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According to Grothendieck, his famous work Esquisse d'un Programme was at least partially inspired by his teaching experience. … The remarkable feature of this example is that here we apparently deal not just with teaching-based research but with teaching-based CHANGE OF THE SUBJECT of reaserch of one of the greatest mathematicians …

According to David Fowler, (Dedekind's theorem: $\sqrt{2}\sqrt{3}=\sqrt{6}$, Amer Math Monthly 99 (1992), 8, 725-733), Dedekind constructed real numbers on Wednesday, November 24, 1858, in the process …

A very similar question was already asked on this site: Teaching homology via everyday examples. …

There is a paper of G. H. Hardy, where this function is studied in great detail: G. H. Hardy, On the integral function $ \Phi_{ a,\alpha,\beta}(z)=\sum x^n/(n+a)^{\alpha n+\beta}$, Quarterly J. Math. …

Suppose you maintain a pond with fish (for profit, of course, this is economics!). When the food is abundant and there are not many fish, the population grows at a constant rate $k>1$ (reproduction ra …

Some of the mathematical advances that influenced people's life in XX century are: Fast Fourier Transform (applications to signal processing, in particular, image processing. Without it modern comput …

Erdos's problem on the length of lemniscates (it is somewhat famous in certain narrow circles). Let $P$ be a polynomial, and consider the set $E=\{ z:|P(z)|=1\}$ in the complex plane. What is the …

Here is another problem on equilibrium points of potentials: suppose that we have infinitely many point masses in $R^3$ (the points do not accumulate). Must there exist a point where the gravitational …

Here is another easy to state problem which is 140 years old but not very famous. Consider the potential of finitely many positive charges: $$u(x)=\sum_{j=1}^n\frac{a_j}{|x-x_j|},\quad x,x_j\in R^3,\q …

Weierstrass's function is the real part of $$\sum_{n=0}^\infty a^nz^{b^n},\quad |z|\leq 1,$$ where $b\geq 2$ is an integer, and $a<1$. It was studied by complex analysis in G. H. Hardy, in a series of …

Here is a typical use in an undergraduate textbook: to prove that for distinct $\lambda_j$ the exponentials $e^{\lambda_jt}$ are linearly independent. It has some applications on the more advanced lev …

I use this example in my teaching in two ways. …

Complete set of Quantum magazine. It exists in good libraries and sometimes can be found on e-bay. The journal existed for 11 years (1990-2001) but apparently there are no enough "bright high school s …

Here is an example of economic interpretation of cohomology from some Russian popular lecture that I read (unfortunately, I do not remember the source; I only remember it was in Russian). Consider a …

There is a nice book, specially written for high school students: V. B. Alekseev, Abel's theorem in problems and solutions. Based on the lectures of V. I. Arnold (to high school students), and also fr …