78
votes

### Theorems that are essentially impossible to guess by empirical observation

One of the most interesting examples that happened recently is the Katz-Sarnak conjecture asserting that the average rank of elliptic curves (ordered by some reasonable height) defined over $\mathbb{Q}...

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76
votes

### Theorems that are essentially impossible to guess by empirical observation

Bootstrap percolation is a two-dimensional two-state cellular automaton with a von Neumann 5-square ("plus") neighborhood where a "white" cell become "black" if it has at ...

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70
votes

### Examples of unexpected mathematical images

Sorry for the duplicate with @paw 's entry , but I'm the discoverer of Sloane's gap and I don't have enough reputation here yet to comment his/her answer...
So here is another explanation of this ...

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60
votes

### The use of computers leading to major mathematical advances II

There is the recent computer-assisted verification of some key statements by Scholze and Clausen about "condensed mathematics". The task has been accomplished by Buzzard, Commelin, and ...

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59
votes

### Theorems that are essentially impossible to guess by empirical observation

Letting $\pi$ be the prime counting function and $\mathrm{Li}$ the logarithmic integral, Littlewood proved in his 1914 article "Sur la distribution des nombres premiers" that the difference $...

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53
votes

### The number $\pi$ and summation by $SL(2,\mathbb Z)$

Here goes $n=2$ a la Fedor Petrov.
Notice that his argument is based on the smart identities $x_z+y_z=|z|$, $z_x=|x|+y_x$ and $z_y=|y|+x_y$ where $x,y$ are 2 vectors in $\mathbb R^2$, $z=x+y$ and $...

- 55.4k

52
votes

### The use of computers leading to major mathematical advances II

Here is an example of type A: Stavros Garoufalidis and Don Zagier have extensive work on refinements of Kashaev's Volume Conjecture (which relates the order of growth of the values of Jones ...

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51
votes

### The number $\pi$ and summation by $SL(2,\mathbb Z)$

Let me write down here a proof that $\sum f(a,b,c,d)=2$, maybe someone sees how this may be generalized for the second moment. I do not.
We denote the vectors $x=(a,b)$ and $y=(c,d)$ and write $g(x,y)...

- 91.9k

47
votes

### Examples of creative experiments by mathematicians in modern days

I am encouraged to give this answer by the comment of the OP "My interest is in creative, non-digital ways of experimenting with mathematical theories, especially aiming for publication".
My ...

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41
votes

### Examples of unexpected mathematical images

Suppose we have a function representated as discrete Fourier series (DFS). Each DFS coefficient has an argument and modulus. But which of them is more important?
This strange question was discussed ...

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35
votes

### What are possible applications of deep learning to research mathematics?

In the context of algebraic geometry, neural networks have become useful tools in the study of Calabi-Yau manifolds. The computation of their topological invariants, metrics and volumes is of ...

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33
votes

### The use of computers leading to major mathematical advances II

Here is an interesting one. Reinforcement learning to generate counter-examples to several open conjectures in combinatorics and graph theory.
https://arxiv.org/abs/2104.14516

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32
votes

### Examples of creative experiments by mathematicians in modern days

I was amused by the work of Scott Aaronson on soap bubbles and Steiner trees. Given $n$ points $p_1$, $p_2$, ..., $p_n$ in $\mathbb{R}^2$, the Steiner tree through these points is the connected planar ...

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32
votes

### Theorems that are essentially impossible to guess by empirical observation

Recent breakthrough work of Ben Green together with an improvement by Zach Hunter imply that there is a red-blue coloring of $[1,n]$ with no red $3$-term progression and no blue $k$-term progression ...

Community wiki

30
votes

### Examples of unexpected mathematical images

The Hofstadter butterfly
plots, as a function of the $y$-coordinate, the spectrum of the
almost Mathieu operator
$H^y:l^2(\mathbb Z)\to l^2(\mathbb Z)$
$$H^y(f)(n)=f(n+1)+f(n-1)+2\cos(2\pi ny)f(n).$$...

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30
votes

### Examples of creative experiments by mathematicians in modern days

Possibly the OP might allow Lehmer's "bicycle chain sieves"
https://en.wikipedia.org/wiki/Lehmer_sieve,
which for several decades were the state of the art in factoring large numbers;
...

Community wiki

29
votes

### Examples of unexpected mathematical images

A long time ago, while attempting to classify certain two-dimensional rational conformal field theories (these are certain quantum field theories which enjoy a particular high level of mathematical ...

29
votes

### Examples of unexpected mathematical images

A recent blog post from google shows what happens if you enhance the parts of an image that triggers image recognition (using neural networks) of certain features.
The results are quite spooky, and ...

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29
votes

### What are possible applications of deep learning to research mathematics?

I have some thoughts on a level of generality that is a bit higher than the question asks for:
One obstacle that faces applications of supervised machine learning to predict properties of ...

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29
votes

### Theorems that are essentially impossible to guess by empirical observation

Goodstein's theorem. For a nonnegative integer $n$, the hereditary base $b$ representation of $n$ is found by writing $n$ in base $b$, then writing the exponents in base $b$, recursively, until the ...

Community wiki

28
votes

### The use of computers leading to major mathematical advances II

A fascinating recent example in category B is the progresses on Kazhdan's property (T) that were made after Ozawa's reformulation of property (T) in terms of semidefinite programming. This has lead in ...

Community wiki

27
votes

### The number $\pi$ and summation by $SL(2,\mathbb Z)$

I add our explanation and the origin of the problem.
To obtain the formula we just need to verify the following lemma (by a straightforward computation).
Magic Lemma. Let $(a,b),(c,d)$ be as in ...

- 4,641

27
votes

### The use of computers leading to major mathematical advances II

What about Giles Gardam's construction of non-trivial units in the mod-2 group algebra of a torsion-free group https://arxiv.org/abs/2102.11818, which solved an 80-year old conjecture, and Alan Murray'...

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26
votes

### The use of computers leading to major mathematical advances II

Duplicating my comment : in 2015 Beeson, Narboux and Wiedijk have formalized all of Euclid in HOL light and Coq https://doi.org/10.1007/s10472-018-9606-x
It allowed them to fix various flaws in Euclid....

Community wiki

24
votes

### What are possible applications of deep learning to research mathematics?

In the category of guessing, there is the Ramanujan Machine. This project got off on the wrong foot with the research mathematics community because their initial announcement made overblown claims (...

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24
votes

### The use of computers leading to major mathematical advances II

For experimental mathematics as that term is usually understood, I would commend to your attention the paper by Roger Behrend, Ilse Fischer and Matjaž Konvalinka, "Diagonally and antidiagonally ...

Community wiki

23
votes

### Examples of unexpected mathematical images

When I was plotting some parametric curves, accidentally I found this one:
$x=t\cos^3(t)$
$y=9t\sqrt{| \cos(t)|}+t\sin(\frac{t}{5})\cos(4t)$
$0<t<\frac{39\pi}{2}$

Community wiki

22
votes

### What computational problems would be good proof-of-work problems for cryptocurrency mining?

Have you considered unknotting knots?
The problem would be finding a sequence of Reidemeister moves for a random link graph that reduces it to the unknot. In some cases, no such sequence would exist, ...

- 3,281

22
votes

### The use of computers leading to major mathematical advances II

Adding to Archie's answer, Marjin Heule and collaborators have proven two major questions in Ramsey theory on the integers using SAT solvers. The 2016 result resolving the boolean Pythagorean triples ...

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