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!
Thursday, June 30, 2016
Wednesday, May 4, 2016
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), ...., pn(q, α)), we have θ0 = p1dq1 + ... + pndqn so -dθ0 =-(dp1∧dq1 + ... + dpn∧dqn) = dq1∧dp1 + ... + dqn∧dpn = 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.
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), ...., pn(q, α)), we have θ0 = p1dq1 + ... + pndqn so -dθ0 =-(dp1∧dq1 + ... + dpn∧dqn) = dq1∧dp1 + ... + dqn∧dpn = 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.
Monday, April 25, 2016
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.
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.
Monday, April 4, 2016
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.
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.
Wednesday, December 30, 2015
Week 1
So, I started this journey by trying to remember what got me
interested in manifold topology with which to begin. I took a physics
course in mechanics, and we went over Lagrangian mechanics, and I was
totally blown away by it: I had no idea what these "generalized
coordinates" about which the professor was talking were. So, I began to
do some digging and wound up taking some more advanced math courses (as I
was a math major who was doing a physics major just to find out about math's major consumer), and I discovered these things called "smooth
manifolds" and "coordinate patches" on them, and I learned that
"generalized coordinates" are just coordinate patches on smooth
manifolds (Why couldn't the physics instructor just have come right out
and said that?).
So, I remembered taking my first Ph.D. course in smooth manifolds, and we learned about smooth maps between smooth manifolds, which I thought were the most beautiful thing about which I had ever learned in my life, and about derivatives of smooth maps between smooth manifolds, Df: TM ---> TN, which are vector bundle morphisms between the tangent bundles of manifolds, which seemed even more beautiful to me than the smooth maps themselves.
So, I wanted to do something with tangent bundles, but the only thing of which I could think to do with them was have a tangent vector field on them aka a first-order ordinary differential equation on them. I knew ODEs on manifolds are a well-studied phenomena, and that I wasn't going to make many contributions there, so I started to think about other things, and I came up with second-order ODEs on manifolds. It turns out, there appears to be a much smaller literature on them, and there appears to be some confusion in the literature on whether you need a Riemannian metric, or at least an affine connection, to even define the acceleration of a curve in a smooth manifold or if you can just do it with the double tangent bundle TTM.
So, that was nice to see a possible niche in the literature, but then I remembered one of the things I found interesting about smooth manifolds and smooth vector bundles was characteristic classes. So, I did some Googling about those and found that they're really defined for principle bundles of smooth manifolds with topological or Lie groups as the fiber (On or SOn in the case of principle bundles associated to vector bundles), and that one then transferred the characteristic classes of the principle bundle back to the associated vector bundle with which one started.
So, I searched YouTube for videos on characteristic classes, and I came across this video, evidently by a CalTech instructor who was giving a simultaneous lecture/class with graduate students at the University of Tokyo via the videos that I was currently watching on YouTube. I jumped in in medias res with this video
and discovered I could hardly follow a thing. He appeared to be defining differential forms (a generalization of vector fields with which I am fluently familiar) and deRham cohomology (another field on which I can practically give lectures) by defining them to be the traces or determinants of powers of differential forms, a formalism that didn't even make sense to me on the surface of it, much less a formalism in which I could do computations - back to "generalized coordinates". So, that was Lecture 7 evidently in a lecture series, so I went back and started with Lecture 1, and I'm not back up to Lecture 7 yet, so I'm not sure how much sense Lecture 7 will make to me yet, but I have learned a lot. I learned about the Laplacian on arbitrary closed Riemannian smooth manifolds and how its solutions are sort-of unique or "our favorite" differential forms representing generators of deRham cohomology; as I had said, I have almost given lectures on deRham cohomology and its relation to integral or group integral handlebody cohomology - they're certainly in my dissertation - but this notion of harmonic representatives - differential forms that satisfy Laplace's equation - of cohomology was news to me.
And, it was beautiful news.
So, next time, I'll outline what I've learned about harmonic differential forms and some examples of them along with the Hodge star (again, news to me and very beautiful mathematics) and normalized harmonic representatives of generators of deRham cohomology. That's all for this week.
So, I remembered taking my first Ph.D. course in smooth manifolds, and we learned about smooth maps between smooth manifolds, which I thought were the most beautiful thing about which I had ever learned in my life, and about derivatives of smooth maps between smooth manifolds, Df: TM ---> TN, which are vector bundle morphisms between the tangent bundles of manifolds, which seemed even more beautiful to me than the smooth maps themselves.
So, I wanted to do something with tangent bundles, but the only thing of which I could think to do with them was have a tangent vector field on them aka a first-order ordinary differential equation on them. I knew ODEs on manifolds are a well-studied phenomena, and that I wasn't going to make many contributions there, so I started to think about other things, and I came up with second-order ODEs on manifolds. It turns out, there appears to be a much smaller literature on them, and there appears to be some confusion in the literature on whether you need a Riemannian metric, or at least an affine connection, to even define the acceleration of a curve in a smooth manifold or if you can just do it with the double tangent bundle TTM.
So, that was nice to see a possible niche in the literature, but then I remembered one of the things I found interesting about smooth manifolds and smooth vector bundles was characteristic classes. So, I did some Googling about those and found that they're really defined for principle bundles of smooth manifolds with topological or Lie groups as the fiber (On or SOn in the case of principle bundles associated to vector bundles), and that one then transferred the characteristic classes of the principle bundle back to the associated vector bundle with which one started.
So, I searched YouTube for videos on characteristic classes, and I came across this video, evidently by a CalTech instructor who was giving a simultaneous lecture/class with graduate students at the University of Tokyo via the videos that I was currently watching on YouTube. I jumped in in medias res with this video
and discovered I could hardly follow a thing. He appeared to be defining differential forms (a generalization of vector fields with which I am fluently familiar) and deRham cohomology (another field on which I can practically give lectures) by defining them to be the traces or determinants of powers of differential forms, a formalism that didn't even make sense to me on the surface of it, much less a formalism in which I could do computations - back to "generalized coordinates". So, that was Lecture 7 evidently in a lecture series, so I went back and started with Lecture 1, and I'm not back up to Lecture 7 yet, so I'm not sure how much sense Lecture 7 will make to me yet, but I have learned a lot. I learned about the Laplacian on arbitrary closed Riemannian smooth manifolds and how its solutions are sort-of unique or "our favorite" differential forms representing generators of deRham cohomology; as I had said, I have almost given lectures on deRham cohomology and its relation to integral or group integral handlebody cohomology - they're certainly in my dissertation - but this notion of harmonic representatives - differential forms that satisfy Laplace's equation - of cohomology was news to me.
And, it was beautiful news.
So, next time, I'll outline what I've learned about harmonic differential forms and some examples of them along with the Hodge star (again, news to me and very beautiful mathematics) and normalized harmonic representatives of generators of deRham cohomology. That's all for this week.
Tuesday, December 29, 2015
Introduction
Hello, all!
This will be my new (professional) blog as I leave graduate school and begin to look for research topics to begin publishing papers.
I have a Ph.D. in differential topology from UWM and am teaching adjunct at Marquette University in Milwaukee, WI. My dissertation is online at the arXiv at this link
My dissertation is pseudo-collars, which are intended to be generalizations of triangulation of Hilbert cube manifolds. Chapman and Siebenmann proved that for aspherical Hilbert cube manifolds that admit Z-compactifications, if they have the same fundamental group, then they are homeomorphic. My dissertation advisor, Craig Guilbaut, along with his co-author Fred Tinsley of Colorado College, seek to prove an analogous result for high-dimensional, but finite dimensional, manifolds. thus proving a long-standing open conjecture called The Borel Conjecture.
So, my dissertation topic is on pseudo-collars, but it's not where my heart is. Honestly, I was just glad Craig was able to find a topic that was in manifold topology and not Geometric Group Theory, which has taken the field of topology research mathematics by storm.
Since I graduated in May, 2015, and wanted to work on a research project, Craig counseled me not to just go after the glory of a "Big Game" conjecture, like The Borel Conjecture, but to work on topics I find "beautiful". (More on that word later.) So, I did quite a bit of soul-searching, and found global analysis and geometric control theory to be topics nearer and more dear to my heart.
So, in a nod to John Baez's "This Week's Finds in Mathematics Physics", I started this blog, mainly as a place for me to think out out loud on my newfound direction in research, global analysis (differential equations on manifolds and related topics, and geometric control theory). As the title of the blog may partially suggest, I hope that by putting pressure on myself to have to update this blog weekly with new "finds" and information, it will keep me at the grindstone of learning about possible research topics/areas. I hope other people will find it interesting as well.
This will be my new (professional) blog as I leave graduate school and begin to look for research topics to begin publishing papers.
I have a Ph.D. in differential topology from UWM and am teaching adjunct at Marquette University in Milwaukee, WI. My dissertation is online at the arXiv at this link
My dissertation is pseudo-collars, which are intended to be generalizations of triangulation of Hilbert cube manifolds. Chapman and Siebenmann proved that for aspherical Hilbert cube manifolds that admit Z-compactifications, if they have the same fundamental group, then they are homeomorphic. My dissertation advisor, Craig Guilbaut, along with his co-author Fred Tinsley of Colorado College, seek to prove an analogous result for high-dimensional, but finite dimensional, manifolds. thus proving a long-standing open conjecture called The Borel Conjecture.
So, my dissertation topic is on pseudo-collars, but it's not where my heart is. Honestly, I was just glad Craig was able to find a topic that was in manifold topology and not Geometric Group Theory, which has taken the field of topology research mathematics by storm.
Since I graduated in May, 2015, and wanted to work on a research project, Craig counseled me not to just go after the glory of a "Big Game" conjecture, like The Borel Conjecture, but to work on topics I find "beautiful". (More on that word later.) So, I did quite a bit of soul-searching, and found global analysis and geometric control theory to be topics nearer and more dear to my heart.
So, in a nod to John Baez's "This Week's Finds in Mathematics Physics", I started this blog, mainly as a place for me to think out out loud on my newfound direction in research, global analysis (differential equations on manifolds and related topics, and geometric control theory). As the title of the blog may partially suggest, I hope that by putting pressure on myself to have to update this blog weekly with new "finds" and information, it will keep me at the grindstone of learning about possible research topics/areas. I hope other people will find it interesting as well.
Subscribe to:
Posts (Atom)