The Julia set of typical quadratic map is locally connected and has measure zero

You need to know: Basic complex analysis, set ${\mathbb C}$ of complex numbers, absolute value $|z|$ of complex number z, boundary of a set, locally connected set, set of Lebesgue measure zero, the notion of (Lebesgue) almost every.

Background: For polynomial $f:{\mathbb C} \to {\mathbb C}$ , let $K_f \subset {\mathbb C}$ be the set of points $z_0\in {\mathbb C}$ such that the sequence $z_0, z_1=f(z_0), z_2=f(z_1), \dots, z_{n+1}=f(z_n), \dots$ is bounded, that is, $|z_n|\leq B, \, \forall n$ for some $B \in {\mathbb R}$ . The boundary $J_f$ of $K_f$ is called the Julia set of f.

The Theorem: On 17th August 2000, Carsten Petersen and Saeed Zakeri submitted to the Annals of Mathematics a paper in which they proved that for almost every complex number c with $|c|=1$ , the Julia set of quadratic polynomial $f(z)=z^2+cz$ is locally connected and has Lebesgue measure zero.

Short context: Julia set is a fundamental concept in the theory of complex dynamics, because it consists of values $z_0$ such that an arbitrarily small perturbation can cause significant changes in the sequence of iterated function values. The Theorem describes the geometry of Julia sets for almost all quadratic polynomials. In fact, Petersen and Zakeri also gave a precise arithmetic sufficient condition on c for the theorem conclusion to hold.

Links: The original paper can be found here.

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The Lagarias-Wang finiteness conjecture is false

You need to know: Matrix, square matrix, (complex) eigenvalue of a square matrix, absolute value of a complex number, $\limsup$ .

Background: The spectral radius $\rho(A)$ of a square matrix A is the largest absolute value of an eigenvalue of A. Let $\Sigma$ be a finite set of $n \times n$ matrices. Let $\rho_k(\Sigma)=\max\{\rho(A_1 A_2 \dots A_k)^{1/k}, \, A_i \in \Sigma, \, i=1,2,\dots,k\}$ . The quantity $\rho(\Sigma) = \limsup\limits_{k\to\infty} \rho_k(\Sigma)$ is called the generalized spectral radius of $\Sigma$ .

The Theorem: On 11th July 2000, Thierry Bousch and Jean Mairesse submitted to the Journal of the AMS a paper in which they proved, among other results, the existence of a finite set $\Sigma$ of matrices (in fact, $\Sigma$ can consist of two $2 \times 2$ matrices) such that $\rho(\Sigma) > \rho_k(\Sigma)$ for all $k \geq 0$ .

Short context: The generalized spectral radius is an important concept useful in a wide range of contexts. It is known that $\rho(\Sigma) \geq \rho_k(\Sigma)$ for all $k \geq 0$ . In 1995, Lagarias and Wang conjectured that, for every $\Sigma$ , there is a k such that $\rho(\Sigma) = \rho_k(\Sigma)$ . This statement became known as the finiteness conjecture. The Theorem disproves this conjecture.

Links: The original paper can be found here.

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The norms of bilinear Hilbert transforms are uniformly bounded

You need to know: Limits, derivative, k-th derivative $z^{(k)}(t)$ , integration, $L^p$ norm $\|z\|_p= \left( \int_{-\infty}^{\infty} |z(t)|^p dt \right)^{1/p}$ .

Background: Function $z:{\mathbb R}\to{\mathbb R}$ is called a Schwartz function if there exist all derivatives $z^{(k)}(t)$ for all $k=1,2,3,\dots$ and all $t\in{\mathbb R}$ , and, for every k and $\gamma\in{\mathbb R}$ , there is a constant $C(k,\gamma)$ such that $|t^\gamma z^{(k)}(t)| \leq C(k,\gamma), \, \forall t\in {\mathbb R}$ . A bilinear Hilbert transform of Schwartz functions f and g is a function $z_{\alpha, \beta}(x) = \lim\limits_{\epsilon\to 0}\int\limits_{\frac{1}{\epsilon}>|t|>\epsilon} f(x-\alpha t)g(x-\beta t) \frac{dt}{t}$ , where $\alpha,\beta \in {\mathbb R}$ are real parameters.

The Theorem: On 2nd June 2000, Loukas Grafakos and Xiaochun Li submitted to Annals of Mathematics a paper in which they proved that for every $2 < p_1 < \infty$ , $2 < p_2 < \infty$ such that $1 < p = \frac{p_1p_2}{p_1+p_2} < 2$ , there is a constant $C = C(p_1, p_2)$ such that $\sup\limits_{\alpha, \beta} \|z_{\alpha, \beta}\|_p \leq C \|f\|_{p_1} \|g\|_{p_2}$ for all Schwartz functions $f, g: {\mathbb R}\to{\mathbb R}$ .

Short context: Bilinear Hilbert transform was introduced by Calderón in 1960, who also asked an important open question about it (which, however, too difficult to formulate it here). The Theorem, in combination with another result, resolves this 40-years old open question. The importance of the Theorem is that the constant C depends only on $p_1$ and $p_2$ , but not on $f,g,\alpha$ , and $\beta$ .

Links: The original paper is available here. See also Section 4.5 of this book for an accessible description of the Theorem.

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The only Banach space isomorphic to each of its subspaces is l^2

You need to know: Vector space, infinite dimensional vector space, Banach space, isomorphic Banach spaces, subspace, closed subspace.

Background: $l^2$ -space is the space of infinite sequences $x=(x_1, x_2, \dots, x_n, \dots)$ equipped with coordinate-wise addition and scalar multiplication and norm $|x| := \sqrt{\sum\limits_{i=1}^\infty x_i^2} < \infty$ .

The Theorem: On 30th March 2000 Timothy Gowers submitted to Annals of Mathematics a paper in which he proved, among other results, that $l^2$ -space is (up to isomorphism) the only infinite-dimensional Banach space which is isomorphic to every infinite-dimensional closed subspace of itself.

Short context: It is easy to see that $l^2$ -space is isomorphic to every infinite-dimensional closed subspace of itself. In his famous 1932 book, Banach asked whether this is the only such example. The Theorem gives a positive answer to this question.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 2.11 of this book for an accessible description of the Theorem.

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Small set of initial points for Newton’s method to find all roots of polynomials

You need to know: Set ${\mathbb C}$ of complex numbers, absolute value of a complex number, polynomials in complex variable, convergence, derivative, natural logarithm $\ln$ .

Background: Let $P_d$ be the set of polynomials of degree d in one complex variable, such that all their roots have absolute values less than 1. For $P \in P_d$ and $z_0 \in {\mathbb C}$ , consider sequence $z_0, z_1=f_P(z_0), z_2=f_P(z_1), \dots$ , where $f_P(z) = z - \frac{P(z)}{P'(z)}$ . If this sequence converges to a root $z^*$ of P, we say that $z_0$ is in the basin of $z^*$ . This is called the Newton’s method for finding roots.

The Theorem: On 24th February 2000 John Hubbard, Dierk Schleicher, and Scott Sutherland submitted to Inventiones mathematicae a paper in which they proved that, for every $d \geq 2$ , there is a set $S_d$ consisting of at most $1.11 d \cdot \ln^2 d$ points in ${\mathbb C}$ with the property that for every polynomial $P \in P_d$ and every root $z^*$ of P, there is a point $s \in S_d$ in the basin of $z^*$ .

Short context: Finding roots of polynomials is one of the basic problems in mathematics, with Newton’s method being one of the most popular methods for its numerical solution. However, its convergence depends on the choice of initial point $z_0$ . The Theorem guarantees that, if we start Newton’s method from all points of $S_d$ , we are guaranteed to find all the roots of any polynomial $P \in P_d$ .

Links: The original paper is available here.

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All critical points of a rational function f are real only if f is equivalent to real

You need to know: Complex numbers, functions in complex variable, complex differentiation, polynomials.

Background: A (complex) rational function f is a ratio of two polynomials with complex coefficients, $f(z)=\frac{P(z)}{Q(z)}$ . If all coefficients of P(z) and Q(z) are real, we say that f is a real rational function. A complex number $z_0$ is a critical point of f if $f'(z_0)=0$ . We say that two rational functions f and g are equivalent if $g(z)=\frac{a f(z)+b}{c f(z)+d}$ for some complex numbers a,b,c,d such that $ad-bc\neq 0$ .

The Theorem: On 25th January 2000 Alexandre Eremenko and Andrei Gabrielov submitted to Annals of Mathematics a paper in which they proved, among other results, that if all critical points of a rational function f are real, then f is equivalent to a real rational function.

Short context: The Theorem equivalently states that, if for polynomials $P(z),Q(z)$ all the solutions of the equation $P(z)Q'(z)-P'(z)Q(z)=0$ are real, then there exist complex numbers $a,b,c,d$ such that $ad-bc\neq 0$ and $aP(z)+bQ(z)$ and $cP(z)+dQ(z)$ are real polynomials. This is a special case of a well-known conjecture of B. and M. Shapiro, made around 1993, which predicts a similar result for any number of polynomials. In a later work, Mukhin, Tarasov, and Varchenko proved this conjecture in general.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 2.1 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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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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