Local Cohomology and Morse Data

This post is next in my series of posts on Morse theory and its various generalizations.  I last talked about “Classical Morse Theory,” (CMT) which studies the how the topology of a (real) smooth manifold is related to critical points of Morse functions defined on that manifold.  Say the manifold is called , and weContinue reading “Local Cohomology and Morse Data”

Local Triviality -> Locally Cone-Like

So, locally cone-like…what did I mean by that?  Let be a complex analytic subset of a complex manifold, a Whitney stratification of , .  Suppose we’ve given a Riemannian metric, r, and denote by the “ball off radius about .” Choosing local coordinates about in , we might as well assume that we’re dealing with theContinue reading “Local Triviality -> Locally Cone-Like”

Local Triviality

In a perfect world, all “naturally occurring” geometric objects in mathematics and physics would have a nice manifold structure, together with a well-behaved ring of functions.  Unfortunately, this is simply not the case.  But how do we proceed?  Do we part completely from the safety of the power techniques of differential topology, and descend intoContinue reading “Local Triviality”

Stability and Genericity

Before I begin, I want to give credit where credit is due: much of the exposition (especially the proofs) of my last post was paraphrased from Guillemin and Pollack’s Differential Topology [1].  One of my favorites. Okay, moving on. We saw last time that Morse functions are pretty neat, and are abundant; “almost all” smooth functionsContinue reading “Stability and Genericity”

A Tour de Morse (theory)

Morse theory is amazing.  Very geometric, more-or-less very intuitive.  You don’t really explore it in detail until you’ve seen a fair bit of differential topology, but if you look closely, you start getting exposed to its core ideas as early as multivariate Calculus. As is the fashion in modern geometry (specifically, algebraic geometry), we studyContinue reading “A Tour de Morse (theory)”

Goals for the (immediate) future

The next few weeks are going to be very (very) busy, what with qualifying exams, more talks, and extremely hard homeworks, so I’m not sure how much I’ll be posting (but, I won’t bail again, last year (I hope, at least…)). So, modulo the existence of free time, I’ve been really wanting to do aContinue reading “Goals for the (immediate) future”

“The” Milnor Fibration: some proof

As it turns out, the Milnor fibration I talked about last time, i.e., the normalized map is diffeomorphic to a much “nicer” fibration: where $\partial latex D_\eta$ is the boundary of  disk about the origin in of radius .  And, this is a (smooth) locally trivial fibration for sufficiently small .  Intuitively, I think thisContinue reading ““The” Milnor Fibration: some proof”

“The” Milnor Fibration: Classical Case

Where were we?   We had just equated the “innocent” question   “How does ‘sit inside’ at ?”     with the more precise(ish)   “How is the real link of at 0 embedded in ?”   Milnor’s genius idea was to realize the complement, , as the total space of a smooth, locally trivialContinue reading ““The” Milnor Fibration: Classical Case”

“The” Milnor Fibration, and Why-You-Should-Care.

I’m back!  After a long period of laziness, I’m back.  Mainly, because the past week, I’ve been kicking myself in the ass for losing basically all my notes over the past few months, and I have to present at the math department’s new seminar in singularity theory.  Aren’t I smart? The topic?  A really usefulContinue reading ““The” Milnor Fibration, and Why-You-Should-Care.”