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The University of Michigan Student Geometry/Topology Seminar |
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The Student Geometry/Topology seminar will be held 3-4 pm on Friday afternoons,
in 3096 East Hall.
Announcement: the last seminar this semester will be Dec. 8. The Dec. 15 talk on Mostow Rigidity will be postponed until next semester for final exams and winter break.
We thank everyone who attended and helped at the Michigan and Michigan State conference this past weekend. It was quite successful and we hope to do it again sometime. Pictures from the conference can be found on the conference website:
MICHIGAN - MICHIGAN STATE CONFERENCE
We are now taking volunteers for speakers next semester! Contact Jose or Aaron
| September 8th | Organizational meeting | |
| September 15th | Marc Krawitz | An introduction to vector bundles and K-theory (abstract) |
| September 22nd | Kevin F. Tucker | Using Cech Cohomology to Classify Vector Bundles (abstract) |
| September 29th | Jose Manuel Gomez-Guerra | Twisted K-theory (abstract) |
October 6th |
Stephanie Jakus | Morse Theory I (abstract) |
| October 13th | Cagatay Kutluhan | Lefschetz Fibrations (abstract) |
| October 20th | Liz Vivas | Fixed point in the interior of a basin of attraction. Holomorphic vs. Diffeomorphisms (abstract) |
| October 27th | Hao Xing | Morse Theory II (abstract) |
November 3rd |
Marshall Williams | Dimension Theory in General Metric Spaces (abstract) |
| November 10th | Aaron Magid | Teichmuller space and surface automorphisms (abstract) |
| November 17th | Johanna Mangahas | Hyperbolicity of the complex of curves: easy torus case (abstract) |
| November 24th | Thanksgiving Break | |
| December 1st | Michigan/Michigan State student seminar December 2-3 | |
| December 8th | Dave Constantine | The Use of the Boundary at Infinity in Rigidity Theory (abstract) |
| December 15th | not meeting this week | |
| December 22nd | Winter Vacation |
Click here
to see the student seminar webpage from Winter 2006.
Click here
to see the student seminar webpage from Fall 2005.
Click here
to see the student seminar webpage from Winter 2005.
Click here
to see the student seminar webpage from Fall 2004.
An introduction to vector bundles and K-theory
Abstract: This talk is the first of a three series talk that will treat topics of K-theory and Twsited K-theory. In this first talk we will set up the basic facts about vector bundles and define the K-theory for a compact space. The K-theory of some particular spaces will be calculated.
Using Cech cohomology to classify vector bundles
This will be the second in a series of talks aimed towards giving an introduction to K-theory. Here, we will revisit the construction of a vector bundle by studying the transition functions given by a trivializing cover of the base space. These give rise to a certain Cech cohomology class, and the goal of this talk is to elucidate this correspondence. Sheaves of groups and Cech cohomology will be introduced and discussed at length. Although we will build upon the definition of a vector bundle given last week, these definitions will be (mostly) self-contained.
Twisted K-theory
This is the last of a three talk series about vector bundles and K-theory. I will explain how in general twistings appear in cohomology theories. Then I will give a description for twisted K-theory using bundles of projective spaces. For doing so I will briefly talk about a classification of such bundles using Cech cohomology. Then I'll use projective bundles to get a nice description for twisted K-theory for twistings corresponding to a third cohomology class.
Morse Theory I
This will be the first in a series of three talks on Morse Theory. This is an introductory talk and promises to be accessible to all graduate students.
Lefschetz Fibrations
Inspired by the so called Lefschetz pencil (a special kind of linear system) construction in algebraic geometry considered by Solomon Lefschetz, Lefschetz fibrations on 4-manifolds became a very important class of structures for our understanding of low dimensional topology in dimension four (especially symplectic 4-manifolds). I will try to give an overview of the idea of Lefschetz fibrations on 4-manifolds through examples and state some of the most important results appeared in the last ten years.
Fixed point in the interior of a basin of attraction. Holomorphic vs. Diffeomorphisms
Let f be an automorphism of C^k with a fixed point at the origin. We call basin of attraction to all the points whose orbit converges to the origin. The behavior of an attracting set varies greatly depending on the eigenvalues of df(0). We will state some results about the topology of this sets and we will prove that in the case of an automorphism with df(0) = Id, the fixed point can't be in the interior of a basin for f holomorphic, but it can be in the interior for f a diffeomorphism.
Morse Theory II
Abstract
Dimension Theory in General Metric Spaces
There are many notions of what the dimension of a seperable metric space should be. We'll talk about some of the most common definitions (topological dimension, Hausdorff dimension, and time permitting, the more recent Assouad Dimension). We'll discuss some elementary facts about the various definitions, and then show that while in general, topological dimension is smaller than Hausdorff dimension (which is in turn smaller than the Assouad dimension), a suitable metric can be found on any space so that the definitions coincide. Along the way, we'll show how to embed any (topologically) n dimensional space into R^(2n+1).
Teichmuller space and a classification of surface automorphisms
We will define Teichmuller space and prove that it is the universal cover of the moduli space of a surface using hyperbolic geometry. Also, we will discuss how the mapping class group acts on the Teichmuller space of a surface and how this can be used to classify elements of the mapping class group. This classification parallels the classification of isometries of the hyperbolic plane, and also corresponds to the Nielsen-Thurston classification of surface automorphisms.
Hyperbolicity of the complex of curves: easy torus case
Curve configurations on a surface S are encoded by a simplicial complex known as its complex of curves K(S). Ten years ago Masur and Minsky proved that the 1-skeleton K_1(S) is Gromov hyperbolic, which we'll define. For the torus, K_1(T) turns out to be the classical Farey graph, which we'll see is 3/2-hyperbolic. We'll mention important applications of the complex of curves to hyperbolic 3-manifolds (resolving the Ending Lamination Conjecture), surface mapping class groups, and Teichmuller geometry.
The Use of the Boundary at Infinity in Rigidity Theory
A 'boundary at infinity' can be defined for non-positively curved spaces. Points on the boundary are equivalence classes of geodesics, where geodesics that stay within a bounded distance of each other as they head out to infinity are equivalent. Surprisingly (to me anyway), although this boundary appears to forget lots of information about the space, it is incredibly useful for proving all sorts of theorems. In addition, many structures we all know and love from different parts of math are used when working with the boundary at infinity - topologies, metrics, measures, quasi-conformal structures, buildings... I'll discuss some of these at a very inroductory level and take a brief look at how the boundary comes into play when proving some big theorems in rigidity theory. I'll also hopefully whet our collective appetite for Felipe's talk on Mostow rigidity the following week.