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

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 2

Øystein Ore proved in 1951 that for $n\gt4$ we have that all elements of $A_n$ are commutators. He subsequently conjectured that this is true for all finite non-abelian simple groups.

In this connection, not long ago I asked if every element of a perfect group is a commutator. It was answered in the negative. I might have caught that $A_5*A_5$ is a counterexample. The smallest such was found to be of order 960.

Ore's conjecture/theorem was proved in 2008, and relies on the classification theorem.

• See M. W. Libeck, E. A. O'Brien, Aner Shalev, Pham Huu Tiep, The Ore conjecture, Journal of the European Mathematical Society, vol. 12, issue 4, 2010. pp. 939-1008. http://doi.org/10.4171/JEMS/220

Interestingly Ore himself was mainly concerned with graph theory.

Answer 3

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 4

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 5

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 6

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 7

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 8

Here is the abstract of Andreas Reinhart, A counterexample to the conjecture of Ankeny, Artin and Chowla, available at https://arxiv.org/html/2410.21864v1

``Let $p$ be a prime number with $p\equiv1\bmod4$, let $\epsilon>1$ be the fundamental unit of ${\bf Z}[{(1+\sqrt p)/2}]$ and let $x$ and $y$ be the unique nonnegative integers with $\epsilon=x+y{1+\sqrt p\over2}$. The Ankeny-Artin-Chowla conjecture states that $p$ is not a divisor of $y$. In this note, we provide and discuss a counterexample to this conjecture.''

Spoiler alert: the counterexample is $p=331914313984493$.

Answer 9

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.

Answer 10

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.