May 5, 2022

Mathematics Colloquium

Sunrose Shrestha, Wes

The topology and geometry of random square-tiled surfacesImagine taking a rubber sheet in the shape of a unit square. Gluing the opposite sides of the square will give you a cylinder of height 1. Now glue the two opposite circles on the edges of the cylinder. What do you get? You get a hollow regular donut (with one hole -- called a torus). If this process is done more generally -- using many squares and gluing edges randomly in pairs -- you get generalized donuts, formally called square-tiled surfaces, with (possibly) many donut holes, formally called genus

 

In this talk, we will see how to study the topology and geometry of a random square-tiled surface.  Along the way, we will also see how the study of these surfaces are motivated by the study of polygonal billiards.

 

April 20, 2022

Topology Seminar

Dami Lee, University of Washington, Seattle

Computation of the Kontsevich--Zorich cocycle over the Teichm{\"u}ller flow

In this talk, we will discuss the dynamics on Teichm{\"u}ller space and moduli space of square-tiled surfaces. For square-tiled surfaces, one can explicitly write down the SL(2,\mathbb{R})-orbit on the moduli space. To study the dynamics of Teichm{\"u}ller flow of the $SL(2,\mathbb{R})$-action, we study its derivative, namely the Kontsevich--Zorich cocycle (KZ cocycle). In this talk, we will define what a KZ cocycle is, and by following explicit examples, we will show how one can compute the Kontsevich--Zorich monodromy. This is part of an on-going work with Anthony Sanchez.

 

April 15, 2022

Special Preliminary Exam

Leyla Yardimci, Wes

Unique Equilibrium States for Geodesic Flows in Non-positive Curvature

 The geodesic flow on a negatively curved manifold is a uniformly hyperbolic system: in particular it has a unique measure of maximal entropy and more generally, unique equilibrium states for Holder continuous potentials. When curvature is assumed to be non-positive, the geodesic flow becomes non-uniformly hyperbolic, and much less is known. For a rank-1 manifold of non-positive curvature, Knieper showed the uniqueness of the measure of maximal entropy but his methods do not generalize to equilibrium states for non-zero potentials.

We will present the main result of Keith Burns, Vaughn Climenhaga, Todd Fisher, and Daniel Thompson's paper in which they use a non-uniform version of Bowen's specification property to establish the existence and uniqueness of equilibrium states for some non-zero potential functions. 

 

April 14, 2022

Mathematics Colloquium

Kim Ruane, Tufts University

Cat (0) Groups

I will give an introduction to CAT(0) groups.  This is a class of groups that is meant to generalize the fundamental groups of compact nonpositively curved Riemannian manifolds.  It’s a bit like studying these groups without knowing any differential geometry.  The approach is more metric geometry and group actions than analysis and differential geometry. I will discuss ways to show a group IS CAT(0) and how to show a group is NOT CAT(0). 

 

April 13, 2022

Topology Seminar

Ivan Levcovitz, Tufts University

Counting Lattices in Products of Trees

BMW (Burger-Mozes-Wise) groups are a class of groups that act geometrically on the product of two infinite, regular trees. For large enough trees, these groups admit the same coarse geometry (they are all pairwise quasi-isometric), yet they can be algebraically quite different. In fact, by a celebrated result of Burger-Mozes, such groups can even be virtually simple. In this talk, I will discuss a random model for generating BMW groups and how we use ideas from this model to solve counting problems related to these groups. This is joint work with Nir Lazarovich and Alex Margolis.

 

 April 7, 2022

Topology Seminar

Corey Bregman, University of Southern Maine

Embedding spaces of split links

The homotopy type of the embedding space of a knot in R^3 or S^3 has been studied extensively, culminating in a complete description by Budney.  However, for links in R^3 less is known. Brendle-Hatcher showed that the space of smooth embeddings of an n-component unlink in R^3 is homotopy equivalent to the space of round unlinks, i.e. where each component is a round circle lying in a plane.  In this talk we generalize their work to the case of a split link with n sublinks, and show that the homotopy type of this embedding space depends only on that of the embeddings of a single sublink in a ball, together with a combinatorial object called the space of essential separating systems.  We also compute the fundamental group.  This is joint work with R. Boyd.  

 

March 30, 2022

Topology Seminar

Ian Gossett, Wes

Induced Subgraphs and Treewidth

The treewidth of a graph is a number that tells us, roughly, how far a graph is from being a tree.  Graphs with small treewidth exhibit certain desirable tree-like properties, so structural characterizations of graphs with small or bounded treewidth are often sought after.  A famous result proved by Robertson and Seymour states that grid minors are canonical obstructions to small treewidth-- graphs have large treewidth if and only if they contain a large grid minor.  We present a recent result (Aboulker et al. 2021) that strengthens the grid minor theorem in the case of H-minor-free graphs: If G is an H-minor-free graph with large treewidth, then G must contain either a large wall or the line graph of a large wall as an induced subgraph.  This result is closely related to a number of recent conjectures, which will be discussed if time permits. 

 

March 24, 2022

Mathematics Colloquium

Rylee Lyman, Rutgers University - Newark

Free groups, graphs, automorphisms, and train track maps

Free groups are fundamental objects in the theory of finitely generated groups. To a topologist, a free group is the fundamental group of a graph. As a group theorist, I'm interested in (outer) automorphisms of free groups (and related groups), which I study by thinking of them as symmetries of graphs. Here the right notion of symmetry is homotopy equivalence. Homotopy equivalences are rather flabby, so to really work with them we need to give them some extra structure, for which the best kind is called a (relative) train track map. Relative train track maps have a beautiful connection with the Perron–Frobenius theory of nonnegative integral matrices. The purpose of this talk is to introduce train track maps, which were first studied by Bestvina and Handel in 1992. Time permitting, I'll discuss some of my contributions to the theory.

 

March 23, 2022

Topolgy Seminar

Arianna Zikos, Wes

Minimal Volume Entropy of Certain Classes of 2-Dimensional Groups

The volume entropy of a space such as a Riemannian manifold or a simplicial complex with a piecewise Riemannian metric is the asymptotic growth rate of the volume of balls in its universal cover with respect to radius. Minimal volume entropy is an invariant which is defined using the volume entropy. Taking this one step further, we can define the minimal volume entropy for groups. Babenko and Sabourau found 2 conditions which guarantee either positive or vanishing minimal volume entropy for simplicial complexes. Bregman and Clay used these conditions to characterize which free-by-cyclic and 2-dimensional right-angled Artin groups have positive or vanishing minimal volume entropy. In this talk we will present the main results of Bregman and Clay’s paper, and Babenko and Sabourau’s conditions that guarantee positive or vanishing minimal volume entropy.

 

March 2, 2022

Topology Seminar

Radhika Gupta, Temple University

Orientable graph maps and stretch factors of free group automorphisms

Given a graph map from a graph to itself, we can associate two numbers to it: geometric stretch factor and homological stretch factor. I will define a notion of orientability for graph maps and use it to characterise when the two numbers are equal. The notion of orientability can be upgraded for (fully irreducible) automorphisms of free groups as well. We'll then talk about free-by-cyclic groups and see how the notion of orientability interacts with the cohomology classes in the BNS invariant of the free-by-cyclic group. This is joint work with Spencer Dowdall and Samuel Taylor. 

 

November 11, 2021

Topology Seminar

Lorenzo Ruffoni, Tufts University

Strict hyperbolization and special cubulation.

Abstract: Gromov introduced some “hyperbolization” procedures to turn a given polyhedron into a space of non-positive curvature, in a way that preserves some of the topological features of the original polyhedron. For instance, a manifold is turned into a manifold. Charney and Davis developed a refined “strict hyperbolization” procedure that outputs a space of strictly negative curvature. This has been used to construct examples of manifolds and groups that exhibit various pathologies, despite having negative curvature. We will describe these procedures, and how to construct geometric actions of the resulting groups on CAT(0) cube complexes. As an application, we find that the groups resulting from strict hyperbolization are virtually linear over the integers. This is joint work with J. Lafont.

 

November 5, 2021

Algebra Seminar

Rylan Gajek-Leonard, UMass Amherst

Iwasawa Invariants of Modular Forms with a p=0

Abstract: Gromov introduced some “hyperbolization” procedures to turn a given polyhedron into a space of non-positive curvature, in a way that preserves some of the topological features of the original polyhedron. For instance, a manifold is turned into a manifold. Charney and Davis developed a refined “strict hyperbolization” procedure that outputs a space of strictly negative curvature. This has been used to construct examples of manifolds and groups that exhibit various pathologies, despite having negative curvature. We will describe these procedures, and how to construct geometric actions of the resulting groups on CAT(0) cube complexes. As an application, we find that the groups resulting from strict hyperbolization are virtually linear over the integers. This is joint work with J. Lafont.

 

October 27, 2021

Topology Seminar

James Farre, Yale University

Long curves and random hyperbolic surfaces

Abstract: Fix a pants decomposition of a closed surface of genus g and build hyperbolic surfaces by gluing hyperbolic pairs of pants along their boundary. The set of all hyperbolic metrics with a pants decomposition having a given set of lengths defines a 3g-3 dimensional immersed torus in the 6g-6 dimensional moduli space of hyperbolic metrics, a twist torus.  Mirzakhani conjectured that as the lengths of the pants tend to infinity, that the corresponding twist torus equidistributes in the moduli space in the measure class of Lebesgue.  In joint work-in-progress with Aaron Calderon, we confirm Mirzakhani’s conjecture.  In the talk, we explain how to import tools in Teichmüller dynamics on the moduli space of flat surfaces with cone points to dynamics on the moduli space of hyperbolic surfaces with geodesic laminations.  

 

October 15, 2021

Algebra Seminar

Christopher Rasmussen, Wesleyan

New bounds for heavenly elliptic curves over quadratic number fields

Abstract:  An abelian variety $A$ over a number field $K$ is called heavenly at the prime $\ell$ if the $\ell$-adic Galois representation on $\ell$-power torsion is upper triangular. This is in essence the ``smallest possible’’ Galois image, and so one expects such behavior to be quite rare. In earlier work with Akio Tamagawa (RIMS), we established a finiteness result on heavenly abelian varieties, but obtained explicit bounds on the prime $ell$ in only a small number of cases. In this talk, I report on work in progress with both Tamagawa and Cam McLeman (University of Michigan, Flint) to give explicit bounds in new cases. Heavenly abelian varieties exhibit unusual behavior on the image of Frobenius at primes of good reduction, and this results in unusual congruence conditions; these conditions are sometimes in conflict with the Weil bounds on trace. In certain cases of low dimension and/or low degree, improved bounds may be obtained.

 

October 14, 2021

Mathematics Colloquium

Chris Hruska, University of Wisconsin-Milwaukee

Hyperbolic and relatively hyperbolic groups

Abstract: In geometric group theory, one studies infinite discrete groups by examining their actions on various (geometric, topological, combinatorial...) spaces.  One particularly appealing setting is the groups that act isometrically on the hyperbolic plane, for instance the fundamental group of a higher genus surface.  Gromov has extended many ideas from that classical setting to a much broader class of groups known as hyperbolic groups.  Somewhat surprisingly Gromov-hyperbolicity can often be detected in the apparently different setting of topological dynamical systems: groups that act by homeomorphisms on a compact metrizable space.  For instance the fundamental group of a surface acts naturally on the circle at infinity of the hyperbolic plane.  Throughout the talk, I will discuss some of these connections between notions of hyperbolicity and topological dynamics on compact metric spaces that arise as certain ``boundaries of groups.''

 

October 6, 2021

Topology Seminar

Thomas Ng, Technion

Controlling growth of subgroups in group extensions

Abstract: Growth of finitely generated groups studies the cardinality of balls as the radius grows.   Precise growth rates are generating set dependent. It is, however, sometimes possible to obtain uniform control over growth rates over all generating sets.  I will discuss both algebraic and geometric tools that relate the subgroup and quotient structure of a group to bounding growth rates.  Using these ideas, we will discuss joint work with Robert Kropholler and Rylee Lyman proving a uniform exponential growth gap for subgroups generated by automorphisms of one-ended hyperbolic groups.

 

October 1, 2021

Algebra Seminar

Zonia Menendez, Wesleyan

Sporadic points on $X_0(n)$

Abstract:  We say a closed point $x$ on a curve $C$ is sporadic if $C$ has only finitely many closed points of degree at most $\deg(x)$. On the Level of Modular Curves that give rise to Sporadic j-invariant (by BELOV) gives a criterion for when images of isolated points remain isolated under an arbitrary morphism of curves. In this talk we will discuss partial results for when this criteria is met for non cuspidal, non- CM points of the curve $X_0(n)$.

 

September 28, 2021

Masters Defense

Masoumeh Soleimani, Wesleyan

Expander graphs

Abstract:  Expander graphs are highly connected and sparse graphs that have a lot of applications in networks and computer science. This property over a graph is equal to another property over a matrix related to the graph. In this talk, we explain two different ways to construct an infinite family of expander graphs. One uses Kazhdan property on a family of groups and gives us Cayley graphs which are expander. Another way a family of expanders can be constructed is by induction and graph products. We extend the definition of an expander for hypergraphs.

 

September 24, 2021

Algebra Seminar

Garen Chiloyan, Wesleyan

Isogeny-torsion graphs - a classification and more

Abstract:  An isogeny graph is a nice visualization of the $\mathbb{Q}$-isogeny class of an elliptic curve defined over $\mathbb{Q}$. A theorem of Kenku shows sharp bounds on the number of distinct, cyclic isogenies that a rational elliptic curve can have (in particular, every isogeny graph has at most 8 vertices). In this talk, we classify what torsion subgroups over $\mathbb{Q}$ can occur at each vertex of a given isogeny graph of elliptic curves defined over the rationals - classifying isogeny-torsion graphs. Then we will briefly talk about (in)finite sets of j-invariants corresponding to isogeny-torsion graphs and ongoing work in the direction of classifying images of p-adic Galois representations at each vertex of an isogeny graph. This is joint work with Álvaro Lozano-Robledo.