Noteworthy, but not so famous conjectures resolved recent years

Question

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) in 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.

Answer 1

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.

Answer 2

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/.

Answer 3

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 .

Answer 4

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."

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

Answer 5

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

Answer 6

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 $n \geq 5$).

Answer 7

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.

Answer 8

“Conway’s knot is not slice”, by Lisa Piccirillo, Annals of Mathematics 191-2 (2020) 581-591, settled a problem (we can call it a conjecture) which was at least four decades old. Read in informal account of the result here: https://www.quantamagazine.org/graduate-student-solves-decades-old-conway-knot-problem-20200519/

Answer 9

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.

Answer 10

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 $2\pi^2$.

Answer 11

Connes' embedding conjecture (from 1976) about the structure of infinite-dimensional von Neumann algebras was shown to be false in the paper

• Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, Henry Yuen, $\mathsf{MIP}^*=\mathsf{RE}$, arXiv:2001.04383.

This was done by a quantum computer science argument, which showed that an interactive proof system with multiple provers sharing quantum entanglement (MIP$^*$) could give proofs for any recursively enumerable (RE) language. The computational power of MIP$^*$ was a long-standing question in complexity theory, but I don't believe anybody thought it was equal to RE until the connection with Connes' conjecture was discovered fairly recently.

The same paper settles Tsirelsen's problem. Boris Tsirelsen stated a theorem without proof in a 1993 survey paper. It was only much later that he was asked about it, and discovered that the simple proof he thought he had didn't work. He posed it as an open problem in 2006, and this paper shows that Tsirelsen's statement is false.

The connection between these problems was already known, so the linked paper only gives a direct proof of the quantum computer science result.

Answer 12

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).)

Answer 13

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

Answer 14

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.

Answer 15

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.

Answer 16

Maryanthe Malliaris and Saharon Shelah proved that the cardinal characteristics $\mathfrak p$ and $\mathfrak t$ are equal, answering a question that goes back at least to the 1970's and probably (with different formulation but the same content) to the 1940's:

• "Cofinality spectrum theorems in model theory, set theory, and general topology." J. Amer. Math. Soc. 29 (2016), 237–297, doi:10.1090/jams830, arXiv:1208.5424, Shelah archive:paper 998).

The definition of $\mathfrak t$ is the shortest possible length $\lambda$ of a well-ordered sequence $(A_\xi:\xi<\lambda)$ of infinite subsets of $\mathbb N$ such that (1) $A_\eta-A_\xi$ is finite whenever $\xi<\eta<\lambda$ and (2) there is no infinite $B\subseteq\mathbb N$ with $B-A_\xi$ finite for all $\xi<\lambda$. That is, $\mathfrak t$ is the smallest length of any inextendible decreasing-mod-finite sequence of inifinte subsets of $\mathbb N$.

$\mathfrak p$ is defined similarly except that, instead of requiring the sequence to decreasing mod finite, one requires only that every finitely many of the $A$'s have an infinite intersection.

It is easy to check that $\aleph_1\leq\mathfrak p\leq\mathfrak t\leq2^{\aleph_0}$. It was also known previously that if $\mathfrak p=\aleph_1$ then $\mathfrak p=\mathfrak t$. (I believe this result is due to Judith Roitman, but I can't find a reference now, not even in my chapter of the "Handbook of Set Theory" where this result is Theorem 6.25. Mea culpa.) I think it was expected that $\mathfrak p<\mathfrak t$ would turn out to be consistent with ZFC, until Malliaris and Shelah proved otherwise. Not only the theorem but the method of proof was surprising, as it involved ideas from model theory, even though the result is purely set-theoretic.

Answer 17

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.

Answer 18

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.

Answer 19

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.

Answer 20

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.

Answer 21

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.

Answer 22

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.

Answer 23

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.

Answer 24

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

Answer 25

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

Answer 26

This MO question The Riemann zeros and the heat equation describes the Newman conjecture. Very briefly, a deformation parameter is introduced into an integral representation of the Riemann zeta function, creating a function of two variables which satisfies the backward heat equation. Newman made the conjecture that any infinitesimal deformation with this extra parameter destroys the Riemann hypothesis:

"This new conjecture is a quantitative version of the dictum that the Riemann hypothesis, if true, is only barely so."

In 2018, Tao and Rodgers were able to use the connection to PDE and posted a proof of the Newman conjecture on the arXiv. Last week, Alexander Dobner, another student of Tao posted a new, purely analytic (and shorter) proof on the arXiv, writing "One final note we make about our proof is that it reveals that Newman's conjecture holds for completely analytic rather than arithmetic reasons.... Thus, our proof of Newman's conjecture is quite different from the proof of Rodgers and Tao which depends fundamentally on knowledge about the gaps between zeta zeros and hence on the arithmetic structure of the zeta function

Answer 27

In the paper

• Friedrich Hirzebruch, Some Problems on Differentiable and Complex Manifolds, Annals of Mathematics Second Series, Vol. 60, No. 2 (1954) pp. 213-236, doi:10.2307/1969629

Hirzebruch collected problems and questions on smooth and complex manifolds presented at a conference the year prior.

For an almost complex manifold $M$ equipped with a hermitian metric, one can form the Laplacian $\Delta_{\bar{\partial}}$. Even though $\bar{\partial}$ need not square to zero on an almost complex manifold, this Laplacian is an elliptic operator and so the kernel of the Laplacian is finite dimensional if $M$ is compact. Denote the dimension of this kernel restricted to $(p,q)$ forms by $h^{p,q}$.

Problem 20 in Hirzebruch's list, attributed to Kodaira and Spencer, asks the following about compact almost complex manifolds:

Let $M^n$ be an almost-complex manifold. Choose an Hermitian structure and consider the numbers $h^{p,q}$ defined as above. Is $h^{p,q}$ independent of the choice of the Hermitian structure? If not, give some other definition of the $h^{p,q}$ of $M^n$ which depends only on the almost-complex structure and which generalizes the $h^{p,q}$ of a complex manifold.

(Note that in the case of an integrable complex structure, the numbers $h^{p,q}$ are metric independent as they are the dimension of the Dolbeault cohomology group $H_{\bar{\partial}}^{p,q}$.)

In 2020, Holt and Zhang posted a preprint (update: the paper is now published in Advances in Mathematics)

• Tom Holt, Weiyi Zhang, Harmonic Forms on the Kodaira–Thurston Manifold, arXiv:2001.10962,

showing that the numbers $h^{p,q}$ are in general metric-dependent. The underlying manifold they work with is the four-dimensional Kodaira–Thurston nilmanifold.

Answer 28

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.

Answer 29

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.

Answer 30

A "hot spot" on a sufficiently regular domain is an interior extremum of the first nonconstant Neumann eigenfunction of the Laplace operator. The Hot Spots conjecture states that hot spots do not exist on convex planar domains.

Chris Judge and Sugata Mondal have settled the Hot Spots conjecture in the affirmative for all Euclidean triangles: Euclidean triangles have no hot spots, Annals of Mathematics 191-1 (2020) 167-211. (preprint)

This conjecture was the subject of Polymath 7.

Answer 31

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.

Answer 32

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.

Answer 33

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.

Answer 34

The Kervaire Invariant One Problem (1969) is a question about which framed manifolds can be converted into spheres via surgery. It's related to the classification of exotic smooth structures on spheres (like Milnor's Fields Medal winning structure on $S^7$ that started the whole field of differential topology by displaying that a homeomorphism need not be a diffeomorphism). After a flurry of work in the 1950s and 1960s, this problem languished with no progress from 1969 until 2009, when it was resolved by Hill, Hopkins, and Ravenel (published in Annals), in all dimensions except 126. The authors have a wonderful new book explaining the proof and the history of the problem. I have some slides where I explain a bit about it (but the importance in differential topology is much more than what I discuss).

Answer 35

In 2019 Anna Erschler and Tianyi Zheng gave a very sharp estimate of the growth of Grigorchuk's first group. Although it was one of the first example of a group of intermediate growth (finitely generated group whose growth is neither polynomial nor exponential), how fast it grows was not really known. In Grigorchuk's original paper, the exponent $\alpha$ in $\mathrm{exp}(Cn^\alpha)$ was only known to lie somewhere between 0.5 and 0.991... Quite a few papers made improvements on the upper bound e.g. Bartholdi brought it down to 0.7675... and Leonov up to 0.504... but until then it remained unknown.

EDIT: if $b_n$ is the cardinality of the ball of radius $n$ in Grigorchuk's first group, Erschler and Zheng proved that $$ \alpha := \lim_{n \to \infty} \frac{ \log \log b_n}{\log n} = \frac{\log 2}{\log \lambda_0} \approx 0.7674 $$ where $\lambda_0$ is the positive real root of the polynomial $x^3-x^2-2x-4$. Note that the group may still grow somehow faster or slower than $\mathrm{exp}{(C n^\alpha)}$, but they identified the dominating term in the growth. Also, since changing the generating set is a bi-Lipschitz map, this is the only part of the growth function that is guaranteed to be independent of the generating set.

Answer 36

The Baez-Dolan Stabilization Hypothesis was posed in 1995. It involves the relationship between weak $n$-categories as $n$ varies. Specifically, if one has an $n+k$ category $C$ and "forgets" to an $n$-category $D$ then $D$ has extra structure. To make sure we didn't forget anything important, we assume $C$ has exactly one object, one 1-cell, one 2-cell, ..., one $(k-1)$-cell, and we are simply reindexing so that the $k$-cells in $C$ become the objects of $D$. For example, if $k=1$ then the objects in $D$ are the morphisms of $C$, so they have a composition law making them into a monoid. If $k = 2$ then the objects of $D$ have both horizontal and vertical composition rules, so by the Eckmann-Hilton argument, the objects of $D$ have the structure of a commutative monoid. If $k \geq 3$, you still get a commutative monoid. The forgetting process stabilized after $k \geq n+2$. The stabilization hypothesis posits that this always happens, in any reasonable model of a weak $n$-category. It was a hypothesis rather than a conjecture because there was no known model at the time of weak $n$-categories.

The stabilization hypothesis has recently been proven in many different models of weak $n$-categories including:

1. Enriched $\infty$-categories (by a remark in Lurie's book (2009), then a 2013 paper of Gepner and Haugseng), published in Advances.

2. Charles Rezk's $\Theta_n$-space model for weak $n$-categories and $(m,n)$-categories, by a 2015 paper of Michael Batanin, published in Proceedings of the AMS.

3. Tamsamani's model, Simpson's higher Segal categories, Ara's $n$-quasicategories, and various models due to Bergner and Rezk, by a 2020 paper by Michael Batanin and me, published in Transactions.

Answer 37

Not sure whether this counts as recent enough:

Robert Cauty proved 2001 the Schauder conjecture that every continuous map of a nonempty compact 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.

Answer 38

Vopenka's Principle is a large cardinal axiom that has several equivalent formulations. Arguably the simplest is the statement

For every proper class of graphs there exists a non-identity homomorphism between two graphs in that class.

Papers on Vopenka's Principle (VP) go back to 1965. In 1988, Adamek, Rosicky, and Trnkova introduced the Weak Vopenka Principle (WVP), proved that VP implies WVP, and asked if WVP implied VP. This was finally answered in 2019 by Trevor Wilson (published in Advances). From the abstract:

Vopenka’s Principle says that the category of graphs has no large discrete full subcategory, or equivalently that the category of ordinals cannot be fully embedded into it. Weak Vopenka's Principle is the dual statement, which says that the opposite category of ordinals cannot be fully embedded into the category of graphs. It was introduced in 1988 by Adamek, Rosicky, and Trnkova, who showed that it follows from Vopenka’s Principle and asked whether the two statements are equivalent. We show that they are not.