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truebaran
  • Member for 8 years, 5 months
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  • Poland
235 votes
36 answers
30k views

What are some counter-intuitive results in mathematics that involve only finite objects?

  • 3,626
188 votes
8 answers
33k views

Are there any series whose convergence is unknown?

  • 2,151
168 votes
6 answers
27k views

Induction on Real Numbers

  • 1,783
142 votes
7 answers
29k views

Elementary proof that $\mathbb{R}^n$ is not homeomorphic to $\mathbb{R}^m$

  • 46.2k
59 votes
1 answer
4k views

Is it possible for a function to be smooth everywhere, analytic nowhere, yet Taylor series at any point converges in a nonzero radius?

56 votes
1 answer
10k views

Difference between simplicial and singular homology?

  • 1,035
51 votes
3 answers
7k views

Roadmap to study Atiyah–Singer index theorem

28 votes
3 answers
9k views

Let $G$ be a Lie group. Show that there is a diffeomorphism $TG \cong G \times T_e G$.

25 votes
1 answer
1k views

How much of an $n$-dimensional manifold can we embed into $\mathbb{R}^n$?

  • 2,660
20 votes
4 answers
2k views

A theorem due to Gelfand and Kolmogorov

19 votes
2 answers
4k views

Spin manifold and the second Stiefel-Whitney class

  • 5,549
14 votes
2 answers
920 views

Which functions are derivatives of some other function?

  • 143
13 votes
2 answers
2k views

Each point of $M$ has a smooth coordinate neighborhood in which the coordinate frame is orthonormal iff $g$ is flat

  • 1,017
13 votes
1 answer
6k views

The dual of the Sobolev space $W^{k,p}$

11 votes
3 answers
3k views

Abstract nonsense proof

  • 9,618
11 votes
1 answer
968 views

Fat geometric realization weakly equivalent to the usual one

11 votes
2 answers
652 views

How many closed subsets of $\mathbb R$ are there up to homeomorphism?

11 votes
2 answers
1k views

Nonisomorphic vector bundles with diffeomorphic total spaces

  • 13.1k
10 votes
3 answers
2k views

Decidability vs Completeness

8 votes
2 answers
311 views

Maximize area of a quadrilateral given three sides

  • 189
7 votes
1 answer
847 views

Fundamental group of open subsets of $\mathbb{R}^n$

  • 4,380
6 votes
2 answers
1k views

$X=\{\cos n + i\sin n : n \in \mathbb{N}\}$ is dense in $\mathbb{S}^1 \subset \mathbb{C}$

  • 3,866
2 votes
1 answer
358 views

A family of events such that all proper subfamilies are independent, but the entire family is not