**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 (Spring 2017):**

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

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

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

**Title:**TBA

**Abstract:**TBA

**Past Talks (Spring 2017) :**

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

**Title: K-Contractible Sets and Group Contractible Sets in 3-Manifolds**Rigidity and the Space of Flags

**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.