Thread: Group Theory

i just wanted to ask this simple question, not based on group theory, but more on what level we approach this topic. (generally speaking)

e.g. Binary Operations ,Homomorphisms, Isomorphism and Cayley’s Theorem
,Factor Groups ,Free Abelian Groups ,Free Groups, Group Presentations

when do we learn all this?

thank you in advance

You learn it if you take a course in abstract algebra at a college or university. It is an upper division course taken by undergraduates who are majoring in mathematics.

More generally, here is a chronology of my own math courses in college/grad school, which may give you a sense of the order in which various courses might be taken:

Freshman: Multivariable Calculus (lower div), Linear Algebra (lower div)

Sophomore: Real Analysis, Abstract Algebra, Math. Topics in Physics

Junior: Complex Analysis, Set Theory, Differential Geom, Linear Algebra

Senior: Point Set Topology, Measure Theory & Functional Analysis, Galois Theory.

1st Yr Grad: Functional Analysis, Differentiable Manifolds, Algebraic Topology, Group Theory, Galois Theory, Commutative Algebra, Algebraic Geometry.

2nd Yr Grad: Topics in Algebraic Topology, Riemannian Geometry, Lie Groups, Lie Algebras, Algebraic Geometry, seminar in Morse Theory.

3rd Yr and on: Miscellaneous topics courses and seminars, but from this point forward most of what I learned was from my advisor's topics courses and from reading papers.

wow.... i was apparently looking through maths olympiad questions and collided with this field in mathematics. + im only 16.... Still a long way to go.

It seems like you are a mathematician(or not) .
Anyway i appreciate your help.

Originally Posted by Heinsbergrelatz

i just wanted to ask this simple question, not based on group theory, but more on what level we approach this topic. (generally speaking)

e.g. Binary Operations ,Homomorphisms, Isomorphism and Cayley’s Theorem
,Factor Groups ,Free Abelian Groups ,Free Groups, Group Presentations

when do we learn all this?

thank you in advance

It is not uncommon for that first course in abstract algebra noted by salsaonline to ocncentrate on groups. It largely depends on the particular school and the inclinations of the instructor, and to some extent the length of the course (quarters vs semesters).

Originally Posted by salsaonline

You learn it if you take a course in abstract algebra at a college or university. It is an upper division course taken by undergraduates who are majoring in mathematics.

More generally, here is a chronology of my own math courses in college/grad school, which may give you a sense of the order in which various courses might be taken:

Freshman: Multivariable Calculus (lower div), Linear Algebra (lower div)

Sophomore: Real Analysis, Abstract Algebra, Math. Topics in Physics

Junior: Complex Analysis, Set Theory, Differential Geom, Linear Algebra

Senior: Point Set Topology, Measure Theory & Functional Analysis, Galois Theory.

1st Yr Grad: Functional Analysis, Differentiable Manifolds, Algebraic Topology, Group Theory, Galois Theory, Commutative Algebra, Algebraic Geometry.

2nd Yr Grad: Topics in Algebraic Topology, Riemannian Geometry, Lie Groups, Lie Algebras, Algebraic Geometry, seminar in Morse Theory.

3rd Yr and on: Miscellaneous topics courses and seminars, but from this point forward most of what I learned was from my advisor's topics courses and from reading papers.

That sequence strikes me as fairly normal, particularly given your specialty.

Question: I have been told recently by facultyat two major state universities that average first-year graduate students are no longer capable of handling what I used to think of as the usual rigorous class in measure and integration. This rather surprises me. Is that your experience as well ?

Originally Posted by DrRocket

Question: I have been told recently by facultyat two major state universities that average first-year graduate students are no longer capable of handling what I used to think of as the usual rigorous class in measure and integration. This rather surprises me. Is that your experience as well ?

I find it unlikely that graduate students are having particular trouble with the measure theory and integration courses. These courses tend to be structured more or less like advanced undergraduate courses. So since the format of these courses is familiar to 1st yr graduates students, I would expect them to do okay in them.

That said, I've never really been in a position to observe how 1st yr students do in measure theory/integration. I took their equivalent when I was in college, and never really looked back.

My overall sense is that the quality of 1st year grad students varies from year to year. I did get assigned as a teaching assistant for Differentiable Manifolds, which is traditionally more challenging to 1st yr grad students than measure theory. And I thought my students performed pretty well. In every new batch there are always going to be a few people who don't belong of course.

I think mathematics faculty can sometimes be over-dramatic when they express disappointment in their students. It's easy for a professor to conflate his idea of how he performed in a course with his idea of how the average student performs. Think back to your time in grad school. Who do you remember the most?--the students who did well, and stayed for 4 or 5 years, or the students who dropped out after the 1st year?

As I see it, the real crisis in graduate math education is that the faculty do very little to prepare their students for careers outside of academia. I think a lot of mediocre graduate students entertain the notion that they will follow in the footsteps of their advisers, without realizing, and without being told that there are simply not enough tenure track positions out there to accommodate everyone. If these students are really so sub-par, why isn't anyone taking the time to pull them aside and provide a realistic assessment of their career options? Instead of grumbling, faculty members would do their students a greater favor by taking an active interest in their future.

Originally Posted by salsaonline

Originally Posted by DrRocket

Question: I have been told recently by facultyat two major state universities that average first-year graduate students are no longer capable of handling what I used to think of as the usual rigorous class in measure and integration. This rather surprises me. Is that your experience as well ?

I find it unlikely that graduate students are having particular trouble with the measure theory and integration courses. These courses tend to be structured more or less like advanced undergraduate courses. So since the format of these courses is familiar to 1st yr graduates students, I would expect them to do okay in them.

That said, I've never really been in a position to observe how 1st yr students do in measure theory/integration. I took their equivalent when I was in college, and never really looked back.

My overall sense is that the quality of 1st year grad students varies from year to year. I did get assigned as a teaching assistant for Differentiable Manifolds, which is traditionally more challenging to 1st yr grad students than measure theory. And I thought my students performed pretty well. In every new batch there are always going to be a few people who don't belong of course.

I think mathematics faculty can sometimes be over-dramatic when they express disappointment in their students. It's easy for a professor to conflate his idea of how he performed in a course with his idea of how the average student performs. Think back to your time in grad school. Who do you remember the most?--the students who did well, and stayed for 4 or 5 years, or the students who dropped out after the 1st year?

As I see it, the real crisis in graduate math education is that the faculty do very little to prepare their students for careers outside of academia. I think a lot of mediocre graduate students entertain the notion that they will follow in the footsteps of their advisers, without realizing, and without being told that there are simply not enough tenure track positions out there to accommodate everyone. If these students are really so sub-par, why isn't anyone taking the time to pull them aside and provide a realistic assessment of their career options? Instead of grumbling, faculty members would do their students a greater favor by taking an active interest in their future.

I don't disagree with your thoughts on career counseling.

However, the issue that I found was not that the entering graduate students did meet usual expectations in the measure and integration class, which is these two cases is a rigorous graduate-level class and not an undergraduate class, but that they no longer taught that material to entering graduate students, but rather had to back up and do "remedial" instruction in analysis. It was not so much a complaint as a change in the classes being taught to first-year students, as compared to when I was in that position.

This complaint was not limited to just one university either, and the universities are separated by well over 1000 miles. If it applied to just one school I would have put it down to a local aberration, either in time or space.

The instruction of first year grad students has been modified to address this shortcomng. In one case it went so far as to produce a new text book (which is at the level I would have expected for such a class based on the course that I took, although we did not have a text and had to produce all of the proofs for ourselves).

I am still taken completely aback by this myself. As far as I am concerned there are really no pre-requisites for measure and integration, beyond mathematical maturity and freshman calculus (the maturity being the hard part). All you need is naive set theory, although a concurrent class in point-set topology would not hurt (for the general notion of locally compact spaces).

I am glad to hear that you have not seen such a shortcoming, so perhaps the situation is not as bad as I was led to believe.

With regard to career counseling, I do not see any reasonable "fix" for professors not counseling students regarding careers outside of academia. They don't know anything about life outside of universities. I'm not sure that any counseling received by a student would be useful, given the ignorance of the potential source.

My impression is equally that, outside of the world of sophisticated investment, there is little appreciation for mathematics in non-academic environments. So a mathematics graduate student is somewhat stuck between two worlds, unless he is able to make the bridge himself. It helps to have some applicable knowledge outside of pure mathematics, so that the student can see the applications for himself. In your case that appears to come from a knowledge of finance and physics. In my case an MS in electrical engineering before pursuing mathematics helped a bit (or maybe I'm just a slow learner and switched fields rather late).

I can tell you that in the company for which I worked before I retired I ran across only about 4 mathematicians, and I was in a position to know all of the senior technical people in that medium-sized aerospace and defense company. None were doing mathematics per se. The closest was one guy who worked with an organization that was basically a bunch of physicists doing advanced engineering involving radar systems.

If you think that mathematics grad students are not adequately counseled regarding non-academic careers, I can assure that nobody is counseled regarding corporate internal politics. That is why you see so many senior executives being fired (or going to jail).

Just out of curiosity and i have asked this before, but is there really post graduate and employed and actual university professor's who use this forum???

Originally Posted by Heinsbergrelatz

Just out of curiosity and i have asked this before, but is there really post graduate and employed and actual university professor's who use this forum???

not any more

There are a couple of us who used to be.

how about you ?

Originally Posted by Heinsbergrelatz

how about you ?

I am one of the "used to be". Salsonline will be a" used to be" shortly, so I counted him when I made my mental list. You may congratulate him on that, he deserves it.

I don't think that there are any others, but you never know for sure.

I am one of the "used to be". Salsonline will be a" used to be" shortly, so I counted him when I made my mental list. You may congratulate him on that, he deserves it.

I don't think that there are any others, but you never know for sure.

ah well thank you for the Information.