Source: Lucas Vieira (LucasVB), Wikimedia Commons
ABOUT: This is where I will post math projects, current interests/studies, and other cool finds I come across in the world of mathematics.
RELATED: Mathematics Book List
08/19/2024
I've recently began trying to learn differential geometry using a couple books and some online notes. I've had the book Differential Geometry in Physics by Gabriel Lugo (can be found here) on my bookshelf for a long time and always thought I'd just use that. I started to take read it and take notes, but quickly ran into some things I disliked. For one, it doesn't have any exercises at the end of the sections/chapters. Also, there are numerous grammatical errors within the first 15 pages and the author makes references to material that has yet to be introduced. I feel that the definitions are not very well motivated and further obfuscated by these remarks regarding more advanced material the reader has yet to see. I haven't totally written the book off, but I opted to study primarily out of another text.
I found a great deal on a hardcover copy (with dust jacket!) of A Course in Differential Geometry and Topology by A. Mishchenko and A. Fomenko from Mir Publishers Moscow. It was a serious treat to be able to get a book from Mir Publishers in hardcover, since so many of these tend to command quite the premium. For anyone interested, many of the titles are available online at the Internet Archive here. The text I am using can be found here. The authors write in a very clear way, using (from what I can tell) mostly standard notation, and they provide many examples. I appreciate the lengths they go to in emphasizing the lessons the reader should draw from the examples. The text also has some exercises, but I have not gotten to the point of attempting any of them. There is a separate problem book to accompany this text as well, but I haven't looked into it. It's also available on the Archive.
I have also been able to find an assortment of online notes to supplement my book, and one set I was able to get printed off, three-hole punched, and put in a binder. To me, a physical copy in my hands will always be superior to a PDF. One thing I have noticed though is I haven't been able to find a lecture series on YouTube that covers diff geo from the perspective of an advanced undergrad/early grad student. I want more than just computational-based differential geometry of curves and surfaces (some discussion of tensor products, manifolds, etc. would be nice), but those topics tend to be found in lectures at the "600 level", i.e., for those a couple years into a Ph.D.
08/26/2024
I've continued to study from the Mir book, but retired Lugo's book. I will need to look through it again, but I am not sure if I even want to keep it. Perhaps it would be best donated to the Richard Sprague Undergrad Lounge at ISU for someone else to find use from it. The first chapter of the Mir book focuses on curvilinear coordinate transformations in Euclidean domains. Typically, the author will use examples from polar, cylindrical, and spherical coordinate transformations. That is, transforming from Cartesian coords. to those systems. The studying was mostly smooth sailing until getting to the construction of the matrix representation of the metric tensor g_{ab} (or in their notation, g_{mp}). They began by exploring the arc length of a parameterized curve under coordinate transformation. From this, they rearranged a double sum into a matrix of functions (the inner product of the partials of the coordinate transformation functions). This took me a while to parse. The notation was a bit opaque for me, but after expanding out the terms and seeing how the sums of products of partials rearrange, I was able to understand the formulation. Following this they introduced the Einstein summation notation, but didn't acknowledge it as such. This is a quirk I've found in old Soviet books, where they will omit or rephrase certain known mathematical entities to favor the USSR. For instance, in one of the books I found (I cannot remember which) they refer to the Euler-Lagrange equation as the Ostrogradsky Equation (the context was not related to Gauss's Theorem, which is alternatively known as Ostrogradsky's Theorem. Wikipedia).
In order to supplement the Mir text, I also got a copy of Erwin Kreyszig's Differential Geometry. This one is focused on curves and surfaces in 3-D, and perhaps has a less theoretical approach, but I thought this would be a good way to ground the content I was reading in the other one. A major plus with the Kreyszig book is that he has solutions for, I believe, all of the problems in the book! That is a huge help when self-studying. I have yet to do the problems from Chapter 1 of the Mishchenko and Fomenko book, but they don't look too crazy.
Originally posted by the mathematician Anthony Bonato on Twitter, I thought this image was particularly illustrative of the relationship between topological spaces, metric spaces, and vector spaces. I particularly enjoy studying the properties of the L^2 space.
I was able to find this post as a source for the image.
Below are slides for a presentation I prepared for the Junior Analysis Seminar at Iowa State University. Here is the presentation itself (redirects to YouTube).
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As part of the Iowa State Mathematical Research Teams (ISMaRT), I worked on an undergraduate research project intended to give undergrads a look at what the mathematical research process was like. I worked with two other math students (who I knew from the ISU Math Club) on generalized versions of the brachistochrone problem. The research itself wasn't anything novel, but surprisingly enough I did find myself running into dead ends where there was either scant resources or, in an exciting way, no resources at all. The literature on the brachistochrone problem itself is plentiful, but various twists on the idea can quickly place someone in much less charted waters. I think this showed me firsthand what it means to be creative in the research process. Some of the versions I found were quite unique. For instance, I was reading a paper from a US Naval Academy physics researcher on what he dubbed the "electrobrachistochrone," where a charged particle moves on a frictionless track under some electrostatic force (as opposed to the usual mass influenced by gravity along a frictionless path).
The project involved several presentations to the two post-docs supervising us on our progress/hurdles we encountered. They were both very keen on providing a more in-depth understanding of the "why" of variational problems, which I appreciated. One of the problems they posed to us was how the brachistochrone will behave compared to a linear or parabolic trajectory. Also, the two coordinators of the project (with whom the post-docs collaborated with on research) asked about the behavior of the brachistochrone as the endpoint tended to infinity. How would the time vary? How would arc length vary? These sort of questions. Some other problems were posed, which I can delve into further at a later time. When it came to deliverables, I took on the role as the numerical analyst for the project. I used MATLAB to simulate the behavior of these different curves and to try and gain some insight into how they behave (which could inform any analytical problem solving attempts). The post-docs said there was a way of dealing with the problems analytically, but I wasn't able to find a good avenue to pursue with that.
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