Any real polynomial can be approximated by hyperbolic real polynomials of the same degree

You need to know: Limit, derivative, polynomial, degree of a polynomial.

Background: The n-th functional power of a function $f:{\mathbb R}\to{\mathbb R}$ is a function $f^n(x)=f(f(\dots f(x)\dots))$ , where f is repeated n times. A point $x_0$ is called periodic for f if $f^k(x_0)=x_0$ for some $k\geq 1$ . The smallest k for which this holds is called period of $x_0$ . A periodic point $x_0$ with period k is called hyperbolic if $|(f^k)'(x_0)| \neq 1$ , and it is called hyperbolic attracting if $|(f^k)'(x_0)|<1$ . For a polynomial $f:{\mathbb R}\to{\mathbb R}$ , and $x_0 \in {\mathbb R}$ , either (i) $\lim\limits_{n \to \infty} |f^n(x_0) - f^n(x^*)|=0$ for some hyperbolic attracting periodic point $x^*$ , or (ii) $\lim\limits_{n \to \infty} |f^n(x_0)|=\infty$ , or (iii) neither (i) nor (ii) happens. Let $S_f$ be the set of all $x_0 \in {\mathbb R}$ for which case (iii) holds. A polynomial f is called hyperbolic if there exist constants $C>0$ and $\lambda>1$ such that $|(f^n)'(x)|>C \lambda^n$ , $\forall n, \forall x \in S_f$ .

The Theorem: On 6th August 2004, Oleg Kozlovski, Weixiao Shen, and Sebastian van Strien submitted to the Annals of Mathematics a paper in which they proved that for every real polynomial $f(x)=\sum\limits_{i=0}^d a_ix^i$ and any $\epsilon>0$ , there exists a hyperbolic real polynomial $h(x)=\sum\limits_{i=0}^d b_i x^i$ such that $|a_i-b_i|<\epsilon$ , $i=0,1,\dots,d$ .

Short context: Hyperbolic polynomials are central objects of study in dynamical systems, and the Theorem solves one of the central problems in this area. Because every “sufficiently smooth” function g can be approximated by polynomials, the Theorem implies that g can be approximated by hyperbolic polynomials, resolving the second part of problem 11 from Smale’s list of problems for the 21st century.

Links: The original paper is available here. See also Section 7.6 of this book for an accessible description of the Theorem.

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Every positive regular solution of the integral equation posed by Lieb is radially symmetric and monotone

You need to know: Euclidean space ${\mathbb R}^n$ , integration over ${\mathbb R}^n$ , space of functions $L^p({\mathbb R}^n)$ (functions $u:{\mathbb R}^n\to{\mathbb R}$ such that $\int_{{\mathbb R}^n}|u|^p < \infty$ ).

Background: For positive integer n and $0 < \alpha < n$ , consider the integral equation $u(x) = \int_{{\mathbb R}^n}\frac{1}{|x-y|^{n-\alpha}}u(y)^{\frac{n+\alpha}{n-\alpha}}dy$ . We call its solution u regular if $u\in L^{\frac{2n}{n-\alpha}}({\mathbb R}^n)$ .

The Theorem: In August 2004, Wenxiong Chen, Congming Li, and Biao Ou submitted to the Communications on Pure and Applied Mathematics a paper in which they proved that every positive regular solution of the integral equation above has the form $u(x) = c\left(\frac{t}{t^2+|x-x_0|^2}\right)^{\frac{n-\alpha}{2}}$ , with some constant $c = c(n, \alpha)$ and some $t > 0$ and $x_0 \in {\mathbb R}^n$ .

Short context: Integral equation above arose in 1983 paper of Lieb on best possible constant in so-called Hardy-Littlewood-Sobolev inequality. It also has connection with a well-known family of semilinear partial differential equations. Lieb posed the classification of all the solutions of this integral equation as an open problem. This problem was open for over 20 years, until was fully solved by the Theorem.

Links: The original paper is available here.

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The number of relative equilibria of the Newtonian 4-body problem is finite

You need to know: Euclidean plane ${\mathbb R}^2$ , rotations, translations, and dilations in the plane, (second) derivative of a function $x:{\mathbb R}\to {\mathbb R}^2$ , uniform rotation, angular velocity.

Background: Let n point particles with masses $m_i > 0$ and positions $x_i \in {\mathbb R}^2$ are moving according to Newton’s laws of motion: $m_j \frac{d^2 x_j}{dt^2} = \sum\limits_{i \neq j} \frac{m_i m_j(x_i - x j)}{r_{ij}^3}, \, 1 \leq j \leq n$ , where $r_{ij}$ is the distance between $x_i$ and $x_j$ . A relative equilibrium motion is a solution of this system of the form $x_i(t) = R(t)x_i(0)$ where $R(t)$ is a uniform rotation with constant angular velocity $v\neq 0$ around some point $c \in {\mathbb R}^2$ . Two relative equilibria are equivalent if they are related by rotations, translations, and dilations in the plane.

The Theorem: On 12th July 2004, Marshall Hampton and Richard Meockel submitted to Inventiones mathematicae a paper in which they proved that, for $n=4$ , there is only a finite number of equivalence classes of relative equilibria, for any positive masses $m_1, m_2, m_3, m_4$ .

Short context: The problem of describing motion of n bodies under gravitation (n-body problem) in space or plane is a fundamental problem in physics and mathematics. In general, the motion can be very complicated even for $n=3$ , but can we at least classify “nice” relative equilibrium motions on the plane? This problem is solved for $n=3$ : in this case, there are always exactly five relative equilibria, up to equivalence. However, for $n\geq 4$ , the question whether the number of relative equilibria is finite is a major open problem, which was included as problem 6 into Smale’s list of problems for the 21st century. The Theorem solves this problem for $n=4$ .

Links: The original paper is available here.

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Typical interval exchange transformation is either rotation or weakly mixing

You need to know: Permutation of $\{1,2, \dots, d\}$ , notation ${\mathbb R}_+^d$ for vectors in ${\mathbb R}_d$ with non-negative components, notation $f^k(x)=f(f(\dots f(x) \dots))$ for k-fold function composition, notation $f^{-k}(A)$ for set $\{x: f^k(x) \in A\}$ , notation $a \equiv b (\text{mod } d)$ if $a-b$ is divisible by d, Lebesgue measure, measurable sets, Lebesgue almost every.

Background: Let $d \geq 2$ be an integer. Permutation $\pi$ of $\{1,2, \dots, d\}$ is called irreducible if $\pi (\{1,\dots,k\}) \neq \{1,\dots,k\}$ for all $1 \leq k<d$ . Given such $\pi$ and $\lambda = (\lambda_1, \dots, \lambda_d)\in {\mathbb R}_+^d$ , an interval exchange transformation $f = f(\lambda, \pi)$ is a map $f:I \to I$ , which divides $I = \left[0,\sum\limits_{i=1}^d \lambda_i\right)$ into sub-intervals $I_i = \left[\sum\limits_{j<i} \lambda_j,\sum\limits_{j \leq i} \lambda_j\right), \, i=1,2\dots,d$ and rearranges the $I_i$ according to $\pi$ (it maps every $x \in I_i$ into $x + \sum\limits_{\pi(j)<\pi(i)} \lambda_j - \sum\limits_{j<i} \lambda_j$ ). $f$ is called weakly mixing if for every pair of measurable sets $A, B \subset I$ , $\lim\limits_{n\to\infty} \frac{1}{n}\sum\limits_{k=1}^{n-1} \left|m(f^{-k}(A) \cap B) -m(A)m(B)\right| = 0$ , where $m$ denotes the Lebesgue measure. Permutation $\pi$ of $\{1,2, \dots, d\}$ is called a rotation if $\pi(i + 1) \equiv \pi(i) + 1$ $(\text{mod } d)$ , for all $i \in \{1,2, \dots, d\}$ .

The Theorem: On 16th June 2004, Artur Ávila and Giovanni Forni submitted to arxiv a paper in which they proved that for every irreducible permutation $\pi$ of $\{1,2, \dots, d\}$ which is not a rotation, and Lebesgue almost every $\lambda\in {\mathbb R}_+^d$ , $f(\lambda, \pi)$ is weakly mixing.

Short context: Interval exchange transformations (IETs in short) are basic examples of measure-preserving transformations $f:I \to I$ (that is, such that $m(f^{-1}(A))=m(A)$ for all measurable $A \subset I$ ), which are central objects of study in dynamical systems. f is called mixing if $\lim\limits_{n\to\infty} m(f^{-n}(A) \cap B) = m(A)m(B)$ for all measurable $A, B \subset I$ , and ergodic if $f^{-1}(A)=A$ implies that $m(A)=0$ or $m(A)=m(I)$ . It is known that every mixing f is weakly mixing, and every weakly mixing f is ergodic. It was known that almost every IET is ergodic, and the Theorem proves a stronger result that almost every non-rotation IET is weakly mixing. Because, by 1980 theorem of Katok, IETs are not mixing, the weak mixing property is the strongest we could hope for.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 7.4 of this book for an accessible description of the Theorem.

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Linear growth of the number of quartic fields with bounded discriminant

You need to know: Prime numbers, infinite product, field, isomorphic and non-isomorphic fields. Also, see this previous theorem description for the concepts of number field, degree of a number field, and discriminant of a number field.

Background: Number fields of degree $n=4$ are called quartic. For $X>0$ , let $N_n(X)$ denotes the number of non-isomorphic number fields of degree n with absolute value of the discriminant at most X. Also, let ${\cal P}$ be the set of prime numbers.

The Theorem: On 7th June 2004, Manjul Bhargava submitted to the Annals of Mathematics a paper in which he proved that the limit $\lim\limits_{X\to\infty}\frac{N_4(X)}{X}$ exists and is equal to $c_4 = \frac{5}{24}\prod\limits_{p\in {\cal P}}\left(1+p^{-2}-p^{-3}-p^{-4}\right) = 0.253...$ .

Short context: Counting number fields up to isomorphism is a basic and important open problem in the area. There is and old folklore conjecture that $\lim\limits_{n\to\infty}\frac{N_n(X)}{X} = c_n>0$ for every fixed n, but, before 2004, this was known only for $n\leq 3$ (see here for the best upper bounds for $N_n(X)$ available for general n). The Theorem proves this conjecture for $n=4$ (quartic fields). In a later work, Bhargava also proved it for $n=5$ .

Links: The original paper is available here. See also Section 5.9 of this book for an accessible description of the Theorem.

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The Dufﬁn–Schaeffer conjecture implies its Hausdorff measure version

You need to know: Set ${\mathbb N}$ of natural numbers, set ${\mathbb R}^+$ of positive real numbers, co-prime integers, monotonic function, infimum, Lebesgue measure, infinite sum.

Background: For a function $g:{\mathbb N}\to {\mathbb R}^+$ , let $S(g)$ be the set of all real numbers $x \in [0,1]$ such that the inequality $\left|x-\frac{a}{b}\right| < \frac{g(b)}{b}$ has infinitely many co-prime solutions $a,b$ . The Duffin–Schaeffer conjecture states that set $S(g) \subset [0,1]$ has Lebesgue measure 1 if and only if $\sum\limits_{n=1}^{\infty} g(n)\frac{\phi(n)}{n}=\infty$ , where $\phi(n)$ is the number of positive integers which are less than n and co-prime with it.

Let $f:[0,\infty) \to [0,\infty)$ be a continuous, non-decreasing function, with $f(0)=0$ . For $\delta>0$ , a $\delta$ -cover of set $S \subset {\mathbb R}$ is a sequence of intervals $I_1, I_2, \dots, I_n, \dots$ of lengths $r_i = |I_i|\leq \delta$ for all i such that $S \subset \bigcup\limits_{i=1}^\infty I_i$ . Let $H_\delta^f(S) = \inf \sum\limits_{i=1}^{\infty}f(r_i/2)$ , where the infimum is taken over all $\delta$ -covers of S. The number $H^f(S)=\lim\limits_{\delta\to 0} H_\delta^f(S)$ is called the Hausdorff f-measure of S. The Hausdorff measure version of the Duffin–Schaeffer conjecture states that if $f(r)/r$ is monotonic, then $H^f(S(g))=H^f([0,1])$ if and only if $\sum\limits_{n=1}^{\infty} f(\frac{g(n)}{n})\phi(n)=\infty$ .

The Theorem: On 2nd June 2004, Victor Beresnevich and Sanju Velani submitted to the Annals of Mathematics a paper in which they proved that the Duffin–Schaeffer conjecture implies its Hausdorff measure version.

Short context: The Duffin–Schaeffer conjecture is a fundamental conjecture in the theory of rational approximation. Its Hausdorff measure version looks much more general and difficult, but the Theorem states that it reduces to the original conjecture. In a later work, Koukoulopoulos and Maynard proved the Duffin–Schaeffer conjecture. By The Theorem, this implies that its Hausdorff measure version is also true.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 6.7 of this book for an accessible description of the Theorem.

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No arithmetic sequence is very well-distributed

You need to know: Prime numbers, logarithm, $O(1)$ notation,

Background: Let ${\cal P}$ be the set of primes. For $x>0$ , let ${\cal P}(x)=\{p\in {\cal P}: p<x\}$ be the set of primes less than x, $\theta(x)=\sum\limits_{p\in{\cal P}(x)} \log p$ , and let $\Delta(x,y) = \frac{\theta(x+y) - \theta(x) - y}{y}$ .

The Theorem: On 1st June 2004, Andrew Granville and Kannan Soundararajan submitted to arxiv and the Annals of Mathematics a paper in which they proved, among other results, the following theorem. Let x be large and y be such that $\log x \leq y \leq \exp\left(\frac{\beta \sqrt{\log x}}{2\sqrt{\log\log x}}\right),$ where $\beta>0$ is an absolute constant. Then there exist numbers $x_+$ and $x_-$ in $(x,2x)$ such that $\Delta(x_+,y) \geq y^{-\delta(x,y)}$ and $\Delta(x_-,y) \leq -y^{-\delta(x,y)},$ where $\delta(x,y) = \frac{1}{\log\log x}\left( \log\left(\frac{\log y}{\log \log x}\right) + \log\log \left(\frac{\log y}{\log \log x}\right) + O(1) \right)$ .

Short context: Function $\theta(x)$ counts primes up to x, each prime p with weight $\log p$ . Famous prime number theorem states that that $\theta(x) \approx x$ for large x, hence the number of weighted primes on interval $(x, x+y]$ is about $\theta(x+y) - \theta(x) \approx y$ . Function $\Delta(x,y)$ measures the quality of this approximation, and the Theorem states that there are intervals with much more and much less primes than average. In author’s words, primes are not very well distributed. In fact, the authors proved a much more general (and surprising) result that the same holds for any “arithmetic sequence”, but the exact definition of this is too difficult to be presented here.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 7.3 of this book for an accessible description of the Theorem.

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The complexity of every irrational algebraic number has superlinear growth

You need to know: Rational and irrational numbers, decimal expansion of a real number, limit inferior $\liminf$ . You also need to know b-ary expansion of real numbers to understand the context.

Background: By complexity function (or just “complexity”) of a real number $\alpha$ we mean the number $p(n)$ of distinct blocks of digits of length n occurring in its decimal expansion. A real number $\alpha$ is called algebraic number if there exists a polynomial P with integer coefficients such that $P(\alpha)=0$ .

The Theorem: On 30th May 2004, Boris Adamczewski and Yann Bugeaud submitted to the Annals of Mathematics a paper in which they proved that the complexity function $p(n)$ of every irrational algebraic number satisfies $\liminf\limits_{n\to\infty}\frac{p(n)}{n} =+\infty$ .

Short context: It is natural to conjecture that the decimal expansion of irrational algebraic numbers such as $\sqrt{2}$ contains all possible digit patterns, that is, $p(n)=2^n$ . However, before 2004, it was only known that $\liminf\limits_{n\to\infty}(p(n)-n) =+\infty$ . The Theorem is a considerable improvement in comparison with this result. It remains true in any b-ary expansion with any base $b\geq 2$ .

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 7.2 of this book for an accessible description of the Theorem.

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O((log n)^1/2)-approximation for the sparsest cut can be done in polynomial time

You need to know: Graph, vertices, edges, polynomial-time algorithm, NP-hard problem.

Background: The sparsest cut problem is to divide the vertices of the graph G into two classes such that to minimise the ratio r of the number of edges between the classes divided by the number of vertices in the smaller class. The optimal ratio $\alpha(G)$ is also known as edge expansion of the graph.

The Theorem: In April 2007, Sanjeev Arora, Satish Rao, and Umesh Vazirani submitted to the journal of the ACM a paper in which they proved the existence of a polynomial-time algorithm that, given any graph G with n vertices, produces a cut with ratio $r=O(\alpha(G) \sqrt{\log n})$ .

Short context: The sparsest cut problem is a fundamental combinatorial problem both for theory and applications, which is NP-hard to solve exactly. In 1988, Leighton and Rao developed a $O(\log n)$ -approximation algorithm for this problem. The Theorem improves the approximation ratio to $O(\sqrt{\log n})$ .

Links: The original paper is available here.

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The Calabi-Yau conjectures are true for embedded surfaces with finite topology

You need to know: Euclidean space ${\mathbb R}^3$ , halfspace of ${\mathbb R}^3$ , topological space, (smooth) surface, boundary of a surface, area of a surface, compact surface, connected surface, homeomorphic surfaces, geodesic on a surface (a locally length-minimising curve).

Background: A surface M is called embedded in ${\mathbb R}^3$ if it can be placed in ${\mathbb R}^3$ without self-intersections. A surface $M \subset {\mathbb R}^3$ is called a minimal surface if every point $p \in M$ has a neighbourhood S, with boundary B, such that S has a minimal area out of all surfaces S’ with the same boundary B. A surface S is called complete if any parametrized geodesic $\gamma: [a, b) \to S$ of S, may be extended into a parametrized geodesic $\gamma': {\mathbb R}\to S$ defined on the entire line ${\mathbb R}$ . A surface is said to have finite topology if it is homeomorphic to a compact surface with a finite number of points removed.

The Theorem: On 9th April 2004, Tobias Colding and William Minicozzi II submitted to arxiv and the Annals of Mathematics a paper in which they proved that the plane is the only complete embedded minimal surface with finite topology in a halfspace of ${\mathbb R}^3$ .

Short context: Minimal surfaces are central objects of study in low-dimensional topology. In 1965, Calabi conjectured that (i) a complete minimal surface in ${\mathbb R}^3$ must be unbounded, and, moreover, (ii) it has an unbounded projection on every line, unless it is a plane. In this generality, this is false: a counterexample to part (ii) is given by Jorge and Xavier in 1980, and to part (i) by Nadirashvili in 1996. However, these examples are not embedded (have self-intersections), and Yau asked in 2000 whether Calabi conjectures are true for embedded surfaces. Because every surface which cannot be covered by any halfspace is unbounded, and has an unbounded projection on every line, the Theorem confirms both conjectures for embedded surfaces with finite topology.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 8.1 of this book for an accessible description of the Theorem.

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