2017 Progress
1) This is the first ArXiv article that I would venture to say I can actually understand some of:
"From Apollonius to Zaremba: Local Global Phenomena in Thin Orbits"
Alex Kontorovich
(Submitted on 27 Aug 2012)
https://arxiv.org/abs/1208.5460
2) I fully understood one page of Discrete Groups, Expanding Graphs, and Invariant Measures by Alex Lubotzky, a book for "mature graduate students." Namely, the definition of a c-expander graph. I don't really know what they look like yet, though.
3) My goal for 2017 is to learn measure theory. By June 1, 2017 I will have a link to a PDF file describing what I understand about measure theory. This will include a brief description of the theory (how it generalizes the notion of a Riemann integral, and will mention how it is used in other fields). [Really, this just means that I want to know what R_{'little ell'}, L^2, and L^p spaces are.
Update: Now that I've seen and heard it explained once, it seems trivial. I need more ambitious goals.
Update 2: By June 1, 2017, I have, in fact, finished reading Rudin's 'Principles' closely for the first time (includes studying, working problems from earlier sections -- my point is that I have finished reading closely the sections on Measure Theory, but have yet to solve problems there.)
4) I'm redoing Rogawski's Calculus, and then doing Gilbert Strang's Linear Algebra after that...
5) Programmed first lines of code, in Python 1/17/2017
Answered a discussion question correctly in Engineering
Differential Equations 1/18/2017
6) I can understand a fair amount of this: https://arxiv.org/pdf/math/0605327v1.pdf
7) and I am working through Rudin's Principles of Mathematical Analysis, slowly.
9) First positive response from a grad program: PhD interview with Penn State for Philosophy, 2/3/2017
10) Sat in on math courses at UChicago, and some big names (!) let me stay.
11) Read my second full article on arXiv, this time on Category Theory. Summary: Leibniz algebras are a generalization of Lie Algebras. read 4/9/2017. https://arxiv.org/pdf/1703.10426.pdf
12) Answered my first discussion question in Manifolds on 5/3/17 (fundamental group of the 2-sphere is the identity!). Correctly solved first problem in manifolds 5/4/17 (p. 40 from Tu).
13) 9 months ago all I knew at this point was that I was going to run out of money, and that if I wanted to continue in academia, I had to get into something good -- and quick! So, I took French I and German I (at the former advice of my philosophy advisor), made sure that I had every back-up plan check done for Philosophy PhD's (I ended up getting into a couple...! :) ) , and then kept going in Math & Comp Sci. This semester (Spring 2017) I took my first Computer Science course (Introduction to Computer Science -- included some Python, assembly language, and examples of machine language), and sat in on more math courses. This April, I signed my first contract with a mathematics department to start my graduate studies. My long-term goal has and will be to continue working towards a PhD in Mathematics, with teaching college (or going into research, or industry!) as my end goals for now. I am feeling good about having a grasp again on Calculus 1-3 (Green's Theorem, Stokes' Theorem, mixed partials are everything!) (it's more fun to read these things from Rudin) and Linear Algebra (went back through Strang's Linear Algebra)-- examples include diagonalizing a matrix using eigenvector matrices and their inverses, spectral analysis of matrices using the size of their eigenvalues, and am getting to Matrices of Linear Transformations again); I'm PUMPED for round two on partial differential equations (I'll be a bit smarter about things this go-around), I'm learning and catching up in abstract algebra [I'm doing one day of algebra after one day of analysis] (groups, modules, PIDS, Integral Domains, Euclidean Domains, Rings, Fields, Galois Theory, and Canonical Forms), and am starting Topology on my own [topology day is after algebra, and before analysis] (Munkres' Topology, would like to also do Hatcher and Guillemin & Pollack, and am starting to understand some of Peter May's Algebraic Topology book and Allen Hatcher's "AT"). Considering starting PDE's again, numerical analysis, and trying to learn more Applied Math.
Right now I'm doing one day of Analysis (Real and Complex: this has included reviewing Rogawski's Calculus for Calculus 1-3 up to Green's Theorem, Stokes Theorem, Divergence, Curl, Jacobians of a linear and non-linear transformation; Rudin's Principles of Analysis [am starting to read some Adult Rudin], Ahlfor's and Freitag & Busam's Complex Analysis), one day of Algebra (Dummit & Foote, Lang's Algebra on Galois Theory, Hungerford's; Insel, Spence, and ____; Hoffman & Kunz\:{e}; etc.)
14) Strong A on midterm, perfect score on final exam in Foundations of Analysis, and fell in love with two theorems in particular which I can't mention here. The best thing, though, was that I just decided that this was going to happen! "Rarely do we accomplish goals that we do not set out for explicitly" - this quote was from an MIT prof on OpenCourseWare for I think a "Teaching Physics" course -- I should really look up the name to cite this correctly. Also, need to update/make a CV in LaTeX.
15) Ohmanohmanohmanohmanohman... Just about understood this one --- need some homotopy in my life. https://arxiv.org/pdf/1705.09323.pdf
16) Understood homotopy for the first time yesterday: https://en.wikipedia.org/wiki/Homotopy
Starting this now: https://www.math.cornell.edu/~hatcher/AT/AT.pdf.
17) Passed my first Qualifying Exam for a Graduate Mathematics Program: June 2017
Moments I am most proud of in mathematics:
1) Realizing that the Laplace Transform of the sine function comes from Euler's identity from Complex Analysis.
6) I can understand a fair amount of this: https://arxiv.org/pdf/math/0605327v1.pdf
7) and I am working through Rudin's Principles of Mathematical Analysis, slowly.
- Constructed the field R from sets of cuts
- Induction proofs involving triangle inequality
8) College math professor said I should want to get my PhD in math and teach college...
(1/30/2017)
8.5) First footnote in a LaTeX article, 2/4/2017 (this is just cool to me, idk why)
(1/30/2017)
8.5) First footnote in a LaTeX article, 2/4/2017 (this is just cool to me, idk why)
9) First positive response from a grad program: PhD interview with Penn State for Philosophy, 2/3/2017
10) Sat in on math courses at UChicago, and some big names (!) let me stay.
11) Read my second full article on arXiv, this time on Category Theory. Summary: Leibniz algebras are a generalization of Lie Algebras. read 4/9/2017. https://arxiv.org/pdf/1703.10426.pdf
12) Answered my first discussion question in Manifolds on 5/3/17 (fundamental group of the 2-sphere is the identity!). Correctly solved first problem in manifolds 5/4/17 (p. 40 from Tu).
13) 9 months ago all I knew at this point was that I was going to run out of money, and that if I wanted to continue in academia, I had to get into something good -- and quick! So, I took French I and German I (at the former advice of my philosophy advisor), made sure that I had every back-up plan check done for Philosophy PhD's (I ended up getting into a couple...! :) ) , and then kept going in Math & Comp Sci. This semester (Spring 2017) I took my first Computer Science course (Introduction to Computer Science -- included some Python, assembly language, and examples of machine language), and sat in on more math courses. This April, I signed my first contract with a mathematics department to start my graduate studies. My long-term goal has and will be to continue working towards a PhD in Mathematics, with teaching college (or going into research, or industry!) as my end goals for now. I am feeling good about having a grasp again on Calculus 1-3 (Green's Theorem, Stokes' Theorem, mixed partials are everything!) (it's more fun to read these things from Rudin) and Linear Algebra (went back through Strang's Linear Algebra)-- examples include diagonalizing a matrix using eigenvector matrices and their inverses, spectral analysis of matrices using the size of their eigenvalues, and am getting to Matrices of Linear Transformations again); I'm PUMPED for round two on partial differential equations (I'll be a bit smarter about things this go-around), I'm learning and catching up in abstract algebra [I'm doing one day of algebra after one day of analysis] (groups, modules, PIDS, Integral Domains, Euclidean Domains, Rings, Fields, Galois Theory, and Canonical Forms), and am starting Topology on my own [topology day is after algebra, and before analysis] (Munkres' Topology, would like to also do Hatcher and Guillemin & Pollack, and am starting to understand some of Peter May's Algebraic Topology book and Allen Hatcher's "AT"). Considering starting PDE's again, numerical analysis, and trying to learn more Applied Math.
Right now I'm doing one day of Analysis (Real and Complex: this has included reviewing Rogawski's Calculus for Calculus 1-3 up to Green's Theorem, Stokes Theorem, Divergence, Curl, Jacobians of a linear and non-linear transformation; Rudin's Principles of Analysis [am starting to read some Adult Rudin], Ahlfor's and Freitag & Busam's Complex Analysis), one day of Algebra (Dummit & Foote, Lang's Algebra on Galois Theory, Hungerford's; Insel, Spence, and ____; Hoffman & Kunz\:{e}; etc.)
14) Strong A on midterm, perfect score on final exam in Foundations of Analysis, and fell in love with two theorems in particular which I can't mention here. The best thing, though, was that I just decided that this was going to happen! "Rarely do we accomplish goals that we do not set out for explicitly" - this quote was from an MIT prof on OpenCourseWare for I think a "Teaching Physics" course -- I should really look up the name to cite this correctly. Also, need to update/make a CV in LaTeX.
15) Ohmanohmanohmanohmanohman... Just about understood this one --- need some homotopy in my life. https://arxiv.org/pdf/1705.09323.pdf
16) Understood homotopy for the first time yesterday: https://en.wikipedia.org/wiki/Homotopy
Starting this now: https://www.math.cornell.edu/~hatcher/AT/AT.pdf.
17) Passed my first Qualifying Exam for a Graduate Mathematics Program: June 2017
Moments I am most proud of in mathematics:
1) Realizing that the Laplace Transform of the sine function comes from Euler's identity from Complex Analysis.
2) Scoring enough points on the first midterm of Complex Analysis to stay in the course when I wasn't guaranteed a spot: hadn't taken the pre-requisites, was driving a half-hour to get to class, and had to score well to stay alive in mathematics.
Running List of Questions in Mathematics:
1) How to turn the columns of a matrix into linear combinations of eigenvectors?
2) How are row operations and similarity related in matrices?
D&F: how is xhx^{-1} = 2Z [need to look up page number for this example]
3) [a big one] how are the eigenvalues of a matrix related to a matrix? What can the eigenvalues of a matrix tell us about a linear transformation? [I feel like I should write a "practice paper" on eigenvalues of linear transformations].
