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 a post that surveys the Milnor fibration in “all” of its various forms, from its conception to modern usage.  I’ve already talked about the “classical” case, i.e., Milnor’s original fibration on the sphere (minus the real link), as well as the fibration “in the ball”, from last time.  In addition to this, I want to talk about

  • The “general” Milnor fibration, associated to a complex analytic function on an arbitrary complex analytic space (for this, I’ll need to talk about the notion of a stratification, and various equisingularity conditions we can impose thereon).
  • A “cohomological” version of the Milnor fibration, which basically says that, for a bounded, constructible complex of sheaves, \mathcal{F}^\bullet, and a complex analytic function f: (X,0) \to (\mathbb{C},0), there is a fundamental system of compact neighborhoods of 0, \{K_n \}_n, such that all the higher direct image sheaves R^i f_*(\mathcal{F}^\bullet|_{K_n}) are locally constant (along with a “stability” result).  For this, I’ll probably need to mention some basic results/theory concerning the derived category, constructible sets, etc.
  • The “vanishing and nearby cycles” of Deligne, associated to a complex analytic function f: (X,0) \to (\mathbb{C},0).  A purely category-theoretic formulation, these produce certain complexes of sheaves whose stalk cohomology at a point is (naturally) isomorphic to the cohomology of the Milnor fiber of f at that point.  Really fucking cool, but I’ll need to do quite a bit of explaining to get to these beasts.

In addition to the last point, I want to (later on, after the above stuff) talk about

  • “Basic” Morse theory
  • Stratified Morse theory
  • Microlocal Morse theory

but that might be quite a lofty goal, at least for the time being.  Who knows, I might wake up smarter some day.

Until next time.


Published by brianhepler

I'm a third-year math postdoc at the University of Wisconsin-Madison, where I work as a member of the geometry and topology research group. Generally speaking, I think math is pretty neat; and, if you give me the chance, I'll talk your ear off. Especially the more abstract stuff. It's really hard to communicate that love with the general population, but I'm going to do my best to show you a world of pure imagination.

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