Other Should I Become a Mathematician?

[Dougggggg]

One of my professors talks frequently about when he had finished the actual mathematical work of his PhD thesis and had solved it. He says something to the extent, "at that moment, I realized I knew something about Mathematics, well the world, that nobody else had ever figured out. It was a great feeling considering the age of the subject dates back before Christ (granted his area of study is a bit younger, Graph Theory)."

I want that moment...

 

[mathwonk]

Here is an example of the technique by B. Segre. the space of cubic surfaces in P^3 and the space of pairs (S,L) where S is a cubic surface and L is a line on S, both have dimension 19. the map (S.L)-->S has degree equal to the number of lines on a general cubic surface, which is in general finite. A reducible cubic made of a plane and a quadric has an infinite number of lines but if we add the data of 6 points on the conic where the plane meets the quadric we enlarge the space of cubic surfaces, adding some new points, but with the space still having dimension 19.

Now the preimage of the triple (S,L,p1,p2,...p6) is the set of lines that lie on the surface S = Q+L and also contain one of the 6 points. There are 12 such lines on the doubly ruled quadric and 15 on the plane, making degree = 27, which is well known to be the right answer. In his book on complex projective varieties Mumford shows this in the traditional way, by proving that every smooth cubic surface is a general enough point of the target, and then counting the 27 lines on the special smooth fermat surface.

 

[Robert1986]

Here's a question for those who are professional mathematicians. When you got your Ph.D. you, of course, wrote a dissertation. I have some questions about this:

1) How and when did you decide that you were going to write your thesis on the topic you picked? Did your advisor help? And if so, how did you pick your advisor?

2) Have you (and ask the same questions about your mathematician friends) stayed in this same general area of research, or have you done something completely different?The reason I ask is that I am graduating in the Fall and I plan (*if* I get in) to go to grad school to get a Ph.D. but one thing I am worried about is finding something to research.

 

[mathwonk]

i had three different advisors, who suggested several different problems. the first couple of problems were solved before me by stronger mathematicians.

the next two I solved, but it turned out they had already ben solved by others, unknown to me and my advisors.

the last one was hard for me but my advisor helped me get the idea. Then one day I heard a famous mathematician was working on it too, but I just tried harder, as hard as I could and I solved it first, by providing a new idea of my own.

I guess the moral of my story is to try to pick a supportive advisor, try to think of a problem that interests you. Best is if you find it yourself, perhaps no one else will be competing with you on it.

And don't give up.

 

[mathwonk]

By the way, these 2 weeks I am a teacher at a camp for brilliant children, called epsilon camp, in Colorado Springs. There are 28 kids aged 8-10 here taking 5 classes a day, and in mine we are going through the first 4 books of Euclid. That is a good chunk of the course I taught to college and grad students in 2009, over a whole semester.

These kids are amazing and I am having a blast. We have already done Euclid's original proof of Pythagoras, and learned to construct a regular pentagon. Tomorrow we will discuss how to do algebra geometrically, and I will try to present a new way to do similar triangles without numbers that I figured out just for this camp.

If you know kids this age, or older, these camps, epsilon camp, and math path, are great for very gifted kids aged 8-18 or so. Look them up on the web. My course notes are there in the student forum, but I guess you cannot access that. Maybe I will ask them to post them publicly, or I will just put them on my web page at UGA later. Of course if you are older and more advanced, and interested, the book by Robin Hartshorne, Geometry: Euclid and beyond, is much better. I got my start trying to emulate his course, and I learned from his book.

 

[qspeechc]

Hi mathwonk, can you recommend any Euclidean (classical?) geometry books that go beyond the stuff we learn in school? I.e. books that assume Euclids Elements and go further? I did note the Hartshorne book you recommend, any others? There is no specific thing I am interested in, I'd just like to know more about classical geometry.

Also, when are you going to start writting and publishing books?

 

[mathwonk]

I don't know what you learned in school, but I recommend starting with Euclid. I myself got a lot more from it than I got in school. A good place to begin is with Hartshorne, as he will refer to Euclid.

Another historical source used by Hartshorne is David Hilbert's Foundations of Geometry.

A nice little paperback that assumes Euclidean geometry and mentions some less well known results is Geometry Revisited, by Coxeter and Greitzer.

But Euclid is the best read for me, then Hartshorne. I recommend reading chapter one of Hartshorne, then Euclid, then continue with Hartshorne as the spirit moves you.

 

[mathwonk]

you will find a few books I have written on my web page at UGA, but they are not published.

 

[Nano-Passion]

you will find a few books I have written on my web page at UGA, but they are not published.

What are your books about? I want to read them if you can supply a link.

 

[qspeechc]

I don't know what you learned in school, but I recommend starting with Euclid. I myself got a lot more from it than I got in school. A good place to begin is with Hartshorne, as he will refer to Euclid.

Another historical source used by Hartshorne is David Hilbert's Foundations of Geometry.

A nice little paperback that assumes Euclidean geometry and mentions some less well known results is Geometry Revisited, by Coxeter and Greitzer.

But Euclid is the best read for me, then Hartshorne. I recommend reading chapter one of Hartshorne, then Euclid, then continue with Hartshorne as the spirit moves you.

I have read Edwin Moise's books Geometry and Elementary Geometry from an Advanced Viewpoint (but when I was an undergraduate).
Thanks for the suggestions.

 

[mathwonk]

moise's second book you mention is one i have seen. i found it more formal and less enjoyable than euclid but it is mathematically excellent. hartshorne (and also euclid) contains much more than moise and should be a lot more fun. but if you mastered moise you know a lot. i still recommend euclid and hartshorne. i think you'll be surprised just how much richer the subject seemed before the modern formalists got hold of it.

 

[qspeechc]

I don't doubt that. The 3-volume edition translated by Heath totals 1502 pages!

https://www.amazon.com/dp/0486600882/?tag=pfamazon01-20
https://www.amazon.com/dp/0486600890/?tag=pfamazon01-20
https://www.amazon.com/dp/0486600904/?tag=pfamazon01-20


Then there is the one volume edition, but edited by some Dana Densmore, at only 527 pages. I don't like the idea of some editor intruding on Sir Heath's work.

https://www.amazon.com/dp/1888009195/?tag=pfamazon01-20

 

[battousai]

I'm about to take an "Honors" Multivariable Calculus class this quarter, as a freshman. I took Calc BC as a senior last year. What should I be expecting?

The book that we'll be using is by Williamson and Trotter

https://www.amazon.com/dp/0130672769/?tag=pfamazon01-20

 

[Kevin_Axion]

Can someone give me a thorough explanation of the differences between applied mathematics and pure mathematics? A school I wish to attend offers two programs which can be found here: http://www.artsandscience.utoronto.ca/ofr/calendar/prg_mat.htm . I'm interested in pursuing research in fields such as neuroscience, and A.I. Which degree is the better approach?

I'm thinking the applied math degree because there are more rigorous probability and computer science courses then the pure mathematics course. For anyone who can't find the information I'll write it out:

Mathematics Specialist

First Year:
Analysis 1, Algebra 1, Algebra 2

Second Year:
Analysis II, Advanced ODE

Third and Fourth Years:

1. Intro to Topology, Groups Rings and Fields, Complex Analysis I, Real Analysis I

2. One of: PDEs; Real Analysis I (Measure Theory)/(Real Analysis I, Real Analysis II)

3. Three of: Combinatorial Methods; Intro to Mathematical Logic, Intro to Differential Geometry, ANY 400-level APM/MAT

4. 2.5 APM/MAT including at least 1.5 at the 400 level (these may include options above not already chosen)
5. Seminar in Mathematics


Applied Mathematics Specialist

First Year:
Analysis I, Algebra I, Algebra II; (Intro to Comp Programming/Intro to CS)/Accelerated Intro to CS

Second Year:
Analysis II, Advanced ODE; Intro to Scientific, Symbolic and Graphical Computation; (Probability and Statistics I, Probability and Statistics II)

Third and Fourth Years:
1. PDEs; Intro to Topology, Groups Rings and Fields, Complex Analysis I, Real Analysis I, Intro to Differential Geometry; Probability

2. At least 1.5 full courses chosen from: Intro to Graph Theory, Intro to Combinatorics, Complex Analysis II, Measure Theory/(Real Analysis II), Differential Geometry; Data Analysis, Time Series Analysis; Numerical Algebra and Optimization, Numerical Approx., Integration, and ODE, Computational PDEs, High-performance computing

3. Two courses from: Mathematical Foundations of QM, General Relativity, Fluid Mechanics, Asymptotic and Perturbation Methods, Applied Non-linear Equations, Combinatorial Methods, Mathematical Finance, Seminar in Mathematics

Thanks!
4. MAT477Y1

 

[Bourbaki1123]

If you're into A.I. you might want some more computer science and statistics (for machine learning) in there, so if you get a minor with one of those programs I would do the second one and minor in comp sci or statistics if possible. Also, you should probably take the graph theory course when you get the option.

 

[mathwonk]

i found heath's scholarly commentary on euclid somewhat tedious. i suggest beginning with the green lion edition of the elements which i think uses heath's translation but omits the extra stuff. the unaltered original is always best.

@battousal: Williamson and Trotter is a wonderful book.

 

[mathwonk]

my class notes from the 2 week epsilon camp course in Euclid's Elements for very bright 8-10 year olds are now up on my web page at UGA.

 

[weld]

How tough is competition to become a perma faculty member at a third tier uni? Are there tons of brilliant postdocs to compete with? How many, 2, 5, 10, 50? How brilliant, just good or very good?

Also, how much time in % do you estimate is spent on doing non-research as a postdoc? Such as lecturing, teaching grad studs, administration, grant writing? I imagine you get like 80% of the time to research, the rest goes to other stuff? Is math particularly different in this regard compared to other fields such as CS, theoretical physics, etc?

 

[Bourbaki1123]

How tough is competition to become a perma faculty member at a third tier uni? Are there tons of brilliant postdocs to compete with? How many, 2, 5, 10, 50? How brilliant, just good or very good?

Also, how much time in % do you estimate is spent on doing non-research as a postdoc? Such as lecturing, teaching grad studs, administration, grant writing? I imagine you get like 80% of the time to research, the rest goes to other stuff? Is math particularly different in this regard compared to other fields such as CS, theoretical physics, etc?

80% might actually be high based on what a couple of my professors have said about not having enough time to work on research. Of course one of them was dean of graduate admissions so he might have had a slightly skewed view of things.

 

[mathwonk]

if you have some ability and do your best you will ultimately be successful.

 

[Kreizhn]

Can someone give me a thorough explanation of the differences between applied mathematics and pure mathematics? A school I wish to attend offers two programs which can be found here: http://www.artsandscience.utoronto.ca/ofr/calendar/prg_mat.htm . I'm interested in pursuing research in fields such as neuroscience, and A.I. Which degree is the better approach?

I have more than a little trepidation about replying to the first part, explaining the difference between applied and pure mathematics. Indeed, almost certainly I'm going to end up stepping on somebody's toes over this.

Essentially pure mathematics is the rigorous development and exploration of mathematical topics and consequences without the worry of applications. Of course, in this sense one needs to be wary of the use of the word "application," something I believe Hardy goes to painful lengths to clarify in his Mathematician's Apology. The expectation of a high level of rigour and a motivation of abstraction are central themes.

Applied mathematics on the other hand focus on applying (surprise) mathematics to the world. Proofs and rigour can still be found, though tend to be less prevalent than in pure mathematics. At higher levels of applied maths research, pure mathematics is still used extensively. The difference is that research in pure math is to advance the mathematical field, while research in applied math is to use those tools to discover something about nature.

If neuro and/or AI are what you really want to do, then applied mathematics is probably the best route. As Bourbaki1123 mentions, comp sci would be very useful for AI, as well as studying combinatorics (of which graph theory may be taken as a subfield). For neuroscience you'll likely want computational experience, as well as extensive exposure to differential equations (ordinary, partial, stochastic, all of 'em).

I'm thinking the applied math degree because there are more rigorous probability and computer science courses then the pure mathematics course. For anyone who can't find the information I'll write it out:

Probability theory can be taught from essentially a measure theory standpoint alone, and can be abstracted as a pure math. However, sometimes learning stuff too abstractly can make it difficult to apply.

 

[Bourbaki1123]

Anyone here to answer for this? pls do it.

The answer lies in your heart.

 

[simplicity123]

What Mathematics do you need for Algebraic topology?

As I plan to focus heavily on topology and algebraic topology this year. I need to get it down as most of motivation for category theory comes from Algebraic topology. I read that group theory is the algebra used in algebraic topology. Group theory is the worst branch of Mathematics along with metric spaces.

 

[Number Nine]

What Mathematics do you need for Algebraic topology?

As I plan to focus heavily on topology and algebraic topology this year. I need to get it down as most of motivation for category theory comes from Algebraic topology. I read that group theory is the algebra used in algebraic topology. Group theory is the worst branch of Mathematics along with metric spaces.

You may not like it, but group theory is hugely, massively, unbelievably important in just about every branch of mathematics. A lot of people find algebra dry because (to use the computer parlance) it's basically the machine language of math, but even a little knowledge of algebra goes a long way.

You heard correctly about group theory and topology. To give you an example: Just like algebra talks about isomorphisms between groups and considers isomorphic groups to be "the same" in an algebraic sense, topology concerns itself with homeomorphisms between topological spaces for similar reasons. The problem is that actually determining whether different kinds of spaces are homeomorphic is difficult, so topologists look for characteristics of topological spaces that are the same for all spaces that are homeomorphic (if these qualities are different, you know that the spaces are not homeomorphic). These qualities are said to be invariant under homeomorphism (sorry if you know all of this already, by the way). Algebraic topology introduces an invariant called the fundamental group of a topological space, which turns out to be incredibly useful in classifying spaces. The theory underlying the fundamental group, obviously, involves (in a sense) attaching a group structure to space through certain methods (look up "homotopy" if you're interested), and turns out be useful in describing all manner of qualities in topology.

You may not like group theory, but if you topology (or category theory) interests you, you won't regret learning it.

 

[simplicity123]

You may not like it, but group theory is hugely, massively, unbelievably important in just about every branch of mathematics. A lot of people find algebra dry because (to use the computer parlance) it's basically the machine language of math, but even a little knowledge of algebra goes a long way.

Group theory isn't that bad. I suppose don't like number theory aspects of it, however I've gotten used to it now. Is galois theory important for algebraic topology, as after I learn't group theory I could go onto that and take a course in it alongside algebraic topology.

You heard correctly about group theory and topology. To give you an example: Just like algebra talks about isomorphisms between groups and considers isomorphic groups to be "the same" in an algebraic sense, topology concerns itself with homeomorphisms between topological spaces for similar reasons. The problem is that actually determining whether different kinds of spaces are homeomorphic is difficult, so topologists look for characteristics of topological spaces that are the same for all spaces that are homeomorphic (if these qualities are different, you know that the spaces are not homeomorphic). These qualities are said to be invariant under homeomorphism (sorry if you know all of this already, by the way). Algebraic topology introduces an invariant called the fundamental group of a topological space, which turns out to be incredibly useful in classifying spaces. The theory underlying the fundamental group, obviously, involves (in a sense) attaching a group structure to space through certain methods (look up "homotopy" if you're interested), and turns out be useful in describing all manner of qualities in topology.

I do understand what you mean. However, it's only really the number theory aspects of groups I hate, isomorphism isn't that bad. Thats why I like topology so much as you can have two objects that look totally different and they are the same topologically. I'm not going to learn Algebraic topology until after Christmas, but make sure I will look up homotopy then. Thanks for the description, even through can't understand most of it as don't know what homotopy is.

You may not like group theory, but if you topology (or category theory) interests you, you won't regret learning it.

What Maths do I need to learn to do Algebraic Geometry?

As I plan to study heavy algebra=group theory+commutative algebra+algebraic topology+lie algebra. However, wondering do you need analysis like functional analysis to do AG? As I remember reading a book and it was saying to study AG you needed to know sheaf theory, complex analysis, differential geometry.

 

[Number Nine]

What Maths do I need to learn to do Algebraic Geometry?

As I plan to study heavy algebra=group theory+commutative algebra+algebraic topology+lie algebra. However, wondering do you need analysis like functional analysis to do AG? As I remember reading a book and it was saying to study AG you needed to know sheaf theory, complex analysis, differential geometry.

Sheaf theory is huge in AG since it's used to understand schemes, which are basically what AG is all about. AG is such a massive and fundamental subject that just about anything you learn will be helpful, but I'd mainly focus on a strong background in algebra (obviously), and maybe trying to pick up a bit of a background in projective geometry (which is interesting enough on its own). If you're interested in a bit of "recreation", I recommend the book Conics and cubics: An introduction to algebraic curves. It's a very elementary text, but you probably haven't encountered the subject before and it's very relevant to algebraic geometry.

 

[mathwonk]

algebraic geometry is the study of solution sets of polynomial equations. these sets are defined algebraically, and possesses a topology and, if the coefficients are complex numbers, also a complex analytic structure. Moreover they occur naturally embedded in projective space. Thus one can use any of those points of view to study them, algebra, projective geometry, algebraic and differential topology, and complex analysis.

Sheaves are a type of coefficients for cohomology groups that were first introduced to study complex manifolds and spaces, and later applied to algebraic varieties.

In the history of algebraic geometry, the first major step far forward was taken by Riemann, who desingularized plane algebraic curves and then studied the complex manifold structure on the desingularization, as well as the possible ways to re embed that manifold back into projective space as an algebraic variety.

Since that time over 150 years ago, the tools of differential forms and homology theory have been essential to the study of algebraic varieties. Before that time, only the simplest curves such as conics could be well studied.

One can begin to study algebraic geometry without knowing all these tools, by looking at examples and seeing gradually the need for more powerful techniques. For this reason of motivation, it is thus recommended to begin with elementary objects such as plane curves, or the Riemann theory of transforming those into complex analysis as "Riemann surfaces".

Beginning books, requiring few tools, include Undergraduate algebraic geometry by Miles Reid, and Riemann surfaces and algebraic curves by Rick Miranda, as well as Basic algebraic geometry by Shafarevich.

Two useful topics often omitted from undergraduate courses in algebra and field theory are the concepts of transcendence degree and integral extensions. Tr.deg. is crucial in algebraic geometry as it plays the role of dimension, and integrality is the ring theoretic version of an algebraic extension of fields.

Studying sheaf theory before plane curves is like studying calculus before plane geometry. Of course both these phenomena do occur in our strange world. I have attached a pdf file: "naive introduction to alg geom".

 

[Bourbaki1123]

algebraic geometry is the study of solution sets of polynomial equations. these sets are defined algebraically, and possesses a topology and, if the coefficients are complex numbers, also a complex analytic structure. Moreover they occur naturally embedded in projective space. Thus one can use any of those points of view to study them, algebra, projective geometry, algebraic and differential topology, and complex analysis.

Sheaves are a type of coefficients for cohomology groups that were first introduced to study complex manifolds and spaces, and later applied to algebraic varieties.

In the history of algebraic geometry, the first major step far forward was taken by Riemann, who desingularized plane algebraic curves and then studied the complex manifold structure on the desingularization, as well as the possible ways to re embed that manifold back into projective space as an algebraic variety.

Since that time over 150 years ago, the tools of differential forms and homology theory have been essential to the study of algebraic varieties. Before that time, only the simplest curves such as conics could be well studied.

One can begin to study algebraic geometry without knowing all these tools, by looking at examples and seeing gradually the need for more powerful techniques. For this reason of motivation, it is thus recommended to begin with elementary objects such as plane curves, or the Riemann theory of transforming those into complex analysis as "Riemann surfaces".

Beginning books, requiring few tools, include Undergraduate algebraic geometry by Miles Reid, and Riemann surfaces and algebraic curves by Rick Miranda, as well as Basic algebraic geometry by Shafarevich.

Two useful topics often omitted from undergraduate courses in algebra and field theory are the concepts of transcendence degree and integral extensions. Tr.deg. is crucial in algebraic geometry as it plays the role of dimension, and integrality is the ring theoretic version of an algebraic extension of fields.

Studying sheaf theory before plane curves is like studying calculus before plane geometry. Of course both these phenomena do occur in our strange world. I have attached a pdf file: "naive introduction to alg geom".

Mathwonk, I'm sort of fishing for motivation to some extent, but I'm wondering how much time (average hours per day/number of months/years) it took for you (as best you can recall) to go from a basic level of understanding of some notions in commutative algebra (part way through Atiyah McDonald or Zariski Samuel or what have you) to a strong, or at least reasonably solid, grasp of the Grothendiek approach and the fundamental results/areas of the field.

 

[mathwonk]

my personal history is probably not relevant but may be instructive anyway.

i began as a star high school math student in the south and got a merit scholarship to harvard. as an undergrad at harvard i could not easily adjust to the need to study everyday and flunked out. i returned and worked hard at studying and attending class and made A's by memorizing proofs in advanced calculus and real analysis and got into brandeis.

I knew almost nothing of algebra, commutative or otherwise, but hung in for a while on talent and tenacity until asked to leave brandeis too.

then i taught for four years and studied differential topology and advanced calculus and returned to grad school at utah. there i studied several variable complex analysis for one year and returned to riemann surfaces the second year.

i wrote a thesis in riemann surfaces and moduli and took a job at UGA. Then I worked hard at learning as much algebraic geometry as possible. i still knew relatively little commutative algebra (and still don't).

i made a living off my grasp of mostly several complex variables, also differential topology, and algebraic topology.

after my third year I went to harvard again as a postdoc and devoted myself to every word dropping from the lips of mumford, griffiths, and hironaka.

those two years gave me a tremendous boost. then i returned to UGA and benefited enormously from collaboration withf my brilliant colleague Robert Varley.

I still hope to master commutative algebra.

 

[sentient 6]

sorry to be off topic, but I was just wondering if anybody had any book suggestions for an introduction to number theory. I have been thinking of getting G.H. Hardy's Intro, but I thought it'd be good to ask before investing.