The dodecahedral conjecture is true

You need to know: Three-dimensional Euclidean space ${\mathbb R}^3$ , volume in ${\mathbb R}^3$ , unit sphere in ${\mathbb R}^3$ , regular dodecahedron, inscribed sphere.

Background: Let $\Lambda$ be any set of points in ${\mathbb R}^3$ with pairwise distance between any of them at least 2. Equivalently, $\Lambda$ is a set of centres of non-intersecting unit spheres. The Voronoi cell $\Omega(\Lambda, v)$ around $v \in \Lambda$ consists of points of space that are closer to v than to any other point $w\in \Lambda$ . Let V* denotes the volume of a regular dodecahedron D* whose inscribed sphere has radius 1. In fact, $V^* = 10\sqrt{130-38\sqrt{5}} = 5.55029...$ .

The Theorem: On 11th November 1998 Thomas Hales and Sean McLaughlin submitted to arxiv and to the Journal of the AMS a paper in which they proved that the volume of any Voronoi cell for $\Lambda$ is at least V*.

Short context: If $\Lambda$ consists on centre O of regular dodecahedron D* and 12 mirror images of O with respect to faces of D*, then the Voronoi cell $\Omega(\Lambda, O)$ is exactly D*, and its volume is V*. In 1943, Toth conjectured that this volume is the smallest possible one. This statement became known as the dodecahedral conjecture. The Theorem confirms this conjecture.

Links: Free arxiv version of the original paper is here, journal version is here.

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There are infinitely many primes of the form x^3+2y^3

You need to know: Prime numbers.

The Theorem: On 29th April 1999, Rodger Heath-Brown submitted to Acta Mathematica a paper in which he proved that there are infinitely many primes of the form $x^3+2y^3$ with integer x, y.

Short context: Let P be a polynomial with integer coefficients in one or more variables. If we substitute integer values instead of variables, will we get infinitely many primes as values of P? This problem is wide open even for simple polynomials like $P(x)=x^2+1$ . Famous Dirichlet’s theorem solves it for linear polynomials, Iwaniec in 1974 resolved the case of quadratic polynomials in 2 variables (which depends essentially on both variables), while Friedlander and Iwaniec proved in 1998 that there are infinitely many primes of the form $x^2+y^4$ with integer x, y. The Theorem resolves this question for polynomial $P(x,y)=x^3+2y^3$ .

Links: The original paper is here.

Optimal approximation ratio for Max-Ek-Sat is 1-1/2^k

You need to know: Boolean variables and their basic operations, class P, class NP, NP-hard problems.

Background: Let $x_1, x_2, \dots, x_n$ be boolean variables. A k-clause is any expression of the form $y_1 \vee y_2 \vee \dots \vee y_k$ , where each $y_i$ is either $x_j$ or $\neg x_j$ for some j.

The Theorem: In August 1997, Johan Hastad submitted to the Journal of the ACM a paper in which he proved, among other results, that for any $\epsilon>0$ and any $k \geq 3$ , it is NP-hard to, given m k-clauses, distinguish the case when all clauses are satisﬁable from the case when at most $(1-2^{-k}+\epsilon)m$ of them are satisﬁable.

Short context: The problem which asks, given m k-clauses, whether it is possible to satisfy them all, is a famous problem which is NP-hard for any $k \geq 3$ . The problem of deciding what the maximal number of clauses we can satisfy is, is known as Max-Ek-Sat. The Theorem is equivalent to the statement that it is NP-hard to approximate Max-Ek-Sat with any constant ratio better than $1-2^{-k}$ . Because it is easy to satisfy $1-2^{-k}$ fraction of clauses by random assignment, this result (a) is the best possible, and (b) suggests that no polynomial time algorithm can beat in approximation ratio the trivial random assignment algorithm.

Links: The original paper is available here.

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The Kepler conjecture is true

You need to know: Three-dimensional Euclidean space ${\mathbb R}^3$ , volume in ${\mathbb R}^3$ , limit superior $\limsup$ .

Background: By open ball $B(A,R)$ in ${\mathbb R}^3$ with center A and radius R we mean the set of points $X \in {\mathbb R}^3$ such that $|AX|<R$ , where $|.|$ denotes the usual distance in ${\mathbb R}^3$ . By “packing of congruent balls” we mean an (infinite) set E of non-overlapping open balls with the same radii. For a fixed point $O \in {\mathbb R}^3$ , denote $\delta(O,E) = \limsup\limits_{R\to\infty}\frac{|B(O,R) \cap E|}{|B(O,R)|},$ where $|.|$ denotes volume in ${\mathbb R}^3$ . It can be shown that $\delta(O,E)$ depends only on E but not on O and is called upper density of E.

The Theorem: On 4th September 1998, Thomas Hales submitted to Annals of Mathematics a paper in which he proved that no packing of congruent balls in ${\mathbb R}^3$ has upper density greater than $\frac{\pi}{3\sqrt{2}}=0.74048...$ .

Short context: Two packings of congruent balls in ${\mathbb R}^3$ (called “cubic close packing” and “hexagonal close packing”) with the same density $\frac{\pi}{3\sqrt{2}}$ have been known for centuries. In 1611, Kepler conjectured that these packings are the densest possible ones. In 1900, Hilbert included this conjecture in his famous list of 23 unsolved problems in mathematics (as a part of the 18-th problem). The Theorem confirms this conjecture. Because the proof uses computer and is difficult for human to verify, Hales initiated a project of computer-assisted verification of the proof. The project was completed in 2015, giving 100% confidence in the correctness of the proof. See here and here for higher dimensional generalisations.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 5.10 of this book for an accessible description of the Theorem.

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Palis conjecture on the arithmetic difference of regular Cantor sets is true

You need to know: Sets of measure zero, Cantor set, regular Cantor set, generic pair of regular Cantor sets.

Background: For any sets $C_1 \subset \mathbb R$ and $C_2 \subset \mathbb R$ , their arithmetic difference is the set $C_1-C_2=\{x-y\,|\,x\in C_1, \, y\in C_2\}$ .

The Theorem: On 31st July 1998, Carlos Gustavo T. de A. Moreira and Jean-Christophe Yoccoz submitted to Annals of Mathematics a paper in which they proved, among other results, that the arithmetic difference of a generic pair of regular Cantor sets of the real line either has measure zero or contains an interval.

Short context: The statement of the Theorem was a conjecture of Palis from 1987. It has applications is the study of dynamical systems.

Links: The original paper is here. See also Section 1.4 of this book for an accessible description of the Theorem.

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Zagier conjecture is true

You need to know: Sum of infinite series.

Background: Multiple zeta function is the expression of the form $\zeta(k_1, k_2, \dots k_r) = \sum\limits_{n_1=1}^\infty \sum\limits_{n_2=n_1+1}^\infty \dots \sum\limits_{n_r=n_{r-1}+1}^\infty \frac{1}{n_1^{k_1} n_2^{k_2} \dots n_r^{k_r}}$ . By $\zeta(\{a, b\}^n)$ we mean $\zeta(a, b, a, b, ..., a, b)$ , where the arguments a and b are repeated n times.

The Theorem: On 29th July 1998, Jonathan Borwein, David Bradley, David Broadhurst, and Petr Lisonek submitted to Transactions of the AMS a paper in which they proved, among other results, that $\zeta(\{1, 3\}^n) = \frac{2\pi^{4n}}{(4n+2)!}$ .

Short context: Zeta function in one variable $\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$ is one of the central functions in the whole of mathematics which has been studied for centuries. For example, famous Euler theorem states that $\zeta(2) = \frac{\pi^2}{6}$ . Euler also derived expressions for $\zeta(1,m)$ . For example, $\zeta(1,2) = \zeta(3)$ , $\zeta(1,3) = \frac{\zeta(4)}{4}=\frac{\pi^4}{360}$ , etc. In 1994, Zagier conjectured a nice 2n-variable generalisation of the last formula. The Theorem confirms this conjecture.

Links: Free arxiv version of the original paper is here, journal version is here.

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Characterisation of ﬁnitely generated groups with word problem in NP.

You need to know: Class P, class NP, NP-complete problems, group, presentation of a group in the form $<S|R>$ where S is a set of generators and R is a set of relations.

Background: A group G is called finitely generated if it has a presentation $<S|R>$ with finite number of generators. If the set of relations is finite as well, G is called finitely presented. A word in group G is any product of generators and their inverse elements. The word problem in G is the problem to decide whether a given word represents an identity element.

For any word a representing the identity element e, there is a sequence of steps that reduces a to the empty word, where each step is a free reduction or insertion or the application of a relator. We call the number of applications of relators in a sequence its cost, and let T(a) be the minimum cost of a sequence that transforms a into e. A function f is called an isoperimetric function of G if $T(a) \leq f(|a|)$ for any a representing e, where $|a|$ is the length of word a. We say that isoperimetric function f is polynomial, if $f(n) = cn^k$ for some constants c and k.

The Theorem: On 20th April 1998, Jean-Camille Birget, Alexander Olshanskii, Eliyahu Rips and Mark Sapir submitted to Annals of Mathematics a paper in which they proved that the word problem of a ﬁnitely generated group H is in NP if and only if H is a subgroup of a ﬁnitely presented group G with a polynomial isoperimetric function.

Short context: Every group H with polynomial isoperimetric function is known to be in NP. The converse is not true: there are groups H with word problem in NP but such that every isoperimetric function grows very fast. The Theorem, however, shows that every such group H can be embedded into a different group G which is finitely presented and has a polynomial isoperimetric function.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 2.9 of this book for an accessible description of the Theorem.

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Almost every real quadratic polynomial is either regular or stochastic.

You need to know: Limits, integration, the concept of (Lebesgue) almost every.

Background: For a quadratic polynomial $f:{\mathbb R}\to {\mathbb R}$ and initial point $x_0 \in {\mathbb R}$ , consider an infinite sequence defined by induction: $x_n=f(x_{n-1})$ , $n=1,2,\dots$ . Let I be the longest interval which f maps into itself. If, for almost every $x_0 \in I$ , the sequence above converges to a cycle, f is called regular. If there exists a function g such that, for almost every $x_0 \in I$ , the proportion of terms in the sequence in any interval $(a,b)$ converges to $\int_a^b g(x)dx$ as $n\to\infty$ , f is called stochastic.

The Theorem: On 15th July 1997, Mikhail Lyubich submitted to arxiv a paper in which he proved that, for almost every $c\in[-2,1/4]$ , polynomial $f(x)=x^2+c$ is either regular or stochastic.

Short context: The sequence $x_n$ defined above is an example of (an orbit of) a dynamical system. The central goal in the study of dynamical systems is to understand the behaviour of almost all orbits for almost all values of the parameter(s). The Theorem achieves this goal for the family of quadratic polynomials, which is the simplest non-trivial case.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 2.7 of this book for an accessible description of the Theorem.

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A finitely presented group can have NP-complete word problem.

You need to know: Class P, class NP, NP-complete problems, group, presentation of a group in the form $<S|R>$ , where S is a set of generators and R is a set of relations.

Background: A group G is called finitely presented if it has a presentation $<S|R>$ with finite number of generators and finite set of relations. A word in group G is any product of generators and their inverse elements. The word problem in G is the problem to decide whether a given word represents the identity element.

The Theorem: On 24th May 1997, Mark Sapir, Jean-Camille Birget, and Eliyahu Rips submitted to the Annals of Mathematics a paper in which they proved, among other results, that there exists a finitely presented group for which the word problem is NP-complete.

Short context: In general, the word problem for a finitely presented group may be well outside of NP. In fact, Novikov proved in 1955 that there are finitely presented groups for which the word problem is undecidable. The Theorem constructs the first example of a finitely presented group whose word problem is in NP but, still, it is unlikely that there is an efficient algorithm for solving it.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 2.8 of this book for an accessible description of the Theorem.

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