Sometimes one simply must force oneself to do something that one needs to do but not necessarily likes to do.
A pure mathematics degree does not guarantee one to get a job, or maybe even worse comparably to other degrees. The inability to use most theoretical knowledge pushes students in pure mathematics far away from the industries. Although arguably, that the boundary between pure mathematics and applied mathematics is not clear at all, the true difference is the mindset of students. The students in applied mathematics will position themselves towards the industry, whereas the students in pure mathematics will preferably learn deeper into the subject and ponder on more theoretical and sophisticated materials. People often still underestimate how far modern mathematics has developed and how hard modern mathematics can be. The truth is brutal: mathematicians in one field may not even be able to understand a talk presented by mathematicians from another field.
In my opinion, the biggest benefit of learning mathematics is to improve one’s analytical and problem-solving skills. Essentially, it teaches me how to study. The problem is that there are no criteria to rate one’s ability to learn scaling 1 to 10, and this is such a useless thing to put it on a resume, at least from my perspective. I do think that I need to transfer this advantage to something more concrete, and good luck for me in the future.
It is finally the time to do this. Algorithms act as the soul in computer science, and it is highly related to mathematics. I plan to dive into a classic textbook Introduction to Algorithms. I am not a beginner, and I do have the confidence to learn this very efficiently. I may also follow the course series in algorithms from MIT OpenCourseWare. Practice is always important and inevitable, so I will proceed with solving problems on LeetCode. A new journey begins now, and wish me the best of luck.
Stochastic Differential Equations
This week I reviewed the concept of Brownian motion and up till solving SDE with the Ito’s Lemma. During the study, I discovered the applications of Geometric Brownian motion in the study of Finance. Cited from John Hull in his book Options, Futures, and Other Derivatives: “Geometric Brownian motion is used to model stock prices in the Black–Scholes model and is the most widely used model of stock price behavior.” The plan for next week is to review Ito’s Lemma and the process to solve SDE. The topic of Geometric Brownian motion also sounds interesting, and it may be an appetizer before my study in Financial Mathematics.
No real progress. This may sound sad but it is common in mathematics research. The good news is that I am now more familiar with the concepts and the settings behind the problem. I also spent time writing all the definitions down just to memorize and make them easy to retrieve from the paper. The plan next week is to finish this process and continue to apply the examples back to the proof.
This week I reviewed the concept of the Lax pair and learned the Painleve Property and the Painleve Algorithm. On Monday’s meeting, my graduate partner showed me a simple Matlab program to illustrate how unstable it is to solve a nonlinear differential equation by ordinary numerical solutions. He also taught me the general picture of the inverse scattering method. The discussion is fun and useful, and I am looking forwards to the next one.
There is so little time but so many things to do.