Update

Despite my hope to the contrary, it would appear that the math I’ve done while here so far as not parlayed into me blogging super-frequently. For what it’s worth: Life is busy. Just in case you were wondering. ^_^

Lately, I’ve been working from home more than I’ve been going to Princeton/IAS. My goal is to change that soon and I actually had a wonderful day at IAS today. I’d like to go tomorrow but I have a work meeting at the least convenient time one can imagine; there’s also no topology seminar at the University tomorrow, so I suppose I’ll be staying in and working again. No harm no foul, I suppose.

So what have I been working on? Well:

  • Universal Circles for Depth-One Foliations of 3-Manifolds. The gist here is: If you have a taut (e.g.) foliation on a 3-manifold, a theorem of Candel says we can find a metric on all the leaves so that they’re hyperbolic. Moreover, by tautness, you can lift to a foliation of the universal cover which is then a foliation whose leaves are hyperbolic discs. A ridiculously deep idea of Thurston was to look at the infinite circle boundaries of these disk leaves and maybe…glue them together? Canonically? And see if that gives insight about things?

    You probably already know how this ends: It’s doable (because he’s Thurston) and it does provide deep insight about the downstairs manifold (see, e.g., the articles by Calegari & Dunfield and/or Fenley, or Calegari’s book…)

    Now, let’s say we do this for certain classes of kind-of-understood-but-still-unknown-enough-to-be-interesting foliations like those of finite depth. Can we get cool manifold stuff by doing this process? I dunno, but maybe.

  • Homologies. My ATE was about Gabai’s work on foliating sutured manifolds, so studying sutured manifolds is something I’m still interested in. One way of doing that nowadays is with this colossal, ridiculously-powerful tool called Sutured Floer homology. So…you know…homology…but when talking with other grad students about the millions of homologies out there and about how nobody really understands what motivates discovers of them, I realized that there was a lot I needed to know before focusing on one homology foreverever. So I’m working on learning stuff about homologies.
  • Geometric Group Theory. Ian Agol is at IAS this year as the distinguished visitor and a lot of his work is on relationships between GGT and 3-manifolds. If you listen to any talk relating those two things, you realize there’s this whole dictionary of words and acronyms like QCERF and LERF and RAAG and Virtually SpecialResidually Finite, etc. etc. I think in order to someday bridge the gap towards doing work like those guys do, I need to know what all these words mean, and what better time to figure that out than right now?! So yea…I’m doing that some, too.
  • Dirac Operators, Spin manifolds,…. At some point soon, I’m going to start working on hypercomplex geometry again, and part of that will be the study of Dirac operators. So far, there are lots of perspectives on those, so we’re going to try to first establish the explicit connections between them and then maybe…do some stuff? I dunno. I also have stuff on Clifford analysis / geometry I want to look at, as well as some more things involving generalized geometries. Lots here.
  • Topological Quantum Computing. This is a pipe dream until I’m able to feed my family and progress on my dissertation. It’s on the radar, though.

Okay, so this was an update! I’ve also been bookmarking some interesting proofs I’ve run across so I’ll know where to look when I decide to expand things here, and…yea.

Oh! And my professional webpage finally exited alpha and went into beta! http://www.math.fsu.edu/~cstover.

And now, Morrrr…se homology. Morse homology. That’s what I’m looking at as a segue into Floer. Another late night ftw!

Later.

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Category Theory: Moving Up, Out

I remember my very first encounter with Category Theory.

I was in my fourth semester (a spring semester) as a master’s student: I had passed my two mandatory abstract algebra classes my first two semesters there and had passed my comprehensive exams during the Fall (my third semester). As was custom, then, I spent my third and fourth semesters taking random “advanced topics” courses aimed at potential doctoral students, and one of the sequences I took was the algebra sequence.

My first semester doctoral-level (or 7000-level as was colloquial there) algebra class was over the classification of finite simple groups and was by far the most difficult class I’d ever taken at the time. Apparently, being a student who doesn’t remotely have the sufficient background and being in a class run by a professor who has unimaginably-greater background – who teaches as if the audience consists of peers – makes for a difficult time. I squeaked out an A.

In the second semester of 7000-algebra, however, things were far less directed. Long story short, it was a potpurri of material, some from algebraic topology, some from homological algebra, and some – about 1/3 of the course, I’d say – from category theory. That was my very first exposure to an area I didn’t otherwise know existed and I remember thinking, This is the most abstract thing that’s ever been devised, and also, There’s no way this will ever be far-reaching outside the realm of mathematics.

I’ve since realized that the first assertion isn’t really true – unsurprisingly since my exposure to other areas has increased drastically since leaving there – but apparently, the second one isn’t either. To be more precise, I stumbled upon this article online which describes a number of non-math areas that have been benefiting – and will continue to benefit – from the use of category theoretic ideas.

It’s really quite amazing to see, but in and of itself is unsurprising given the fact that category theory itself was devised to provide unity among the wide variety of subdisciplines of mathematics. As a pure mathematician, I always tried to find a balance between being interested in too broad a range of topics and being too narrow with my scope; the spread of category theory invites us all to analyze that aspect of ourselves. To borrow a quote from David Spivak’s exposition (available on the arXiv),:

It is often useful to focus ones study by viewing an individual thing, or a group of things, as though it exists in isolation. However, the ability to rigorously change our point of view, seeing our object of study in a different context, often yields unexpected insights. Moreover this ability to change perspective is indispensable for effectively communicating with and learning from others. It is the relationships between things, rather than the things in and by themselves, that are responsible for generating the rich variety of phenomena we observe in the physical, informational, and mathematical worlds.

Here’s to you, category theory!

The Intersection of the Collection of Dreams and the Collection of Maths, or Dynkin π-λ Systems

I’ve posted before about how easily my sleep can be dominated by math stuff after a hard day or thirty of being cooped up in an office, grinding away at theorems and postulates and proofs with hardly a break in the mix.

Over the summer, the same thing happens after only a medium-hard day or three.

I woke up twice this morning, about two hours apart, and both times I was thinking about a random piece of mathematics not related to anything I’ve been actively studying recently. When I finally awoke a third time – this time, for good – I of course couldn’t remember it at all.

Then, finally, I sat in silence and forced my synapses to make connections they didn’t want to make and eventually, after a solid twenty minutes of mental strain, it all came flooding back in.

This is an exposition about so-called Dynkin (π-λ) Systems and the corresponding Dynkin π-λ Theorem. Feel free to stick around.

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Online reading seminar for Zhang’s “bounded gaps between primes”

Dr. Terence Tao has arranged for an online reading seminar to go through Dr. Yitang Zhang’s recent proof of the “Bounded Gaps Conjecture.” To say that this is a wonderful opportunity to pick something valuable up regarding a field that’s very hot right now in the research community would be the ultimate understatement.

What's new

In a recent paper, Yitang Zhang has proven the following theorem:

Theorem 1 (Bounded gaps between primes) There exists a natural number $latex {H}&fg=000000$ such that there are infinitely many pairs of distinct primes $latex {p,q}&fg=000000$ with $latex {|p-q| \leq H}&fg=000000$.

Zhang obtained the explicit value of $latex {70,000,000}&fg=000000$ for $latex {H}&fg=000000$. A polymath project has been proposed to lower this value and also to improve the understanding of Zhang’s results; as of this time of writing, the current “world record” is $latex {H = 4,802,222}&fg=000000$ (and the link given should stay updated with the most recent progress.

Zhang’s argument naturally divides into three steps, which we describe in reverse order. The last step, which is the most elementary, is to deduce the above theorem from the following weak version of the Dickson-Hardy-Littlewood (DHL) conjecture for some $latex {k_0}&fg=000000$:

Theorem 2 ($latex {DHL[k_0,2]}&fg=000000$) Let $latex {{\mathcal H}}&fg=000000$ be an…

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Algebraic Geometry Observation I: Algebraic Varieties

In order to define any algebraic geometry structures (a sheaf, for example), one has to first understand what an algebraic variety is. And thus:

Observation I. It’s damn-near impossible to find someone who gives a straightforward definition of an algebraic variety straight off the bat.

Instead, most authors tend to define an affine algebraic variety – first as the common zero set of a collection \{F_i\}_{i\in I} of complex polynomials in \mathbb{C}^n and later as a “variety that can be embedded in affine space as a Zariski-closed set” (Smith et. al., An Invitation to Algebraic Variety). Then, half a book later or more (it’s on page 144 of the aforementioned book), it’s said that an (abstract) algebraic variety is a topological space with an open cover consisting of sets homeomorphic to affine algebraic varieties which are glued together by so-called transition functions that are morphisms in the category of affine algebraic varieties.

This of course requires knowledge of category theory, the Zariski topology, etc. etc.

As of now, this 30+ minutes of searching has gotten me through about 3/4 of a page in Chris Elliott’s online manuscript concerning D-modules.

le sigh