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 into… Continue reading Local Triviality

Over the past few decades, Morse theory has undergone many generalizations, into many different fields.  At the moment, I only know of a few, and I understand even fewer. Well, let’s begin at the beginning: Classical Morse theory (CMT) Stratified Morse theory (SMT) Micro-local Morse theory (MMT) The core of these theories is, of course,

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 functions

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 study… Continue reading A Tour de Morse (theory)

“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 useful… Continue reading “The” Milnor Fibration, and Why-You-Should-Care.