Unanswered Questions
7,693 questions with no upvoted or accepted answers
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Hironaka's proof of resolution of singularities in positive characteristics
Recent publication of Hironaka seems to provoke extended discussions, like Atiyah's proof of almost complex structure of $S^6$ earlier...
Unlike Atiyah's paper, Hironaka's paper does not have a ...
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On the first sequence without triple in arithmetic progression
In this Numberphile video (from 3:36 to 7:41), Neil Sloane explains an amazing sequence:
It is the lexicographically first among the sequences of positive integers without triple in arithmetic ...
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Does every triangle-free graph with maximum degree at most 6 have a 5-colouring?
A very specific case of Reed's Conjecture
Reed's $\omega$,$\Delta$, $\chi$ conjecture proposes that every graph has $\chi \leq \lceil \tfrac 12(\Delta+1+\omega)\rceil$. Here $\chi$ is the chromatic ...
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A = B (but not quite); 3-d arrays with multiple recurrences
Many years ago, I discovered the remarkable array (apparently originally discovered by Ramanujan)
1
1 3
2 10 15
6 40 105 105
24 196 700 1260 945
...
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Is there anything to the obvious analogy between Joyal's combinatorial species and Goodwillie calculus?
Combinatorial species and calculus of functors both take the viewpoint that many interesting functors can be expanded in a kind of Taylor series. Many operations familiar from actual calculus can be ...
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What does the theta divisor of a number field know about its arithmetic?
This question is about a remark made by van der Geer and Schoof in their beautiful article "Effectivity of Arakelov divisors and the theta divisor of a number field" (from '98) (link).
Let ...
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36
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3-colorings of the unit distance graph of $\Bbb R^3$
Let $\Gamma$ be the unit distance graph of $\Bbb R^3$: points $(x,y)$ form an edge if $|x,y|=1$.
Let $(A,B,C,D)$ be a unit side rhombus in the plane, with a transcendental diagonal, e.g. $A = (\alpha,...
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35
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Orthogonal vectors with entries from $\{-1,0,1\}$
Let $\mathbf{1}$ be the all-ones vector, and suppose $\mathbf{1}, \mathbf{v_1}, \mathbf{v_2}, \ldots, \mathbf{v_{n-1}} \in \{-1,0,1\}^n$ are mutually orthogonal non-zero vectors. Does it follow that $...
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Grothendieck's "List of classes of structures"
In Lawvere's article Comments on the Development of Topos Theory, the author writes:
Similarly, Grothendieck and others unerringly recognized which kinds of mathematical structures are 'preserved ...
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32
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The easily bored sequence
If we want to compare the repetitiveness of two finite words, it looks reasonable, first of all, to consider more repetitive the word repeating more times one of its factors, and secondarily to ...
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Vertex coloring inherited from perfect matchings (motivated by quantum physics)
Added (19.01.2021): Dustin Mixon wrote a blog post about the question where he reformulated and generalized the question.
Added (25.12.2020): I made a youtube video to explain the question in detail.
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32
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Minimal number of intersections in a convex $n$-gon?
For a convex polygon $P$, draw all the diagonals of $P$ and consider the intersection points made by those diagonals. Let $f(n)$ be the minimal number of such intersections where $P$ ranges over all ...
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Is there a Mathieu groupoid M_31?
I have read something which said that the large amount of common structure between the simple groups $SL(3,3)$ and $M_{11}$ indicated to Conway the possibility that the Mathieu groupoid $M_{13}$ might ...
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A Combinatorial Abstraction for The "Polynomial Hirsch Conjecture"
Consider $t$ disjoint families of subsets of {1,2,…,n}, ${\cal F}_1,{\cal F_2},\dots {\cal F_t}$ .
Suppose that
(*)
For every $i \lt j \lt k$
and every $R \in {\cal F}_i$, and $T \in {\cal F}_k$,
...
CommunityBot
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31
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"Three great cocycles" in Complex Analysis as cohomology generators
In his lecture notes, C. McMullen discusses "the three great cocycles" in Complex Analysis: the derivative $$f\mapsto\log f',$$ the non-linearity $$f\mapsto (\log f')'dz$$
and the Schwarzian ...
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