Contents: Math 414/514 is an introduction to Analysis on Euclidean spaces (). The emphasis is theoretical, as opposed to the more computational approach of calculus. From the Course Description on the Department’s site:

Introduction to fundamental elements of analysis on Euclidean spaces including the basic differential and integral calculus. Topics include: infinite series, sequences and series of function, uniform convergences, theory of integration, implicit function theorem and applications.

Grading: Based on homework. No late homework is allowed. Collaboration is encouraged, although you must turn in your own version of the solutions, and give credit to books/websites/… you consulted and people you talked/emailed/… to.

I do not want to have exams in this course. However, an important component of being proficient in mathematics is a certain amount of mental agility in recalling notions and basic arguments. I plan to assess these by requesting oral presentations of solutions to some of the homework problems throughout the term. If I find you lacking here, it will be necessary to have an exam or two. The final exam is currently scheduled for Wednesday, December 18, 2013, 12:00 – 2:00 pm.

I will use this website to post additional information, and encourage you to use the comments feature. If you leave a comment, please use your full name, which will simplify my life filtering spam out.

On occasion, I post links to supplementary material on Google+. Circle me and let me know if you are interested, and I’ll add you to my Analysis circle. As with this blog, I encourage you to comment there.

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One Response to 414/514 – Advanced calculus aka Analysis I – Syllabus

[…] for all closed measure zero sets , then is strong measure zero. (Since this was intended for my analysis course, and I do not see how to prove Pawlikowski’s argument without some appeal to results in […]

(1) Patrick Dehornoy gave a nice talk at the Séminaire Bourbaki explaining Hugh Woodin's approach. It omits many technical details, so you may want to look at it before looking again at the Notices papers. I think looking at those slides and then at the Notices articles gives a reasonable picture of what the approach is and what kind of problems remain […]

The description below comes from József Beck. Combinatorial games. Tic-tac-toe theory, Encyclopedia of Mathematics and its Applications, 114. Cambridge University Press, Cambridge, 2008, MR2402857 (2009g:91038). Given a finite set $S$ of points in the plane $\mathbb R^2$, consider the following game between two players Maker and Breaker. The players alternat […]

Yes. This is a consequence of the Davis-Matiyasevich-Putnam-Robinson work on Hilbert's 10th problem, and some standard number theory. A number of papers have details of the $\Pi^0_1$ sentence. To begin with, take a look at the relevant paper in Mathematical developments arising from Hilbert's problems (Proc. Sympos. Pure Math., Northern Illinois Un […]

I am looking for references discussing two inequalities that come up in the study of the dynamics of Newton's method on real-valued polynomials (in one variable). The inequalities are fairly different, but it seems to make sense to ask about both of them in the same post. Most of the details below are fairly elementary, they are mostly included for comp […]

Let $C$ be the standard Cantor middle-third set. As a consequence of the Baire category theorem, there are numbers $r$ such that $C+r$ consists solely of irrational numbers, see here. What would be an explicit example of a number $r$ with this property? Short of an explicit example, are there any references addressing this question? A natural approach would […]

First of all, $f(z)+e^z\ne 0$ by the first inequality. It follows that $e^z/(f(z)+e^z)$ is entire, and bounded above. You should be able to conclude from that.

Yes. The standard way of defining these sequences goes by assigning in an explicit fashion to each limit ordinal $\alpha$, for as long as possible, an increasing sequence $\alpha_n$ that converges to $\alpha$. Once this is done, we can define $f_\alpha$ by diagonalizing, so $f_\alpha(n)=f_{\alpha_n}(n)$ for all $n$. Of course there are many possible choices […]

I disagree with the advice of sending a paper to a journal before searching the relevant literature. It is almost guaranteed that a paper on the fundamental theorem of algebra (a very classical and well-studied topic) will be rejected if you do not include mention on previous proofs, and comparisons, explaining how your proof differs from them, etc. It is no […]

No, the rank of a set $x$ is the least $\alpha$ such that $x\in V_{\alpha+1}$. Note that if $\alpha$ is limit, any $x\in V_\alpha$ belongs to some $V_\beta$ with $\beta

[…] for all closed measure zero sets , then is strong measure zero. (Since this was intended for my analysis course, and I do not see how to prove Pawlikowski’s argument without some appeal to results in […]