Codimension k Stratum: A Manifold with Corners

  • Thread starter seydunas
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In summary, the conversation is about understanding the concept of "manifold with corners whose codimension k stratum" from the lecture notes by Hutchings on Morse Homology. The speaker is struggling to understand the sentence and asks for help. The other person mentions finding a definition for codimension k-stratum and also asks for hints on proving the "Compactification of moduli space" from the lecture notes.
  • #1
seydunas
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Hi

What do you understand from this sentence: manifold with corners whose codimension k stratum? i am reading the lecture notes by Hutchings, Morse Homology. When i see the compactification theorem of modili spaces, i read a this sentence but i understood nothing.
 
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  • #2
I've been wondering about that also. Post the answer here if you find out please.
 
  • #3
Hi,

i found a definition of codimension k-stratum. We define the codimension k-stratum of X to be the set X_k of points x in X with a chart f : U---> R^n-k x [0, infinity)^k such that at least one of the last k-coordinates of f(x) is zero.
Note that X_0 is just the interior of X.

I wonder that if you prove the "Compactification of moduli space" in Hutchings Lecture note, can you give me some hints, or direct detail.
 

1. What is a codimension k stratum?

A codimension k stratum is a type of mathematical object that can be described as a manifold with corners. It is a subset of a higher-dimensional space that is defined by a specific number of boundary constraints, or "corners". The dimension of the space is reduced by k, hence the term "codimension".

2. How is a codimension k stratum different from a regular manifold?

A codimension k stratum differs from a regular manifold in that it has a lower dimension due to the presence of boundary constraints. Regular manifolds do not have any corners or boundary constraints, and are instead smooth and continuous in all directions. Additionally, a codimension k stratum can have a more complex structure and topology compared to a regular manifold.

3. What are some examples of codimension k strata?

Some examples of codimension k strata include the boundary of a polytope, the set of singular points on a surface, and the set of critical points in optimization problems. These are all subsets of higher-dimensional spaces that are defined by a specific number of boundary constraints or "corners".

4. How are codimension k strata used in mathematics?

Codimension k strata are used in a variety of mathematical fields, including algebraic geometry, topology, and optimization. They are particularly useful in understanding the structure and behavior of higher-dimensional spaces and can help solve problems in these fields by providing a more concrete and geometric perspective.

5. What are some challenges in studying codimension k strata?

One of the main challenges in studying codimension k strata is their complex and often non-linear structure. This can make it difficult to analyze and understand their properties, as well as to develop algorithms for computing with them. Additionally, the presence of boundary constraints can make it challenging to generalize results from regular manifolds to codimension k strata.

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