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Conjectures play important role in development of mathematics. Mathoverflow gives an interaction platform for mathematicians from various fields, while in general it is not always easy to get in touch with what happens in the other fields.

Question What are the conjectures in your field proved or disproved (counterexample found) recent years, which are noteworthy, but not so famous outside your field ?

Answering the question you are welcome to give some comment for outsiders of your field which would help to appreciate the result.

Asking the question I keep in mind by "recent years" something like a dozen years before now, by a "conjecture" something which was known as an open problem for something like at least dozen years before it was proved and I would say the result for which the Fields medal was awarded like a proof of fundamental lemma would not fit "not so famous", but on the other hand these might not be considered as strict criteria, and let us "assume a good will" of the answerer.

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    $\begingroup$ In number theory, the Sato-Tate conjecture about elliptic curves over $\mathbf Q$ was a problem from the 1960s and Serre's conjecture on modularity of odd 2-dimensional Galois representation was a conjecture from the 1970s-1980s. Both were settled around 2008. (For ST conj., the initial proof needed a technical hypothesis -- not part of the original conj. -- of a non-integral $j$-invariant, which was later removed in 2011.) For those not familiar with these problems, their solutions use ideas coming out of the proof of Fermat's Last Theorem. And 2008 is now almost 12 years ago? Time flies... $\endgroup$
    – KConrad
    Feb 2 '19 at 22:35
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    $\begingroup$ @KConrad Why not turn this into an answer? $\endgroup$
    – Wojowu
    Feb 2 '19 at 22:40
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    $\begingroup$ What about disproved conjectures—where people have found counterexamples? Are those not worthy of being noted? $\endgroup$
    – Peter Shor
    Feb 3 '19 at 2:06
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    $\begingroup$ @PeterShor do you mean not worthy of being noted or noteworthy of being... knotted? Well anyway, you could ask a separate (not seperate) question. Maybe mathoverflow.net/questions/138310/… would be a good candidate for an answer to it. $\endgroup$
    – KConrad
    Feb 3 '19 at 4:30
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    $\begingroup$ @PeterShor I mean to INclude disproved conjectures, thank you for your remark, I will edit question accordingly $\endgroup$ Feb 3 '19 at 8:53

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In 2016, Andrew Suk (nearly) solved the "happy ending" problem; that is, he proved (On the Erdős-Szekeres convex polygon problem, J. Amer. Math. Soc. 30 (2017), 1047-1053, doi:10.1090/jams/869, arXiv:1604.08657) that $2^{n+o(n)}$ points in general position guarantee the existence of $n$ points in convex position which improves the upper bound of $4^{n-o(n)}$ given by Erdős and Szekeres in 1935 and nearly matches the lower bound of $2^{n-2}+1$ given by Erdős and Szekeres in 1960 which they conjectured to be optimal.

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The famous Nussbaum conjecture stated that every continuous map of a closed ball in a Banach space with a compact iterate (i.e. the iterate has relatively compact range) has a fixed point. Again Robert Cauty (see my previous post) proved it 2015 in the positive by showing that even a Lefschetz type fixed point theorem for maps with compact iterates holds:

  • Cauty, Robert, Un théorème de Lefschetz–Hopf pour les fonctions à itérées compactes, Crelle Journal für die reine und angewandte Mathematik 2017 (729), https://doi.org/10.1515/crelle-2014-0134

The conjecture was formulated in about 1970.

As Robert Nussbaum once pointed out, the attractivity of this conjecture lied in the fact that it is apparently so simple to prove, and that it can in fact be shown relatively easily under mild additional hypotheses (differentiability is such an “obviously” sufficient hypothesis, or that the map is even condensing, or that the range of some iterate has a locally nice topological structure, ...), but the longer one works on the problem, the harder it seems, and the less likely that one does not need any additional hypothesis. Many novelties in the field were inspired by proofs under such additional hypotheses.

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The Baez-Dolan corbordism hypothesis or conjecture which states that the higher corbordism category is the free symmetric higher monoidal category on a single object was formalised by Lurie and proven in his paper classifying topological field theiries in 2008.

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Not sure whether this counts as recent enough:

Robert Cauty proved 2001 the Schauder conjecture that every continuous map of a nonempty commpact convex subset of a topological vector space (not necessarily locally convex!) has a fixed point:

  • Cauty, Robert, Solution du problème de point fixe de Schauder, Fundamenta Mathematica 170, 2001, 231-246.

Although some problems have been found in the original paper, it seems that they could all be fixed.

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