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I'm a grad student pursuing to Master's degree. My major is Applied Mathematics. Here I call my advisor Mr. Iggy. (Not the real name)

The branch of math that Mr. Iggy researches is Algebraic Topology. As such, the undergraduate courses he opens include Algebra (I and II) and Topology (I and II). I took all such courses, yet there are always too few students that takes Topology II.

I declared early (when I was junior) that I would be a grad student under Mr. Iggy, and everyone in the department agreed. As a consequence, I was able to attend seminars opened by him early. So far, I attended two times.

In the first seminar, I made a presentation about Tychonoff's Theorem, which is outside of the scope of Topology II. In the second seminar, I made a presentation about the typeclasses of Haskell(the programming language)'s Prelude.

Soon I entered the grad school and became Mr. Iggy's student. In this year's first semester, I took his grad course, Algebraic Topology I. And in the next (second) semester, I will take Computational Homotopy Theory. In the meantime (this summer vacation), he will re-open his seminars. And this is where I have a problem.

Mr. Iggy advised me (and his other students) to prepare for taking Computational Homotopy Theory and share what I've studied, so his students will learn the course "well". I was quite doubtful of this because I had many topics I wanted to research on my own, and because I thought seminars were for sharing progresses of such researches. I told him this doubt, yet he just said "Seminars are just for learning," and "You won't be able to complete such researches when you're just in Master's course."

I'm quite confused as for the first quote. And as for the second quote, I can't figure out what he's expecting me to research when the department lacks Doctor's course. Should I just follow his instructions?

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    Isn't algebraic topology a bit removed from applied mathematics? I realize this probably doesn't have much relevance to your main concerns and question, but others (besides me) might be curious as to whether you meant "pure" instead of "applied". Jul 14 at 10:23
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    @DaveLRenfro The keypoint of the department is to, while learning pure mathematics, research how it can be applied for real-world problems. To demonstrate, other courses Mr. Iggy might open include Applied Geometry and Computational Topology. Jul 14 at 11:57
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    @DaveLRenfro "Applied topology" is a rather active field (I even specialize in it!), though has only really taken off in the last 20 years. In fact algebraic topology is a cornerstone more specifically, it's easier to discretize and get onto a computer than some other things. Jul 14 at 14:08
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Generally speaking, the purpose of a seminar is to "disseminate knowledge". Now, depending on which kind of knowledge is being "disseminated", there can be two types of seminars: research seminars and teaching seminars. Regardless of their type, seminars can be beneficial for two distinct groups: the presenter and the audience.

Research seminars: here, the purpose is showcasing one's own research - i.e., disseminate new knowledge. It is assumed that the audience has enough expertise to understand the content of the seminar - which is unlikely to happen if the audience is composed (mostly) of grad/undergrad-students. Benefits:

  • the presenter can benefit by receiving interesting questions by the audience, which can help guide their future research; they can also benefit from potential collaboration requests from the audience. Of course, they can also benefit by receiving some sort of compensation.
  • the audience can benefit by getting some inspiration for future research, by learning something new, or by "staying updated" with the current state of the art.

Teaching seminars: here, the purpose is to disseminate existing knowledge to the audience - maybe as a digression from a given course main content. Benefits:

  • the presenter can benefit either by compensation, or by the "spotlight" received during the seminar; of course, the benefit can also be the "fullfilment of one's duties".
  • the audience will benefit by learning something that could expand their perspective on a given subject, e.g., some realistic applications of a course's topics, or some deeper analyses on some topics that would be otherwise difficult to explain (and to evaluate, due to their complexity) in the course.

To answer your question

you must decide which type of seminar you are going to hold. I assume it is a "teaching seminar", hence try to get the most out of it yourself (as the presenter), and try to also make sure that the audience gets the most out of it.

Finally, I want to clarify a misconception that you may have. In your post, you state:

I had many topics I wanted to research on my own,

...although "any" form of research is "good", "good (and interesting) research" is the one where a person spends months, years or even decades (!) of time. Research does not mean "read a book that is not covered in a class and present its findings". This is what your advisor meant by his sentence.

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  • There is another advantage to teaching: Explaining something (and preparing for that), often improves your own understanding. "If you can't explain it simply, you don't understand it well enough." (A. Einstein)
    – Roland
    Jul 15 at 7:49
  • @Roland you are correct, but I think that what you say applies to both cases (teaching and research). When giving "research seminars" on some topic I had to find some ways to condense all information in a comprehensible way - and that inadvertedly led me to understand the problem better!
    – P. Shark
    Jul 15 at 9:11
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You are correct that seminar is often used for discussing research progress and looking for new approaches to research, but the term isn't universally used in that way. It can also mean a somewhat less formal course in which the students in the seminar present key elements related to the topic of the "seminar". The professor is there to evaluate and give feedback. I'll guess that he is using the term in this sense.

A seminar in the first sense would be peopled by those with very narrow interests and would usually have more than one professor and a few students. The professors might even outnumber the students and do much/most of the presentation and lead discussions. In the second sense of the term, there is probably one professor and the intent is to give students the "opportunity" to prepare and present topics so as to prepare them for a key factor of their future career.

I have been in seminars in the first sense a couple of times as a doctoral student but in the second sense only once, as an undergraduate.

Yes, it is good to follow the instructions of the professor. Your own research can be discussed separately with your advisor, who just happens to be the same person.

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  • I updated seminars to describe the term's use in both senses.
    – Buffy
    Jul 14 at 19:34

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