An explicit construction of polynomials with optimal condition numbers

You need to know: Basic complex analysis, set ${\mathbb C}$ of complex numbers, absolute value $|z|$ of complex number z, polynomials in complex variable, root of a polynomial, derivative $P'(z)$ of polynomial $P(z)$ , binomial coefficients ${N\choose i}=\frac{N!}{i!(N-k)!}$ .

Background: For polynomial $P(z)=\sum_{i=0}^N a_i z^i$ with complex coefficients $a_i$ , its Weil norm is $||P|| = \sqrt{\sum_{i=0}^N {N\choose i}^{-1}|a_i|^2}$ , and its (normalised) condition number is $\mu(P) = \max\limits_{z \in {\mathbb C}: P(z)=0} \left(\sqrt{N}\frac{||P||(1+|z|^2)^{N/2-1}}{|P'(z)|}\right)$ .

The Theorem: On 4th March 2019, Carlos Beltrán, Ujué Etayo, Jordi Marzo, and Joaquim Ortega-Cerdà submitted to arxiv a paper in which they proved the existence a constant C and an explicit formula, which, given any integer $N>0$ , produces a polynomial $P_N$ of degree N such that $\mu(P_N) \leq C \sqrt{N}$ .

Short context: A fundamental problem in mathematics is numerical computation of roots of polynomials. Because in numerical studies the coefficients of polynomials are known only approximately, the central issue is to understand what effect on the roots has a slight perturbation of the coefficients. In 1993, Shub and Smale introduced the condition number $\mu(P)$ and observed that for polynomials with small $\mu(P)$ the root computation is stable with respect to the coefficients perturbations. They also proved that a random polynomial P of degree N has $\mu(P)\leq N$ with probability at least $1/2$ , and posed the problem to construct an example of such polynomials explicitly. The Theorem solves this problem. The bound $\mu(P_N) \leq C \sqrt{N}$ is the best possible up to a constant factor.

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

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The hot spots conjecture is true for Lip domains

You need to know: Euclidean plane ${\mathbb R}^2$ , orthonormal coordinate system, smooth (that is, differentiable) function in 2 variables, first and second order partial derivatives, notation $\Delta f = \frac{\partial^2 f}{\partial x_1^2}+\frac{\partial^2 f}{\partial x_2^2}$ , directional derivative $\frac{\partial f}{\partial n}$ for a vector n, normal vector, Lipschitz function.

Background: Let $\Omega \subset {\mathbb R}^2$ be a Lipschitz domain, that is, a bounded domain with boundary $\partial\Omega$ such that every $x \in \partial\Omega$ has a neighbourhood U such that $\partial\Omega \cap U$ is the graph of a Lipschitz function in some orthonormal coordinate system. A bounded, open, connected Lipschitz domain is called a Lip domain if it is given by $\Omega = \{(x1, x2): f_1(x_1)<x_2<f_2(x_1)\}$ , where $f_1$ , $f_2$ are Lipschitz functions with constant 1. The second Neumann eigenvalue $\mu_2=\mu_2(\Omega)$ is the smallest positive real number such that there exists a not identically zero, smooth function $u : \Omega \to {\mathbb R}$ that satisfies the equation $\Delta u =-\mu_2 \cdot u$ on $\Omega$ and the boundary condition $\left.\frac{\partial u}{\partial n}\right|_{\partial \Omega} \equiv 0$ at the smooth points of $\partial\Omega$ , where n denotes the outward pointing normal vector (see here for an equivalent definition of $\mu_2$ ). A function u that satisfies these conditions is called the second Neumann eigenfunction for $\Omega$ .

The Theorem: On 17th December 2001, Rami Atar, and Krzysztof Burdzy submitted to The Journal of the AMS a paper in which they proved that if $\Omega \subset {\mathbb R}^2$ is a Lip domain, then the second Neumann eigenfunction attains its extrema at the boundary of $\Omega$ .

Short context: The conjecture that the second Neumann eigenfunction attains its extrema at the boundary of $\Omega$ is known as the hot spot conjecture. In simple English, it states that if a flat piece of metal is given an initial heat distribution which then flows throughout the metal, then, after some time, the hottest point on the metal will lie on its boundary. The conjecture was proposed by Rauch in 1974. In 1999, Burdzy and Werner constructed a domain (with holes) for which the conjecture fails, but it still believed to be true for domains without holes. The Theorem confirms the conjecture for all Lip domains. In a later work, the conjecture has also been proved for all triangles.

Links: The original paper is available here.

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Kesten’s theorem holds for random Kronecker sequences on the torus T^d

You need to know: Matrix, determinant of a matrix, notation $AC$ for the image of set $C$ under linear transformation defined by matrix $A$ , notation $\times$ for direct product of sets, notation “ $x \mod 1$ ” for the real number $y\in[0,1)$ such that $x-y$ is an integer, notation ${\mathbb P}$ for the probability, selection uniformly at random.

Background: Let $d>0$ be an integer, $I=\{1,2,\dots,d\}$ , $T=[0,1)$ , and $T^d$ be set of vectors $x=(x_1, \dots, x_d)$ with $x_i \in T$ for every $i \in I$ . Let $U$ be the set of vectors $u=(u_1, \dots, u_d)$ such that $v_i\leq u_i \leq w_i$ for every $i \in I$ , where $v_i, w_i$ are fixed such that $0<v_i<w_i<1/2$ for every $i \in I$ . For each $u\in U$ , let $C_u$ be the set of vectors $y=(y_1, \dots, y_d)$ such that $|y_i|\leq u_i$ for every $i \in I$ . For a (small) $\eta>0$ , let $G_\eta$ be the set of $d\times d$ matrices with determinant $1$ and real entries $a_{ij}$ , such that $|a_{ii}-1|<\eta$ for all $i$ and $|a_{ij}|<\eta$ for all $i\neq j$ . Let $X=T^d \times T^d \times U \times G_\eta$ . For $\nu = (\alpha, x, u, A) \in X$ and integer $N>0$ , let $M(\nu, N)$ be the number of integers $1\leq m \leq N$ such that $(x+m\alpha) \mod 1 \in A C_u$ , and let $D(\nu, N)=M(\nu, N) - 2^d (\prod_{i=1}^d u_i) N$ .

The Theorem: On 19th November 2012 Dmitry Dolgopyat and Bassam Fayad submitted to arxiv a paper in which they proved that if $\nu$ is selected in $X$ uniformly at random, then, for all real $z$ , $\lim\limits_{N\to\infty}{\mathbb P}\left(\frac{D(\nu,N)}{(\ln N)^d}\leq z\right)=F(\rho_d z)$ , where $\rho_d$ is a constant depending only on $d$ , and $F(z)=\frac{\arctan(z)}{\pi}+\frac{1}{2}$ .

Short context: For every irrational $\alpha$ , it is known that sequence $\alpha, 2\alpha, \dots, m\alpha, \dots$ (called the Kronecker sequence) is uniformly distributed on $[0,1)$ , and the same is true for shifted sequence $x+\alpha, \dots, x+m\alpha, \dots$ . More formally, if $M(N)$ is the number of terms of this sequence (with $1\leq m\leq N$ ) belonging to some interval $[a,b) \subset [0,1)$ , then $\lim\limits_{N\to\infty}\frac{M(N)}{N}=b-a$ . Quantity $D(N)=M(N)-(b-a)N$ measures how fast this convergence happens. In 1962, Kesten established the limiting distribution of $D(N)$ , after appropriate scaling, provided that $x$ and $\alpha$ are selected at random. The Theorem establishes a multidimensional version of this result.

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

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Flat Littlewood polynomials exist

You need to know: Polynomials, degree of a polynomial, set ${\mathbb C}$ of complex numbers, absolute value $|z|$ of complex number $z$ .

Background: A polynomial $P(z)$ of degree n in complex variable z is called a Littlewood polynomial if $P(z) = \sum_{k=0}^n \epsilon_k z^k$ , where $\epsilon_k \in \{-1,1\}$ for all $0\leq k \leq n$ .

The Theorem: On 22nd July 2019, Paul Balister, Béla Bollobás, Robert Morris, Julian Sahasrabudhe and Marius Tiba submitted to arxiv and Annals of Mathematics a paper in which they proved the existence of constants $\Delta>\delta>0$ such that, for all $n\geq 2$ ,
there exists a Littlewood polynomial $P(z)$ of degree n with $\delta\sqrt{n} \leq |P(z)| \leq \Delta\sqrt{n}$ for all $z \in {\mathbb C}$ with $|z|=1$ .

Short context: Polynomials satisfying the condition of the Theorem are called flat polynomials, hence the Theorem states that flat Littlewood polynomials exist. It answers a question of Erdos from 1957, and confirms a conjecture of Littlewood made in 1966.

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

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There exist finitely generated coarsely non-amenable groups that are coarsely embeddable into l^2

You need to know: Symmetric difference $A \Delta B = (A \setminus B) \cup (B \setminus A)$ of sets $A$ and $B$ , notation $|A|$ for the number of elements in finite set $A$ , notation $\times$ for set product, set ${\mathbb N}$ of natural numbers, metric space $(X; d)$ where $X$ is a set and $d$ is a metric, infinite dimensional Hilbert space $l^2$ , group, generating set of a group, finitely generated group, word metric of a group with respect to a generating set.

Background: We say that metric space $(X; d_X)$ is coarsely embeddable into metric space $(Y; d_Y)$ is there exists a map $f:(X; d_X)\to(Y; d_Y )$ such that $d_Y(f(x_n); f(y_n)) \to \infty$ if and only if $d_X(x_n; y_n) \to \infty$ for all sequences $(x_n)_{n\in {\mathbb N}}$ and $(y_n)_{n\in {\mathbb N}}$ in $X$ . We say that a finitely generated group $G$ is coarsely embeddable into $(Y; d_Y )$ if it is so for the word metric with respect to a finite generating set $S$ . This property does not depend on the choice of $S$ .

A metric space $(X; d)$ is called uniformly discrete if there exists a constant $r>0$ such that, for any $x,y \in X$ , we have either $x=y$ or $d(x,y)>r$ . A uniformly discrete metric space $(X; d)$ is coarsely amenable if for every $\epsilon>0$ and $R>0$ , there exist a constant $S>0$ and a collection of finite subsets $\{A_x\}_{x \in X}$ , $A_x \subseteq X \times {\mathbb N}$ for every $x\in X$ , such that (a) $|A_x \Delta A_y|/|A_x \cap A_y|\leq \epsilon$ when $d(x,y) \leq R$ , and (b) $A_x \subseteq B(x,S) \times N$ . A finitely generated group $G$ is called coarsely amenable if it is so for the word metric with respect to a finite generating set $S$ , and is called coarsely non-amenable otherwise.

The Theorem: On 19th June 2014, Damian Osajda submitted to arxiv a paper in which he proved the existence of finitely generated coarsely non-amenable groups that are coarsely embeddable into the infinite dimensional Hilbert space $l^2$ .

Short context: The notion of coarse amenability is a week version of amenability, and has many equivalent formulations and applications. In particular, it is known that any finitely generated coarsely amenable groups is coarsely embeddable into $l^2$ . The question whether the converse is true (Are groups coarsely embeddable into $l^2$ coarsely amenable?) is a natural question raised by a number of researchers. The Theorem provides a negative answer.

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

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The set of non-weakly mixing IETs has Hausdorff codimension at most 1/2

You need to know: Notation $f^k(x)=f(f(\dots f(x) \dots))$ for k-fold function composition, notation $f^{-k}(A)$ for set $\{x: f^k(x) \in A\}$ , Lebesgue measure, measurable sets, Lebesgue almost every, Hausdorff dimension of a set.

Background: Let $d \geq 2$ be an integer, and let $\Lambda_d \subset {\mathbb R}^{d-1}$ be the set of vectors $\lambda=(\lambda_1,\dots,\lambda_{d-1})$ such that $0<\lambda_1<\dots<\lambda_{d-1}<1$ . For $\lambda \in \Lambda_d$ , let $f_\lambda^*:[0,1)\to[0,1)$ be a function that divides $[0,1)$ into subintervals $[0,\lambda_1), [\lambda_1, \lambda_2), \dots, [\lambda_{d-1},1)$ and rearranges them in the opposite order. Function $f:[0,1)\to[0,1)$ is called weakly mixing if for every pair of measurable sets $A, B \subset [0,1)$ , $\lim\limits_{n\to\infty} \frac{1}{n}\sum\limits_{k=1}^{n-1} \left|m(f^{-k}(A) \cap B) -m(A)m(B)\right| = 0$ , where $m$ denotes the Lebesgue measure. Let $\text{dim}(S)$ and $\text{codim}(S)=d-1-\text{dim}(S)$ denote Hausdorff dimension and Hausdorff codimension of set $S \subset \Lambda_d$ , respectively.

The Theorem: On 2nd January 2018, Jon Chaika and Howard Masur submitted to arxiv a paper in which they proved that for every odd $d\geq 3$ , the set $S$ of all $\lambda$ such that $f_\lambda^*$ is not weakly mixing has $\text{codim}(S)=1/2$ .

Short context: Interval exchange transformations (IETs in short) are functions defined similarly to $f_\lambda^*$ but such that subintervals $[0,\lambda_1), [\lambda_1, \lambda_2), \dots, [\lambda_{d-1},1)$ are rearranged according to an arbitrary permutation $\pi$ , see here for details. In 2004, Ávila and Forni proved that almost all IETs are weakly mixing. In 2016, Avila and Leguil proved a stronger result that set S of non-weakly mixing IETs is not even full-dimensional, or, in other words, has a positive codimension. This raises the problem of determining this codimension. The Theorem solves this problem for permutation $\pi = (1,\dots,d)\to(d,\dots,1)$ with odd $d$ . It is a corollary of a more general result establishing inequality $\text{codim}(S) \leq 1/2$ for a broad class of permutations $\pi$ .

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

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For every Banach space, its Rademacher type and Enflo type coincide

You need to know: Metric space, Banach space, notation $\|.\|$ for the norm in the Banach space, independent random variables, expectation (denoted by E).

Background: Let $\epsilon_1, \epsilon_2, \dots$ be a sequence of indepedndent random variables, each equal to $+1$ or $-1$ with equal probabilities. We say that a Banach space $X$ has Rademacher type $p\in[1,2]$ if there exists a constant $C\in (0,\infty)$ such that inequality $E\left\|\sum\limits_{i=1}^n \epsilon_j x_j\right\|^p \leq C^p \sum\limits_{j=1}^n\|x_j\|^p$ holds for all $n\geq 1$ and all $x_1, \dots, x_n \in X$ . For $\epsilon=(\epsilon_1, \dots, \epsilon_n)$ , and index $j$ , denote $\epsilon^{j-}$ the vector $(\epsilon_1, \dots, -\epsilon_j, \dots, \epsilon_n)$ . We say that a Banach space $X$ has Enflo type $p$ if there exists a constant $C\in (0,\infty)$ such that inequality $E\|f(\epsilon)-f(-\epsilon)\|^p \leq C^p \sum\limits_{j=1}^n E\|f(\epsilon)-f(\epsilon^{j-})\|^p$ holds for all $n\geq 1$ and every function $f:\{-1,1\}^n \to X$ .

The Theorem: On 13th March 2020, Paata Ivanisvili, Ramon van Handel, and Alexander Volberg submitted to arxiv a paper in which they proved that a Banach space $X$ has Rademacher type $p\in[1,2]$ if and only if $X$ has Enflo type $p$ . In other words, Rademacher type and Enflo type coincide.

Short context: The notion of Rademacher type is one of the central and useful concepts in the theory of Banach spaces, and researchers have long tried to extend it to the general metric space. However, the original definition use addition in $X$ substantially, and is difficult to extend. In 1978, Enflo introduced a definition of type which is straightforward to extend to metric spaces, and conjectured that, for Banach spaces, the two notions of type coincide. If a Banach space $X$ has Enflo type $p$ , then, applying the definition to function $f(\epsilon)=\sum_{j=1}^n \epsilon_j x_j$ , it is easy to see that $X$ has Rademacher type $p$ as well. However, the converse direction of the conjecture remained open for decades, despite intensive study and a number of partial results. The Theorem proves the conjecture in full.

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

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Non-collision singularities exist in a planar Newtonian 4-body problem

You need to know: Euclidean plane ${\mathbb R}^2$ , (second) derivative of a function $x:{\mathbb R}\to {\mathbb R}^2$ .

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 $x_j$ . The motion of particles is determined by the initial conditions: their masses and their positions and velocities at time $t=0$ .

The Theorem: On 29th August 2014, Jinxin Xue submitted to Acta Mathematica a paper in which he proved that, for $n=4$ , there is a non-empty set of initial conditions, such that all four points escape to infinity in a finite time, avoiding collisions.

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, studied by many authors, see, for example, here. In general, the motion can be very complicated even for $n=3$ , but can we at least understand under what initial conditions the solution to the system of Newton equations (presented above) is well-defined for all $t\geq 0$ ? This may be not the case for two reasons: (a) collision happened, and (b) there are no collisions but a point escape to infinity in a finite time. Case (b) is known as non-collision singularity. In 1897, Painlevé proved that there are no such singularities for $n=3$ , but conjectured their existence for all $n>3$ . In 1992, Xia proved this conjecture (for motions in ${\mathbb R}^3$ ) for $n\geq 5$ . The Theorem establishes the remaining (and the hardest) case $n=4$ .

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

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The Julia set of the Feigenbaum polynomial has zero Lebesgue measure

You need to know: Basic complex analysis, set ${\mathbb C}$ of complex numbers, absolute value $|z|$ of complex number z, boundary of a set, area (that is, Lebesgue measure) of a set $A \subset {\mathbb C}$ , notation $f^n$ for the n-th iterate of function $f:{\mathbb C}\to {\mathbb C}$ .

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 $z_0, z_1=f(z_0), \dots, z_{n}=f(z_{n-1})=f^n(z_0), \dots$ 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.

We say that point $z_0\in {\mathbb C}$ is periodic if $f^n(z_0)=f(z_0)$ for some positive integer n, and the smallest n such that this holds is called the period of $z_0$ . For each n, let $r_n$ be the (unique) real number such that $0$ is a periodic point with period $2^n$ of polynomial $f(z)=z^2+r_n$ . It is known that the limit $r_\infty=\lim\limits_{n\to\infty} r_n$ exists and numerically equal to $-1.401155...$ . The polynomial $f(z)=z^2+r_\infty$ is called the Feigenbaum polynomial.

The Theorem: On 22nd December 2017, Artem Dudko and Scott Sutherland submitted to arxiv a paper in which they proved that the Julia set of the Feigenbaum polynomial has zero Lebesgue measure.

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 Julia set is known to have zero area for almost all but not all quadratic polynomials. The Feigenbaum polynomial is important in so-called renormalization theory in dynamics, and the question whether its Julia set has zero area was a long-standing open question. The Theorem resolves it affirmatively.

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

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Almost all orbits of the Collatz map attain almost bounded value

You need to know: Set ${\mathbb N}$ of positive integers, logarithm, limits.

Background: The Collatz map $\text{Col}: {\mathbb N}\to {\mathbb N}$ is defined by (i) $\text{Col}(n)=3n+1$ when $n$ is odd and (ii) $\text{Col}(n) = n/2$ when n is even. For any $n\in{\mathbb N}$ , let $\text{Col}^k(n)$ denote the k-th iterate of $\text{Col},$ let $\text{Col}^{\mathbb N}(n)=\{n, \text{Col}(n), \text{Col}^2(n), \dots\}$ be the Collatz orbit, and let $\text{Col}_{\text{min}}(n) = \inf\limits_{k \in {\mathbb N}} \text{Col}^k(n)$ denote the minimal element of the Collatz orbit $\text{Col}^{\mathbb N}(n)$ .

We say that subset $A\subseteq {\mathbb N}$ has logarithmic density 1, if $\lim\limits_{x\to \infty}\left(\frac{1}{\log x}\sum\limits_{k\in A, k\leq x}\frac{1}{k}\right)=1$ . We say that a property P holds for almost all $n \in {\mathbb N}$ (in the sense of logarithmic density) if the subset of ${\mathbb N}$ for which P holds has logarithmic density 1.

The Theorem: On 8th September 2019, Terence Tao submitted to arxiv a paper in which he proved that for any function $f: {\mathbb N} \to {\mathbb N}$ such that $\lim\limits_{n\to \infty} f(n)=+\infty$ , one has $\text{Col}_{\text{min}}(n) < f(n)$ for almost all $n \in {\mathbb N}$ (in the sense of logarithmic density).

Short context: The famous Collatz conjecture predicts that $\text{Col}_{\text{min}}(n)=1$ for all $n\in{\mathbb N}$ , but it remains well beyond reach of the current methods. As a partial progress, Terras proved in 1976 that $\text{Col}_{\text{min}}(n)<n$ for almost all n. In 1979, Allouche improved this to $\text{Col}_{\text{min}}(n)<n^\theta$ for almost all n, for any fixed constant $\theta>0.869$ . In 1994, Korec proved the same result for any $\theta>\frac{\log 3}{\log 4}=0.792...$ . The Theorem implies that one can take any $\theta>0$ , and, more generally, $\text{Col}_{\text{min}}(n)<f(n)$ for almost all n and any function f going to infinity with n. For example, one can take $f(n)=\log\log\log\log n$ .