Garrett Alston - Riemann Surfaces and Galois Theory
Let me start by saying that this talk should be
accessible to anyone
who undertands the next two sentences.
The function \sqrt{z} cannot be defined on the punctured unit disk
D^2\{0} in the complex plane. However, we can take two copies of the
punctured disk, make branch cuts in both, glue the cuts together,
and
get a Riemann surface on which the square root function is defined.
This construction can be generalized: Given any algebraic function
on a
Riemann surface X, we can construct another Riemann surface Y that
gives a branched cover Y->X. Moreover, the function field K(Y) is a
field extension of K(X), and the Galois group of the field extension
is
isomorphic to the group of deck transformations of the cover.
In the first part of the talk we will talk about Riemann surfaces,
covering spaces, algebraic functions, and Galois theory.
In the second part we will talk about the theory from the viewpoint
of
algebraic geometry. As an application, we will talk about Hurwitz's
theorem, and construct all genus 2 curves as branched covers of P^1.
Links
Mike Lock - Khovanov Homology
Khovanov Homology is a knot and link invariant that
was developed in the
late 90's by Mikhail Khovanov. It can be regarded as a
categorification of the
Jones Polynomial. It arises from a completely combinatorial
construction that
associates a graded vector space to every possible smoothing of the
knot or link
diagram. It's combinatorial nature allows for very structured
relationships
between the Khovanov homology of certain families of knots.
I will begin my talk by defining and proving the invariance of
the Jones
Polynomial (this will help introduce the idea behind the construction
of Khovanov
homology). Then I will construct the cochain groups and the
corresponding
differentials the Khovanov cochain complex. Lastly, I will present a
result
giving a formula for the Khovanov homology of (2,n) Torus Knots and
mention the
proof technique.
Tony Ache - Some basic properties of the Ricci Tensor
Given a Riemannian metric g I will start by defining what its Ricci curvature is (we will denote it by Ric(g)) and then we will see what are some basic difficulties we might encounter when trying to solve the system Ric(g)=r for some unknown metric g and a given Ricci Candidate r. We will also discuss some applications of this PDE viewpoint to the study of 3-Manifolds (Including an old result due to R. Hamilton). This talk is based on Chapter 5 of the book "Einstein Manifolds" by Arthur Besse.
Tony Ache - More awesome propertiess of the Ricci Tensor [Part II]
This is a continuation of last week's talk
Jo Nelson - Blowing up and down is fun
Blowing up a point in C^n basically involves gluing in a copy of (n-1) dimensional projective space in place of the point in a smooth sort of way. I will explain how this is done smoothly and how it causes our blown up space to be a complex line bundle over C^n. The definition of a line bundle will be reiterated at this point in my talk. Then I will go onto talk about some of the interesting properties that we obtain when we view our blown up space as a line bundle over the original space and the geometry behind what is going on. I will move onto the example of blowing up a point in CP^2 and show how blowing up is an interesting (i.e. nontrivial) thing to do to a manifold. Blowing down can be though of as the inverse operation to blowing up and this will be discussed. If time and audience interest permits I will talk about blowing up and down in the symplectic category in a vague handwavy way as to be accessible to non-symplectic geometers. This is a quite interesting case to consider, since there will be interesting (i.e. nontrivial) relationship between the symplectic form on our original manifold to our blown up form (as to make it continue to be symplectic). The conclusion will involve the statements of 4 dimensional applications of blowing up and down that allow one to solve super hard problems in symplectic geometry in a novel way.
Patrick Curran - Cartan's method of moving frames
I want to give an overview of Cartan's method of moving frames. The method will give us a way to tell if 2 submanifolds of a homogeneous space M=G/H are equivalent (up to the action of G on M). It should also give us some differential invariants (e.g curvature, torsion, arc-length for curves; the Gauss and mean curvature for surfaces)
Michael Childers - A brief introduction to Kähler geometry
Kahler manifolds will be defined and their curvature will be found. Some major (mostly resolved) problems in Kahler geometry will be discussed. Examples will give a feel for what a Kahler manifold is. If necessary, complex manifolds and structures will be defined.
Michael Childers - The calabi-yau conjecture: what's the deal?
THis is a continuation of last week's talk.
Rachel Davis - Curves and Algebraic Geometry
Fulton in Algebraic Topology- A First Course writes that it is an important fact from the theory of algebraic curves that, after suitable algebraic transformations, every algebraic curve can be put in a certain form which includes that the curve has at most nodes for singularities. Hartshorne spends a whole section showing that every algebraic curve can be embedded in P^2 with at most nodes for singularities. I will try to show an outline of the theorems that Hartshorne uses leading up to this fact. His treatment includes the use of some important formulas from algebraic geometry such as Riemann-Roch and Hurwitz's Theorem so I will spend some time talking about these ideas.
Sarah Matz - Findingthe Worst pseudo-Anosov Mixer on a surface of genus g
I'll give a brief overview of stirring with rods, and what it means to have pseudo-Anosov mixing (as well as what I mean by "worst"). Then I'll move to a surface of genus g, and describe the algorithm for finding the worst pseudo-Anosov map. The process uses a wide variety of tools including homology, Newton's formula, the Euler-Poincare formula, and Lefschetz numbers.