How does my future converge? Where does it converge to?
As time goes by, the year of graduation comes close. I have to face the question that I have pondered over for a while. Or rather a question, it is like a decision or just a thing.
Unavoidably, I will say that I will consider mathematics as a major part of my future. I love it, and I can make a living of it, hopefully. My main goal is to have a happy life, earn some money, and maybe leave my footprints in this world. To do this, I know for a fact that undergraduate mathematics is far from enough. Then what should I do after graduation? What should I consider as my future career? What is the plan? I hardly have time to think over these right now, but I hope that I am looking forward and struggling in the direction.
After the concepts of integral domains, principal ideal domains and unique factorization domains were introduced, the Gauss’ Lemma and the Eisenstein Criterion finished Chapter 12 of the textbook. We then talked about the topic of Quadratic Number Fields and worked with some connections towards lattices.
Last week, we began to talk about the fields. The definitions of algebraic and transcendental elements were specified, and we talked about the degree of a field extension. The homework this week will mainly focus on ruler and compass constructions, and I guess that will be a lot of fun.
During some time in this course, I felt like a course of number theory would be tremendously helpful, but I had never had a chance to do so. Playing with numbers is sometimes frustrating, and I experienced that before when doing some math puzzles.
The things became to be more series and sophisticated after the topic of Sobolev spaces entered the chat. For this part of the course, we followed the chapter 5 of the book Partial Differential Equations by L. C. Evans, which I am still not quite sure why. I am still reading the Ordinary Differential Equations by Arnold, but I was reading Chapter 3 about the linear system. That is so far off. Professor Evans from Berkeley also wrote the notes that we are using for the Stochastic Differential Equations reading course. The materials are well-written and wonderful, and it requires time to fully understand.
There is still one thing I wanted to mention. Last week we talked about the Sobolev Inequality, and Professor Kuli mentioned that in differential geometry, a field that I was willing to study but didn’t, it is equivalent to the Soperimetric Inequality. There lies a beauty of mathematics, when you connect different fields of mathematics. I didn’t get into the statement since I haven’t learned differential geometry, but I will surely learn differential geometry as well as partial differential equations sooner or later.
Stochastic Differential Equations:
We began to discuss the topic of Stochastic Differential Equations. First of all, the definition and properties of Itô calculus are introduced. Although both Evan’s and Oksendal’s book talk about the connections with Martingales, we didn’t focus on that may be due to the reason of the varieties of audience. Tomorrow will finally discuss how to solve a one-dimention Stochastic Differential Equation. This reading group is great, and professor Schreiber even found some examples and problems to help us understand the materials.
The more I study, I realize that this topic plays an important role in differential areas. I am really glad that I was informed and was able to join the group. Let’s go!!
What do I love to experience a class from a master is that he has its preference in teaching. Professor Tracy taught extremely fast, and I need to learn from his notes, the textbook reading, and his lectures with some extra materials. In these weeks we talked about the continuous Markov Chain, the Kolmogorov forward and backward equations, Martingales, and the Martingale Convergence Theorem. I also consulted Wikipedia and learned the complete statements, even the connection to convergence.
The most exciting thing is that he represented some of his research papers to us to read! One is about the general solution integral formula for the Asymmetric Simple Exclusion Process(ASEP) on the integer lattice. The outline is clear and easy to read, but the proofs are rigorous and sophisticated. The other one used the previous results to obtain three asymptotic results for ASEP. Since the integral from the previous results involved the Fredholm determinant, the analysis of the spectrum of a matrix came in play. Professor Tracy seemed to plan to talk about the simpler 3D case of the first result, but since I know about all the prerequisites to be able to understand the papers, I will read them through in the future.
The reading of ordinary differential equations is still going. Both of my research projects come to a bottleneck. I hope that I can fight through this. Bless me.