Should I Become a Mathematician?

[STEMucator]

I have been following this thread for quite a while. After getting a business diploma I realized I was more interested in economics, which led to mathematics and here I am 3 years later with soon to be bachelor degree in mathematics. I started out with almost no knowledge of math, but I worked my way all the way through the good and hard times.

I am going to study my masters in September(in my country bachelors degree is 3 years and gradschool/masters is 2 years). I have a dilemma now.
I applied for grad school in a top 50 university in the world in my country and they told me I was not qualified because I didn't have enough measure theoretical statistics. I was automatically accepted into the masters program in my current university which is ranked a bit lower than the top 50 university.

Right know I am 25 years old and I could just go directly into the master programme at my current school and finish in two years. Or I could spend one extra year trying to take an extra course on measure theoretical statistics and try to get into the top 50 university next year. However, by doing so I will graduate one year later (and I am not that young any longer)...

I wonder if anyone has been in this position before and maybe they could tell me what kind of benefits I could get by doing either the first choice or the second...

Concepts are equal if you go to university A, or B. The definition of the derivative of a function won't change. You shouldn’t care about the school you choose because you can still can be the best you can be. Don’t worry about popularity rankings, worry about improving yourself every day by working hard, and doing what you enjoy most.

 

[Nous]

I have been following this thread for quite a while. After getting a business diploma I realized I was more interested in economics, which led to mathematics and here I am 3 years later with soon to be bachelor degree in mathematics. I started out with almost no knowledge of math, but I worked my way all the way through the good and hard times.

I am going to study my masters in September(in my country bachelors degree is 3 years and gradschool/masters is 2 years). I have a dilemma now.
I applied for grad school in a top 50 university in the world in my country and they told me I was not qualified because I didn't have enough measure theoretical statistics. I was automatically accepted into the masters program in my current university which is ranked a bit lower than the top 50 university.

Right know I am 25 years old and I could just go directly into the master programme at my current school and finish in two years. Or I could spend one extra year trying to take an extra course on measure theoretical statistics and try to get into the top 50 university next year. However, by doing so I will graduate one year later (and I am not that young any longer)...

I wonder if anyone has been in this position before and maybe they could tell me what kind of benefits I could get by doing either the first choice or the second...

I haven’t done any graduate level math work, so I can’t comment on that specifically, but generally it is just as important to find a thesis supervisor who is working in an area that interests you. If your current school has a professor whose research aligns with your interests and goals, I would likely give it a higher decision weight.

If you are doing a course-based master degree, then this wouldn’t likely apply.

 

[Astronuc]

A Path Less Taken to the Peak of the Math World, or an Unusual or Unlikely Path to Mathematics
https://www.quantamagazine.org/a-path-less-taken-to-the-peak-of-the-math-world-20170627/

Huh’s math career began with much less acclaim. A bad score on an elementary school test convinced him that he was not very good at math. As a teenager he dreamed of becoming a poet. He didn’t major in math, and when he finally applied to graduate school, he was rejected by every university save one.

Nine years later, at the age of 34, Huh is at the pinnacle of the math world. He is best known for his proof, with the mathematicians Eric Katz and Karim Adiprasito, of a long-standing problem called the Rota conjecture.

Karim Adiprasito, June Huh, Eric Katz
Hodge Theory for Combinatorial Geometries
https://arxiv.org/abs/1511.02888
From the abstract, the authors state, "We prove the hard Lefschetz theorem and the Hodge-Riemann relations for a commutative ring associated to an arbitrary matroid M. . . ."

I went looking for more information on the Rota conjecture, and apparently there is more than one.
https://en.wikipedia.org/wiki/Rota's_conjecture

 

[member 587159]

A Path Less Taken to the Peak of the Math World, or an Unusual or Unlikely Path to Mathematics
https://www.quantamagazine.org/a-path-less-taken-to-the-peak-of-the-math-world-20170627/


Karim Adiprasito, June Huh, Eric Katz
Hodge Theory for Combinatorial Geometries
https://arxiv.org/abs/1511.02888
From the abstract, the authors state, "We prove the hard Lefschetz theorem and the Hodge-Riemann relations for a commutative ring associated to an arbitrary matroid M. . . ."

I went looking for more information on the Rota conjecture, and apparently there is more than one.
https://en.wikipedia.org/wiki/Rota's_conjecture

Nice article. Thanks for sharing!

 

[kay bei]

@mathwonk I read in many places about the importance of geometric intuition in advanced mathematics. If you read some articles and interviews with Misha Gromov, or historical articles on Rochlins and Alexandrovs geometric schools, they mention there are different ways of approaching non standard problems, which includes algebraic and geometric.

It is said that Misha, Rochlin and Alexandrov made heavy use of geometric intuition. They are just to name a few but there are many more mathematicians who say this is an important part of their research, including those studying abstract algebra. I don't understand how this can be possible in a course such as Abstract Algebra. I wish there was some books, or examples, at least at a lower level which show the geometric approach to solving problems in abstract mathematics.

Is this something you also believe in? Have you noticed other mathematicians and including yourself, making use of geometric ideas to help solve their research questions and to explore mathematical ideas, to guide the algebraic formulation... Please let me know because there seems to be a lot of mystery about this online and no one provides any examples of this anywhere on how they do that.

Dirac said in an interview that he does mathematics geometrically, he said to another physicists, "how do you think about de-Sitter space?" and Dirac said I think of it geometrically. Other physicists at the time of Maxwell also commented on his geometric intuition, stating he thinks in this way.

Please, if you know of any guides, references, books, articles, videos, hidden-gems, that show this way of mathematical investigation, please suggest and I would also like to hear your comments on the use of this approach in the abstract subjects at the undergraduate, graduate and research level.

Thank you for making this thread! :)

 

[mathwonk]

the following book springs to mind:

https://www.amazon.com/dp/0133198316/?tag=pfamazon01-20

I am an algebraic geometer and think mainly geometrically.

in these notes, there is an appendix on p. 32 called the geometry of rings. that may interest you.

https://www.math.uga.edu/sites/default/files/inline-files/8000b.pdf

 

[kay bei]

the following book springs to mind:

https://www.amazon.com/dp/0133198316/?tag=pfamazon01-20

I am an algebraic geometer and think mainly geometrically.

in these notes, there is an appendix on p. 32 called the geometry of rings. that may interest you.

https://www.math.uga.edu/sites/default/files/inline-files/8000b.pdf

@mathwonk what do you mean by thinking geometrically? Could you explain somehow? I don't understand how a geometrical picture can be associated to any space higher than 3 dimensions. And how can you get a geometric picture for some algebraic structures? I don't see any geometry in that appendix as I read it? No picture, no visuals, the words used do not convey geometry to me. How is that geometric thinking? It all seems still very abstract and algebraic. At what level is that appendix pitched at? I am looking for something for a beginner, say for example either High School Senior or starting 1st year of university. It would be great if there was a book which teaches this way of thinking at a much lower level and slowly develop the thinking to more complex situations and abstractions. There is no clear transition and explanation of how to carry the geometric thinking over to other fields of mathematics which arent usually associated with geometry, such as combinatorics, abstract algebra... That book by Shifrin does not have good reviews, have you read it? Does it meet the requirements of geometric thinking as you define it?

 

[CGandC]

I've studied physics with applied math and only recently I've started learning pure math. Whilst studying, I was being driven mad by not knowing what went wrong in some of my proofs and at some nights I would see nightmares about being stuck in a circular proof or being required to prove something immediately whilst I didn't have any clue how. Sometimes I wouldn't be able to sleep well just because I've been doing proofs all day. and I've been asking people on forums for help - one of the greatest advantages of the Internet. This has changed my life ( for the better ), I'm now a completely different person from what I've been before learning logic and proof theory, I think differently now ( Although I am an still kind of an anti-social introvert ). I wish I'd somehow study pure math back when I was a kid, maybe It'd change my life forever. I feel like pure math gave me new skills in reasoning and almost everywhere I look or do it feels like pure math is there. However, I think one cannot learn properly either Applied math or Pure math without the other - at least that's how I feel about it.

 

[homeomorphic]

I am going to have to come back from the dead, I guess, since this is up my alley.

One point is that there is a distinction between "geometry" and "visualization" and the terms are somewhat ambiguous. I will answer under the interpretation that both are desirable. At times, we may need to be flexible with regard to what our idea of geometry is. This flexibility can be motivated by trying to proceed from lower-dimensional or otherwise simpler examples before tackling the general case. When that approach is taken, you can use analogy to extend from the more visual cases to more generality.

For group theory, check out Nathan Carter's Visual Group Theory for a more visual approach. He emphasizes Cayley diagrams, as a way of visualizing the structure of a group. For more geometric subject matter, you might want to take a look at a subject where group theory is applied to a more geometric subject, such as Coxeter theory. For that, you might like the book, Mirrors and Reflections: The Geometry of Finite Reflection Groups. A nice class of examples of symmetry groups is the symmetry groups of Platonic solids, such as the icosahedron. John Baez had some interesting notes about that, but I couldn't dig them up right off the bat. Another interesting book is Abel's Theorem in Problems, which is based on the notes of V. I. Arnold, who was one of the top geometric thinkers up until his death several years ago.

Rings and fields are a little less geometric, but you might gain some insight by reading about Euclid's algorithm straight from Euclid himself in Book VII of The Elements. Keep in mind the problem of finding a greatest common length that would fit into two given lines evenly. Taking that line of thought into its modern form will illuminate a decent chunk of an introductory abstract algebra course, as far as commutative rings are concerned. Another insight that's often not presented in more abstract texts is the intuition behind the Chinese remainder theorem in terms of the problem of counting soldiers or similar examples. I think you can search math overflow or math stack exchange for that kind of discussion. Often, doing a search with the terms "intuition" or "motivation" will turn up what you are looking for.

The algebraic geometry perspective that expands on analytic geometry by studying zero sets of polynomials should also be enlightening here (for ring theory, in particular), but that is more Mathwonk's territory. There are definitely more beginner-friendly books on algebraic geometry out there, but I never had time to make it through any of them, and I forget which one I was trying to find the time to read a while back.

Another thing that could be mentioned is that there is also such a thing as "algebraic intuition" and "motivation". Those are also things, apart from visualization that will help build an understanding of algebra. A lot of that is knowing examples. Knowing a little bit of number theory, for example, will help to make the subject seem less like pie in the sky nonsense, dreamed up by mathematicians, for no other reason than that they thought it was more interesting than twiddling their thumbs (as presented by most modern algebra books). In that connection, you can gain a lot by reading the books of John Stillwell on the history of math or his Elements of Algebra. Often, he also has a more geometric approach than usual, although it's not his main focus, which is motivating things through their history.

Next, I'll say a couple things about fields. Those are classified according to their characteristic, which could either be the 0 (for extensions of the rationals), in which case, we're in more geometric territory already, or a finite field. This isn't a topic I know that much about, but there are some interesting connections with geometry lurking here. Problems like enumerating all the projective spaces over (look up the Fano plane, for example), or coding theory (which deals with, among other things, sphere-packing problems in the finite-field world) come to mind. Some concepts from good old-fashioned real or complex projective geometry may continue to apply to the case of finite fields.

Another topic would be linear algebra. I worked out most of the intuition for myself with a little help from professors (mostly filling in some gaps from my real analysis prof's explanations). One possible geometric approach could be to try to study the subject concurrently with computer graphics. You can use the visuals from studying the real number case to help remember and understand more general treatments that you'd find in Abstract Algebra, as long as you are careful and can keep straight what might not apply in a more general case (counter-examples help).

 

[homeomorphic]

Oh, by the way, my math book intuition is telling me that Mathwonk's book suggestion may be a good one in the long run, despite the negative reviews, partly because I trust his taste to a certain extent. However, the bad reviews suggest there are hurdles towards appreciating it, and it probably is best used as more of a supplemental source. Some books may be better for deepening your understanding of a subject that you already know than for learning it the first time. I tend to think a good book is a good book, but there are cases where that doesn't hold.

 

[mathwonk]

The fact you want high school level material is helpful. The notes i linked are higher level. For high schoolers, I suggest this: consider the graph of a cubic polynomial. The fact that the degree of the polynomial is 3, connects up to the fact that the maximum number of intersections of the graph with a straight line is 3. the degree of the polynomial is algebra, the number of intersections is geometry.

Notice that there are 5th degree polynomials which have at most 3 intersections with a straight line, a reflection of the fact that some intersections may have complex coordinates!

And may I say that the reason for the negative reviews of Ted's book is that it is hard to read for students with weak reading skills. But it has a lot to offer those who are willing to work at it. good luck!

 

[homeomorphic]

In case anyone is interested, today, I came across the Algebraic Geometry book for beginners that I didn't have time to get through: C.G. Gibson, Elementary Geometry of Algebraic Curves. It's hard to get more elementary than that as a starting point, but you would want to study linear algebra and multivariable calculus first, and then you could look up a few other facts as needed from abstract algebra. Should have some good examples of geometry meeting abstract algebra at a lower undergrad-level.

The thing about "high-school-level" or first-year university is that Abstract Algebra itself is typically considered above that level.

 

[symbolipoint]

The thing about "high-school-level" or first-year university is that Abstract Algebra itself is typically considered above that level.

Never been listed as such in any community college catalogs, at least none that I ever read. Abstract Algebra was never a course shown as any lower-division college or university course; but maybe this was a detail missed if not looking for actual course requirements for a Mathematics Major student. Does this depend on where in the world someone studies? Something of course offerings changed in the last decade?

 

[MidgetDwarf]

Delete please.

 

[homeomorphic]

A long time ago, I tutored a student who mentioned they had a special program where they learned group theory in high school (in the US). The book Abel's Theorem in Problems was originally taught to high school students in Russia. In terms of formal prerequisites, abstract algebra could be considered high school level, since I think the theorems are generally proven from scratch in a first course, but it's generally thought that students should have a little more mathematical maturity and examples under their belt before studying it. I'm not sure what the situation is in other countries.

It could be jumping the gun to want to jump straight into Abstract Algebra without first studying at least linear algebra, some calculus, maybe some set theory. And if you are someone who doesn't want to be blasted with abstract generalities right off the bat, it probably would be a good idea to throw in some number theory and complex analysis (which might have real analysis, in turn, as a prerequisite, which is typically considered more difficult than abstract algebra, but also somewhat less abstract, since it's pretty much just proving calculus). Complex is overkill, but the problem is people who haven't taken it might not have a decent understanding complex numbers, with which to understand certain concepts. Anyway, the overall point is, if you want things to be less abstract, it probably means you should wait even longer to study the more abstract subjects, rather than trying to jump straight in, the reason being that you will have a wider range of examples to draw from, rather than just having to take mathematician's word for it that the abstractions make sense.

 

[mathwonk]

It is hard to make general statements like this without some confusion. If I start an abstract algebra class with the definition of a group: namely, a set with an associative binary operation which has an identity and inverses , then few young persons will likely catch on.

But if I say: consider a cube, and imagine all ways of rotating it, so as to keep its center fixed, and so as to have each face wind up in the position of another face. How many such rotations are possible? Given any rotation, can you see how to rotate back to the original position?

Then I have a better chance of capturing the attention and imagination if a young student, especially if I display a model of a cube, or better hand out models to each student, and let them handle them and practice rotating them.

Moral: you can teach anything to any age student, if you express yourself in a language the student understands.

 

[zhenyong]

Not from the US. I have a Bachelor of Science in Physics and a Master of Engineering Science.

Mid-way through my undergraduate study, I realized that I am more interested in pure math than physics and wanted to apply for a PhD in pure math after my graduation. I had done research and written a thesis related to math (related to general relativity) for my undergraduate final year project under the supervision of a math lecturer. Unfortunately, after graduation I could not find a math research project for master’s degree and had to apply for a masters in another field. I ended up working on a project related to signal processing.

My undergraduate CGPA is not great (2.92/4.0). I worked hard in my master’s study and managed to publish a conference paper and a journal paper (one more currently requiring revisions for publication, and one more has not been submitted yet). This allowed me to graduate earlier. Now that I have completed my master’s degree, I want to apply for a PhD in pure math. I had consulted several lecturers/professors from my university as well as other universities at different countries, and I had been getting mixed opinions. I was told that transitioning to pure math is not that hard, I was told that my background is more qualified for applied math instead, some told me that I should consider PhD programs in the US since they generally offer coursework that can bridge the gaps in my background. I applied to five universities for PhD in applied math last year. I got rejected from four of them, the remaining one is still under review. This made me feel that my achievements in my master’s study are not helping me in the admissions at all.

But deep down, I am still more interested in pure math. Given my background, is it possible for me to get admitted to a PhD in pure math? I am aware that my background is not qualified for pure math. From what I have learned, it seems that I could either:

1. Take the GRE general and math subject test (uncertain due to the pandemic) and apply to grad schools in the US or
2. Apply to a master’s in pure math in other countries.

Did anyone have a similar experience? Or this is just a pipe dream?

 

[hackenabush]

A Path Less Taken to the Peak of the Math World, or an Unusual or Unlikely Path to Mathematics
https://www.quantamagazine.org/a-path-less-taken-to-the-peak-of-the-math-world-20170627/


Karim Adiprasito, June Huh, Eric Katz
Hodge Theory for Combinatorial Geometries
https://arxiv.org/abs/1511.02888
From the abstract, the authors state, "We prove the hard Lefschetz theorem and the Hodge-Riemann relations for a commutative ring associated to an arbitrary matroid M. . . ."

I went looking for more information on the Rota conjecture, and apparently there is more than one.
https://en.wikipedia.org/wiki/Rota's_conjecture

I was born hard of hearing and so for me school was not very easy. I failed every math course from elementary to high school. Once I got into one of the local universitiesI wanted to get better at math. So I took a course going back over the basics which really helped. After that I was able to complete math up to Calculus 2.

 

[symbolipoint]

I was born hard of hearing and so for me school was not very easy. I failed every math course from elementary to high school. Once I got into one of the local universitiesI wanted to get better at math. So I took a course going back over the basics which really helped. After that I was able to complete math up to Calculus 2.

I seriously like and understand your post there. Not everyone understands and would do as you did. You do see and have demonstrated to yourself, at least, that EFFORT is often too much undervalued.

 

[MidgetDwarf]

Not from the US. I have a Bachelor of Science in Physics and a Master of Engineering Science.

Mid-way through my undergraduate study, I realized that I am more interested in pure math than physics and wanted to apply for a PhD in pure math after my graduation. I had done research and written a thesis related to math (related to general relativity) for my undergraduate final year project under the supervision of a math lecturer. Unfortunately, after graduation I could not find a math research project for master’s degree and had to apply for a masters in another field. I ended up working on a project related to signal processing.

My undergraduate CGPA is not great (2.92/4.0). I worked hard in my master’s study and managed to publish a conference paper and a journal paper (one more currently requiring revisions for publication, and one more has not been submitted yet). This allowed me to graduate earlier. Now that I have completed my master’s degree, I want to apply for a PhD in pure math. I had consulted several lecturers/professors from my university as well as other universities at different countries, and I had been getting mixed opinions. I was told that transitioning to pure math is not that hard, I was told that my background is more qualified for applied math instead, some told me that I should consider PhD programs in the US since they generally offer coursework that can bridge the gaps in my background. I applied to five universities for PhD in applied math last year. I got rejected from four of them, the remaining one is still under review. This made me feel that my achievements in my master’s study are not helping me in the admissions at all.

But deep down, I am still more interested in pure math. Given my background, is it possible for me to get admitted to a PhD in pure math? I am aware that my background is not qualified for pure math. From what I have learned, it seems that I could either:

1. Take the GRE general and math subject test (uncertain due to the pandemic) and apply to grad schools in the US or
2. Apply to a master’s in pure math in other countries.

Did anyone have a similar experience? Or this is just a pipe dream?

Not necessarily. The not so good gpa hurts. Have you taken any pure math courses, or work through any books? Just from what you wrote, you may need to do post-baccularate in mathematics (pure), or take a few classes to meet the minimum requirements to get into a masters program in mathematics (US universities).

As it stands, you are more qualified to go onto an applied masters program, then a pure math MS program.

Both routes will require you to pay out of pocket and take courses. Public State Schools tend to be cheaper... Do you have the money to spend? Get a work visa and go to school/work full time? Sometimes conditional acceptance to a program is offered. Wont necessarily be a top school, but it would allow you to learn a bit, and hopefully you use it as opportunity to become a better student and make connections. Then, maybe onto a PhD program...

I believe I have messaged you with information of programs, including an applied math/engineering bridge program.

 

[mathwonk]

@zhenyong: I am out of the loop for a long time, but if you love math, I think you have a possible chance. That said, you don't really know if you love math until you learn some non trivial math. The foundational knowledge needed for graduate work in pure math, is linear algebra, with theory, and advanced calculus, several variables, including the implicit function theorem, which in my experience is actually quite rare.

But some books that are good background for graduate studies might include say Spivak's Calculus, and his Calculus on Manifolds, and maybe an algebra book like Mike Artin's Algebra. Oh yes, a topology book would help too, like something by Munkres. Have you read the early posts in this huge thread? going back to 2006 or so? This question is surely discussed at length there. Maybe in the very first post #1... Actually I recommend you at least skim the first 50-100 posts in this thread, roughly through page 2 or 4, or maybe a little more.

As to getting admitted to grad school, you mainly have to convince them you are qualified. I myself just walked into the math dept of University of Washington, Seattle, asked to take the PhD prelims, (after preparing of course for some months), passed them, and was offered admission, and a fellowship.

My point is, for a pure math PhD program, I think it matters more what you know, than what grades you have in undergraduate school. E.g. my undergrad GPA was mediocre, but I later learned that material by studying it and teaching it. However, when applying for lower level public school teaching jobs, I was asked for my undergrad GPA, and did not get those jobs. So when looking for positions, I recommend not to aim either too high or too low, if you want to be appreciated for what you have to offer. Maybe the distinction here is that admission to an educational program may hinge mainly on whether you can handle the work, while getting a job often requires holding a specific degree. I.e. even if you lack paper qualifications, you can sometimes convince people in person that you are qualified, assuming that you are, but that may not suffice to get certain jobs.

one possibly relevant comment about being admitted and even supported by a pure math Phd program: these programs often have a certain amount of money to give out to support students, and because pure math is not a particularly lucrative career, they often do not have enough qualified applicants to use up the available money. since they must give out this money every year or else lose it, they will sometimes admit simply the most qualified applicants they have, even if not extremely well qualified. Unfortunately this is sometimes bad for the admitted students at the bottom of the pile, because although they may be admitted, they may not succeed.

But I am just saying that admission to a pure math Phd program is somettimes easier than you might think because it is less competitive since the salary afterwards is lower. E.g. the school, Univ of Washington, where I was offered admission to the pure math PhD program based only on a test performance, nowadays does not even entertain applications to their undergraduate comp sci program from anyone out of state, no matter how well qualified. This is because those graduates can expect good salaries afterwards. It is possible they are still more accepting of pure math applicants for undergrad and grad programs, but I don't actually know that.

Good luck finding the right program for you at this point in your journey. A personal interview with someone sympathetic and knowledgeable, who can assess your background and potential, like a math professor, may be useful in identifying what that should be.

 

[Ritu bhabhi]

If you like to dedicate your entire life to mathematics, only then you should become a mathematician. But remember, probability dictates that our mind changes at an instant depending on many factors. Only go forth with mathematics if you think you have what it takes to be dazzled by its beauty.