81
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 ...
80
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}...
62
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 $...
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 ...
57
votes
What mathematical problems can be attacked using DeepMind's recent mathematical breakthroughs?
This is a bit speculative, and perhaps too challenging for an undergraduate project, but I wonder if an AlphaGeometry type approach might be possible for the task of automatically upper bounding sums ...
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54
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 ...
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 ...
44
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 ...
37
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 ...
34
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
34
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 ...
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 ...
31
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 ...
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;
...
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 ...
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 ...
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'...
26
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 (...
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....
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 ...
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, ...
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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 ...
21
votes
Examples of unexpected mathematical images
I was computing various (2D orthogonal) projections of the Leech polytope (the $196\,560$ points forming the smallest shell of the Leech lattice focusing on those which exhibit various symmetries (viz....
21
votes
What are possible applications of deep learning to research mathematics?
Neural networks might help to speed up computations in the monster group
$\mathbb{M}$, which is the largest finite sporadic simple group. Such a
network could be in some sense a (rather large) cousin ...
21
votes
The use of computers leading to major mathematical advances II
An example of type B (I think) is Marijn Heule's program to reduce the size of graphs satisfying a given property. As an application it allowed him to find 5-chromatic unit distance graphs smaller ...
20
votes
May $p^3$ divide $(a+b)^p-a^p-b^p$?
We explain the pattern observed by Joe Silverman, deducing
the existence of infinitely many solutions, some of which even have
$p^5 | (a+b)^p - a^p - b^p$.
Lemma. If $n \equiv 1 \bmod 3$ then the ...
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19
votes
Examples of creative experiments by mathematicians in modern days
Psychologist Frank Rosenblatt built the first neural networks in 1957/8. Today it is trivial to build a neural network using software, but Rosenblatt built a neural network using analogue hardware.
...
18
votes
What are possible applications of deep learning to research mathematics?
Some recent work on neural formal theorem proving (already mentioned in the question, but the examples give a sense of the state of the art):
Generating correctness proofs with neural networks
...
18
votes
Theorems that are essentially impossible to guess by empirical observation
Probably Shannon's theorem(s) about existence of good (error-correcting) codes is an example: "most"/"random" codes achieve close to channel capacity, but explicit description (and ...
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