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### 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 ...

### 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 ...

### 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 ...

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

### 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 ...

### 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 ...

### Examples of unexpected mathematical images

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

### 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; ...

### 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 ...

### 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 ...

### 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 ...

### 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 ...

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 ...

### 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 ...
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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'...

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....

### 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 (...

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 ...

### Examples of unexpected mathematical images

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