The Study Seminar -- Winter 2007
Thursdays 3:10-5:00 -- East Hall 2866

PAST SEMINARS: FALL 06

To schedule a talk, or for more information, please contact Pekka Pankka (pankka 'at' umich.edu).

Thursday, April 26.

Speaker: Saara Lehto, University of Helsinki.
Title: Doubling for General Sets (after Buckley, Hanson and MacManus)

Abstract: When can a set E of positive Lebesgue measure on the real line be sent
to a set of zero Lebesgue measure by a quasisymmetric map?
Equivalently, when can we find a doubling measure on the real line that
gives E zero measure?

Buckley, Hanson and MacManus study the question for Cantor sets
constructed by removing intervals of relative size c_n on the level n
step in the construction. They give a complete answer to the question
in terms of summability criteria for the sequence (c_n).

Thursday, April 19.

Speaker: Jasun Gong, UofM.
Title: Structure of Null Sets in the Plane and Applications (after Alberti, Csörnyei, and Preiss) (cont.)

Abstract: In this talk we will discuss selected results from a recent paper of G. Alberti, M. Csörnyei, and D. Preiss, as well as various questions posed by the authors. In particular we will define the notion of a "weak tangent field" to a (Lebesgue) null set in the plane and present a few applications, including some results in geometric measure theory, as well as the "Rank-1 Property" for functions of bounded variation. (cont.)

Thursday, April 12.

Speaker: Jasun Gong, UofM.
Title: Structure of Null Sets in the Plane and Applications (after Alberti, Csörnyei, and Preiss)

Abstract: In this talk we will discuss selected results from a recent paper of G. Alberti, M. Csörnyei, and D. Preiss, as well as various questions posed by the authors. In particular we will define the notion of a "weak tangent field" to a (Lebesgue) null set in the plane and present a few applications, including some results in geometric measure theory, as well as the "Rank-1 Property" for functions of bounded variation.

Thursday, April 5.

Speaker: Mario Bonk, UofM.
Title: Manifolds that are homeomorphic, but not diffeomorphic VIII.

Abstract: We continue the sequence of lectures devoted to John Milnor's work on 7-spheres.

Thursday, March 29.

Speaker: Daniel Meyer, UofM.
Title: Pasting together Julia sets; a worked out example of mating (cont.)

Abstract: After the paper by Milnor by the same title. Given two monic polynomials z^n+... one can "mate" them together along the circle at infinity, where one polynomial acts on each hemisphere. The resulting map may be "equivalent" to a rational map. The equator then gets mapped to a a curve that is dense in the Julia set. I will describe the constuction, as well as some fascinating aspects about the invariant curve. All spheres in this talk will have dimension at most 2, and thus will be standard.

Thursday, March 22.

Speaker: Daniel Meyer, UofM.
Title: Pasting together Julia sets; a worked out example of mating

Abstract: After the paper by Milnor by the same title. Given two monic polynomials z^n+... one can "mate" them together along the circle at infinity, where one polynomial acts on each hemisphere. The resulting map may be "equivalent" to a rational map. The equator then gets mapped to a a curve that is dense in the Julia set. I will describe the constuction, as well as some fascinating aspects about the invariant curve. All spheres in this talk will have dimension at most 2, and thus will be standard.

Thursday, March 15.

Speaker: Mario Bonk, UofM.
Title: Manifolds that are homeomorphic, but not diffeomorphic VII.

Abstract: We continue the sequence of lectures devoted to John Milnor's work on 7-spheres.

Thursday, March 8.

Speaker: Pekka Pankka, UofM.
Title: Manifolds that are homeomorphic, but not diffeomorphic VI.

Abstract: We continue the sequence of lectures devoted to John Milnor's work on 7-spheres.

February 22. and March 1.

No seminar.

Thursday, February 15.

Speaker: Pekka Pankka, UofM.
Title: Manifolds that are homeomorphic, but not diffeomorphic V.

Abstract: In Study Seminar's sequence of lectures devoted to understanding John Milnor's work on 7-spheres, I will discuss the construction and elementary properties of characteristic classes of vector bundles in terms of curvature forms. (continued)

Thursday, February 8.

Speaker: Pekka Pankka, UofM.
Title: Manifolds that are homeomorphic, but not diffeomorphic IV.

Abstract: In Study Seminar's sequence of lectures devoted to understanding John Milnor's work on 7-spheres, I will discuss the construction and elementary properties of characteristic classes of vector bundles in terms of curvature forms.

Thursday, February 1 .

Speaker: Mario Bonk, UofM.
Title: Manifolds that are homeomorphic, but not diffeomorphic III.

Abstract: I'll review some basic notions of differential geometry such as vector bundles and connections on vector bundles as a preparation for a discussion of characteristic classes (Chern classes and Pontryagin classes). (continued)

Thursday, January 25.

Speaker: Mario Bonk, UofM.
Title: Manifolds that are homeomorphic, but not diffeomorphic II.

Abstract: I'll review some basic notions of differential geometry such as vector bundles and connections on vector bundles as a preparation for a discussion of characteristic classes (Chern classes and Pontryagin classes).

Thursday, January 18.

Speaker: Mario Bonk, UofM.
Title: Manifolds that are homeomorphic, but not diffeomorphic.

Abstract: In a landmark paper published in 1956, John Milnor proved that there are smooth manifolds homeomorphic to the 7-sphere that are not diffeomorphic. This work would later earn Milnor the Fields Medal. In the Study Seminar a sequence of lectures will be devoted to understanding Milnor's work. An introduction into relevant topics such as vector bundles, connections, characteristic classes, and Morse theory will be given. The talks will be accessible to everyone with a basic knowledge of the theory of manifolds.

Thursday, January 11.

Speaker: Kai Rajala, University of Jyväskylä.
Title: The sharp quantitative isoperimetric inequality (after Fusco, Maggi and Pratelli).

Abstract: We study how the so-called Fraenkel asymmetry of a Borel set in a euclidean space can be controlled by its isoperimetric deficit in a sharp manner. Fraenkel asymmetry measures how large portion of the set can be covered by a ball with same measure, while the isoperimetric deficit measures the ballness of the set in the sense of the isoperimetric inequality.

Thursday, January 4.

Speaker: Kevin Whyte, University of Illinois at Chicago.
Title: Large scale geometry and topology

Abstract: The study of metric spaces on the large scale has emerged from the overlap of geometry, topology, and group theory. I will discuss the foundations and motivations of the subject, and some of the results that are milestones of the subject. One of these major threads in the understanding of classical geometries from the coarse perspective; for the three dimensional geometries this program was recently finished. At another end of the subject are the examples of infinite groups produced by combinatorial group theorists. Surprisingly, some recent techniques seem to be relevant to some of both extremes.