An earlier answer by Andrey Rekalo suggested the Banach fixed point theorem. I want to elaborate on that answer with a particular direction to take this theorem: fractals.

The proof of BFPT is so easy that, without a "flashy" application, it's tempting for students to dismiss it as trivial. But it pairs extremely well with the observation that, if $M$ is a "reasonable" metric space, then appropriate combinations of contraction mappings on $M$ yield interesting contraction mappings on an appropriate hyperspace of $M$. In particular, we have the following:

$(\star)\quad$ If $F_1,...F_n$ are contraction mappings on the plane $\mathbb{R}^2$ (with the usual metric), then the map $$X\mapsto \bigcup_{1\le i\le n}F_i[X]$$ is again a contraction mapping on the space of compact subsets of $\mathbb{R}^2$ equipped with the Hausdorff metric.

The BFPT then kicks in and says that "solutions exist" to the corresponding equations describing certain sets. It's a fun exercise, then, to describe fractals such as the Sierpinski triangle "algebraically." See e.g. this survey paper by Natoli.

In terms of fitting this into 30 minutes, the major hurdle is the definition of the Hausdorff metric, but I think this can be handwaved with a reasonable picture. (And there's no need to prove $(\star)$ in the presentation itself; it can be given as a fun exercise.)

An earlier answer by Andrey Rekalo suggested the Banach fixed point theorem. I want to elaborate on that answer with a particular direction to take this theorem: fractals.

The proof of BFPT is so easy that, without a "flashy" application, it's tempting for students to dismiss it as trivial. But it pairs extremely well with the observation that, if $M$ is a "reasonable" metric space, then appropriate combinations of contraction mappings on $M$ yield interesting contraction mappings on an appropriate hyperspace of $M$. In particular, we have the following:

$(\star)\quad$ If $F_1,...F_n$ are contraction mappings on the plane $\mathbb{R}^2$ (with the usual metric), then the map $$X\mapsto \bigcup_{1\le i\le n}F_i[X]$$ is again a contraction mapping on the space of compact subsets of $\mathbb{R}^2$ equipped with the Hausdorff metric.

The BFPT then kicks in and says that "solutions exist" to the corresponding equations describing certain sets. It's a fun exercise, then, to describe fractals such as the Sierpinski triangle "algebraically." See e.g. this survey paper by Natoli.

In terms of fitting this into 30 minutes, the major hurdle is the definition of the Hausdorff metric, but I think this can be handwaved with a reasonable picture. (And there's no need to prove $(\star)$ in the presentation itself; it can be given as a fun exercise.)

An earlier answer by Andrey Rekalo suggested the Banach fixed point theorem. I want to elaborate on that answer with a particular direction to take this theorem: fractals.

The proof of BFPT is so easy that, without a reasonably-surprising application, it's tempting for students to dismiss it as trivial. But it pairs extremely well with the observation that, if $M$ is a "reasonable" metric space, then appropriate combinations of contraction mappings on $M$ yield interesting contraction mappings on an appropriate hyperspace of $M$. In particular, we have the following:

$(\star)\quad$ If $F_1,...F_n$ are contraction mappings on the plane $\mathbb{R}^2$ (with the usual metric), then the map $$X\mapsto \bigcup_{1\le i\le n}F_i[X]$$ is again a contraction mapping on the space of compact subsets of $\mathbb{R}^2$ equipped with the Hausdorff metric.

The BFPT then kicks in and says that "solutions exist" to the corresponding equations describing certain sets. It's a fun exercise, then, to describe fractals such as the Sierpinski triangle "algebraically." See e.g. this survey paper by Natoli.

In terms of fitting this into 30 minutes, the major hurdle is the definition of the Hausdorff metric, but I think this can be handwaved with a reasonable picture. (And there's no need to prove $(\star)$ in the presentation itself; it can be given as a fun exercise.)

An earlier answer by Andrey Rekalo suggested the Banach fixed point theorem. I want to elaborate on that answer with a particular direction to take this theorem: fractals.

The proof of BFPT is so easy that, without a "flashy" application, it's tempting for students to dismiss it as trivial. But it pairs extremely well with the observation that, if $M$ is a "reasonable" metric space, then appropriate combinations of contraction mappings on $M$ yield interesting contraction mappings on an appropriate hyperspace of $M$. In particular, we have the following:

$(\star)\quad$ If $F_1,...F_n$ are contraction mappings on the plane $\mathbb{R}^2$ (with the usual metric), then the map $$X\mapsto \bigcup_{1\le i\le n}F_i[X]$$ is again a contraction mapping on the space of compact subsets of $\mathbb{R}^2$ equipped with the Hausdorff metric.

The BFPT then kicks in and says that "solutions exist" to the corresponding equations describing certain sets. It's a fun exercise, then, to describe fractals such as the Sierpinski triangle "algebraically." See e.g. this survey paper by Natoli.

In terms of fitting this into 30 minutes, the major hurdle is the definition of the Hausdorff metric, but I think this can be handwaved with a reasonable picture. (And there's no need to prove $(\star)$ in the presentation itself; it can be given as a fun exercise.)