The University of Michigan Student Geometry/Topology Seminar
Fall 2005

The Student Geometry/Topology seminar is held 4-5 pm on Friday afternoons. We will meet in room 3088 of East Hall through September, then in room 3096.
If you would like to give a talk, just send an email to vavriche at umich dot edu.

Schedule of Talks

September 16th Organizational meeting
September 23rd Jose Gomez A non-embedding theorem for projective spaces
September 30th Aaron Magid Seifert Surfaces And Other Knot Invariants (abstract, reference for the talk)
October 7th Ben Schmidt Security in Spaces (abstract)
October 14th Eric Zupunski Very basics of 3-manifold topology (abstract)
October 21st Diane Vavrichek Hierarchical accessibility of finitely presented groups (abstract)
October 28th Hao Xing Witten genus, modular form and irreducible representation of Monster (abstract)
November 4th Kevin Wildrick From Diameter to Length in Abstract Metric Spaces (abstract)
November 11th Alan Stapledon An Embedding Theorem for Projective Spaces
November 18th Indira Chatterji (OSU) TBA (This talk is for the regular Topology Seminar, but at our usual time and place.)
November 25th No meeting Thanksgiving break
December 2nd Dave Anderson Intersections on manifolds and pseudomanifolds
December 9th Johanna Mangahas Scott core theorem for 3-manifolds with finitely generated, indecomposable fundamental groups (abstract)
December 16th No meeting Have a good holiday

Click here for the webpage of the "Future Directions in 3-Manifolds" conference, that was held in Ann Arbor, October 15-18. (Email Diane if you know of any other conferences we should link to here.)
Click here to see the student seminar webpage from Spring 2005.
Click here to see the student seminar webpage from Fall 2004.
Abstracts

September 30th, Aaron Magid, Seifert Surfaces And Other Knot Invariants
A Seifert surface for a knot is an orientable surface with boundary equal to the knot. We will prove that every knot has a Seifert surface. From this we can define the genus of a knot and prove that the composition of two nontrivial knots can never be trivial. That is, knots cannot 'cancel' under composition. Finally, we will look at Seifert matrices and see how these relate to other knot invariants such as the Alexander polynomial.
Links to an expansion of what Aaron talked about are available here in PDF, DVI, or Tex format.

October 7th, Ben Schmidt, Security in Spaces
In these uncertain times, security has become an issue. I'll discuss recent work of Gutkin describing why the president should feel safe to visit the torus but not the higher genus surfaces!

October 14th, Eric Zupunski, Very basics of 3-manifold topology
In this talk, I plan to present basic objects in 3-manifold topology (such as irreducible manifolds and Heegaard splittings), in the hopes of helping the listener better understand some of the talks at the upcoming conference in 3-manifold topology.

October 21st, Diane Vavrichek, Hierarchical accessibility of finitely presented groups
A hierarchy of a 3 dimensional manifold is a sequence which gives a decomposition of the manifold by splitting it along nontrivial connected sums, and incompressible surfaces. If the manifold is Haken, this sequence is finite. We will present a topological proof of an analogous result for finitely presented groups, due to Delzant and Potyagailo.

October 28th, Hao Xing, Witten genus, modular form and irreducible representation of Monster
Genera is a ring homomorphism from oriented cobordism ring to complex number. It plays an important rule in Atyiah-Singer Index Theorem and Hizebruch Signature Theorem. In both these two theorems, the genera are special case of an interesting family: Elliptic genus, which connects with elliptic curve. Besides these classical application, Landweber, Stong, Ochanine and Witten studied elliptic genus in the late 80s of last century. From physics point of view, Witten discovered the Witten genus, which can be explained as index of Dirac operator on loop space. Different from classical index as a number, the loop index is a modular form. Witten genus has lots of application in topology and mathematical physics. Based on that, Grameme Segal proposed to study elliptic cohomogy and "elliptic object"; Mike Hopkins invents Topological modular form. Moreover, Witten genus has interesting connection with irreducible representation of Monster. Hizebruch proposed a prize question on looking for a 24 dimension Monster manifold, on which one can construct lots of representation of Monster group. The first part of the prize question has been solved by Mark Mahowald and Mike Hopkins in 2001 using stable homotopy theory and topological modualr form.

November 4th, Kevin Wildrick, From Diameter to Length in Abstract Metric Spaces
Metric spaces with no curves of finite length arise naturally in complex dynamics, hyperbolic geometry, and geometric measure theory. However, under certain volume growth conditions, bounds on diameter can be used to bound length. We will give motivation for such results and prove a theorem of Semmes along these lines. Time permitting, we will consider applications of this theorem to quasisymmetric uniformization problems.

December 9th, Johanna Mangahas,Scott core theorem for 3-manifolds with finitely generated, indecomposable fundamental groups
Every 3-manifold M with finitely generated fundamental group has a "compact core": a compact submanifold N such that inclusion of N in M induces an isomorphism of fundamental groups. We'll see the proof for the special case where pi_1(M) is indecomposable. As a corollary, finitely generated 3-manifold groups are finitely presented. Prof. Scott published all these results in 1973.

Back to UM Math seminars page.