Recently, some classmates and I were lamenting the fact that our classmates in other disciplines had almost no conception of what we did, despite the large mathematics population a …

Even professors in academic departments other than mathematics---never mind other educated people---do not know that such a field as mathematics exists. Once a professor of medici …

Let $z \in \mathbb{C} \backslash \lbrace 1 \rbrace$ with $|z| = 1$. We consider the following infinite series, which necessarily converges: $$S(z) := \sum_{n = 1}^{\infty}\frac{z^n …

I am a TA for a multivariable calculus class this semester. I have also TA'd this course a few times in the past. Every time I teach this course, I am never quite sure how I should …

I am teaching Calc I, for the first time, and I haven't seriously revisited the subject in quite some time. An interesting pedagogy question came up: How misleading is it to regard …

Is a continuos and injective mapping from circle to a jordan curve is surjective ?

How does a depressed graduate student go about recovering his enthusiasm for the subject and the question at hand? Edit: I am not that grad student; it is a very talented friend o …

I'm collecting advanced exercises in geometry. Ideally, each exercise should be solved by one trick and this trick should be useful elsewhere (say it gives an essential idea in som …

Dear All, I'm teaching Calculus and my students asked me to calculate the integral of x*tan(x). I spent quite a lot of effort to do this, but I'm now even not sure if the integra …

hi.. i have a matrix like this: { 1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 1/12, 1/13, …

The study of reciprocity laws is a centerpiece of modern mathematics. Of the last ten Fields Medalists, two of them (Ngô Bảo Châu and Laurent Lafforgue) were awarded Fields Medals …

This summer, I will be teaching an introductory course in graph theory to talented high school seniors. The intent of the course is not to establish proficiency in graph theory, pe …

Are there any books that present theorems as problems? To be more specific, a book on elementary group theory might have written: "Thereom: Each group has exactly one identity" and …

Let $f(x,y)$ be a a real valued function on an open subset of $\mathbf{R}^2$ with continuous partial derivatives $\frac{\partial^2 f}{\partial x^2}$ and $\frac{\partial^2}{\partial …

I am developing an introductory topology course for undergraduates, and I am wondering what topics to cover. At my institution, real analysis is not a prerequisite for the course, …