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1500 questions
57
votes
8 answers
Two (probably) equal real numbers which are not proved to be equal?
Can someone give me a nice example of two computable real numbers which are believed but not proved to be equal?
I never really understood the assertion that "the reals do not have decidable equality" until I saw concrete examples such as this gem.…
Kevin Buzzard
- 40,559
57
votes
43 answers
What are some mathematical sculptures?
Either intentionally or unintentionally.
Include location and sculptor, if known.
Gerald Edgar
- 40,238
57
votes
16 answers
What are examples of books which teach the practice of mathematics?
One may classify the types of mathematics books written for students into two groups: books which merely teach mathematics (i.e., they present theorems and proofs, ready-made, as it were) and those books which teach one the art of mathematics (i.e.,…
Allawonder
- 335
57
votes
2 answers
Recent observation of gravitational waves
It was exciting to hear that LIGO detected the merging of two black
holes one billion light-years away. One of the black holes had 36
times the mass of the sun, and the other 29. After the merging the
mass was that of 62 suns, with the rest…
Richard Stanley
- 49,238
57
votes
4 answers
Advice for PhD Supervisors
My first PhD student is having his viva tomorrow. Hence, I began contemplating a bit about the whole process of supervising. One thing I realized is that while there seems to be plenty of advice for PhD students I cannot recall ever seeing advice…
Yiftach Barnea
- 5,208
- 2
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57
votes
11 answers
Interesting results in algebraic geometry accessible to 3rd year undergraduates
On another thread I asked how I could encourage my final year undergraduate colleagues to take an algebraic geometry or complex analysis courses during their graduate studies.
Willie Wong proposed me following idea - to show them some interesting…
ifk
- 1,034
57
votes
7 answers
Maryam Mirzakhani's works
Maryam Mirzakhani has made several contributions to the theory of moduli spaces of Riemann surfaces.
Mirzakhani was awarded the Fields Medal in 2014 for "her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli…
user68866
57
votes
7 answers
What are surprising examples of Model Categories?
Background
Model categories are an axiomization of the machinery underlying the study of topological spaces up to homotopy equivalence. They consist of a category $C$, together with three distinguished classes of morphism: Weak Equivalences,…
Greg Muller
- 12,679
57
votes
6 answers
Escape the zombie apocalypse
Consider zombies placed uniformly at random over $\mathbb{R}^2$ with asymptotic density $\mu$ zombies/area. You are placed at a random point and can move with speed $1$. Zombies move with speed $v\leq 1$ straight towards you, what is the probability…
TROLLHUNTER
- 730
57
votes
3 answers
Italian school of algebraic geometry and rigorous proofs
Many of the amazing results by Italian geometers of the second half of the 19th and the first half of the 20th century were initially given heuristic explanations rather than rigorous proofs by their discoverers. Proofs appeared only later. In some…
algori
- 23,231
57
votes
2 answers
What arithmetic information is contained in the algebraic K-theory of the integers
I'm always looking for applications of homotopy theory to other fields, mostly as a way to make my talks more interesting or to motivate the field to non-specialists. It seems like most talks about Algebraic $K$-theory mention that we don't know…
David White
- 29,779
57
votes
3 answers
Number of elements in the set $\{1,\cdots,n\}\cdot\{1,\cdots,n\}$
Let $A_n=\{a\cdot b : a,b \in \mathbb{N}, a,b\leq n\}$. Are there any estimates for $|A_n|$? Will it be $o(n^2)$?
Kamalakshya
- 887
56
votes
3 answers
What are the benefits of viewing a sheaf from the "espace étalé" perspective?
I learned the definition of a sheaf from Hartshorne—that is, as a (co-)functor from the category of open sets of a topological space (with morphisms given by inclusions) to, say, the category of sets. While fairly abstract at the outset, this seems…
Simon Rose
- 6,250
56
votes
6 answers
What do Weierstrass points look like?
As somebody who mostly works with smooth, real manifolds, I've always been a little uncomfortable with Weierstrass points. Smooth manifolds are totally homogeneous, but in the complex category you are just walking around, minding your own business,…
Matt Noonan
- 3,984
56
votes
6 answers
Is the Mendeleev table explained in quantum mechanics?
Does anybody know if there exists a mathematical explanation of the Mendeleev table in quantum mechanics? In some textbooks (for example in "F.A.Berezin, M.A.Shubin. The Schrödinger Equation") the authors present quantum mechanics as an axiomatic…
Sergei Akbarov
- 7,274