8
votes
How to think of algebraic geometry in characteristic p?
The basic strategy you give is correct: To first approximation, algebraic geometry over $\overline{\mathbb F_p}$ is like algebraic geometry over $\mathbb C$ except for extra phenomena. Usually, I ...
- 141k
5
votes
Accepted
Heuristic interpretations of the PA-unprovability of Goodstein's Theorem
You've shown how to prove, in PA, the statement "the Goodstein sequence starting with $p$ terminates" for any given $p$. But once $p$ is given, that statement has a proof in PA that just ...
- 72.4k
4
votes
Accessible literature on fractional dimensions of subsets of $\mathbb R^n$
Erin Pearse's Introduction to dimension theory and fractal geometry may well be suited for this purpose. It introduces the various ways to define and measure a fractional dimension (box counting, ...
- 183k
4
votes
Heuristic interpretations of the PA-unprovability of Goodstein's Theorem
To my way of thinking, the arguments you mention seem not to distinguish sufficiently between the content of Goodstein's theorem as a universal claim $\forall p$ and Goodstein's theorem as a ...
- 230k
4
votes
How to think of algebraic geometry in characteristic p?
Since you've mentioned $\mathbb{A}^1$, we can look at group schemes $G \subset \mathbb{G}_a$ sitting inside the additive group scheme, and contrast what is going on over characteristic 0 versus over ...
- 41
4
votes
Hilbert's approach to Riemann hypothesis using Fredholm's work
Since this question was bumped to the front page, I might address
Q1: Can someone provide historical references for it?
This goes back to André Weil, who writes in [1] that Ernst Hellinger, a student ...
- 183k
3
votes
Completing half of Hilbert's program: Foundations that are conservative over Peano Arithmetic
The foundation in this answer is probably a bit weird to work in, but I think it is interesting meta-mathematically and would at least be interesting to compare other foundations too. Perhaps it is bi-...
- 6,298
3
votes
Online, evolving, collaborative foundational text projects
The 1Lab is an online, collaborative reference for univalent mathematics formalised in Cubical Agda.
3
votes
When is one 'ready' to make original contributions to mathematics?
There is certainly much sense/truth in other answers and comments... if we are careful to put them in context.
E.g., Weil and Serre were two of the most scholarly of mathematicians in the 20th century....
2
votes
Golden ratio in contemporary mathematics
The golden ratio, $\phi:=\frac{1+\sqrt{5}}{2}$, plays a key role in minimizing the volume of the axis-aligned bounding box (AABB) that contains the minimum-link polygonal chain which joins the $8$ ...
2
votes
Colloquial catchy statements encoding serious mathematics
When I learned some algebraic geometry for the first time, the teacher gave this paraphrase of Tolstoy:
"All smooth points are smooth in the same way; each singular point is singular in its own ...
1
vote
Accessible literature on fractional dimensions of subsets of $\mathbb R^n$
The book Fractal Geometry - Mathematical Foundations and Applications by Kenneth Falconer may be what you are looking for. As far as I recall, everything is done in $\mathbb{R}^n$ and he tries to keep ...
- 56
1
vote
What kind of computer tools topologists/geometers use to visualize the objects they deal with?
Nate Eldredge's comment is the one to take on board. Imagine a square 2" wide. Draw a circle of radius 1" at each corner. In the middle you can draw a circle touching all four circles, whose ...
- 451
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