Other Should I Become a Mathematician?

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Becoming a mathematician requires a deep passion for the subject and a commitment to problem-solving. Key areas of focus include algebra, topology, analysis, and geometry, with recommended readings from notable mathematicians to enhance understanding. Engaging with challenging problems and understanding proofs are essential for developing mathematical skills. A degree in pure mathematics is advised over a math/economics major for those pursuing applied mathematics, as the rigor of pure math prepares one for real-world applications. The journey involves continuous learning and adapting, with an emphasis on practical problem-solving skills.
  • #3,271


I have wanted to be a pure mathematician since I was about 15. I thought I was quite good at mathematics till I got to university and saw how talented some of my fellow classmates are. For instance many of them do no study besides attending class, and spend most of their time socialising/drinking/gaming yet still out perform me in many assesment tasks. This is quite frankly soul crushing as I spend most of my time thinking about mathematical things and it really highlights my lack of natural ability.

The thing is I still love the subject. Yet everytime one of my cocky friends tells me how well he has done in the latest exam or assignment I feel crushed and betrayed my the subject I love so much. I used to believe that genius was 90% Hardwork, but now I see that most of these sayings are just political correctness gone wrong.

Is it possible to even get a phd let alone tenure in pure mathematics when you lack that "spark" of genius?

I see you keep ignoring what I am telling you about knowing the tricks of the trade and of learning in general.

Part of your problem seems to be that you are thinking about different things than your friends. They problem just hit the homework and don't really question things, whereas you question things.

Here's one of the tricks I had up my sleeve in undergrad that I still use when I can, that, in particular, explains how I would avoid having to study much outside class sometimes (except to come up with my own explanations of things where necessary). I didn't take notes. Instead, during the lecture, I kept repeating what was said in the lecture up to that point in time in my mind. Always summarizing the lecture in my mind, while continuing to listen to the next part. A few things I didn't understand, I would set aside for meditating on later. After the lecture, I would go over it in my mind again. Sometimes, this meant I was already done studying after my session of reflection on the lecture was over. I would do it during the day while I was going about my business, eating, driving, etc. The usual rate of retention from lectures is 10%, whereas, if I concentrate, I can often recall 100% of the content (though not the specifics of how it was delivered). Even two years from now, if I so desire, just by a little review as necessary. If I wanted to, I could rehash the entire lecture. This would work best if the lecture was fairly conceptual in nature, and thus more memorable.

Another trick that I had was just reading Visual Complex Analysis. I think if someone reads it and understands a good portion of that book and its message, they would outperform someone with otherwise equal ability by a long shot in all their subsequent classes.
 
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  • #3,272


homeomorphic said:
I see you keep ignoring what I am telling you about knowing the tricks of the trade and of learning in general.

Part of your problem seems to be that you are thinking about different things than your friends. They problem just hit the homework and don't really question things, whereas you question things.

Here's one of the tricks I had up my sleeve in undergrad that I still use when I can, that, in particular, explains how I would avoid having to study much outside class sometimes (except to come up with my own explanations of things where necessary). I didn't take notes. Instead, during the lecture, I kept repeating what was said in the lecture up to that point in time in my mind. Always summarizing the lecture in my mind, while continuing to listen to the next part. A few things I didn't understand, I would set aside for meditating on later. After the lecture, I would go over it in my mind again. Sometimes, this meant I was already done studying after my session of reflection on the lecture was over. I would do it during the day while I was going about my business, eating, driving, etc. The usual rate of retention from lectures is 10%, whereas, if I concentrate, I can often recall 100% of the content (though not the specifics of how it was delivered). Even two years from now, if I so desire, just by a little review as necessary. If I wanted to, I could rehash the entire lecture. This would work best if the lecture was fairly conceptual in nature, and thus more memorable.

Another trick that I had was just reading Visual Complex Analysis. I think if someone reads it and understands a good portion of that book and its message, they would outperform someone with otherwise equal ability by a long shot in all their subsequent classes.

Homeomorphic I am not ignoring you, I just cannot apply your techniques to my life. If I may say so you have an exceptional memory and mathematical ability, even if I do the things you mention it is to no avail. I cannot recall an entire lecture for the life of me, I will forget the intracies of a proof as soon as I finish reading it, regardless if it is in a book or on a blackboard. I think you just have the spark of genius which I spoke of, something I will never have, so there is no point in me trying to follow your advice (I have been trying over the last few days with no success). I wish I could just walk away from mathematics, to become a Doctor or an engineer or something less intellectually ambitious but whenever I have considered it, it has left me feeling empty.
 
  • #3,273


Group_Complex said:
I have wanted to be a pure mathematician since I was about 15. I thought I was quite good at mathematics till I got to university and saw how talented some of my fellow classmates are. For instance many of them do no study besides attending class, and spend most of their time socialising/drinking/gaming yet still out perform me in many assesment tasks. This is quite frankly soul crushing as I spend most of my time thinking about mathematical things and it really highlights my lack of natural ability.

The thing is I still love the subject. Yet everytime one of my cocky friends tells me how well he has done in the latest exam or assignment I feel crushed and betrayed my the subject I love so much. I used to believe that genius was 90% Hardwork, but now I see that most of these sayings are just political correctness gone wrong.

Is it possible to even get a phd let alone tenure in pure mathematics when you lack that "spark" of genius?

Every field requires a particular skillset where you need to have some kind of exceptional ability in a few particular things.

Everybody has particular things that they are good at: some are great at dealing with people and can understand what makes people tick but they are horrible at analyzing situations devoid of emotion or personality, while others can look at something objectively in a kind of brutally honest manner but may not really understand other people that well.

The point I'm trying to make is that there are many things that have different requirements and we do have quite a lot of different avenues to pursue.

If you are not exceptional in one thing and you don't go on to that thing, don't take it personally: find the place where you can really do your thing well and become good at that.

Also I want to say that if you are absolutely set on doing a particular thing, then just remember that you can be flexible and pursue the options that are very close to that thing so much that it's hard to differentiate in many respects.

There are tonnes of careers that utilize the same kinds of skills and provide the same kinds of work that the one you originally envisioned that you haven't already thought about, and you may be surprised at how enjoyable those may be.

I would talk to as many people as you can about different options and get a feel of the kinds of people and skills that they employ and consider those options that are as close to what you are set on so that at least these things give you something to think about.
 
  • #3,274


Homeomorphic I am not ignoring you, I just cannot apply your techniques to my life. If I may say so you have an exceptional memory and mathematical ability, even if I do the things you mention it is to no avail. I cannot recall an entire lecture for the life of me, I will forget the intracies of a proof as soon as I finish reading it, regardless if it is in a book or on a blackboard. I think you just have the spark of genius which I spoke of, something I will never have, so there is no point in me trying to follow your advice (I have been trying over the last few days with no success). I wish I could just walk away from mathematics, to become a Doctor or an engineer or something less intellectually ambitious but whenever I have considered it, it has left me feeling empty.

I may have an exceptional memory and good, but not great mathematical ability, but how did I get there? A lot of people outperform me in classes, too, all the time. True, in some of my undergrad classes, I was way ahead of anyone else. In grad school, I just feel retarded all the time, compared to the best students, and especially compared to the professors.

If reviewing stuff isn't working, try reviewing it every day. The key is that you have to practice remembering. The main concept is that if you want to remember, you have to try to recall things WITHOUT LOOKING. Actually, maybe it would be easier to try to apply some of things in subjects other than math first. In math, it's compounded by the difficulty of being able to conceptualize well. I don't know if you like languages. Try to just start small. Take one Spanish (or your favorite language) word, and try to focus on that one word. Just be determined that you will never forget it. Review it every day. You'll never forget it. I think everyone has the ability to put facts into long term memory. Think about it. There are some things you just don't forget. Why? What is it that makes those things memorable?

This might be an eye-opener.

http://www.ted.com/talks/joshua_foer_feats_of_memory_anyone_can_do.html

Note that this kind of memory isn't that useful in math because I think understanding is more important. However, it is a big hint as far as what is possible.
 
  • #3,275


chiro said:
Every field requires a particular skillset where you need to have some kind of exceptional ability in a few particular things.

Everybody has particular things that they are good at: some are great at dealing with people and can understand what makes people tick but they are horrible at analyzing situations devoid of emotion or personality, while others can look at something objectively in a kind of brutally honest manner but may not really understand other people that well.

The point I'm trying to make is that there are many things that have different requirements and we do have quite a lot of different avenues to pursue.

If you are not exceptional in one thing and you don't go on to that thing, don't take it personally: find the place where you can really do your thing well and become good at that.

Also I want to say that if you are absolutely set on doing a particular thing, then just remember that you can be flexible and pursue the options that are very close to that thing so much that it's hard to differentiate in many respects.

There are tonnes of careers that utilize the same kinds of skills and provide the same kinds of work that the one you originally envisioned that you haven't already thought about, and you may be surprised at how enjoyable those may be.

I would talk to as many people as you can about different options and get a feel of the kinds of people and skills that they employ and consider those options that are as close to what you are set on so that at least these things give you something to think about.

My interests are in academic mathematics, I would not be happy working in any other capacity.
 
  • #3,276


hi groupcomplex

- The thing is I still love the subject. Yet everytime one of my cocky friends tells me how well he has done in the latest exam or assignment I feel crushed and betrayed my the subject I love so much. I used to beliee that genius was 90% Hardwork, but now I see that most of these sayings are just political correctness gone wrong.

Well, there is exceptional talent, and then there are people who do put in 45 hours a week and get outstanding grades too, and then there are the other 80% of people...where anything can and does happen.

You can do 75% of anything the 'talented' people do, if you put in the hours, and slowly climb the ladder, mastering course after course... Mathwonk made some comments about this months ago, and i was quiet surprised at all the hope and enthusiasm he offered for people who struggle, or don't feel they got any natural talent for stuff. [Maybe someone can find the message number for that one]


Another thing is, for some of these students, they might seem leaps and bound ahead of you, but that isn't any guarantee they will be choosing math as a career or that they might do less well later on, or stop where you'll be taking way more math classes than they are. Depth is important, as well as knowing the ideas [especially in physics], and in some ways that may be more important than the problem solving long term, or talent.

And did these people read the subject beforehand? or do they just focus on the absolute minimum of essentials for good grades, with some talent and some studying...

here it could be they got a different box of tools, they got a toolkit months or years before you started yours in the first week of classes...

or they study differently and work the problems differently... or who knows...--------hi homeomorphic

- Here's one of the tricks I had up my sleeve in undergrad that I still use when I can, that, in particular, explains how I would avoid having to study much outside class sometimes (except to come up with my own explanations of things where necessary). I didn't take notes. Instead, during the lecture, I kept repeating what was said in the lecture up to that point in time in my mind. Always summarizing the lecture in my mind, while continuing to listen to the next part. A few things I didn't understand, I would set aside for meditating on later.

I knew one math teacher, who actually said for people to stop scribbling with notetaking and just follow his lectures and absorb it, and he said that he'd be following the textbook closely so there's no need for 'notes'. He didnt say that to all his classes though...

As long as you're reading the book and doing the problems, i think it should work...
It wouldn't work in classes with 'no text' like in the days of Oppenheimer's Quantum courses where people were rushing to copy down the blackboard and it was near impossible to catch up before he would erase stuff and go on...[Schiff's book in the 40s 50s was said to be largely based on those] Those were the days of notetaking!

Personally i think in many cases, notetaking is done, because the teacher didnt pick a deep enough textbook, or he didnt add 1-3 other supplementary texts to make that unnecessary, or he didnt toss out photostats of outlines and notes and summaries to his classes so they don't *need* to take notes.

Often i would find that there was a 50s 60s or 70s textbook that explained things more like the teacher's style and if you browsed that book with the 'curriculum/syllabus gunk', you'd probably get 70% more out of the damn class.

Where there are 'notes' there is somewhere in the uni-curse, an older textbook that said it better and waaaaaaaay less sloppy!and some classes, are geared where the teacher wants to 'demonstrate' and then you read the text, and a lot of others where you read pages xx to xx, and then you come to class.
 
  • #3,277
try not to get suckered by that game of " my friend never studies but got a higher score than me".

so what? do the subject for the joy of learning it. if need be make some new friends with a better attitude toward learning.

and eventually those people who do not study will fail, no matter how brilliant.
 
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  • #3,278


@mathwonk,

1) Do you think Brandeis being a "lesser" math dept. should be a concern re: their certificate program?

2) Thoughts on the math GRE? It sounds like that's pretty relevant given that it provides some kind of objective measure, no?

3) Thoughts on what I could do during the 9 months between applying to grad school and matriculating?

Thanks again for all the help,
Mariogs
 
  • #3,279


Has anyone worked with any of K.A. Stroud's texts? Particularly, I'm asking about his Vector Analysis text.
 
  • #3,280


I think there's 12 books in the Stroud and Booth and they use the 1960s Programmed Instruction style of teaching [which half the time is done right]

Engineering Mathematics came out about 1970 and it's like in a fifth and sixth edition recently, and I'm not sure if the artwork or newest changes are for the better...

-------
some comments in my notes

[both Engineering Mathematics and Advanced Engineering Mathematics are a great help for Differential Equations in ways Boyce and DiPrima are not]

[remarkable work]

[I have studied numerous mathematics texts, and I can say with absolute certainty that this is the finest mathematics text I have ever found.Unlike virtually every other technical math book out there(calculus, differential equations, integral equations, statistics etc)this book provides more than the dreary, boring, purely analytic approach (algebra,limits etc) that tends to practically wipe out true understanding. In my calculus class I hear questions whose answers are extremely masked by the highly esoteric mathematical bull@#$@, but which present themselves so easily with a simple picture. This book provides those pictures, but more remarkably it is written in such a way that people want to work through it - compared to those other books. In addition, this book has been shown (via experimentation) to significantly increase test scores - compared to standard lecture approach.]

---------
Linear Algebra
Differential Equations
Vector Analysis - 2005 - 448 pages
Complex Variables
Engineering Mathematics - Sixth Edition - 1200 pages
Advanced Engineering Mathematics - Fourth Edition - 1280 pages
Essential Mathematics for Science and Technology/Foundation Mathematics
Further Engineering Mathematics
Laplace Transforms
Fourier Series and Harmonic Analysis
Mathematics for Engineering Technicians
Engineering Mathematics Through Applications - Kuldeep Singh [only book in the series not written by Stroud and Booth]
---------
so it's a set of 12 books which started in 1970...

and from what i gather, it belongs up there with Schaums, and REA, and now Stroud...
--------

There was an early early 70s [seemed about 1967-1971] that was a paperback 5 Volume set on learning calculus by programmed instruction i saw once in the library that looked great, anyone remember the author, or the publisher, exact name, or hell, comments about that one?

programmed instruction books are rare, some are well done, and it's really a lot of work to do it properly, and i always cringe when people say oh computers do it better and stuff, but stroud is one of the better ones out there that is still in print, and still well liked.

i got one spiral bound one for electrodynamics for like a physics 12 or college physics course, but i aint seen any others...Hope this is helpful...

can't offer a detailed criticism though...
 
  • #3,281
Brandeis a "lesser" math dept? lesser than what? it certainly offered me more than i could handle from Monsky, Mayer, Brown, Buchsbaum, Palais, Matsusaka, H. Levine, J. Levine, Rossi, Auslander, Seeley, Spivak, Sherman, Wells, Vasquez, ...and I have no reason to believe it has slipped from those days, even if I do not personally know most of the young people there today. It is ranked around #40 by US News but the problem with such rankings is that Brandeis is better as a math dept than US News is as a magazine.
 
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  • #3,282


Does anyone have a list of undergrad textbooks used at Brandeis?
I didnt see much of a syllabus offered to 'non-students'...

It's always a shame since i always judge schools by how 'accessible' their syllabus is and how far back they show it, or keep their old course stuff 'online'... It's always been a surprise for years now on the web that schools and teachers always wipe the old slates clean and then put up the ominous:
'Textbook: TBA'

I'm always on the search for schools where they use older oddball textbooks, or dump a ton of "suggested" texts after the one required one...For me, the syllabus is the key to it all, if i get funny vibes, i run for the hills...
once you got that, then i think the teachers, school reputation, etc etc counts.
For others, most any texts will do, as long as you get dumped with 'extra' homework and challenges... [which i think works maybe for the higher up classes...]
 
  • #3,283


Well take it with a grain of salt, but in my spare time a whlle i mixed up a bunch of rankings for unis just for my own fun...

anyhoo Brandeis is probably in the top 40 schools for higher math...for people who like lists...
here we go:
201 Brandeis University - Waltham, MA
[#64 Top End Physics]
[#40 Top End Mathematics]
[#68 Chemistry Top End]

------
8 Princeton University - [#1 Top End Mathematics]
1 Harvard University - [#2 Top End Mathematics]
2 Stanford University - [#2 Top End Mathematics]
3 University of California, Berkeley - [#2 Top End Mathematics]
5 Massachusetts Institute of Technology - [#2 Top End Mathematics]
9 University of Chicago - [#6 Top End Mathematics]
6 California Institute of Technology - [#7 Top End Mathematics]
11 Yale University - [#7 Top End Mathematics]
7 Columbia University - [#9 Top End Mathematics]
22 University of Michigan, Ann Arbor - [#9 Top End Mathematics]
32 New York University - [#9 Top End Mathematics]
-----
13 University of California, Los Angeles - [#12 Top End Mathematics]
12 Cornell University - #13 Top End Mathematics]
17 University of Wisconsin–Madison - [#14 Top End Mathematics]
38 University of Texas at Austin - [#14 Top End Mathematics]
69 Brown University - Providence, RI - [#14 Top End Mathematics]
28 University of Minnesota, Twin Cities - [#17 Top End Mathematics]
15 University of Pennsylvania - [#18 Top End Mathematics]
25 University of Illinois at Urbana-Champaign - [#18 Top End Mathematics]
30 Northwestern University - Evanston, IL - [#18 Top End Mathematics]
-----
19 Johns Hopkins University - [#21 Top End Mathematics]
31 Duke University - Durham, NC - [#21 Top End Mathematics]
37 University of Maryland, College Park - [#21 Top End Mathematics]
14 University of California, San Diego - [#24 Top End Mathematics]
16 University of Washington - [#24 Top End Mathematics]
55 Rutgers University - Piscataway, NJ - [#24 Top End Mathematics]
167 State University of New York at Stony Brook - Stony Brook, NY - [#24 Top End Mathematics]
39 University of North Carolina at Chapel Hill - [#28 Top End Mathematics]
45 Pennsylvania State University-University Park - [#28 Top End Mathematics]
67 Purdue University - West Lafayette, IN - [#28 Top End Mathematics]
93 Indiana University - Bloomington, IN - [#28 Top End Mathematics]
99 Rice University - Houston, TX - [#28 Top End Mathematics]
-----
59 Carnegie Mellon University - Pittsburgh, PA - [#33 Top End Mathematics]
62 Ohio State University - Columbus, OH - [#33 Top End Mathematics]
80 University of Utah - Salt Lake City, UT - [#33 Top End Mathematics]
9 University of California, Davis - [#36 Top End Mathematics]
104 Georgia Institute of Technology - Atlanta, GA - [#36 Top End Mathematics]
182 University of Illinois at Chicago - [#36 Top End Mathematics]
308 City University of New York City College - New York, NY - [#36 Top End Mathematics]
-----
29 Washington University in St. Louis - [#40 Top End Mathematics]
78 University of Arizona - Tucson, AZ - [#40 Top End Mathematics]
92 University of Virginia - Charlottesville, VA - [#40 Top End Mathematics]
201 Brandeis University - Waltham, MA - [#40 Top End Mathematics]
47 University of California, Irvine - #44 Top End Mathematics]
86 Michigan State University - East Lansing, MI - [#44 Top End Mathematics]
89 Texas A&M University - College Station, TX - #44 Top End Mathematics]
280 University of Notre Dame - Notre Dame, IN - [#44 Top End Mathematics]
34 University of Colorado - [#48 Top End Mathematics]
35 University of California, Santa Barbara - [#48 Top End Mathematics]
42 Vanderbilt University - Nashville, TN - [#48 Top End Mathematics]
74 Boston University - Boston, MA - [#48 Top End Mathematics]
103 Dartmouth College - Hanover, NH - [#48 Top End Mathematics]
111 North Carolina State University - Raleigh, NC - [#48 Top End Mathematics]
198 Virginia Polytechnic Institute and State University [Virginia Tech] - Blacksburg, VA - [#48 Top End Mathematics]
-----
46 University of Southern California - Los Angeles - [#55 Top End Mathematics]
116 The University of Georgia - Athens, GA - #55 Top End Mathematics]
51 University of Pittsburgh - PA - [#58 Top End Mathematics]
58 University of Florida - Gainesville, FL - [#58 Top End Mathematics]
227 Rensselaer Polytechnic Institute - Troy, NY - [#58 Top End Mathematics]
277 University of Missouri - Columbia, MO - [#58 Top End Mathematics]
281 University of Oregon - Eugene, OR - [#58 Top End Mathematics]
444 Northeastern University - Boston, MA - [#58 Top End Mathematics]
-----
132 University of Iowa - Iowa City, IA - [#55 Top End Mathematics]
94 Arizona State University - Tempe, AZ - [#64 Top End Mathematics]
136 University of Massachusetts Amherst - Worcester, MA - [#64 Top End Mathematics]
158 Iowa State University - Ames, IA - [#64 Top End Mathematics]
215 Louisiana State University - Baton Rouge, LA - [#64 Top End Mathematics]
268 University of Kansas - Lawrence, KS - [#64 Top End Mathematics]
500+ Claremont Graduate University Claremont, CA - [#64 Top End Mathematics]
-----
79 University of Rochester - Rochester, NY - #70 Top End Mathematics]
125 University of California, Riverside - Riverside, CA - [#70 Top End Mathematics]
155 Florida State University - Tallahassee, FL - [#70 Top End Mathematics]
178 University of Delaware - Newark, DE - [#70 Top End Mathematics]
193 University of Tennessee - Knoxville, TN - [#70 Top End Mathematics]
100 Emory University - Atlanta, GA - [#75 Top End Mathematics]
121 Tufts University - Medford, MA - [#75 Top End Mathematics]
126 University of California, Santa Cruz - [#75 Top End Mathematics]
270 University of Kentucky - Lexington, KY - [#75 Top End Mathematics]
326 Kansas State University - Manhattan, KS - [#75 Top End Mathematics]
345 Syracuse University - Syracuse, NY - [#75 Top End Mathematics]
347 Temple University - Philadelphia, PA - [#75 Top End Mathematics]
357 Tulane University - New Orleans, LA - [#75 Top End Mathematics]
379 University of Oklahoma - Norman, OK - [#75 Top End Mathematics]
-----
187 University of Nebraska - Lincoln, NE - [#84 Top End Mathematics]
234 State University of New York at Buffalo - [#84 Top End Mathematics]
244 The University of New Mexico - Albuquerque - [#84 Top End Mathematics]
266 University of Houston - Houston, TX - [#84 Top End Mathematics]
296 Washington State University - Pullman, WA - [#84 Top End Mathematics]
500+ SUNY-Binghamton Binghamton, NY - [#84 Top End Mathematics]
-----
87 Case Western Reserve University - Cleveland, OH - [#90 Top End Mathematics]
112 Oregon State University - Corvallis, OR - [#90 Top End Mathematics]
152 Colorado State University - Fort Collins, CO - [#90 Top End Mathematics]
170 The University of Connecticut - Storrs, CT - [#90 Top End Mathematics]
233 State University of New York at Albany - [#90 Top End Mathematics]
286 University of South Carolina - Columbia, SC - [#90 Top End Mathematics]
400 Auburn University - Auburn, AL - [#90 Top End Mathematics]
432 Lehigh University - Bethlehem, PA - [#90 Top End Mathematics]
500+ Oklahoma State University Stillwater, OK - [#90 Top End Mathematics]
500+ Rutgers, the State University of New Jersey-Newark Newark, NJ - [#90 Top End Mathematics]
-----
18 University of California, San Francisco - [Not in the Top End Mathematics]
33 Rockefeller University - [Not in the Top End Mathematics]
-----
-----
-----and for perspective
world-wide

-----
4 University of Cambridge, England - [#5 World Ranking Mathematics]
44 University of Paris 11 [Paris-Sud 11 University], France - [#6 World Ranking Mathematics]
40 University of Paris 6 [Pierre and Marie Curie University], France - [#7 World Ranking Mathematics]
10 University of Oxford, England - [#8 World Ranking Mathematics]
[example] - 5 Massachusetts Institute of Technology - [#9 World Ranking Mathematics]
-----
77 Moscow State University, Russia - [#23 World Ranking Mathematics]
115 Tel Aviv University, Ramat Aviv, Israel - [#25 World Ranking Mathematics]
[example] - 38 University of Texas at Austin - [#26 World Ranking Mathematics]
-----
24 Kyoto University, Japan - [#33 World Ranking Mathematics]
98 University of Bonn, Germany - [#34 World Ranking Mathematics]
382 University of Paris Dauphine [Paris 9], France - [#34 World Ranking Mathematics]
[example] - 19 Johns Hopkins University - (Rowland) Baltimore, MD - [#35 World Ranking Mathematics]
[example] - 6 California Institute of Technology - [#37 World Ranking Mathematics]
-----
196 University of Warwick, England - [#40 World Ranking Mathematics]
23 ETH Zurich [Swiss Federal Institute of Technology], Switzerland - [#42 World Ranking Mathematics]
27 University of Toronto, Canada - [#43 World Ranking Mathematics]
143 University of Pisa, Italy - [#44 World Ranking Mathematics]
[example] - 25 University of Illinois at Urbana-Champaign - [#45 World Ranking Mathematics]
26 Imperial College London [The Imperial College of Science, Technoloy and Medicine], England - [#46 World Ranking Mathematics]
70 Ecole Normale Superieure - Paris, France - [#47 World Ranking Mathematics]
61 University of Bristol. England - [#48 World Ranking Mathematics]
110 National University of Singapore, Kent Ridge, Singapore - [#49 World Ranking Mathematics]
142 University of Paris Diderot [Paris 7], France - [#50 World Ranking Mathematics]
-----
20 The University of Tokyo, Japan - Tied #50-75 World Ranking Mathematics]
52 University of Utrecht, Holland - [Tied #50-75 World Ranking Mathematics]
60 Australian National University, Australia - [Tied #50-75 World Ranking Mathematics]
64 Hebrew University of Jerusalem, Israel - [Tied #50-75 World Ranking Mathematics]
114 Technion-Israel Institute of Technology, Haifa, Israel - [Tied #50-75 World Ranking Mathematics]
200 Autonomous University of Madrid, Spain - [Tied #50-75 World Ranking Mathematics]
206 Ecole Polytechnique, France - [Tied #50-75 World Ranking Mathematics]
224 Peking University, Peking, China - [Tied #50-75 World Ranking Mathematics]
236 The Chinese University of Hong Kong, Hong Kong - [Tied #50-75 World Ranking Mathematics]
340 Scuola Normale Superiore - Pisa, Italy - [Tied #50-75 World Ranking Mathematics]
363 University of Bielefeld, Germany - [Tied #50-75 World Ranking Mathematics]
385 University of Rennes 1, France - [Tied #50-75 World Ranking Mathematics]
500+ Humboldt University of Berlin, Berlin, Germany - [Tied #50-75 World Ranking Mathematics]
[example] - 34 University of Colorado - Boulder, CO - [Tied #50-75 World Ranking Mathematics]
[example] - 45 Pennsylvania State University - University Park, PA - [Tied #50-75 World Ranking Mathematics]
-----
21 University College London, England - [Tied #75-100 World Ranking Mathematics]
41 University of Manchester, England - [Tied #75-100 World Ranking Mathematics]
54 University of Zurich, Switzerland - [Tied #75-100 World Ranking Mathematics]
66 McGill University, Canada - [Tied #75-100 World Ranking Mathematics]
106 Louis Pasteur University [Strasbourg I], France - [Tied #75-100 World Ranking Mathematics]
120 Tokyo Institute of Technology, Japan - [Tied #75-100 World Ranking Mathematics]
139 University of Milan, Italy - [Tied #75-100 World Ranking Mathematics]
141 University of Muenster, Germany - [Tied #75-100 World Ranking Mathematics]
147 University of Tuebingen, Germany - [Tied #75-100 World Ranking Mathematics]
168 Technical University of Denmark, Denmark - [Tied #75-100 World Ranking Mathematics]
188 University of New South Wales, Australia - [Tied #75-100 World Ranking Mathematics]
229 RWTH Aachen University, Germany - [Tied #75-100 World Ranking Mathematics]
231 Simon Fraser University, Canada - [Tied #75-100 World Ranking Mathematics]
349 The Hong Kong Polytechnic University, Hong Kong - [Tied #75-100 World Ranking Mathematics]
383 University of Provence [Aix-Marseille 1], France - [Tied #75-100 World Ranking Mathematics]
------


when i came across some ranking that were interesting, i'd make up a list...

example:

2 Stanford University
Stanford University Stanford, CA

[#4 Best Undergraduate Teaching]
[#1 Top End Physics]
[#4 Atomic and Molecular Physics/Optics and Lasers]
[#6 Solid State Physics]
[#5 Relativity/Gravitation/Cosmology]
[#6 Particle Physics/Quantum Field Theory/String Theory]
[#4 Quantum Physics]
[#1 Aerospace Engineering]
[#2 Mechanical Engineering]
[#3 Civil Engineering]
[#1 Electrical Engineering]
[#3 Geophysics and Seismology]
[#2 Top End Mathematics]
[#9 Algebra and Number Theory]
[#9 Applied Mathematics]
[#4 Geometry]
[#6 Mathematical Logic]
[#8 Topology]
[#1 Statistics]
[#1 Chemistry Top End]
[#4 Physical Chemistry]
[#8 Inorganic Chemistry]
[#3 Organic Chemistry]
[#3 Cell Biology]
[#6 World Ranking Physics]
[#4 World Ranking Mathematics]
[#4 World Ranking Chemistry]
[#2 World Ranking Engineering Techology]

so if i care about a fluid dynamics text or quantum mechanics or physical chem or transistor books or geometry texts, i know where to peek...

the lists are out there, but there's a lot of funny ones, but at least knowing roughly what the ballpark is like out there is sort of fun to peek at, minus wasting a month of gut lining making up yer list...

interesting to see how the european unis rate to US ones, and how physics or math changed say in germany after the war...
 
  • #3,284


Well take it with a grain of salt, but in my spare time a whlle i mixed up a bunch of rankings for unis just for my own fun...

anyhoo Brandeis is probably in the top 40 schools for higher math...


for people who like lists...
here we go:



201 Brandeis University - Waltham, MA
[#64 Top End Physics]
[#40 Top End Mathematics]
[#68 Chemistry Top End]

------
8 Princeton University - [#1 Top End Mathematics]
1 Harvard University - [#2 Top End Mathematics]
2 Stanford University - [#2 Top End Mathematics]
3 University of California, Berkeley - [#2 Top End Mathematics]
5 Massachusetts Institute of Technology - [#2 Top End Mathematics]
9 University of Chicago - [#6 Top End Mathematics]
6 California Institute of Technology - [#7 Top End Mathematics]
11 Yale University - [#7 Top End Mathematics]
7 Columbia University - [#9 Top End Mathematics]
22 University of Michigan, Ann Arbor - [#9 Top End Mathematics]
32 New York University - [#9 Top End Mathematics]
-----
13 University of California, Los Angeles - [#12 Top End Mathematics]
12 Cornell University - #13 Top End Mathematics]
17 University of Wisconsin–Madison - [#14 Top End Mathematics]
38 University of Texas at Austin - [#14 Top End Mathematics]
69 Brown University - Providence, RI - [#14 Top End Mathematics]
28 University of Minnesota, Twin Cities - [#17 Top End Mathematics]
15 University of Pennsylvania - [#18 Top End Mathematics]
25 University of Illinois at Urbana-Champaign - [#18 Top End Mathematics]
30 Northwestern University - Evanston, IL - [#18 Top End Mathematics]
-----
19 Johns Hopkins University - [#21 Top End Mathematics]
31 Duke University - Durham, NC - [#21 Top End Mathematics]
37 University of Maryland, College Park - [#21 Top End Mathematics]
14 University of California, San Diego - [#24 Top End Mathematics]
16 University of Washington - [#24 Top End Mathematics]
55 Rutgers University - Piscataway, NJ - [#24 Top End Mathematics]
167 State University of New York at Stony Brook - Stony Brook, NY - [#24 Top End Mathematics]
39 University of North Carolina at Chapel Hill - [#28 Top End Mathematics]
45 Pennsylvania State University-University Park - [#28 Top End Mathematics]
67 Purdue University - West Lafayette, IN - [#28 Top End Mathematics]
93 Indiana University - Bloomington, IN - [#28 Top End Mathematics]
99 Rice University - Houston, TX - [#28 Top End Mathematics]
-----
59 Carnegie Mellon University - Pittsburgh, PA - [#33 Top End Mathematics]
62 Ohio State University - Columbus, OH - [#33 Top End Mathematics]
80 University of Utah - Salt Lake City, UT - [#33 Top End Mathematics]
9 University of California, Davis - [#36 Top End Mathematics]
104 Georgia Institute of Technology - Atlanta, GA - [#36 Top End Mathematics]
182 University of Illinois at Chicago - [#36 Top End Mathematics]
308 City University of New York City College - New York, NY - [#36 Top End Mathematics]
-----
29 Washington University in St. Louis - [#40 Top End Mathematics]
78 University of Arizona - Tucson, AZ - [#40 Top End Mathematics]
92 University of Virginia - Charlottesville, VA - [#40 Top End Mathematics]
201 Brandeis University - Waltham, MA - [#40 Top End Mathematics]
47 University of California, Irvine - #44 Top End Mathematics]
86 Michigan State University - East Lansing, MI - [#44 Top End Mathematics]
89 Texas A&M University - College Station, TX - #44 Top End Mathematics]
280 University of Notre Dame - Notre Dame, IN - [#44 Top End Mathematics]
34 University of Colorado - [#48 Top End Mathematics]
35 University of California, Santa Barbara - [#48 Top End Mathematics]
42 Vanderbilt University - Nashville, TN - [#48 Top End Mathematics]
74 Boston University - Boston, MA - [#48 Top End Mathematics]
103 Dartmouth College - Hanover, NH - [#48 Top End Mathematics]
111 North Carolina State University - Raleigh, NC - [#48 Top End Mathematics]
198 Virginia Polytechnic Institute and State University [Virginia Tech] - Blacksburg, VA - [#48 Top End Mathematics]
-----
46 University of Southern California - Los Angeles - [#55 Top End Mathematics]
116 The University of Georgia - Athens, GA - #55 Top End Mathematics]
51 University of Pittsburgh - PA - [#58 Top End Mathematics]
58 University of Florida - Gainesville, FL - [#58 Top End Mathematics]
227 Rensselaer Polytechnic Institute - Troy, NY - [#58 Top End Mathematics]
277 University of Missouri - Columbia, MO - [#58 Top End Mathematics]
281 University of Oregon - Eugene, OR - [#58 Top End Mathematics]
444 Northeastern University - Boston, MA - [#58 Top End Mathematics]
-----
132 University of Iowa - Iowa City, IA - [#55 Top End Mathematics]
94 Arizona State University - Tempe, AZ - [#64 Top End Mathematics]
136 University of Massachusetts Amherst - Worcester, MA - [#64 Top End Mathematics]
158 Iowa State University - Ames, IA - [#64 Top End Mathematics]
215 Louisiana State University - Baton Rouge, LA - [#64 Top End Mathematics]
268 University of Kansas - Lawrence, KS - [#64 Top End Mathematics]
500+ Claremont Graduate University Claremont, CA - [#64 Top End Mathematics]
-----
79 University of Rochester - Rochester, NY - #70 Top End Mathematics]
125 University of California, Riverside - Riverside, CA - [#70 Top End Mathematics]
155 Florida State University - Tallahassee, FL - [#70 Top End Mathematics]
178 University of Delaware - Newark, DE - [#70 Top End Mathematics]
193 University of Tennessee - Knoxville, TN - [#70 Top End Mathematics]
100 Emory University - Atlanta, GA - [#75 Top End Mathematics]
121 Tufts University - Medford, MA - [#75 Top End Mathematics]
126 University of California, Santa Cruz - [#75 Top End Mathematics]
270 University of Kentucky - Lexington, KY - [#75 Top End Mathematics]
326 Kansas State University - Manhattan, KS - [#75 Top End Mathematics]
345 Syracuse University - Syracuse, NY - [#75 Top End Mathematics]
347 Temple University - Philadelphia, PA - [#75 Top End Mathematics]
357 Tulane University - New Orleans, LA - [#75 Top End Mathematics]
379 University of Oklahoma - Norman, OK - [#75 Top End Mathematics]
-----
187 University of Nebraska - Lincoln, NE - [#84 Top End Mathematics]
234 State University of New York at Buffalo - [#84 Top End Mathematics]
244 The University of New Mexico - Albuquerque - [#84 Top End Mathematics]
266 University of Houston - Houston, TX - [#84 Top End Mathematics]
296 Washington State University - Pullman, WA - [#84 Top End Mathematics]
500+ SUNY-Binghamton Binghamton, NY - [#84 Top End Mathematics]
-----
87 Case Western Reserve University - Cleveland, OH - [#90 Top End Mathematics]
112 Oregon State University - Corvallis, OR - [#90 Top End Mathematics]
152 Colorado State University - Fort Collins, CO - [#90 Top End Mathematics]
170 The University of Connecticut - Storrs, CT - [#90 Top End Mathematics]
233 State University of New York at Albany - [#90 Top End Mathematics]
286 University of South Carolina - Columbia, SC - [#90 Top End Mathematics]
400 Auburn University - Auburn, AL - [#90 Top End Mathematics]
432 Lehigh University - Bethlehem, PA - [#90 Top End Mathematics]
500+ Oklahoma State University Stillwater, OK - [#90 Top End Mathematics]
500+ Rutgers, the State University of New Jersey-Newark Newark, NJ - [#90 Top End Mathematics]
-----
18 University of California, San Francisco - [Not in the Top End Mathematics]
33 Rockefeller University - [Not in the Top End Mathematics]
-----
-----
-----


and for perspective
world-wide

-----
4 University of Cambridge, England - [#5 World Ranking Mathematics]
44 University of Paris 11 [Paris-Sud 11 University], France - [#6 World Ranking Mathematics]
40 University of Paris 6 [Pierre and Marie Curie University], France - [#7 World Ranking Mathematics]
10 University of Oxford, England - [#8 World Ranking Mathematics]
[example] - 5 Massachusetts Institute of Technology - [#9 World Ranking Mathematics]
-----
77 Moscow State University, Russia - [#23 World Ranking Mathematics]
115 Tel Aviv University, Ramat Aviv, Israel - [#25 World Ranking Mathematics]
[example] - 38 University of Texas at Austin - [#26 World Ranking Mathematics]
-----
24 Kyoto University, Japan - [#33 World Ranking Mathematics]
98 University of Bonn, Germany - [#34 World Ranking Mathematics]
382 University of Paris Dauphine [Paris 9], France - [#34 World Ranking Mathematics]
[example] - 19 Johns Hopkins University - (Rowland) Baltimore, MD - [#35 World Ranking Mathematics]
[example] - 6 California Institute of Technology - [#37 World Ranking Mathematics]
-----
196 University of Warwick, England - [#40 World Ranking Mathematics]
23 ETH Zurich [Swiss Federal Institute of Technology], Switzerland - [#42 World Ranking Mathematics]
27 University of Toronto, Canada - [#43 World Ranking Mathematics]
143 University of Pisa, Italy - [#44 World Ranking Mathematics]
[example] - 25 University of Illinois at Urbana-Champaign - [#45 World Ranking Mathematics]
26 Imperial College London [The Imperial College of Science, Technoloy and Medicine], England - [#46 World Ranking Mathematics]
70 Ecole Normale Superieure - Paris, France - [#47 World Ranking Mathematics]
61 University of Bristol. England - [#48 World Ranking Mathematics]
110 National University of Singapore, Kent Ridge, Singapore - [#49 World Ranking Mathematics]
142 University of Paris Diderot [Paris 7], France - [#50 World Ranking Mathematics]
-----
20 The University of Tokyo, Japan - Tied #50-75 World Ranking Mathematics]
52 University of Utrecht, Holland - [Tied #50-75 World Ranking Mathematics]
60 Australian National University, Australia - [Tied #50-75 World Ranking Mathematics]
64 Hebrew University of Jerusalem, Israel - [Tied #50-75 World Ranking Mathematics]
114 Technion-Israel Institute of Technology, Haifa, Israel - [Tied #50-75 World Ranking Mathematics]
200 Autonomous University of Madrid, Spain - [Tied #50-75 World Ranking Mathematics]
206 Ecole Polytechnique, France - [Tied #50-75 World Ranking Mathematics]
224 Peking University, Peking, China - [Tied #50-75 World Ranking Mathematics]
236 The Chinese University of Hong Kong, Hong Kong - [Tied #50-75 World Ranking Mathematics]
340 Scuola Normale Superiore - Pisa, Italy - [Tied #50-75 World Ranking Mathematics]
363 University of Bielefeld, Germany - [Tied #50-75 World Ranking Mathematics]
385 University of Rennes 1, France - [Tied #50-75 World Ranking Mathematics]
500+ Humboldt University of Berlin, Berlin, Germany - [Tied #50-75 World Ranking Mathematics]
[example] - 34 University of Colorado - Boulder, CO - [Tied #50-75 World Ranking Mathematics]
[example] - 45 Pennsylvania State University - University Park, PA - [Tied #50-75 World Ranking Mathematics]
-----
21 University College London, England - [Tied #75-100 World Ranking Mathematics]
41 University of Manchester, England - [Tied #75-100 World Ranking Mathematics]
54 University of Zurich, Switzerland - [Tied #75-100 World Ranking Mathematics]
66 McGill University, Canada - [Tied #75-100 World Ranking Mathematics]
106 Louis Pasteur University [Strasbourg I], France - [Tied #75-100 World Ranking Mathematics]
120 Tokyo Institute of Technology, Japan - [Tied #75-100 World Ranking Mathematics]
139 University of Milan, Italy - [Tied #75-100 World Ranking Mathematics]
141 University of Muenster, Germany - [Tied #75-100 World Ranking Mathematics]
147 University of Tuebingen, Germany - [Tied #75-100 World Ranking Mathematics]
168 Technical University of Denmark, Denmark - [Tied #75-100 World Ranking Mathematics]
188 University of New South Wales, Australia - [Tied #75-100 World Ranking Mathematics]
229 RWTH Aachen University, Germany - [Tied #75-100 World Ranking Mathematics]
231 Simon Fraser University, Canada - [Tied #75-100 World Ranking Mathematics]
349 The Hong Kong Polytechnic University, Hong Kong - [Tied #75-100 World Ranking Mathematics]
383 University of Provence [Aix-Marseille 1], France - [Tied #75-100 World Ranking Mathematics]
------


when i came across some ranking that were interesting, i'd make up a list...

example:

2 Stanford University
Stanford University Stanford, CA

[#4 Best Undergraduate Teaching]
[#1 Top End Physics]
[#4 Atomic and Molecular Physics/Optics and Lasers]
[#6 Solid State Physics]
[#5 Relativity/Gravitation/Cosmology]
[#6 Particle Physics/Quantum Field Theory/String Theory]
[#4 Quantum Physics]
[#1 Aerospace Engineering]
[#2 Mechanical Engineering]
[#3 Civil Engineering]
[#1 Electrical Engineering]
[#3 Geophysics and Seismology]
[#2 Top End Mathematics]
[#9 Algebra and Number Theory]
[#9 Applied Mathematics]
[#4 Geometry]
[#6 Mathematical Logic]
[#8 Topology]
[#1 Statistics]
[#1 Chemistry Top End]
[#4 Physical Chemistry]
[#8 Inorganic Chemistry]
[#3 Organic Chemistry]
[#3 Cell Biology]
[#6 World Ranking Physics]
[#4 World Ranking Mathematics]
[#4 World Ranking Chemistry]
[#2 World Ranking Engineering Techology]

so if i care about a fluid dynamics text or quantum mechanics or physical chem or transistor books or geometry texts, i know where to peek...

the lists are out there, but there's a lot of funny ones, but at least knowing roughly what the ballpark is like out there is sort of fun to peek at, minus wasting a month of gut lining making up yer list...

interesting to see how the european unis rate to US ones, and how physics or math changed say in germany after the war...
 
  • #3,285


as for textbooks, recall that Mike Spivak was at Brandeis when he wrote both his calculus book and his differential geometry series. So those books which are considered the gold standard in both subjects were written specifically for courses at Brandeis. When I was there I also attended some undergraduate classes in algebra where the lecturer was adapting the famous books by Bourbaki to his class.for some current textbooks, consult individual instructor's webpages, e.g.

http://people.brandeis.edu/~cherveny/

http://people.brandeis.edu/~hsultan/

http://people.brandeis.edu/~jbellaic/teaching.html

http://people.brandeis.edu/~bernardi/index.php?page=teaching

http://people.brandeis.edu/~kleinboc/
 
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  • #3,286


What was Mike Spivak's inspiration for his calculus and diff geometry books? I am always curious what they *used* when they were in school, and sometimes the list of what they thought were great books, or not-so great books... when they were starting out...

I think what i remember most about Spivak's book was i saw it offered once, by one teacher in one class for calculus at the local uni, and didnt see it before or after... I didn't know which book it was other than it was plain looking and 'furry' and i was really impressed with the back that had tons of comments about dozens and dozens of texts, and i thought, wow 3 sentences about Hardy's Pure Mathematics... or a line or two about Courant...

always liked textbooks, in first year that would slide in some recommended books that way...
 
  • #3,287


i want to be a mathematician, but i don't hve the natural talent for it..


we need mathematicans today to solve the great underlying mysteries of math today.
 
  • #3,288


- i want to be a mathematician, but i don't hve the natural talent for it..

In your last course, did you do 'every problem' in each chapter?I used to be skeptical, of the advice i once got, but you can likely get a B almost all the time if you burn 10-15 hours a week on the problems...

and without any talent i think you could crank through and pass 80% of any math class for a degree, if not more...

heck if you burn 300 hours on a complete textbook, maybe you'll create a 'toolbox' for talent... -----

skill is what comes with practice, start small, wring out 101% out of one chapter of your math or physics book... flip a coin and try the next chapter later on...don't rush a textbook, and don't cheat yourself not doing 90% of the problems. If you can read the whole book, do all the problems, all it takes is a enormous amount of time...

but what you do learn will be pretty damn solid.
 
  • #3,289


"What was Mike Spivak's inspiration for his calculus and diff geometry books?"

We would need to ask Mike this, but he went to Harvard where they used Courant and Hardy when I was there. Also when I taught from his book I noticed some of the proofs were similar to ones used in Courant, so my personal conjecture then was that Courant was at least one inspiration for his Calculus.

We really should ask him this question though.

The diff geom book is much more ambitious and comprehensive, and from reading it, it seems to be inspired by the original sources it references, such as Gauss and Riemann, perhaps Weil for the local proof of Poincare duality. I have not read many of the more modern sources, but they seem to include Cartan,... (Mike was also Milnor's student.)
 
  • #3,290


- We would need to ask Mike this, but he went to Harvard where they used Courant and Hardy when I was there. Also when I taught from his book I noticed some of the proofs were similar to ones used in Courant

Hardy would be a rough ride if you had to go through it quickly...then again any analysis text is...if you need to push through half in 12-15 weeks

I always wondered what people used from the middle 60s onwards, when courant was still going strong, was Apostol an instant classic when it came up like in 57-58 or did it take years for it to catch on?

It would be great if you could recall all the textbooks you went though during your undergrad years, i know you trickled bits and pieces here through the years...of the main ones...- The diff geom book is much more ambitious and comprehensive, and from reading it, it seems to be inspired by the original sources it references, such as Gauss and Riemann, perhaps Weil

actually that was the most interesting part of the Brandeis links you tossed me about 8 of the 22 or so faculty had homepages with texts, and it was the differential geometry courses that most impressed me, and i actually had one more textbook to add to my list...

damn, Wulf something an oxford text...and it was interesting to see how they would use three books in tandem, and i only knew of two of the books being used 'together'...

so it was nice to see what books people can read at roughly the same time when taking a first or second class...

-------------

a. Brandeis in some classes used Erdmann's book on Lie Algebras, since all you need to tackle that textbook is linear algebra. And *then* then go into Humphreys and Fulton.

[though some feel that to tackle humphreys it's best to read a. Herstein b. Hoffman and Kunze, which are both considerably harder and would take a lot of time]

b. Brandeis used Lang's Real and Functional Analysis text with Zimmer at the same time

c. Brandeis for a second class in Real Analysis did a. Lang's Real and Functional Analysis b. Rudin's Real and Complex Green book c. Real Analysis by Kolmogorov and Fomin - Dover

d. surprised they used the 70s early 80s Mardsen and Tromba for vector calculus, which is generally well disliked, but if you put a ton of effort into it or supplement it, it's better. To the plus it's got detailed explanations, doesn't shove definitions at you, has meaningful illustrations, but it can be a confusing text and the cause of many many headaches. And it's not the smoothest for self-study either... [my guess is people read thomas and finney and marsden and tromba together]

e. Brandeis uses Fraleigh and Gallian for Abstract Algebra [my notes show it's an easier hop to start with Fraleigh, then read Artin and then read Dummitt]

f. Brandeis for Topology uses Hatcher-Greenberg-Munkres together, likely after the main Munkres text]

g. Brandeis also used Hocking for Topology, though i think for the course only do chapter 1 2 and then hop to 5. It's a clear book and good for self-study i hear.

h. Rolfsen's book on Knots and Links [Harvard would use cromwell as a main text, and then supplement it with Livingstone/Gilbert/[and more lightly]/Burde/Rolfsen/Kauwach]

i. Brandeis for Diff Geo I goes a. Spivak b. Warner c. Milnor d. Bott and Tu

j. Brandeis for Diff Geo II goes a. doCarmo [Riemann Geometry] b. more Spivak c. Milnor and Stasheff d. Roe

k. [yet other courses for Diff Geo II use doCarmo and use Petersen's Riemannian book and Lee's Riemannian book with Warner]

l. If you're reading Warner's book Foundations of Differential Manifolds and Lie Groups, Wulf Rossmann's Oxford book is great to read with it. [and well ideally you'd need to read a. Lee b. the other Lee book c. a bit of Messay d. a bit of Boothby e. a bit of Warner] So basically one new book on the list when one tackles Lee's Introduction to Topological Manifolds and Lee's Introduction to Smooth Manifolds. Assuming i actually finish a topology and differential geometry book that is...

m. Brandeis also liked Pressley's Differential Geometry book which is an easy read, much like Erdmann's on Lie Algebra...

n. one odd thing for Diff Geo was a. Gallot-Hullin-Lafontaine b. Spivak c. Milnor. I would assume for a second course...

o. they are big fans of D'angelo's proof book which seems friendlier than most of the others, which probably helps people later on so they don't go to pieces with Analysis...

p. Falcon's Fractal Geometry book, which is one of the best ones out there was used in some second year grad school course on 'Hausdorf Dimension'...

q. Markov Chains - they use Norris with Lawler as two of the main texts sometimes [though they seem to like adding miserable textbooks on financial economic stuff on the reading list too, since it's trendy or a good career option] Mind you, Hoel's books and Lawler look like real gems [I'm all for the great 70s books still in print by Howard Mifflin in Statistics and Stochastic processes,like Hoel's without any need for being in the 27th edition either, though they are pricey, they are still IN PRINT, and they are easy gentle reads.] [yup Mifflin and Hoel did a great job in the early 70s comming out with an easy book on probability, an easy book on statistics and an easy book on stochastic processes, all in print with no need for useless updating either to look new. They are fine as they are. I think Stanford in the 70s was big on those three textbooks]

-----

Anyhoo, that's some of my notes, and some of the stuff you helped me out with by suggesting the Brandeis teacher homepage links... much appreciated!
 
  • #3,291


RJinkies said:
***This is RJinkies quote to Mathwonk, I cut the rest off***

It would be great if you could recall all the textbooks you went though during your undergrad years, i know you trickled bits and pieces here through the years...of the main ones...

I would also be really interested. Though I'm just learning the basics now on my slow pace, but it would be really interesting resource.

Infact, it would be great if you someday would like to write more of a guide to learning mathematics. Like from elementary to end of undergraduate studies. Like about the order of topics (e.g. when would be useful to study topology and other things like that..), and which are good books for people with different talents. And books for those of us who really have no talent at all but study it anyway for the fun of it..

Of course it would be a lot of work and I already appreciate all the help you have given. This whole thread was really interesting to read but the info is little scattered around.. : )
 
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  • #3,292
when i was at harvard the instructors gave book recommendations, but never followed them. the books were just for your outside reading. the lecturers composed their own version of the material. to be honest most of the time the lectures were significantly better than anything in the best books, but not always as detailed.

freshman calc: john tate, recommended texts: Courant, and G.H Hardy: Pure Mathematics, and Foundations of Analysis by E. Landau.

sophomore algebra: garrett birkhoff, text: survey of modern algebra by birkhoff and maclane.

sophomore calculus: i forget what book, maybe Taylor, but the book was apparently chosen by a committee and the professor was contemptuous of at least some of it, e.g. lack of proof of implicit functiion theorem.

sophomore diff eq. herman gluck; text: earl coddington. the best part of this course occurred at the end, when prof gluck departed from the routine stuff in the book and presented a beautiful proof of the existence theorem for solutions of first order ode's, by the contraction lemma for complete metric spaces.

sophomore algebra, instructor newcomb greenleaf, texts: linear algebra and matrix theory, by evar nering; fundamental concepts of higher algebra by a. adrian Albert, galois theory by emil artin. also notes by Andrew Gleason on linear algebra available from the dept.

complex variables, text by Henri Cartan. except when taught by Ahlfors, who used his own book.

junior:
advanced calculus: lynn loomis, official text: calculus of several variables by wendell fleming, but the lectures followed more closely the book Foundations of modern analysis by Jean Dieudonne'; including sturm liouville theory, supplemented by lectures on content theory and a lovely presentation of vector geometry via the group of motions in intrinsic euclidean geometry. much of this course is now recorded in the book by Loomis and Sternberg.

another reference text for this course was advanced calculus by spencer, steenrod, and nickerson,.

a very useful course on introductory analysis taught by george mackey with no text. a good book now on related material is his text on complex analysis: Lectures on the theory of functions of a complex variable.

senior:
real analysis taught by lynn loomis, no textbook, it covered abstract measure theory as in the book of Halmos, and some Banach algebras and stone weierstrass theorem. much too abstract to be really useful. some of my friends who understand the material now recommend the book by zygmund and wheeden at least for the integration theory.

algebraic topology taught by raoul bott, text: algebraic topology by spanier. most people today recommend the book by allen hatcher.

I want to remind that i did not learn much from this somewhat harsh and user unfriendly first exposure to significant mathematics. but i admit those who worked harder did.

very few of my courses in graduate school even recommended any texts at all, everything was in the lectures. the only books even referred to in any grad school course were the hand written notes on algebraic topology and differential forms by friedlander, griffiths, and morgan, the book course in arithmetic by serre, and some seminars read the books on several complex variables by gunning and rossi, and by hormander, and the lectures on riemann surfaces by gunning, and the book Topology from the differentiable viewpoint, by John Milnor.when i began recovering my math career after my first unsuccessful attempt at mastering it from some of the books and courses mentioned above, i learned more from spivak's calculus and calculus on manifolds, and frederick greenleaf's book on one complex variable. i also read lectures on algebraic topology by marvin greenberg, and books by chinn and steenrod, and william massey. i also liked hurewicz' book on ode, and hurewicz and wallman on dimension theory, kelley on general topology, and lang's analysis I. modern algebra by van der waerden was also frequently helpful but not always, as was algebra by lang.

I personally find it hard to find many algebra books that are really user friendly, but there is one significant exception, the book Algebra by Michael Artin is quite wonderful. Many people recommend Dummitt and Foote and it does have many good qualities, but I have some criticisms of it.

I do not like books that are written to show off how clever the author is rather than to make the material look easy and clear, and many algebra books seem to fail this test to me, along with books like rudin's analysis.

Virtually everyone likes Geometry of Algebraic Curves, vols I and II, for that more specialized subject, carefully written over 30 years by, (in the interest of full disclosure, my good friends), E. Arbarello, M. Cornalba, P. Griffiths, and J. Harris.
 
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  • #3,293


@mathwonk:

I was wondering, does one need to get into a "super" university like Harvard to learn a lot of math, or what?
 
  • #3,294
no. the advantages of harvard are a deservedly good reputation and lots of money, hence they attract a strong interesting student body, outstanding professors, high quality living quarters, a prestigious reputation for the school and its degree, expensive equipment and facilities, opportunities available in boston, horrible winters...oops that's a negative.

harvard does not necessarily offer better advice to people struggling, or more personal attention.

one learns by hard work on material that has been clearly presented. if you actually read the books listed above that my harvard instructors recommended but that i did not read, you will obviously be miles ahead of me and many other harvard students.
 
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  • #3,295


freshman calc: Courant/Hardy/Landau - Foundations of Analysis

how long were people using Hardy? I got the impression both Courant and Hardy did well into the early and mid 60s, though i think hardy faded a bit quicker. Esp with so many people trying to replace both with all the newer 60s texts.

What did you think of Hardy and Landau?

Hardy seems like a pretty rough ride for anyone taking math after the Space Race.
I think anyone reading it would go through it at a glacial pace, and i wonder if anyone finished the damn thing...

Landau looks cool, totally minimalist, and puzzling as Babylonian cuniform...

-------

sophomore calculus: i forget what book,

oh damn, that's the best part...

Was it more of Courant? and was there another vector book?



were any of these possibly on the reading list, or recommended by the teachers?

1952 Kaplan - Advanced Calculus - Addison-Wesley
1955 AE Taylor - Ginn
1957 Apostol [I'd think you'd remember that one]
1959 Nickerson Spencer and Steenrod - van Nostrand
1961 Olmstead - Appleton-Crofts
1964 Protter and Morrey - Addison-Wesley [all these would probably be after you took your degree/classes]
1964 Smirnov - Addison-Wesley
1965 Buck - McGraw-Hill [actually that's probably the second edition, there was probably a first edition 1957-1963ish]
1965 Fleming - Addison-Wesley
1967? Spivak - WA Benjamin?
1968 Loomis and Sternberg - Addison-Wesley- free pdfs at his website
1970 Rossi - Addison-Wesley [oh oh another Brandeis person]


[I'm not sure if missed anyone from 1955-1980s there, but if there's any famous forgotten text from the 50s 60s 70s, tell me someone]
[oh hell tell me about the terrible ones too!]

my feeling there wasnt really anything out in the 70s... just Thomas and Finney clones and 15% of the books just mentioned...

---

I get the feeling that Apostol and Buck soaked up most of the sales at the high end, and Thomas and Finney for the rest]

when did the first Spivak come out? wasnt that like in 1967 I assume you read it after your degree, and the other book he did i think was 1965 on manifolds..
[or did you zoom through it after your degree and before grad school]

I always found it interesting where i'd struggle with a mainstream book and then eons later, find it more approachable [or find the easy and hard books on the same subject more approachable]


I used to think that you liked Loomis before, but it was more 'something you went through' but wouldn't really recommend... [when you clarified things a while later]

----------------


sophomore algebra - linear algebra and matrix theory - nering
fundamental concepts of higher algebra - aa Albert

What did you think of Nering?

I assume that was a fixed up edition of Albert's 1930's abstract algebra books
[Modern Abstract Algebra - Chicago 1937]
[Introduction to Algebraic Theories - Chicago 1941 - more an introduction to the other book]

Linear Algebra didnt really seem to take off till the 50s/60s, or bits of it in a Calculus III part of the text...
[or they dropped it being called Theory of Equations like using that famous Uspensky book and made it way easier and modern looking in the mid 60s]
[maybe it was all the mainframes doing Linear Programming that got it popular in the schools]

1951 Wade - The Algebra of vectors and matrices - Addison-Wesley
1952 Perlis - Theory of Matrices - Addison-Wesley
1952 Stoll - LInear Algebra and Matrix Theory - McGraw-Hill
1964 Bickley-Thompson - Matrices and their Meaning - van Nostrand 1964


-------------

- complex variables, text by Henri Cartan

so the pures went cartan and the applied went to churchill? [or did anyone do the easiest thing and read churchill first?]

Kaplan did a big Addison-Wesley on Complex too in 1953...

---------

advanced calculus: official text: calculus of several variables by wendell fleming, but the lectures followed more closely the book Foundations of modern analysis by Jean Dieudonne

Did you take adv calculus at two different times, or was fleming out that early?

[I got the impression that Courant and Spivak and Fleming were the best of the texts from the good ole days from you]


----------

senior: real analysis taught by lynn loomis, no textbook, it covered abstract measure theory as in the book of Halmos

Halmos came out in 1950 and probably the closest in style is Bruckner.

I remember seeing a strange set of analysis books at Simon Fraser, they used Goldberg [Wiley 1976] and Bruckner [Prentice-Hall 1996]

Goldberg looked stiff, but i heard it's pretty traditional and a touch gentler as far as dry analysis books go, but it's sure a rare one, musta been popular in the mid 70s and with the MAA and got tossed into obscurity when Rudin got pushed more and more...

[I still find Binmore or Colin Clark [The Theoretical Side of Calculus] as the two easier books out there]

and didnt Marsden write a pretty gentle and wordy Analysis text? It seemed the book to read before tackling Hardy]

------------

algebraic topology taught by raoul bott, text: algebraic topology by spanier. most people today recommend the book by allen hatcher.

How did you find Bott's texts? [Bott and Tu]



Spanier... well i was going to say, amazon, but i peeked and it's from the chicago list of books...

[Spanier is the maximally unreadable book on algebraic topology. It's bursting with an unbelievable amount of material, all stated in the greatest possible generality and naturality, with the least possible motivation and explanation. But it's awe-inspiring, and every so often forms a useful reference. I'm glad I have it, but most people regret ever opening it.]

--------

I want to remind that i did not learn much from this somewhat harsh and user unfriendly first exposure to mathematics

people say that Caltech's course probably 'teaches' more, but if you throw teaching out the window, Harvard is the most difficult one...

I found these notes 'somewhere' and it had to deal with Rudin's textbook ...

-------

[Harvard 55ab takes about 50 hrs a week of study]
[Thoughts on the flaws of Harvard 55]
[After having chosen Caltech over Princeton and Harvard to pursue a math major, I feel strongly that the math department's main feeder course here - Math 5 - is by far the strongest of the various courses at top universities which are taken by the strongest math students. It's main virtue is that it is long enough (a year) to do something serious, and that it does it in a thorough methodical way, building up steadily to huge, important theorems that you actually understand fully by the time you get to them.]
[I know that the 'stronger than the others' claim is true for sure in comparison to Princeton, since I actually took their math major feeder courses when I was a high school senior. (Problems there: teaching quality haphazard, too-advanced material rushed through so that even the brightest students are lost, though Jordan Ellenberg's Math 214 was a well-known and beautiful exception - but he's not there anymore.) And yes, I think Math 5 here is stronger even than Harvard's Math 55. While Harvard's famous course covers a lot of esoteric and advanced topics, it does so with very little unity and requires overwhelming amounts of outsdie reading so that even the best students miss 30% or so of the ideas.]
[After a year and a half at Caltech, I knew everything that a Math 55 graduate knew, but various comments I've heard make it pretty clear that most of them come out with a "scattered" feeling - they've been exposed to a lot but don't have a particularly unified picture. Math 5 keeps to a more manageable area and explores it more deeply, and so one comes away with some very tangible and coherent knowledge.]
[Those are my feelings on the subject.]


and...

[Caltech Math108a - used Rudin and Carothers and Elias Stein Complex Book - 2 real+1complex]
[the combo of the three is better than Harvard 55]
[Loomis and Sternberg's book used to be used for Harvard 55ab]

and

[I think this book is inappropriate for use as an undergraduate textbook. Its use at the introductory graduate level is defensible, but I see no reason to choose this book when better ones are available. Apostol's Analysis book is at a similar level but has much richer discussion and is more comprehensive. For a book slightly more elementary than that, I would recommend Taylor and Mann. Like I said above--as a sequel to this or similar books, I think the Rudin "Real and Complex Analysis" book is absolutely wonderful. This book does have one purpose for which I found it to be very well-suited: it is useful to work through, perhaps only once, to review the subject and solidify your understanding of the material. But its value as such does not warrant purchasing it at the obscene price.]

-------

- almost all algebra books seem to fail this test to me

[high school or abstract?]

a. Gallian
b. Fraleigh
c. Beachy and Blair
d. Allenby
e. Saracino
f. Pinter
g. Childs

those 7 i think are the easiest ones on my list, and the first two are probably 'well-known'


how did you find Paul Cohn's books [1970s-1990s]

[i think one of his introductory books was fixed up considerably with the newer editions]

not sure what to think though, since it's not used that much in any of the syllabuses out there [or anymore]

--------

- along with books like rudin's analysis.
- Many people recommend Dummitt and Foote and it does have many good qualities but I have several criticisms of it.

What texts are you somewhat [or completely] sour on?

It's rare to actually hear people criticize a popular book, or classic [in whole or part]


Heck, the first time i saw Apostol's texts i said, man, none of this is really necessary... but i was impressed at how huge the books were, and thought man it would be one hell of a school that used these as 60 weeks of 'an introduction to calculus'...

but I'm sure if one tackled a mini calculus course or had a book to read in parallel, it would be much better. But as a first and only textbook, oh i shuddered, but i definitely spent a good 30 minutes at it in the 1980s saying, wow this is surreal, it's the hardest calculus book i seen.

much later on, i added it to my 'shopping list'


----------

I added three books to the list too..

Nering is a new one...
Mackey's complex text
and Arbarello...

your stories definitely do get better the more we hear them mathwonk!
much appreciated
 
  • #3,296


well here's one more story about my sophomore calc book and why i don't remember the name. after getting a D- in freshman honors calc part 2 from john tate (a course i had only attended once a month, during my slow decline before eventually getting kicked out for a year), when i returned in the fall i had to take non honors sophomore calc, taught as it happens also by tate.

tate was a great prof, but in the non honors course he had to use the book chosen by the departmental calculus committee instead of picking his own. So it was one of those routine mediocre books they use at places that are not harvard, reasonable but not too challenging (Taylor?). the course was ridiculously easy in comparison to the previous year's course, and although i did not work or attend much and seldom handed in hw, i was still passing as i recall.

one day in discussing the implicit function theorem in class on a day when i was there, tate read disgustedly from the book's treatment: "the proof of this result is beyond the scope of this book". He slammed the book on the desk and said loudly "well it's not beyond the scope of this course!" and went over to the board.

Then he stopped, looked back at the offending book lying on the desk, strode quickly back, grabbed the book and slammed it into the trash can with both hands.

Then at the end of the class, he went back, calmly retrieved the book from the trash and assigned homework from it.
 
  • #3,297


mathwonk said:
well here's one more story about my sophomore calc book and why i don't remember the name. after getting a D- in freshman honors calc part 2 from john tate (a course i had only attended once a month, during my slow decline before eventually getting kicked out for a year), when i returned in the fall i had to take non honors sophomore calc, taught as it happens also by tate.

tate was a great prof, but in the non honors course he had to use the book chosen by the departmental calculus committee instead of picking his own. So it was one of those routine mediocre books they use at places that are not harvard, reasonable but not too challenging (Taylor?). the course was ridiculously easy in comparison to the previous year's course, and although i did not work or attend much and seldom handed in hw, i was still passing as i recall.

one day in discussing the implicit function theorem in class on a day when i was there, tate read disgustedly from the book's treatment: "the proof of this result is beyond the scope of this book". He slammed the book on the desk and said loudly "well it's not beyond the scope of this course!" and went over to the board.

Then he stopped, looked back at the offending book lying on the desk, strode quickly back, grabbed the book and slammed it into the trash can with both hands.

Then at the end of the class, he went back, calmly retrieved the book from the trash and assigned homework from it.

This needs to be said.
 
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  • #3,298


another reason for not remembering the name of the sophomore calc book may be that i did not own a copy and just borrowed one to read the day before the test. i thought that was cool, then.
 
  • #3,299


I got mixed feelings about tossing proofs or epsilons into a first calculus course, and i thnk the new math did 'kill off' the Syl Thompsons, JE Thompsons, and the easy to read, easy to understand calculus texts common till the late 40s/early 50s [Granville Longley Smith as well, which i liked browsing in the library, when texts were built so you could read it all, and follow it all]

And well, there should be a point made where honours calculus and regular calculus has to do some trade-offs, a math teacher does need to know what is essential and what is 'merely details'.

[Bueche made that point in his introduction to his College Physics text where sometimes you *need* to push the essential ideas and do it well sometimes].

But, it's hard to say, how good/awful the book is, for some book is challenging enough, which could be the *audience* of the book... Remember that in the majority of cases the math or physics course is just a 'feeder' for engineering or basic requirements for some 'other course'. It's not math for mathematicians or physics for physicists... though i think actually it might be nicer in some cases for people to jump through the hoop twice, with an easy book and then a super detailed book.

There's a lot of Taylor's but i don't think it was AE Taylor...

Sherwood and Taylor did their prentice-hall book in the 40s and it was definitely in the top 10 books for the 1945-1950 period.

the early 40s is when the last edition of Horace Lamb's Calculus book, which was probably THE long winded calculus text paired with Hardy's Pure Mathematics, and the late 40s is when the last tweak of Longley Smith came out after 50 plus years of handholding... [it was a popular one for teaching in the US Military too]

and then Taylor and Mann did Advanced Calculus in 1955 and was/is still going in a third edition into the 1980s...

-------

Taylor and Mann [1ed 1955 2ed 1972? 3ed 1983]

[Excellent Clarity of Presentation]
[This book has a clarity unparalleled among books covering similar topics. While it contains an extensive amount of prose, it is still fairly compact: the book explains each result, the motivation for it, and points out possible pitfalls and considerations. Examples are well-chosen, proofs are easily followed. The order of the book is a bit chaotic, but it's written in such a way that it is easy to skip around in it.]
[My only complaint about this book is that I wish it covered a bit more material. This book might not go quite as far as some people might want, especially for a two-semester sequence or for courses at the graduate level.]
[I would recommend this book to anyone who already knows calculus and wants to learn (the more rigorous topic of) analysis on their own, or anyone selecting a textbook for an undergraduate advanced calculus course. This book also makes a good reference, and I was happy to permanently add it to my collection. For a more advanced book covering topics beyond those covered in this book, I would recommend Apostol's analysis book.]
---
[Worth every penny]
[This is the advanced calculus text I used at University of Washington while getting my BS in mathematics. I loved it then, and I've just purchased another copy to use for review. It's extremely well written. If you're looking for a good second year calculus text, this one's it.]
---
[Wonderfully Masterful]
[I am no expert in the area of Mathematical Analysis, but I am an avid reader of any book that pertains the subject. I found this book in my schools mathematics lounge and could not resist reading it from cover to cover. This book is of the quality of such authors as Buck, Widder, Courant, and Rudin. As another reviewer has noted, this book is definitely worth every penny. It is not dry or to pedantic as some of the other afore mentioned authors, yet it is not simple and lacking in content. Of course like any quality Advanced Calculus book it requires the reader to have mathematical maturity as well as patience and the drive to self-explore the concepts. If one cannot follow simple examples and from those examples formulate their own, they may want to review the very basics of mathematics or consider a different major. I would highly recommend this book to advanced undergraduates or beginning gradutes students as a reference book or for self study.]

------

Anyhoo it is surprising that Har would use in the early 60s a mainstream calculus text that wanted a minimum of proofs...

-----

Actually here's a good question, what would be the ideal textbook and supplementary texts that you'd pick Mathwonk for 1960 Harvard, for honours and mainstream calculus?

[I thought of the question when i thought, gee i wonder if Thomas would be a way better choice for the non-honours class than the 'unknown textbook']

---------

I'm thinking

a. Franklin - McGraw-Hill 1953
b. Thomas - Addison-Wesley [2ed 1953 3ed 59-61ish] [before it was Thomas and Finney]

a. Courant Blackie/Interscience 1938
b. Kaplan [for Calculus 3/4] Addison-Wesley 1952
[maybe Taylor for the second class]
[maybe Apostol for both classes]


I just wonder if back then you'd find Thomas too easy, and Apostol too challenging...

i found it interesting that there wasnt too much choice till the New math days really when good and bad textbooks on calculus [and high school and second year] just exploded

Courant was used from the depression till the Space Race and was still pretty strong 60-65 for books... and then the creepier gold courant/fritz john book came out, which was neater and weirder, basically courant bowed down to the new math pressures [heh] and well most people like it, with mixed feelings, but almost *always* prefer the original

I think he started the second edition unneccessary textbook change hype *grin*
 
  • #3,300
for me it was not so much the book, as when i started to take learning seriously, but some books like spivak went out of their way to reach me before i knew how to study. i.e. no book is too hard for a serious student, but some books reach out to the clueless.
 
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