Take two…

Well- not much happened this fall…  OK, a lot happened, but we were all too busy to tell anyone about any of it!

The biggest lesson I learned was not to try teaching two classes while taking a full load of graduate credits. And buying a house.  And moving…  Yes, I took on a bit too much, and my posts necessarily suffered.

Winter quarter will be different! Continue reading

A New View of Analysis:

“Think about what things do rather than about how they are defined.”

That’s a nice piece of advice from Dr. Erdman in response to my question about ideals in class today. Having already taken abstract algebra, the image of an ideal in my head is as an algebraic structure with certain properties:

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Real Analysis: A New Beginning!

Hello everyone!

The text has been updated! So has the syllabus for those of us who want to get a jump on the homework.

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Why Order is Important…

Not a lot of time this morning, but if you are interested in seeing how order is used in set theory, John Armstrong is doing a nice series of the creation of numbers at The Unapologetic Mathematician. Today he talks about Archimedean fields and how the real numbers are in the ‘sweet spot’ between being too big (any larger and they wouldn’t be Archimedean) and too small (any smaller and they would not be Dedekind Complete).

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Free Real Analysis Book

The 3 of us are taking Real Analysis this year, and our professor has his own book online, “A Companion to Real Analysis.” It’s dense, contains no proofs, and is generally intense. But, those are good things. I wouldn’t use it by itself, but it provides you a great linear progression, moving from basic set theory, functions, and order, then into Algebra, Vector Spaces, Topology, Continuity, and Normed Linear Spaces. Clearly it’s not complete (he’s constantly updating), as it is only to be used in the first part of a year long course. But, it’s proven to be a solid, and rigorous, introduction to the topic.

[It also contains a digression on Categories, but if you have no prior knowledge of them, you’re libel to be lost for a while.]

Quote:

Most of (real) analysis is a study of the interplay of algebra and topology. Thus far in these notes there has been quite a bit of algebra, but no topology. Now we begin to remedy this unhappy situation.

-Saij

New Analysis and PDE Journal

Terence Tao has joined the editorial board of a new Analysis and PDE journal. If you have any papers that you’re looking to submit, consider this one. Journals in all academic fields are notoriously expensive (read unaccesable), and that serves to be a clog in the arteries of progress. There is a growing consensus among Mathematicians, scientists, and academics that information doesn’t belong in the hands of major publishing houses who require you to sell you their soul just to read what’s inside.

Last year, the entire editorial board of the Elsevier journal Topology resigned, and this year the Springer journal K-theory, suffered the same fate. I say good.

I’m glad to see that the Mathematical Sciences Publishers makes available more (potentially) high quality journals that won’t break the bank. Other titles include Geometry & Topology, Algebra & Number Theory, and Algebraic & Geometric Topology.

In the modern world of blogs and PDF Files, the only service the journals provide is their editorial board. We’ll see how the future plays out, and what it does to modern science.

–Saij

The Lyceum Mathematikoi Test Post

This is a test post.  This post is only a test.  Do not pass go, do not collect two-hundred dollars mod n.