I'm teaching an undergrad course in real analysis this Fall and we are using the text "Real Mathematical Analysis" by Charles Pugh. On the back it states that real analysis involves no "applications to other fields of science. None. It is pure mathematics." This seems like a false statement. My first thought was of probability theory. And isn't PDE's sometimes considered applied math? I was wondering what others thought about this statement.
[Math] Real analysis has no applications
real-analysissoft-questionteaching
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Our department has had a lot of discussions about resources. I rather like the following ideas and good practices that have been implemented or proposed here:
The Calculus Room. Instead of having each calculus TA sit in his or her office and wait for students, they are assigned times in one big help room called "The Calculus Room". This is a much more efficient system that helps more students per hour of labor.
We have a grade distribution system called "MyUCDavis" that I use a lot. Students can see all recorded grades quickly, throughout the entire quarter. They can also see their rank on each test (and HW) and a histogram of test scores. I imagine that Moodle can do something like this too. I like it because the students can know where they stand, and because I don't like face-to-face questions about grades. Also, I of course recommend posting homework and solution sets on the web, but these days that should go without saying.
WebWork. We just started using WebWork after a negative experience with another system. It is not perfect, but it is a genuinely helpful educational tool. Enough human attention to homework is better in principle, but grading homework in calculus can easily deteriorate to the point that WebWork is better.
Cheap/free textbooks. This is more about saving the students' money than ours, but in the face of a 30% fee hike in one year, we are eager to create goodwill. When a good choice is available, I like the model of using a book that is both sold in print and has a free or nearly free PDF. (Or maybe we can arrange to print and bind such a book at the copy shop.) The first really good book in this model that we used was Hatcher, Algebraic Topology, but more recently there are others. I respect ideas such as wiki-books and teaching with Wikipedia as experiments and supplements, but they are not presently a good substitute for a tried-and-true, structured textbook.
Slowly, incrementally try to raise standards. For instance, we recently shifted our 3-quarter intro analysis so that the first quarter is lower division. The first quarter is taught in the style of Spivak's classic, Calculus. (But Spivak is expensive. As of this year, we use Thomson, Bruckner, Bruckner, because it's a very nice textbook, and the PDF is only one dollar.) More commonly, we just revise the syllabus of this or that course to make it more interesting. We do not have a two-track system for good students vs bad students and I suspect that I wouldn't want it. The students are free to take harder or easier courses within a certain range.
I am tempted to stump for the centrality of Galois theory in modern mathematics, but I feel that this subject is too close to my own research interests (e.g., I have worked on the Inverse Galois Problem) for me to do so in a truly sober manner. So I will just make a few brief (edit: nope, guess not!) remarks:
1) Certainly when I teach graduate level classes in number theory, arithmetic geometry or algebraic geometry, I do in practice expect my students to have seen Galois theory before. I try to cultivate an attitude of "Of course you're not going to know / remember all possible background material, and I am more than willing to field background questions and point to literature [including my own notes, if possible] which contains this material." In fact, I use a lot of background knowledge of field theory -- some of it that I know full well is not taught in most standard courses, some of it that I only thought about myself rather recently -- and judging from students' questions and solutions to problems, good old finite Galois theory is a relatively known subject, compared to say infinite Galois theory (e.g. the Krull topology) and things like inseparable field extensions, linear disjointness, transcendence bases....So I think it's worth remarking that Galois theory is more central, more applicable, and (fortunately) in practice better known than a lot of topics in pure field theory which are contained in a sufficiently thick standard graduate text.
2) In response to one of Harry Gindi's comments, and to paraphrase Siegbert Tarrasch: before graduate algebra, the gods have placed undergraduate algebra. A lot of people are talking about graduate algebra as a first introduction to things that I think should be first introduced in an undergraduate course. I took a year-long sequence in undergraduate algebra at the University of Chicago that certainly included a unit on Galois theory. This was the "honors" section, but I would guess that the non-honors section included some material on Galois theory as well. Moreover -- and here's where the "but you became a Galois theorist!" objection may hold some water -- there were plenty of things that were a tougher sell and more confusing to me as a 19 year old beginning algebra student than Galois theory: I found all the talk about modules to be somewhat abstruse and (oh, the callowness of youth) even somewhat boring.
3) I think that someone in any branch of pure mathematics for whom the phrase "Galois correspondence" means nothing is really missing out on something important. The Galois correspondence between subextensions and subgroups of a Galois extension is the most classical case and should be seen first, but a topologist / geometer needs to have a feel for the Galois correspondence between subgroups of the fundamental group and covering spaces, the algebraic geometer needs the Galois correspondence between Zariski-closed subsets and radical ideals, the model theorist needs the Galois correspondence between theories and classes of models, and so forth. This is a basic, recurrent piece of mathematical structure. Not doing all the gory detail of Galois theory is a reasonable option -- I agree that many people do not need to know the proofs, which are necessarily somewhat intricate -- but skipping it entirely feels like a big loss.
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As it happens, I just finished teaching a quarter of undergraduate real analysis. I am inclined to rephrase Pugh's statement into a form that I would agree with. If you view analysis broadly as both the theorems of analysis and methods of calculation (calculus), then obviously it has a ton of applications. However, I much prefer to teach undergraduate real analysis as pure mathematics, more particularly as an introduction to rigorous mathematics and proofs. This is partly as a corrective (or at least a complement) to the mostly applied and algorithmic interpretation of calculus that most American students see first.
Some mathematicians think, and I've often been tempted to think, that it's a bad thing to do analysis twice, first as algorithmic and applied calculus and second as rigorous analysis. It can seem wrong not to have the rigor up-front. Now that I have seen what BC Calculus is like in a high school, I no longer think that it is a bad thing. Obviously I still think that the pure interpretation is important. On the other hand, both interpretations together is also fine by me. I notice that in France, calculus courses and analysis courses are both called "analyse mathématique". I think that they might separate rigorous and non-rigorous calculus a bit less than in the US, and it could be partly because of the name.
In fact, it took me a long time to realize how certain non-rigorous explanations guide good rigorous analysis. For instance, the easy way to derive the Jacobian factor in a multivariate integral is to "draw" an infinitesimal parallelepiped and find its volume. That's not rigorous by itself, but it is related to an important rigorous construction, the exterior algebra of differential forms.
Finally, I agree that Pugh's book is great. As the saying goes, you shouldn't judge it by its cover. :-)