Lectures on Contact Topology in Singularity Theory

Thời gian: 15:15 đến 16:45 ngày 24/02/2017, 10:15 đến 11:45 ngày 27/02/2017, 09:00 đến 11:00 ngày 02/03/2017,

Địa điểm: C2-714

Báo cáo viên: Masaharu Ishikawa (Tohoku University)

Tóm tắt:

1. The simplest example: complex Morse singularity

A contact structure is a hyperplane distribution on a manifold which is everywhere non-integrable. The simplest example is the standard contact structure on the 3-sphere. Its Reeb vector field can be regarded as the monodromy vector field of the Milnor fibration of a complex Morse singularity of two variables. The mutual positions of the standard contact structure, the Reeb vector field and the Milnor fibers give us the notion of ``contact structures supported by open book decompositions’’. In this first lecture, I will introduce the standard contact structure on the 3-sphere, describe the Milnor fibers and explain these relationships explicitly.

2. Contact structures supported by Milnor fibrations

In the second lecture, I will give the definition of ``contact structures supported by open book decompositions’’, and introduce an observation for weighted homogeneous singularities by Giroux and a result of Caubel, Nemethi, and Popescu-Pampu in general case. The lecture will starts from basic definitions and theorems as contact manifolds, Darbouxtheorem and Gray theorem.

3. Legendrian graphs and quasipositive surfaces

In the last lecture, I will talk about Legendrian knots and graphs. A Legendrian knot is a simple closed curve in a contact manifold which is tangent to the contact structure. A Legendrian graph is a graph embedded in a contact manifold whose edges are Legendrian. A Legendrian knot/graph in the standard contact structure on the 3-sphere can be represented by a front. A ribbon surface of a Legendrian graph is a compact surface which retracts to the graph and whose framing is given by the contact structure. For example, the Milnor fiber is realized by a ribbon surface canonically. On the other hand, Rudolph introduced a notion of quasipositive surfaces. We can prove that, in the 3-sphere, a contact structure supported by an open book decomposition is standard if and only if the fiber surface is quasipositive. In this lecture, I will introduce their definitions and explain the outline of the proof.

Chương trình:

24/2 (Thứ sáu):

+ 14:00-15:00 Prof. Mutsuo Oka

+ 15:15-16:45 Prof. Masaharu Ishikawa

27/2 (Thứ hai):

+ 9:00-10:00 Prof. Mutsuo Oka

+ 10:15-11:45 Prof. Masaharu Ishikawa

2/3 (Thứ năm):

+ 9:00-11:00 Prof. Masaharu Ishikawa