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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 outsideres of your field which would help to appreciate the result.

Asking the question I keep in mind by "recent years" something like dozen years before now, by "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 Fields medal was awarded like proof of fundamental lemma would not fit "not so famous", but on the other hand these might not be considered as strict criterias, 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 at 22:35
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    $\begingroup$ @KConrad Why not turn this into an answer? $\endgroup$ – Wojowu Feb 2 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 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 at 4:30
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    $\begingroup$ @KConrad you are hearly welcome to convert comment to an answer, time borderline 12 years is not strict $\endgroup$ – Alexander Chervov Feb 3 at 8:58

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Karim Adiprasito proved the g-conjecture for spheres in a preprint that was posted in December of last year: https://arxiv.org/abs/1812.10454.

This was probably considered the biggest open problem in the combinatorics of simplicial complexes. See Gil Kalai's blog post: https://gilkalai.wordpress.com/2018/12/25/amazing-karim-adiprasito-proved-the-g-conjecture-for-spheres/.

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Konstantin Tikhomirov recently proved that the probability that a random $n\times n$ Bernoulli matrix $M_n$ with independent $\pm 1$ entries, and $$\mathbb{P}[M_{ij}=1]=p,\quad 1\leq i,j\leq n,$$ is singular is $$ \mathbb{P}[M_n~\mathrm{is~singular}]=(1-p+o_n(1))^n $$ for any fixed $p\in (0,1/2].$

This problem was considered by Komlos, Kahn-Komlos-Szemeredi, Bourgain, Tao-Vu etc., so I am unsure if it qualifies in terms of being not-so-famous.

Nevertheless it was exciting reading about it in Gil Kalai's blog here .

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Hinged dissections exist. (See 3-piece dissection of square to equilateral triangle? for an animation of Dudeney's famous equilateral-triangle-to-square hinged dissection.)

Abbott, Timothy G., Zachary Abel, David Charlton, Erik D. Demaine, Martin L. Demaine, and Scott Duke Kominers. "Hinged dissections exist." Discrete & Computational Geometry 47, no. 1 (2012): 150-186. Springer link.

"Abstract. We prove that any finite collection of polygons of equal area has a common hinged dissection. That is, for any such collection of polygons there exists a chain of polygons hinged at vertices that can be folded in the plane continuously without self-intersection to form any polygon in the collection. This result settles the open problem about the existence of hinged dissections between pairs of polygons that goes back implicitly to 1864 and has been studied extensively in the past ten years."


          HingedFig6
The proof is not simple—as hinted by the above figure—but it is constructive.

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The homological conjectures in commutative algebra using perfectoid methods. A survey on many recent developments written by André can be found here.

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A remarkable example is the Gaussian correlation conjecture (which only recently became the Gaussian correlation inequality). The formulation is very simple:

For arbitrary centered Gaussian measure, any two convex symmetric sets are positively correlated.

It was formulated over 60 years ago (in the above general form, in 1972) and since then had been attacked by many mathematicians. Despite its apparent simplicity, only several partial results had been obtained before its complete proof in 2014.

What is remarkable is that the proof was quite simple and came from a retired statistician Thomas Royen, whose previous scientific output was not very noticeable. Moreover, the article was turned down by some scientists. It seems that the true reasons were that the author was not well known, and the article itself did not look serious (you can find its first non-LaTeX version here). Finally, it was published by some predatory "Far East" journal. Unsurprisingly, it took about two years for the proof to come to the public attention, and for its author to become famous.

Unfortunately, the story brings out some unpleasant features of the scientific community: hypocrisy and prejudice.

More on the story here.

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Ladner's theorem states that there exist $\mathsf{NP}$-intermediate problems when $\mathsf{P}\neq\mathsf{NP}$. However, the problem constructed in Ladner's proof is rather 'unnatural'. The question arises of whether any 'natural' examples of problems can be $\mathsf{NP}$-intermediate.

The Dichotomy Conjecture of Feder and Vardi (first stated here) states that, under the assumption that $\mathsf{P}\neq\mathsf{NP}$, the computational problems known as constraint satisfaction problems (CSPs for short) are either $\mathsf{NP}$-complete or belong to $\mathsf{P}$.

The consensus in the community (last I knew) is that Dmitriy Zhuk (https://arxiv.org/abs/1704.01914) and Andrei Bulatov (https://arxiv.org/abs/1703.03021) have independently proven the conjecture to be true. Their proofs cap a decades long approach of applying universal algebra to the question.

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    $\begingroup$ Amateur mathematician here. How does the first paragraph relate to the rest of this post? The Dichotomy Conjecture being proved neither proves nor disproves the question of "natural NP-intermediate examples", does it? It proves there are no CSPs, but surely there are "natural" non-CSP problems? $\endgroup$ – BlueRaja Feb 3 at 13:58
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    $\begingroup$ @BlueRaja You are correct. The stated result essentially says that if we want to find NP-intermediate problems, then we have to look into problems more complicated than CSPs. $\endgroup$ – Wojowu Feb 3 at 15:04
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A tensor $T \in V^{\otimes n} = V \otimes V \otimes \dotsb \otimes V$ is symmetric if $T$ is invariant under permutations of the factors $V$ in the tensor product (here $V$ is a finite-dimensional vector space). The tensor rank of $T$ is the least number of terms in a decomposition of $T$ as a sum of simple (aka decomposable, aka pure, aka rank one...) terms, i.e., ones of the form $v_1 \otimes \dotsb \otimes v_n$. The symmetric tensor rank of $T$ is the least number of terms in a decomposition of $T$ as a sum of terms of the form $v^{\otimes n} = v \otimes v \otimes \dotsb \otimes v$, i.e., $v_1 = \dotsb = v_n = v$. Evidently the symmetric tensor rank of a symmetric tensor is greater than or equal to its tensor rank.

Comon raised the question of whether every symmetric tensor has symmetric tensor rank equal to its tensor rank, say over the complex numbers. It is true for $n=2$, which is the statement that the (ordinary matrix) rank of a symmetric matrix is equal to the rank of the quadratic form represented by that matrix. "Comon's conjecture" is the assertion that the answer to Comon's question is positive; it's not completely clear to me whether Comon made this assertion (as opposed to just asking the question), but anyway that name has been used. Besides $n=2$, some other cases were known.

Recently Shitov has given a counterexample over the complex numbers.

Shitov, Yaroslav, A counterexample to Comon’s conjecture, SIAM J. Appl. Algebra Geom. 2, No. 3, 428-443 (2018). ZBL1401.15004.

The counterexample is fairly complicated. Shitov takes $n=3$ and $\dim V = 800$, and constructs a symmetric tensor whose rank is $903$, but whose symmetric rank is greater than $903$. (The symmetric rank of Shitov's example is not known.) Shitov's counterexample works over the complex numbers. No counterexample is known over the real numbers, or any other field. So Comon's conjecture is still open for the reals and other fields, but I wouldn't bet on it holding.

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    $\begingroup$ When I first read at the end "no real counterexample is known" I thought "wait, you just said that there is a counterexample". Then I read further and understood what you meant by a real counterexample. $\endgroup$ – KConrad Feb 16 at 7:53
  • $\begingroup$ @KConrad Thanks for the feedback. I guess I should try to avoid saying "real X" for "X defined over the real numbers", at least when it's meant for a wider audience. $\endgroup$ – Zach Teitler Feb 18 at 17:19
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    $\begingroup$ Imagine what students starting to learn higher math might think when they hear the term "simple complex Lie group/algebra". I remember in the first week of college being shown Serre's A Course in Arithmetic, which had such a gentle-sounding title and then I was baffled by the first two sentences: "Let $K$ be a field. The image of $\mathbf Z$ in $K$ is an integral domain, hence isomorphic to $\mathbf Z$ or to $\mathbf Z/p\mathbf Z$, where $p$ is prime...". Now it's all obvious, but not at that time. $\endgroup$ – KConrad Feb 18 at 18:43
  • $\begingroup$ @KConrad, make it "an open closed subset of a reduced irreducible representation of a simple complex Lie group". $\endgroup$ – Michael Feb 18 at 21:59
  • $\begingroup$ @Michael Please, think of the children! $\endgroup$ – Zach Teitler Feb 19 at 16:29
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The Audin conjecture in symplectic topology, posed in 1988 by Audin in her famous paper on Lagrangian immersions, asserts that all Lagrangian tori in the standard symplectic vector space have minimal Maslov number 2. This was recently proven by Cieliebak and Mohnke:

https://arxiv.org/abs/1411.1870

That paper nicely summarises the history of the conjecture:

"This question was answered earlier for n = 2 by Viterbo [57] and Polterovich [54], in the monotone case for n ≤ 24 by Oh [52], and in the monotone case for general n by Buhovsky [12] and by Fukaya, Oh, Ohta and Ono [28, Theorem 6.4.35], see also Damian [22]. A different approach has been outlined by Fukaya [27]. The scheme to prove Audin’s conjecture using punctured holomorphic curves was suggested by Y. Eliashberg around 2001. The reason it took over 10 years to complete this paper are transversality problems in the non-monotone case."

Edit: It occurred to me that this paper is probably now published; indeed it appeared in Inventiones in 2017. Here is the DOI:

https://doi.org/10.1007/s00222-017-0767-8

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S. T. Yau conjectured in the 80's that every compact Riemannian 3-manifold should contain infinitely many different minimal surfaces (smooth, closed). This was proved last year by Antoine Song.

Song built on a long story of breakthroughs in the area by Fernando Marques and Andre Neves using Min-Max theory. Another earlier big result was the solution of the Willmore Conjecture about embedding minimal tori: The Willmore energy $\int_{\Sigma}H^2$ of any smoothly immersed torus in $\mathbf{R}^3$ is at least $2\pi^2$.

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The strong no loop conjecture for quiver algebras $A$ states that a simple module $S$ with $Ext_A^1(S,S) \neq 0$ has infinite projective dimension. It was proven here https://www.sciencedirect.com/science/article/pii/S0001870811002714 . The more general conjecture for Artin algebras is still open.

(The result can be used to check for finite global dimension of endomorphism algebras, see for example Does this algebra have finite global dimension ? (Human vs computer).)

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In https://arxiv.org/abs/1812.02448, Tadayuki Watanabe announced a disproof of the Smale conjecture in dimension 4. In particular, he shows that the inclusion $O(5) \hookrightarrow \mathrm{Diff}(S^4)$ is not a homotopy equivalence. This was the last remaining dimension in which it was not known whether the inclusion $O(n+1) \hookrightarrow \mathrm{Diff}(S^n)$ was a homotopy equivalence (it is for $n \leq 3$ and it is not for $n \geq 5$).

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Graph theory / Discrete dynamics: In 2007, A. Trahtman proved the Road Coloring Conjecture, which had been posited 37 years earlier by R. Adler and B. Weiss.

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The Hall-Paige conjecture, first posed in 1955 by Marshall Hall and L. J. Paige, is the following:

A finite group $G$ has a complete mapping if and only if its Sylow $2$-subgroups are not cyclic.

Note that a complete mapping is a bijection $\phi : G \to G$ such that the function given by $\psi(g) = g \phi(g)$ is also a bijection. The above statement was shown to be necessary by Hall and Paige, but its sufficiency remained open until very recently; in 2009, it was shown to be sufficient to only check the cases when $G$ is a finite simple group, and the same year all finite simple groups except for $J_4$ were shown to satisfy the conjecture. John Bray then dealt with this final case in unpublished work, and Peter Cameron was able to convince him (see this) to publish these noteworthy calculations many years later; the final proof of the Hall-Paige conjecture, together with some consequences of it regarding synchronicity in groups, was written up in 2018 and can be found as a preprint on the arXiv.

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The following comes directly from Gabriel Peyré's excellent twitter feed:

The Weierstrass function is continuous if $a<1$ but nowhere differentiable if $ab>1$. The Hausdorff dimension of its graph was conjectured by Mandelbrot in 1977 and proved by Shen in 2016.

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some important conjectures in matroid theory, for instance the Rota conjecture on excluded minors (by Geelen, Gerards and Whittle, still unpublished, note claiming proof here) and the log-concavity conjecture (also due to Rota) for the characteristic polynomial (arxiv.org/abs/1511.02888). The method of the latter had several applications to solve more problems in matroid theory.

edit: let me add to that Liu's counterexample to the extension space conjecture

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Kiran Kedlaya finished the proof of Deligne's conjecture (1.2.10) made in La conjecture de Weil, II, which is definitely "noteworthy", and perhaps "not so famous" compared to the original Weil conjectures.

Colloquium talk: Companions in etale cohomology.

Annotated reading list for the working seminar at the IAS on the proof.

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Recently, Dasgupta, Kakde and Ventullo proved Gross's conjecture on the value at zero of the $p$-adic $L$-function constructed by Cassou-Noguès, and Deligne and Ribet. The article, On the Gross-Stark Conjecture, was published in Annals of Mathematics in 2018, and can be found here. Here is the abstract:

In 1980, Gross conjectured a formula for the expected leading term at $s = 0$ of the Deligne-Ribet $p$-adic $L$-function associated to a totally even character $\psi$ of a totally real feld $F$. The conjecture states that after scaling by $L(\psi\omega^{-1},0)$, this value is equal to a $p$-adic regulator of units in the abelian extension of $F$ cut out by $\psi\omega^{-1}$. In this paper, we prove Gross's conjecture.

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The Weibel conjecture about negative K-groups was proven in 2018 by Moritz Kerz, Florian Strunk, Georg Tamme.

The conjecture states that if $X$ is a Noetherian scheme of Krull dimension $d$, the negative K-groups $K_i(X)$ vanish when $i<-d$. Moreover $\mathbb{A}^1$-invariance also holds in that range, that is $$K_i(X)\to K_i(X\times\mathbb{A}^r)$$ is an isomorphism for $i\le -d$.

The paper where they solve the conjecture is particularly remarkable because they use methods from derived algebraic geometry to solve a problem with apparently no relation to it.

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Tyler Lawson's recent proof that the Brown-Peterson spectrum $BP$ at the prime 2 has no $E_∞$-ring structure. This was later generalized at odd primes, using similar methods, by Andrew Senger.

The proof proceeds via a detailed study of secondary power operations for ring spectra, which is valuable in itself.

This result suggests that $BP$ should have no natural "geometric model", since such models often endow the corresponding cohomology theory with an $E_∞$-ring structure.

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Manolescu refuted the Triangulation Conjecture. The paper is

Pin(2)-equivariant Seiberg-Witten Floer homology and the Triangulation Conjecture, J. Amer. Math. Soc. 29 (2016), 147-176, doi:10.1090/jams829, arXiv:1303.2354

And you can read a blog post about it at Low Dimensional Topology.

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    $\begingroup$ Is this really "not so famous" (as per the question)? (Also, link-only answers tend to be frowned upon since the link could die in the future.) $\endgroup$ – Noah Schweber Feb 14 at 18:14
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A Margulis spacetime is the quotient of the Minkowski space by a free proper orientation-preserving isometric action of a free group of rank at least two.

From Danciger, Kassel, and Guéritaud:

"Based on a question of Margulis, Drumm–Goldman conjectured in the early 1990s that all Margulis spacetimes should be tame, meaning homeomorphic to the interior of a compact manifold."

In a series of paper, I believe Choi, Drumm, and Goldman, and independently Danciger, Kassel, and Guéritaud resolved this conjecture affirmatively.

Links:

  1. Topological tameness of Margulis spacetimes, by Suhyoung Choi, William Goldman
  2. Tameness of Margulis space-times with parabolics, by Suhyoung Choi, Todd Drumm, William Goldman
  3. Geometry and topology of complete Lorentz spacetimes of constant curvature, by Jeffrey Danciger, François Guéritaud, Fanny Kassel
  4. Margulis spacetimes via the arc complex, by Jeffrey Danciger, François Guéritaud, Fanny Kassel
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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 an additional technical hypothesis -- not part of the original conjecture -- of a non-integral $j$-invariant, which was later removed in 2011.) For those not familiar with these problems, their solutions build on ideas coming from the proof of Fermat's Last Theorem.

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Here are two examples from operator theory/operator algebras. Both were open problems for more than forty years. The first example is remarkable because it is was quite well known, it is so simple to state and was elusive for so much time. The second example is notable because of its importance and because it was open since Arveson's seminal "subalgebras" paper.

  1. In 2015 Kreg Knese proved that von Neumann's inequality holds for triples of $3 \times 3$ contractions (this breakthrough followed important work of Lukasz Kosinski). The introduction to Knese's paper explains it all well (I blogged about it, in case you want a version with some more superlatives).

  2. In 2013 Davidson and Kennedy proved the existence of "sufficiently many boundary representations" in every operator system, which was an open problem raised by Arveson in 1969. Here is their paper on arxiv (here is a blog post I wrote on this, geared towards non-specialists). Davidson and Kennedy's solution came five years after Arveson himself settled the problem for separable operator systems (following important work of Dritschel and McCullough), which was also an exciting development.

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Turan's book On a new method of analysis and its applications focuses on bounds on power sums. The quantity $$ T(m,n)=\inf_{|z_k|=1} \max_{\nu=1,\ldots,m} \left| \sum_{k=1}^n z_k^\nu\right|, $$ for various choices of $m,n$ has been of interest since then. The case $m\sim n^{B}$ has recently been settled by Andersson, using a character sum estimate due to Katz, in the paper available on arXiv here. The result essentially states that $$ T(m,n)\asymp \sqrt{n}, $$ if $m=\lfloor n^B \rfloor,$ if $B>1$ is fixed. This was also an open problem by Montgomery in his Ten lectures on the interface between analytic number theory and harmonic analysis.

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