 2014


Click here for the quick-look schedule.


9:00 Registration

9:15 Welcome/Overview/Conferences, Dan Margalit and Panel

10:30 Tea

10:45 Talks I, Tara Brendle and Priyam Patel

12:00 Lunch

2:00 Seminar, Jason Manning

3:00 Tea

3:30 Abstracts, Tarik Aougab

4:30 Beamer / Inkscape, Justin Lanier and Shane Scott


9:00 Etiquette, Tara Brendle and Priyam Patel

10:30 Tea

10:45 Habits, Jason Manning and Rebecca Winarski

12:00 Lunch

2:00 Seminar: The level four braid group, Tara Brendle

The so-called integral Burau representation gives a symplectic representation of the braid group. In this talk we will discuss the resulting congruence subgroups of braid groups, that is, preimages of the principal congruence subgroups of the symplectic group. In particular, we will show that the level 4 congruence braid group is equal to the group generated by squares of Dehn twists. One key tool is a "squared lantern relation" amongst Dehn twists. This is joint work with Dan Margalit.

3:00 Tea

3:30 Job Process, Aaron Abrams, Tara Brendle, and Kelly Delp

4:30 CV / Web / Teaching Statements, Kelly Delp and Aaron Abrams

6:30 Sage, Vince Lucarelli


9:00 Publishing, Tarik Aougab and Jason Manning

10:30 Tea

10:45 Careers, Panel: Aaron Abrams, Ruth Charney, Kelly Delp, Jesse Johnson, Vince Lucarelli, Shaffiq Welji

12:00 Open BBQ

12:15 Lunch n’ Learn: Random functions on finite sets, Vincent Lucarelli

This talk begins with a description of employment opportunities for mathematicians at the National Security Agency and the agency's primary missions. The second half of the talk introduces random functions on finite sets, properties of these functions, and concludes with an application.

1:00 Lunch n’ Learn: Google, Jesse Johnson

2:00 Grants and Research Statements, Ruth Charney, John Etnyre, and Jason Manning

3:00 Photo + Tea

3:30 Colloquium: Finding hyperbolic-like behavior in non-hyperbolic spaces, Ruth Charney

In the early '90s, Gromov introduced a notion of hyperbolicity for geodesic metric spaces. The study of groups of isometries of such spaces has been an underlying theme in much of the work in geometric group theory since that time. Many geodesic metric spaces, while not hyperbolic in the sense of Gromov, nonetheless display some hyperbolic-like behavior. I will discuss a new invariant, the Morse boundary of a space, which captures this behavior. (Joint work with Harold Sultan and Matt Cordes.)

The tea and colloquium will be held in the Clary Theater at the Student Success Center.

4:30 Student Talks I

Contracting Boundaries of CAT(0) Groups, Devin Murray

There are a number of different notions for the boundary of a non-positively curved group. In many cases, they are powerful tools in understanding the algebraic structure of the group as well as the dynamics of group actions. Charney and Sultan introduced a new boundary for CAT(0) spaces called the contracting boundary which has a number of properties that make it particularly well suited for studying groups which act isometrically on CAT(0) spaces. I will introduce the main definitions and present a number of results about the contracting boundary of CAT(0) groups including some results about subgroups, dynamics and a topological classification of hyperbolic CAT(0) groups.

Quasiconvex Hierarchies for Relatively Hyperbolic Non-Positively Curved Cube Complexes, Eduard Einstein

Non-positively curved (NPC) cube complexes are important tools in low dimensional topology and group theory and play a prominent role in Agol's proof of the Virtual Haken Conjecture. Constructing a hierarchy for a NPC cube complex is a powerful method of decomposing its fundamental group essential to the theory of NPC cube complex theory. When a cube complex admits a hierarchy with nice properties, it becomes possible to use the hierarchy structure to make inductive arguments. I will explain what a quasiconvex hierarchy of an NPC cube complex is and briefly discuss some of the applications. We will see an outline of how to construct a quasiconvex hierarchy for a relatively hyperbolic NPC cube complex and some of the hyperbolic and relatively hyperbolic geometric tools used to ensure the hierarchy is indeed quasiconvex.

Conjugacy Geodesics in Coxeter Groups, Aaron Calderon

Coxeter groups are groups generated by reflectional symmetries of mathematical objects. These groups and the spaces on which they act are an important source of examples in group theory and topology and range from tilings of manifolds to homology spheres to CAT(0) cube complexes. In this talk I will discuss results on language theoretic properties of the set of geodesics on the Cayley graphs of Coxeter groups and a generalization (due to Tits) called extended Coxeter groups. These properties are related to many classical problems in geometric group theory, including automaticity, rationality of the group's growth series and the solvability of the word problem.

1-Dimensional Boundaries of Groups with Isolated Flats, Matthew Haulmark

In 2000 Kapovich and Kleiner proved that if G is a one-ended hyperbolic group that does not split over a two-ended subgroup, then the boundary of G is either a Menger curve, a Sierpinski carpet, or a circle. In this talk I will discuss CAT(0) spaces, their boundaries, and what it means for a group to be CAT(0). I will also provide a generalization of the Kapovich and Kleiner theorem to the isolated flats setting.

6:30 Pizzaque


7:30 Chair Yoga, Vita LoFria

9:00 Student Talks II

Relative currents and Loxodromic elements in the relative free factor complex, Radhika Gupta

I will define relative currents with respect to a free factor system. Using north-south dynamics of relative currents we will show which outer automorphisms act on the complex of free factors relative to a free factor system.

Divergence in Coxeter Groups, Ivan Levcovitz

The divergence function measures the rate in which pairs of geodesics stray apart in a given metric space. Cayley graphs of Coxeter groups, in particular, exhibit a rich class of divergence functions. This talk will introduce divergence functions and Coxeter groups and then will discuss new developments and applications. No background on these topics will be assumed.

Distortion for Abelian Subgroups of Out(Fn), Derrick Wigglesworth

In this talk I will introduce the notion of (un)distorted subgroups. Then I will give an outline of the ingredients used to prove that abelian subgroups of Out(Fn) are undistorted.

10:30 Tea

10:45 Student Talks III

The Weinstein Conjecture, Bahar Acu

The Weinstein conjecture states that, on a compact contact manifold, any Reeb vector field carries at least one periodic orbit. The conjecture was proven for all closed 3-dimensional manifolds by Taubes, but it is still open in higher dimensions. In this talk, we will provide the motivation behind this conjecture along with related results in certain cases and iterate the ideas and the machinery used in dimension 3 to higher dimensions to prove the Weinstein conjecture on a special class of Weinstein fillable contact manifolds.

The Milnor Fiber of the Braid Arrangement, Michael Dougherty

In the 1960s, Milnor proved that certain intersections of balls with complex hypersurfaces can be expressed as fiber bundles. In particular, the complement of an arrangement of complex hyperplanes is a fiber bundle over the circle. However, in even the “nicest” such setting, the braid arrangement, the homology of the fiber is unknown in general. To tackle this problem, we can leverage the combinatorial structure of noncrossing partitions to create a geometrically appealing simplicial complex for the Milnor fiber.

The augmentation category map induced by exact Lagrangian cobordisms, Yu Pan

To a Legendrian knot, one can associate an A-infinity category, the augmentation category. An exact Lagrangian cobordism between two Legendrian knots gives a functor of the augmentation categories of the two ends. We study the functor and establish a long exact sequence relating the corresponding Legendrian cohomology categories of the two ends. As applications, we prove that the functor between augmentation categories is injective on objects, and find new obstructions to the existence of exact Lagrangian cobordisms. The main technique is a recent work of Chantraine, Dimitroglou Rizell, Ghiggini and Golovko on Cthulhu homology.

12:30 Lunch

2:00 Seminar: Characterizing stable subgroups, Tarik Aougab

In both group theory and geometry, "negatively curved behavior" is extremely useful when it exists. For instance, many computational problems that are generally intractable, such as the word problem for finitely generated groups, become solvable when we restrict to the class of hyperbolic groups. Unfortunately, one naturally comes across many interesting groups and spaces that are not hyperbolic. It then becomes useful to ask: what subgroups (or subspaces) look negatively curved even though the ambient space might have large flat regions? We will discuss several different notions of negative curvature and how to detect them in a geodesic metric space. We will then use these notions to characterize and find "stable subgroups" of the mapping class group and of the outer automorphism group of the free group. These are hyperbolic subgroups that satisfy strong convexity properties and they turn out to be a natural generalization of the "convex cocompact" Kleinian groups. This is joint work with Matt Durham and Sam Taylor.

3:00 Tea

3:30 Talks II, Tarik Aougab and Dan Margalit

4:30 Student Talks IV

Dehn functions of subgroups of right-angled Artin groups, Ignat Soroko

The question of what is a possible range for the Dehn functions (a.k.a. isoperimetric profile) for certain classes of groups is a natural and interesting one. Due to works of many authors starting with Gromov, we know a lot about the isoperimetric profile for the class of all finitely presented groups. Much less is known for many natural subclasses of groups, such as subgroups of CAT(0) groups or of right-angled Artin groups. In the joint work with Noel Brady, we prove that polynomials of arbitrary degree are realizable as Dehn functions of subgroups of right-angled Artin groups. Our examples are built in several stages.

A basis for the space of order 5 chord diagrams, Allison Stacey

In the field of Vassiliev Knot Invariants, chord diagrams play a key role in that they contain all the necessary information for the invariants. Modulo the four-term relations, they form a graded algebra graded by the number of chords. The order of the bases for each grade is known and bases are not unique. I will show a basis I found for the space and a rigorous proof that it is indeed a basis. I discovered it through the connection to closed Jacobi diagrams which are trivalent graphs with a circle around the outside, so I will also be showing my proposed basis for order 5 closed Jacobi Diagrams. I wrote up my calculations as art pieces so there will also be a lot of art shown in this talk.

Heegard Splittings and Pseudo-Anosov Maps (Paper by Hossein Namazi and Juan Souto), Sunny Xiao

Let M and M' be two 3-dimensional handlebodies whose boundaries are identified with a surface $S$ of genus at least two. The handlebodies can be glued along their boundaries using a homeomorphism f of S, and a natural question that arises is when (if ever) does the 3-manifold obtained via this gluing procedure admit a hyperbolic structure. Using the theory of the deformation space of hyperbolic structures on 3-manifolds, Namazi and Souto answer this question by giving sufficient conditions on f. In this talk, I will present their construction and, time permitting, we will also learn about some of the topological consequences of their results that could not be obtained assuming the mere existence of such a negatively curved metric.

Arithmetic progressions in the primitive length spectrum, Nicholas Miller

There have been a host of prime geodesic theorems over the past several decades displaying a surprising analogy between the behavior of primitive, closed geodesics on hyperbolic manifolds and the behavior of the prime numbers in the integers. For instance, just as the prime number theorem dictates the asymptotic growth of the number of primes less than n, there is an analogous asymptotic growth for primitive, closed geodesics of length less than n. In this talk, I will give a brief review of the relevant definitions and go on to give the history of this analogy. I will then discuss some recent work extending this relationship to give the geodesic analogue of the Green--Tao theorem on arithmetic progressions in the prime numbers.


9:00 Student Talks V

Hidden Symmetries and Commensurability of 2-bridge link complements, William Worden

The canonical triangulations and symmetry groups of 2-bridge link complements are well understood and relatively easy to describe. We exploit this fact to show that non-arithmetic 2-bridge link complements have no hidden symmetries (i.e., symmetries of a finite cover that do not descend to symmetries of the link complement itself), and are pairwise incommensurable. I will discuss commensurability, hidden symmetries, and canonical (Delaunay) triangulations of cusped hyperbolic manifolds. With this background material in hand, I will then give a rough sketch of our proof that 2-bridge links have no hidden symmetries. This work is joint with Christian Millichap.

A geometric construction of thin subgroups in SU(2,1), Joseph Wells

In a linear algebraic group, a lattice is a discrete subgroup with finite covolume; these objects have been studied quite extensively over the past century for their interesting algebraic and geometric properties. In recent years, in particular due to the work of Sarnak, there has been renewed interest in similar type of subgroup called a thin group, which is an infinite-index subgroup of a lattice having same Zariski-closure as the lattice. In this talk, I'll discuss recent work toward constructing thin subgroups of SU(2,1) by way of complex hyperbolic geometry.

Homological Stability and Instability of Configuration Spaces of manifolds., Megan Maguire

In it's weakest form, we say that a family of topological spaces is homologically stable if for fixed i the ith homology groups of X_n and X_{n+1} are isomorphic for n sufficiently large. Notions of homological stability have been investigated for a wide range of topological families, including Hurwitz spaces, moduli of curves, and configuration spaces. Arnol'd first proved integral homological stability for the unordered configuration spaces of R^2. This was extended to open (connected, orientable, finite type) manifolds by McDuff and Segal (independently), and recently Church (via the method of representation stability), followed by Randal-Williams (via a method more akin to Segal), proved rational homological stability for all (connected, orientable, finite type) manifolds. Using the tools of Totaro, we compute the Betti numbers, both stable and unstable, of the unordered configuration spaces of some example spaces, such as a genus 1 Riemann surface and CP^3, and prove a vanishing theorem about the unstable homology a la Church, Farb, and Putman (joint with Melanie Wood).

10:30 Tea

10:45 Student Talks VI

On links homotopic, but not isotopic to the unlink, Bakul Sathaye

The Whitehead link is a well-known example of a 2-component link which is link homotopic, but not isotopic, to the unlink. However, there aren't any known examples of such links with more than 2 components. In this talk, we describe a family of such links with more than 2 components. Further, all their sublinks are isotopic to the unlink. The same methods also help us construct links, which further have all the Milnor invariants to be zero. At the end we briefly present an application of this result in constructing a family of new examples of 4-manifolds which have singular metric with non-positive curvature, but do not support a smooth Riemannian structure with non-positive sectional curvature.

Electricfying Teichmuller Spaces, Ashley Weber

Take a space X to be any geodesice metric space. Farb defines a related space called the electric space, X_elt, which collapses certain subsets to a sphere of radius 1. Electricfying a space can have some interesting consequences. For example, the electric Teichmuller space is delta-hyperbolic and quasi-isometric to the curve complex; this fact was used (and proved) in the original proof by Massar and Minsky that the curve complex is delta-hyperbolic. I will define the electric Teichmuller space and prove that this space is quasi-isometric to the curve complex.

Cyclic Branched Covers of the Sphere and the Liftable Mapping Class Group, Tyrone Ghaswala

Given a (possibly branched) covering space between surfaces, we can ask the following question: Which elements of the mapping class group of the base space have representatives that lift to homeomorphisms of the total space? These mapping classes form a subgroup called the liftable mapping class group. Birman and Hilden first considered this question in the case of the 2-sheeted cover of the sphere by a surface. The covering they considered arises by taking the quotient of a closed surface by the action of a fixed hyperelliptic involution. In this case, they answered the question above with a resounding "everything lifts"! In this talk we will investigate the liftable mapping class group for other families of cyclic branched covers over the sphere.

12:05 Reflections and Farewell, Dan Margalit