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

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 2

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 3

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 4

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 5

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 6

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 7

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 8

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.