Week 39: DeRham (Homology and) Cohmology with Coefficients in the Canonical Line Bundle

So, I studied differential topology from Kosinski's book and cellular homology from Geoghegan's book and, for a long time, I thought handlebody homology was the be-all and end-all for homology theories for smooth manifolds. Then that changed, basically with the Asian professor from video series embedded in my Week 01 posts in the 'blog, and I began to think that DeRham cohomology and "DeRham homology" were the be-all and the end-all homology and cohomology theories, at least for orientable, closed, connected, (smooth) Riemannian manifolds.

So, I've been toying around with a construction for a while, based on an idea from a paper by my dissertation advisor, Craig Guilbault. In a paper he co-authors with Fred Tinsley, they discuss homology (and cohomology, implicitly) with "twisted integer" coefficients.

This gave me the idea to try DeRham cohomology (and homology, in a meaningful sense of the phrase, using Poincare duals of differential forms) with "twisted real" coefficients, that is H^*_DR(M; ℝ [ℤ₂]) with "group field" coefficients; one essentially works in the double-cover of the manifold -- which is 2 disjoint copies of the manifold, if the initial manifold is orientable, or the orientation double-cover, if the initial manifold is non-orientable -- and does all of one's calculations there, then one "divides by 2".

I talked with Craig, and in Hatcher's book, there is a notion of homology with "twisted integer" coefficients that is different from both the ordinary, untwisted homology and from the twisted "group field" coefficients homology of the double cover; evidently, all 3 homologies fit into a long exact sequence. Davis and Kirk's book gives a nice treatment of the topic of homology with twisted coefficients in general. This coincides with the notion of De Rham cohomology with coefficients in the canonical line bundle over a manifold (which gives the cohomology with twisted coefficients from Davis and Kirk in the case M is non-orientable and the usual cohomology of M in the case M is orientable - just what the doctor ordered!) from Bott and Tu's book.

So, I discovered a way to rescue an orientation form in the case that M is non-orientable with De Rham cohomology. Now, I just need a way to recover torsion from De Rham cohomology, and I'll be in business with an all-purpose version of De Rham cohomology.

Week 27: A New Job

I got hired on May 25, 2016 at the Milwaukee School of Engineering as a full-time adjunct faculty member ("lecturer" is how they technically classify me, I guess; I'm hoping to move up to "adjunct assistant professor" quickly, if my teaching goes well and then hopefully to "assistant professor" or "associate professor" if I can get a research program off the ground). I'm going to be teaching 2 sections of Calc I, 2 sections of a calculus-based probability and statistics course, and, as a recent addition, 1 section of a Calc IV course (they run their Calculus courses differently there; this course covers multiple integral in the plane and 3-space - but not line or surface integrals - and infinite series).

So, I got the Calc I book the day I signed the contract and the statistics book a few days later, and I've been LaTeX'ing up my lecture notes for the Calc I and statistics course since then, so I haven't had any time to devote to my "old" research (I have one paper on which I just gave a presentation at WGT earlier this month that's almost finished and another one on 1-sided h-cobordisms with non-split total group of the side with the more complicated fundamental group that I haven't really started working on) or my "new" research (the topic of this blog).

I have a week's worth of lecture notes done in each of the Calc I and statistics courses, and hope to get a jump start on the Calc IV course this weekend. With those in the books, I'm hoping I can settle into a schedule of working on lecture notes, typing up old papers and working on old research, and working on new research, all the while tutoring at my part-time tutoring gig. But, until then, I'm going to be a little light on research.

But, it's great news for me that I got this full-time gig. The job market is really dismal for math Ph.D.'s right now, as nearly as I can figure. The job postings I was looking at on Vitae right before I landed the MSOE gig were requiring out-of-state hires to pay for their own travel expenses to apply for the job; who can afford that? I mean, even if you're desperate and willing to do that, you can really only afford doing that 2 or 3 times before you're out of money. I can only guess that they're going only to get local candidates to apply after a few months or a year of that as an industry-wide policy; if they were able to find viable candidates with only a local job search, they really shouldn't have been advertising in Vitae with which to begin. I just don't see that as an equilibrium industry-wide policy, but time will tell.

Anyways, I'm very excited about my new job, and I'm pouring all my energies into getting ready for that in the Fall. It's kinda slow going LaTeX'ing up my lecture notes, so I may need to punt on that and start doing chalk-talk lecture notes in a few weeks if I don't think I will be able to finish the semester in Beamer slides in time for the end of the semester. They gave me a spiffy laptop on which they said I can install Ubuntu Linux if I want, and they use this spiffy cloud storage system Box that plays well with Linux (Marquette only used OneDrive for their cloud storage, so I had to "roll my own" cloud storage, especially after my hard drive on my desktop-server crashed in the Fall of last year), so I'm just really loving this new job - and, it hasn't even started yet!

Week 19: The Symplectic Form and Symplectic Manifolds

So, in my undergraduate biology text, there was a chapter entitled, "Biology, Having Found Its Holy Grail, Drinks Deeply From It" (at least, that's how I remember the chapter title), and that sums up my last week. I found a book at the library, Foundations of Mechanics by Abraham and Marsden (the same Abraham as in Transversal Mappings and Flows by Abraham and Robbin), and it has a wonderful exposition of symplectic manifolds as it pertains to Lagrangian and Hamiltonian mechanics. It turns out, associated to the cotangent bundle, T^*(Q), of a configuration space (or smooth manifold, as we differential topologists like to call them) Q, there is a canonical 1-form -θ, and its exterior derivative, ω = -dθ, which is the canonical symplectic form associated to T^*(Q). All the hullabaloo associated to symplectic and contact geometry stems from this fact.

So, it took me all of Monday (05/02/2016) after class to wrap my brain around the definition of the canonical 1-form and symplectic form: evidently, if (q, u^*, v, w) denotes the general point of TT^*(Q), then θ = u^*(v) (using the canonical flip on TT(M), around which took me 5 months to wrap my brain - nice to see some of that theory pay off), and ω is then -dθ. A technically precise, but difficult to understand, definition of the canonical 1-form is that it is (u^*)o(Dτ^*_Q)(q,u^*), where τ^*_Q is the cotangent bundle projection map. To say the same thing slightly differently, as it is tradition to use q to represent an arbitrary point of Q ("generalized coordinates") and α to represent an arbitrary (co)tangent vector at q and using a c-patch φ with φ(q, α) = (q₁(q, α), ..., q₁(q, α), p₁(q, α), ...., p_n(q, α)), we have θ₀ = p₁dq₁ + ... + p_ndq_n so -dθ₀ =-(dp₁∧dq₁ + ... + dp_n∧dq_n) = dq₁∧dp₁ + ... + dq_n∧dp_n = ω - neat.

So, I also discovered this great video series on YouTube on symplectic and contact geometry:



The first video has a nice overview of symplectic geometry, most of which I already knew, but the second video takes a dog-leg into things more "theory of symplectic manifold"-oriented and less "building up to geometric control theory"-oriented, so I'm not sure how much more of the series I'm going to watch.

So, that's it for this week. I hope to start on Geometric Control of Mechanical Systems by Bullo and Lewis soon, and then onto trying to try to publish something in the field.

Week 18: Symplectic and Contact Geometry

So, I finally hit paydirt and discovered symplectic and contact geometry a few weeks ago already, I guess. Symplectic manifolds are essentially time-independent ("autonomous") second-order ordinary differential equations on manifolds, and contact manifolds are essentially time-dependent ("non-autonomous") second-order ordinary differential equations on manifolds; both arise from Lagrangian mechanics in physics.

This, together with geometric control theory, is the holy grail of applications of differential manifolds for which I have been searching my whole life. The subject goes through Lagrangian mechanics, but from a differential topologist's point of view. I thought global analysis was the starting point for geometric control theory, but global analysis is essentially just first-order ordinary differential equations on manifolds, and second-order ODEs on manifolds (SODEs) are H***-and-gone from first-order ones; there's no Poincare-Hopf theorem, for a start.

I found a wonderful, free, paper/book online, Finsler-Lagrange Geometry by Bucataru and Miron, and I have been working through it the last for weeks, at this point, I guess. Like Transversal Mappings and Flows by Abraham and Robbin, it has some wonderful material on differential manifolds and their tangent and double-tangent bundles in the first chapter, but some of it is a bit of a math-out as I get to the final 3/4's of that chapter; I will definitely need to work through some examples of the transition functions from the change-of-coordinate-patch formulas - hopefully, it won't take me the 5 weeks it took to understand the canonical flip on the double-tangent bundle from Abraham and Robbin.

The material is just delightful. I finally understand the covariant derivative of a vector field along a curve (it can be defined with just a non-linear connection, but also surely with a Levi-Civita connection on a Riemannian manifold). I like the pedagogy of defining parallel transport along smooth (and, extended to piecewise-smooth) path and covariant derivative first, then defining the connection and adding things like symmetry (torsion-free), linearity, and homogeneity later; you can really see that you are differentiating a vector field along a curve by parallel transporting the nearby vectors back to the tangent space at a given point p and then doing the Newton quotient in that tangent space to get the derivative vector. This makes the notion of the second derivative of the intrinsic velocity vector field of a curve intuitive and makes you see that the curves with the covariant derivative of the velocity vector field being zero are the natural candidates for geodesics.

I still have to clean up the math-out from the end of chapter 1, and I really need to understand vertical and horizontal vectors in the double-tangent bundle - The Geometry of Jet Bundles by Sanders is just wonderful for these things, because he states them for arbitrary vector bundles, so you can compare and contrast with other authors who only cover the tangent and double-tangent bundle.

It'll still be a few chapters before I get to actual Lagrangians and sympletic/contact forms, but I really feel like I've found my calling in life: I truly find this mathematics "beautiful", and that is for what my major professor asked me to search.

Week 15: The Canonical Flip

So, I've been trying to read through Transversal Mappings and Flows by Abraham and Robbin (Joel Robbin was my linear algebra professor at UW back when I was an undergrad, and he literally wrote the book on the subject: we used a manuscript that I still have to this day that eventually became a book Matrix Algebra Using MINImal MATlab. The manuscript had about a third of it devoted to pure linear algebra, over abstract finite dimensional real and complex vector spaces, not using matrices; the theorems were mainly restatements of theorems from the matrix algebra portion of the book, and referenced their matrix algebra counterparts, but this portion of the book was written like manifold theory using manifold that were all diffeomorphic to R^n, and it really set the tone for a lot of math I was to learn later) in my efforts to learn global analysis (I've had a really hard time finding a good introduction to the subject, and this one seems as good as any, maybe better than most). The book talks about the double tangent bundle and its relations to second order ordinary (autonomous) differential equations, and front and center is the "canonical flip" on the double tangent bundle.

The Wikipedia article on the double tangent bundle also mentions the canonical flip, but I couldn't make heads or tails of it. I have been trying to read through Robbin's presentation of it since, oh, roughly since I made my last blog post, in December 2015, but I just couldn't wrap my head around it.

Then, finally, it came to me yesterday as I was riding the elevator back up to my parents' condo after having pored over the section on the double tangent bundle in Abraham and Robbin at a coffeshop for the umpteen-millionth time. In the book The Geometry of Jet Bundles by D. L. Saunders (a book on partial differential equations on manifolds, yet another book on global analysis I'm nominally trying to work through in my efforts to transition from manifold topology [MSC 57] to global analysis [MSC 58] and beyond that to geometric control theory [no MSC code yet, AFAIK]), he goes over in detail that, for an arbitrary vector bundle \pi: E -> M, in considering the tangent bundle to E, D\pi: TE -> TM, we have that for the horizontal vector bundle - that is, if (p, v1, v2, v3) is an arbitrary point of TE, the horizontal vector bundle is {(p, v2) | (p, v1, v2, v3) \in TE} - is canonically vector bundle isomorphic to TM - and, I understood the canonical flip! It just interchanges TM = {(p, v1) | (p, v1, v2, v3) \in TE} \subseteq TE with the horizontal vector bundle {(p, v2) | (p, v1, v2, v3) \in TE}.

So, persistence and hard work does sometimes pay off.

Now, Abraham and Robbin go on to define a second-order (autonomous) ordinary differential equations as dual sections of \tau(TM) and D\tau(M), and they, just before that, show that D: \Gamma(\tau(M) -> \Gamma(D\tau(M) and \omega \circ D: \Gamma(\tau(M) -> \tau(TM) are linear (over real vector bundles), so if \xi: M -> TM is an first-order (autonomous) ordinary differential equation on M, then (say) (1/2)D\xi + (1/2)(\omega \circ D\xi) is a second-order (autonomous) ordinary differential equation on M - nifty stuff.