**Seminar: Geometry and Topology**

Welcome to the Geometry and Topology seminar of the Florida State University. We meet on Tuesdays, at 3:35 pm in 201 LOV. If you interested in giving a talk feel free to contact me.

**Upcoming Talks (Fall 2017):**

**November 28, 2017:**Thanittha Kowan (Florida State University)

**Title:**Cannon's Conjecture

**Abstract:**We will introduce Cannon's length-area method and discuss its application to solving Cannon's Conjecture. We will discuss other ideas that Cannon believes may be applicable to solving the conjecture.

**December 5, 2017:**Opal Graham (Florida State University)

**Title:**TBA

**Abstract:**TBA

**Past Talks:**

**November 14, 2017:**Philip L Bowers (Florida State University)

**Title:**A Menger Redux: Embedding Metric Spaces Isometrically in Euclidean Spaces

**Abstract:**In the period 1928-31, Menger proved and refined a three-part characterization of those metric spaces that isometrically embed in the n-dimensional Euclidean space. His proof was very abstract and phrased in the general setting of abstract congruence systems. Computational geometers have used this characterization in various applications and have appreciated the power of abstraction in proving such a useful result, but at the same time have sought for a more friendly geometric proof. In this talk I will describe Menger?s result in clear geometric language and give an indication of the proof of the second and most interesting part of the characterization.

**November 14, 2017:**Philip L Bowers (Florida State University)

**Title:**Expansion Complexes and Combinatorial Hierarchy

**Abstract:**In 1997, Bowers and Stephenson constructed an infinite, reflective, and conformally regular pentagonal tiling of the plane by describing how to build an expansion complex from the pentagonal subdivision rule of Cannon, Floyd, and Perry. Recently we introduced very general expansion complexes defined from arbitrary subdivision rules and explored some of their salient properties. In particular, we articulated simple conditions on subdivision rules that guarantee that the associated expansion complexes generate infinite hierarchical families of complexes in a common local isomorphism class. These combinatorial hierarchies sometimes can be used to determine the conformal type of the associated conformal tilings. Also, the analysis offers a constructive method for generating all periodic hierarchies, which have been important in Cannon, Floyd, and Perry?s program for proving the Cannon Conjecture. In this talk, I will offer a walking tour of finite subdivision rules, expansion complexes, and hierarchical families of planar polygonal complexes. In a later talk, Thanittha Kowan will give a walking tour of the Cannon Conjecture.

**October 17, 2017:**Kathleen Petersen (Florida State University)

**Title:**Weierstrass points on character varieties

**Abstract:**Character varieties are moduli spaces of hyperbolic structures on 3-manifolds, and are complex algebraic sets. I'll introduce Weierstrass points, which are intrinsic flex points on algebraic sets, and discuss some preliminary results about Weierstrass points on character varieties.

**October 24, 2017:**Jay Leach (Florida State University)

**October 3, 2017:**Alice Le Brigant (University of Bordeaux)

**Title:**Computing distances and optimal matchings between manifold-valued curves

**Abstract:**This talk will be concerned with the study of curves lying in a certain manifold, and the construction of a convenient framework to do statistics on such curves in a way that does not depend on their parameterizations. This can be done in two steps. First, by equipping the space of these curves with a Riemannian metric measuring the difference between their velocities. Second, by applying an algorithm that computes the optimal matching between two parameterized curves. We will show simulations on curves lying in manifolds of constant sectional curvature, namely the plane, the hyperbolic plane and the sphere.

**September 26, 2017:**Alexander Reznikov (Florida State University)

**Title:**Distribution of minimal energy points on compact manifolds

**Abstract:**Let's place many particles on a 2-dimensional sphere and on a 2-dimensional torus and assume the particles repel each other according to the potential 1/r^s. What are we going to see in an hour? Turns out that on the sphere we will always see a relatively uniform distribution of the particles. On the torus, however, the situation depends on the parameter s. For example, if s is small, we will not see any particles on a large portion of the torus. And if s is large, we will again see the uniform distribution. We will give an overview of these results and discuss how the distribution of minimal energy points depends on the geometry of our set. We will also look at some open problems in this area. (The talk will be accessible to graduate students who know what is a measure)

**September 19, 2017:**Sergio Fenley (Florida State University)

**Title:**Partially hyperbolic diffeomorphisms

**Abstract:**This will be an introductory/survey talk in this topic. These diffeomorphisms are currently an extremely important area within dynamical systems. They are diffeomorphisms so that the tangent bundle splits into 3 invariant subbundles: stable, unstable, and center. The bundles vary continuously, with constant dimension. A lot of the talk will be concentrated on the case of 3-dimensional manifolds. We will talk about the integrability of the stable and unstable bundles - a very classical result. We will also talk about the possible non unique integrability of the center or center stable (unstable) bundles. This is related to the non unique integrability of differential equations.

**April 25, 2017:**Zhe Su (Florida State University)

**Title:**The Square Root Velocity Framework for Curves in a Homogeneous Space.

**April 18, 2017:**Aamir Rasheed (Florida State University)

**Title:**Essential embeddings and immersions of surfaces in a 3-manifold

**Abstract:**In this talk we will discuss a theorem which gives a sufficient condition for two embedded surfaces in an irreducible 3-manifold to be isotopic. We shall also discuss some generalizations. For example, when are two essential immersions of surfaces homotopic.

**April 11, 2017:**John Bergschneider (Florida State University)

**Title:**K-Contractible Sets and Group Contractible Sets in 3-Manifolds

**Abstract:**A subset W of a closed n-manifold M is K-contractible, where K is a connected complex of dimension less then or equal to n-1 if the inclusion map from W to M factors homotopically though a map to K. A manifold M is of K-category less then or equal to m if it can be covered by m K-contractible open sets. In this talk we discuss a cutting lemma and how it applies to K-contractible sets in 3-manifolds.

**April 4, 2017:**Jakob Møller Andersen (Florida State University)

**Title:**Rigidity and the Space of Flags

**Abstract:**This will be an introduction to an ongoing project that I am working on in collaboration with S. Preston and M. Bauer. To give some background we're going to review some rigidity theory for surfaces in R^3. We'll define bendable and infinitesimally bendable surfaces, give some examples and end with a classical result on rigidity of convex surfaces. The corresponding theory for curves in euclidean space is rather trivial: curves are always bendable. It is therefore natural to study the space of all such bendings, which can be realized as the space of all unit-speed parametrizations. This can be show to be an infinite dimensional manifold, and one can study Riemannian metrics on it. The simplest L^2 type metric can be interpreted as a kinetic energy, and for certain boundary conditions the geodesics correspond to motions of whips (without gravity). Inspired by this theory, we are trying to analyze the space of all bendings of a simple surface, a flat square, which can also be identified as the configuration space of a physical flag. An L^2 metric on this space will still correspond to kinetic energy, and the geodesics can be interpreted as the motion of a flag (without gravity). Our aim is to show that this space is also an infinite dimensional manifold, perhaps with a set of singularities, and show existence of short-time solutions of the geodesic equation. I will present some partial results in this direction.

**March 28, 2017:**Thanittha Kowan (Florida State University)

**Title:**Expansion complexes for subdivision operators

**Abstract:**I will talk about constructions of expansion complexes under mild restrictions on a subdivision operator t. Specially, I will discuss a key theorem of this work saying that if we add a delta-bound condition on t, then there is a t-aggregate of an expansion complex for t.

**March 7, 2017:**Eric Klassen (Florida State University)

**Title:**Comparing Shapes of Curves and Surfaces II

**Abstract:**Shape is an important attribute of objects, their images, graphs of functions, etc. Understanding shape is important in many applied fields. For example, the shape of an internal organ might be an indicator of how healthy it is, while the shape of a handwriting sample might indicate whether a signature is forged or authentic. While humans have an intuitive ability to perceive and compare shapes, it is an interesting challenge to develop mathematical formulations to make these comparisons precise, and to implement them on computers. In this talk, I will give a brief survey of some useful techniques which involve Riemannian geometry, Lie groups, and functional analysis.

**February 28, 2017:**Jay Leach (Florida State University)

**Title:**A-polynomials of some 2-twist knots

**Abstract:**My talk will be on the A-polynomials of some specific 2-bridge knots and on a bound for the coefficients on the edges of the A-polynomial.

**February 21, 2017:**Eric Klassen (Florida State University)

**Title:**Comparing Shapes of Curves and Surfaces

**Abstract:**Shape is an important attribute of objects, their images, graphs of functions, etc. Understanding shape is important in many applied fields. For example, the shape of an internal organ might be an indicator of how healthy it is, while the shape of a handwriting sample might indicate whether a signature is forged or authentic. While humans have an intuitive ability to perceive and compare shapes, it is an interesting challenge to develop mathematical formulations to make these comparisons precise, and to implement them on computers. In this talk, I will give a brief survey of some useful techniques which involve Riemannian geometry, Lie groups, and functional analysis.

**February 14, 2017:**Leona Sparaco (Florida State University)

**Title:**Character Varieties of Some Families of Hyperbolic Link Complements

**Abstract:**Let M be a hyperbolic manifold. The SL2(C) character variety of M is essentially the set of all representations pi_1(M) --> SL2(C) up to trace equivalence. This algebraic set is connected to geometric properties of the manifold M. In this talk we will look at some properties of the character variety of M when M is a link complement with a non-trivial symmetry.

**January 31, 2017:**Amod Agashe (Florida State University)

**Title:**The cohomology groups of certain quotients of products of upper half planes and upper half spaces

**Abstract:**We shall discuss the cohomology groups of compact quotients of products of upper half planes and upper half spaces (which are models for hyperbolic three spaces) under the action of certain types of groups. By Hodge theory, these groups are related to the space of harmonic differential forms on the products of upper half planes and upper half spaces that are invariant under the group action. We shall describe this space by showing that it is a direct sum of two subspaces, the universal and cuspidal subspaces, thus generalizing a theorem of Matsushima and Shimura. We will see, in particular, that the cohomology groups are often trivial. This is a part of joint work with Lydia Eldredge.