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As explained in the comments, if $k$ is a field, then $k$-algebra homomorphisms $k^X\to k$ are in bijection with $|k|^+$-complete ultrafilters on $X$ (that is, ultrafilters closed under $|k|$-fold int …

In a Swedish math competition for high-school teachers, where I was on the problem committee, I contributed with this problem (swedish). For those that are not familiar with Swedish, the problem is …

Road coloring theorem and its proof by Avraham Trahtman are very accessible.

I have problems to verify your calculations with Macaulay 2, concretely the fiber over $t = 0$ (resp. $a=0$ as its called by me). I used the description given at the OP for the ideal $\mathcal{I}$ an …

Although this is not directly about Riemannian geometry, his paper Arc structure of Singularities (1995) certainly classifies as sufficiently geometrical. By all accounts, this paper was actually conc …

The Essential John Nash (ed. Kuhn and Nasar) contains the full text of nine of Nash's papers along with some editorial introductions and an autobiographical essay by Nash.

Let me just summarize the comment thread in case someone more knowledgeable is willing to intervene. Artie observes that The Cox Ring of a Del Pezzo surface by Batyrev--Popov shows that every effecti …

I'm posting some references given to me by Jon Chaika as a CW answer, which addresses the question of whether we can get a set $X_\alpha$ such that the restriction to $X_\alpha$ is Hölder continuous. …

You are not detailed about the setting, but in many situations, say for an equilibrium measure of a Hölder continuous function on a repeller, the Hausdorff dimension of your union is the maximum (or s …

"Teach the subject before its applications." Some important constructions seem quite pointless until you understand the rationale for them. For example, I recall finding the lectures in freshman line …

There are many applications of numerical invariants (LS-category) to differential geometry, but let me give one pretty example to resolution of polynomial equations. In the paper "On the equation of …

Using Priestley duality for distributive lattices and compact, totally disconnected ordered topological spaces, many purely algebraic questions have been solved using quite simple topological tools. F …

There are many examples coming from topological field theories, though I'm not quite sure that it's the sort of example you're looking for. At first it seems that TFT's are about using algebraic stru …

Being a sheaf in the étale topology is insufficient. Begin with the locally ringed space $(\text{Spec}\mathbb{Z},\mathcal{O}_{\text{Spec}\mathbb{Z}})$. Let $$i:X\hookrightarrow \text{Spec}\mathbb{Z} …

The whole field of geometric group theory very much uses geometric and topological arguments to prove group theory facts. Stallings for example used topological arguments to prove that only free group …