The Student Geometry/Topology seminar will be held 3-4 pm on Friday afternoons, in 3096 East Hall.

For information, or to volunteer to speak on any topic you wish, contact Felipe (felipear AT umich DOT edu) or Dave (constand AT umich DOT edu)

Schedule of Talks

January 4th	organizational meeting
January 11th	Johanna Mangahas	Pseudo-Anosov mapping classes, hands-on (abstract)
January 18th	Diane Vavrichek	Algebraic Geometry over Groups (abstract)
January 25th	Aaron Magid	Simultaneous Uniformization: Connections between Hyperbolic 3-Manifolds and Teichmuller Space (abstract)
February 1st	Ben Schmidt	Questions about geodesics in Riemannian manifolds (abstract)
February 8th	Ricardo Portilla	Buildings (abstract) (picture)
February 15th	Nina White	Lie Groupoids and Oribifold Groupoids (abstract) (picture)
February 22nd		no meeting - Winter Break
February 29th		no meeting - Winter Break
March 7th	Jose Gomez	Invariants on orbifolds (abstract)
March 14th	Kyle Hofmann	Meromorphic functions on compact Riemann surfaces (abstract)
March 21st	no meeting	see Bryden Cais speak during this time in the Sumner Myers Colloquium (food will follow!)
March 28th	Dave Constantine	Dynamics and Number Theory (abstract)
April 4th	Qian Yin	Hirschs Immersions of Manifolds (abstract)
April 11th

Abstracts

Johanna Mangahas

Pseudo-Anosov mapping classes, hands-on

We'll walk through Thurston's construction of pseudo-Anosov mapping classes from composition of Dehn twists.

Diane Vavrichek

Algebraic Geometry over Groups

We will introduce some of the basic objects and results in this new area of geometric group theory.

Aaron Magid

Simultaneous Uniformization: Connections between Hyperbolic 3-Manifolds and Teichmuller Space

There will be no bumping or self-bumping in this talk. Our goal will be to state and sketch a proof of Bers' simultaneous uniformization theorem, which parameterizes the convex cocompact hyperbolic structures one can place on the manifold S x I, where S is a closed surface of genus at least 2. We will define the Teichmuller space of a surface, a Beltrami differential, and state the measurable Riemann mapping theorem (Ahlfors-Bers), which is a key tool in the proof of simultaneous uniformization. We will also review what it means to have a hyperbolic structure on the manifold S x I, and how the hyperbolic structure is related to conformal structures on the boundary surfaces.

Ben Schmidt

Questions about geodesics in Riemannian manifolds

Geodesics in a Riemannian manifold M are curves that locally minimize energy between pairs of points. Their behavior in the large, however, depends more upon the topology and geometry of M. I'll discuss some questions that aim to relate global aspects of geodesics to the geometry of M.

Ricardo Portilla

Buildings

We will give an brief introduction to buildings and their connections with different geometries.

Nina White

Lie Groupoids and Oribifold Groupoids

Orbifolds are usually introduced as a generalization of manifolds-- instead of locally looking like $\mathbb{R}^n$, they locally look like quotients of $\mathbb{R}^n$ by the linear action of a finite group. In this talk, I'll take a different approach, following the first three sections of an introductory paper by Ieke Moerdijk, in which the theory of orbifolds is developed via (certain kinds of) groupoids. Using groupoids allows us to work with orbifolds via global objects, instead of via local objects like charts . In the second lecture (by Jose), we'll see why this approach is a bridge to \emph{global} invariants of an orbifold such as its homotopy-type and K-theory.

Jose Gomez

Invariants on Orbifolds

This talk is a continuation of introductory series in Orbifolds. We will discuss some invariants on Orbifolds such as orbifold fundamental group and orbifold K-theory. If time allows, we will have some discussion on twisted orbifold K-theory.

Kyle Hofmann

Meromorphic functions on compact Riemann surfaces

A Riemann surface is a one-dimensional complex manifold. Riemann surfaces arise naturally by analytically continuing functions in the complex plane. For example, a function such as the square root cannot be defined on the complex plane because when you follow it along a loop around the origin, it does not come back to where it started. Instead, the square root function naturally lives on a Riemann surface called the Riemann surface of the square root. On this Riemann surface, the square root becomes a meromorphic function, and this implies that it looks like a covering map except at finitely many points, called branch points, where it looks like z -> z^n for some n. This description represents the Riemann surface as what is called a branched cover of CP^1. Any Riemann surface that admits a meromorphic function looks like a branched cover of CP^1.

It turns out that every Riemann surface is a branched cover of CP^1. We'll explain why this is so for compact Riemann surfaces, and as time permits, we'll discuss why this implies that all Riemann surfaces are algebraic varieties.

Dave Constantine

Dynamics and Number Theory

In this talk I'll discuss a few ways that geometry and dynamics can yield number theoretical results. The focus will be on the proof of the Oppenheim Conjecture using Ratner's Theorems. If time allows I'll talk about recent progress on the Littlewood Conjecture by Einsiedler, Lindenstrauss and Katok.

Qian Yin

Hirschs Immersions of Manifolds

We will see how an open n-dimensional manifold with trivial tangent bundle can be immersed in the Euclidean space with the same dimension.