Author: Rose_Zh

Career Path Interview (Spring 2018, Part I)

Interviewee: H.L.
Major: Mathematics
Year: Senior

1) Why did you choose maths as your major?
H.L.: I have always been interested in numbers, patterns and how things operate. However, upon entering college, I was introduced to political science, economics, philosophy, anthropology and other subjects, which caught so much of my attention that I considered majoring in one of those fields. Nevertheless, the attention was only momentary. Mathematics remained my favorite, and was hence chosen as my major. As I advanced, mathematics grew on me, and became my ultimate intellectual pursuit.

2) What are the areas of math that you have studied?
H.L.: Like most undergraduate math students, I took courses in calculus, linear algebra, real/complex analysis, abstract algebra, topology, ordinary/partial differential equations, discrete structures, set theory, etc. In terms of more advanced materials, I studied elements of measure theory, distribution theory and probability, functional analysis, Fourier analysis, approximation theory, analysis on PDE, matrix theory, computational group theory, ring, module, field, Galois theory, elements of representation theory, categorical language, algebraic topology, homological algebra, etc. I am also introduced to other branches of mathematics including mathematical logic and differential geometry.

3) Do you have any comment on these areas?
H.L.: Given my current maturity, I may not be qualified to give any deep comment about these subjects. However, the impression I have is that many topics in analysis and abstract algebra often resort to the study of linear algebra. Just to name a few, the concept of duality is one of the principal elements in the study of module theory, which is itself a generalized study of vector space over a field. In functional analysis, which is roughly speaking the study of infinite-dimensional vector spaces equipped with topological or metric structure, whose fundamental elements also include the Riesz representation theorems and dual of Hilbert and Banach spaces. In PDE, the solvability of second-order elliptic equations is firstly established by the existence of weak solutions, which turned out to be a consequence of what is known as the Fredholm alternative given the condition of a compact linear operator involved in the partial equation. There are also other topics in algebraic analysis where application of matrix theory is ubiquitous.

4) Which area is your favourite?
H.L.: Thus far, I am mostly interested in measure theory, functional analysis, and PDE.

5) What’s you plan after college?
H.L.: I recently applied to several Ph.D. programs. I wish to pursue graduate study primarily in the area of mathematical analysis.

6) Have you ever considered working as opposed to grad school?
H.L.: I have wanted to pursue an academic career since I was in high school. However, if things do not turn out well, I am open to opportunities in math-related industries.

7) Did your math research experience affect your perspective? If so, in what way?
H.L.: My participation in summer 2017 REU at Cornell University, and in directed reading courses at Berkeley definitely helped form my perspective on mathematical research. Prior to these experiences, I did not know what a life of a researcher would entail. It was not just about learning and being exposed to a vast amount of materials relevant to the research project, but also about honing my communication skills. By frequently discussing problems and concepts that I struggled to understand with other group members, I learned how to organize and present mathematical ideas in such a way that not only my partners, but also participants working on other projects, can understand. As a result, I received intuitive feedback, which helped me form fresh insights into difficult problems. Additionally, I find it very helpful and motivating to attend graduate seminars organized by faculty members as well as graduate students at Berkeley, even though my comprehension of the presented materials may not be adequate and may even be, a lot of times, absent. However, these seminars provide intuitions in terms of how materials I learned in graduate courses are used and developed. Thereby, they portrait a clearer picture of what the disciplines look like at the forefront and how important it is to communicate with other mathematicians about new ideas on unsolved problems.

8) Do you mind sharing with us your long-term career goal?
H.L.: Upon obtaining a Ph.D degree, I hope to get a postdoctoral and then a faculty position at some research university to fulfil my dream of teaching and conducting original research in mathematics.

GRE Math Subject Test Preparation Strategies

Overview:

  • Official guide & official test practice book
  • Format:
    The test consists of one section which is 2 hours and 50 minutes long.
  • When is the test offered?
    In April, October and November each year.
    Note: The test registration deadline for the November one is usually before the time when the result of the October test comes out. So, it may be a good idea to register for both if one is in urgent need of a good score.
  • Nearest Location:
    Berkeley High.
  • Graduate programs that require GRE Math Subject Test score:
    Math & Applied Math.
  • Graduate programs for which a good Math Subject score would be helpful:
    Financial Mathematics, Physics, Statistics, etc..

Test preparation strategies:

  • START EARLY so that you will have enough time to review all the materials, and will be able to take the subject test again (which many people do) if necessary.
  • Start with a diagnostic exam: identify the topics that you are unfamiliar with, and master them.
  • Review ALL the concepts mentioned in the official guide, and relevant problem-solving techniques.
  • Practice a lot, practice often.
  • Mock exams: do the full length past exams under TIMED conditions, at a place where you are unlikely to be disturbed. (Speed matters!)
  • When practicing, mark the questions that you can’t solve even if you manage to guess the correct answer.

Test prep books:

  • Princeton Review: fairly exhaustive, but contains typos and errors.
  • Research & Education Association (REA): covers more topics than the actual test does, contains 6 practice exams, and is great for those who have a good score already but want an even better one.

Where to find past exams:

  • Google keywords like “old gre practice”, and the following forms shall be available online: Form 0568, 1268, 8767, 9367, and 9768.
    Note: If a URL is no longer valid, use the Wayback Machine to retrieve it.
  • Some are available at: http://www.wmich.edu/mathclub/gre.html (This webpage has some other information on test preparation as well.)
  • ETS has one past exam available on its website.

What if I don’t know about a certain topic yet?

  • Take a relevant course.
  • Learn the basics by reading a textbook and doing the exercises. (Here is a book-list for reference; you may search the topic in the second column.)

Other resources:

We [Lilian and Rose] are more than willing to share our Subject Test experience, and help you with your test preparation, so please come by our advising hours! Besides, I [Rose] have written up some test preparation notes for my own sake, and will gladly share them on an individual basis.