If N is a product of two prime powers, map (x,y)->x^Ny^N is surjective on every finite non-abelian simple group

You need to know: Prime number, group, identity element, finite group, notation $g^{-1}$ for the inverse of group element $g$ , subgroup.

Background: A subgroup S of group G is called normal, if $g\cdot a\cdot g^{-1}\in S$ for any $a \in S$ and $g \in G$ . A group G is called simple if it does not have any normal subgroup, except of itself and $\{e\}$ , where $\{e\}$ is the set consisting on identity element only. A group G is called abelian if $g\cdot h = h \cdot g$ for every $g \in G$ and $h \in G$ , and non-abelian otherwise.

The Theorem: On 1st May 2015, Robert Guralnick, Martin Liebeck, Eamon O’Brien, Aner Shalev, and Pham Tiep submitted to Inventiones Mathematicae a paper in which they proved the following result. Let p, q be prime numbers, let a, b be non-negative integers, and let $N=p^aq^b$ . Then every element g of every finite non-abelian simple group G can be written as $g=x^Ny^N$ for some $x,y \in G$ .

Short context: Given group G, let w be a map mapping pair $x,y \in G$ into $w(x,y)=x^Ny^N$ . Let $w(G)$ be the set of all $g\in G$ such that $g=w(x,y)$ for some $x,y\in G$ . Map w is called surjective on G if $w(G)=G$ . In this language, the Theorem states that w is surjective on every finite non-abelian simple group G. Earlier, this result was proved for commutator map $w(x,y)=x^{-1}y^{-1}xy$ , see here. Also, it is known that $w(G)\cdot w(G)=G$ for much broader class of maps w, see here.

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

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The topological Tverberg conjecture is false

You need to know: Eucledean space ${\mathbb R}^d$ , continuous map $f: {\mathbb R}^m \to {\mathbb R}^d$ , image of a set under map f, polytope, face of a polytope, convex hull, prime numbers, power of a prime.

Background: An m-simplex, denoted $\Delta_m$ , is an m-dimensional polytope which is the convex hull of its $m+1$ vertices. The “topological Tverberg conjecture” states that for given integers $r\geq 2$ , $d \geq 1$ , $m \geq (r-1)(d+1)$ , and for any continuous map $f: \Delta_m \to {\mathbb R}^d$ , there are r pairwise disjoint faces of $\Delta_m$ whose images have a point in common.

The Theorem: On 1st February 2015, Florian Frick submitted to arxiv a paper in which he proved that the topological Tverberg conjecture is false for any r that is not a power of a prime and any dimension $d \geq 3r+1$ .

Short context: The topological Tverberg conjecture was formulated by Bárány, Shlosman and Szűcs in 1981, and was considered by many experts as a major open problem, a holy grail of topological combinatorics. The case when f is an affine map is equivalent to Tverberg’s 1966 theorem (hence the name of the conjecture). The conjecture was also known to hold when r is a prime or a power of a prime, and was commonly believed to be true in general. The Theorem refutes the conjecture.

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

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Tarski’s circle squaring problem can be solved with only measurable pieces

You need to know: Euclidean plane ${\mathbb R}^2$ , Lebesgue measurable subsets of ${\mathbb R}^2$ (that is, subsets for which the area is well-defined), notation $A+v$ for the translation of set $A\subset {\mathbb R}^2$ by a vector $v\in {\mathbb R}^2$ .

Background: We call two sets $A,B \subset {\mathbb R}^2$ equidecomposable by translations if there are partitions $A = A_1 \cup \dots \cup A_m$ and $B = B_1 \cup \dots \cup B_m$ , such that $B_i = A_i + v_i$ , $i=1,\dots,m$ , for some vectors $v_1, \dots, v_m \in {\mathbb R}^2$ . If, moreover, all $A_i$ (and thus $B_i$ ) are Lebesgue measurable, we say that A and B are equidecomposable by translations with Lebesgue measurable pieces.

The Theorem: On 25th January 2015, Łukasz Grabowski, András Máthé, and Oleg Pikhurko submitted to arxiv a paper in which they proved that a circle and a square of the same area on the plane are equidecomposable by translations with Lebesgue measurable pieces.

Short context: In 1990, Laczkovich, answering a 1925 question of Tarski, proved that circle and a square of the same area are equidecomposable by translations. The pieces $A_i$ and $B_i$ in Laczkovich’s proof are not Lebesgue measurable, and he left the the problem whether or not the circle can be squared with measurable pieces as an interesting open question. The Theorem resolves this question affirmatively. Of course, there is nothing special about circle and square, and the Theorem holds in all dimensions, and for any pair of bounded sets with non-empty interiors of the same Lebesgue measure subject to technical condition that the boundaries of the sets have dimensions lower that the sets.

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

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Short and long averages of a bounded multiplicative function are effectively related

You need to know: Notation ${\mathbb N}$ for the set of positive integers, coprime integers.

Background: A function $f:{\mathbb N} \to [-1,1]$ is called multiplicative if $f(1)=1$ and $f(ab)=f(a)f(b)$ holds for all coprime $a,b\in {\mathbb N}$ .

The Theorem: On 19th January 2015, Kaisa Matomäki and Maksym Radziwiłł submitted to arxiv a paper in which they proved the existence of absolute constants $B,C$ (one can take $B=20000$ ) such that for any multiplicative function $f:{\mathbb N} \to [-1,1]$ , for any $2\leq h \leq X$ , and any $\delta>0$ , inequality $\left|\frac{1}{h}\sum\limits_{x\leq n \leq x+h}f(n) - \frac{1}{X}\sum\limits_{X\leq n \leq 2X}f(n) \right|\leq \delta + B\frac{\log\log h}{\log h}$ folds for all but at most $CX\left(\frac{(\log h)^{1/3}}{\delta^2h^{\delta/25}} + \frac{1}{\delta^2(\log X)^{1/50}}\right)$ integers $x\in[X,2X]$ .

Short context: Many functions of central importance in number theory (for example, the Möbius function and the Liouville function) are multiplicative. For many applications, it is important to estimate average value $\frac{1}{h}\sum\limits_{x\leq n \leq x+h}f(n)$ of such function in short intervals of length h. The Theorem states that this is approximately equal to the average $\frac{1}{X}\sum\limits_{X\leq n \leq 2X}f(n)$ over “long” interval $[X,2X]$ , which is much easier to estimate. This has a lot of applications, some of them are derived in the same paper.

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

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The largest gap between consecutive primes below X grows at least as log X log_2 X log_4 X/log_3 X

You need to know: Prime numbers.

Background: Let $p_n$ denotes the n-th prime. For $X>2$ , let $G(X)=\sup\limits_{p_n \leq X}(p_{n+1}-p_n)$ be the largest gap between consecutive primes below X. For $X>16$ , let $F(X)=\frac{\log X \log \log X \log \log \log \log X}{\log \log \log X}$ . With notation $\log_r (X)$ for the r-th fold logarithm, this simplifies to $F(X)=\frac{\log X \log_2 X \log_4 X}{\log_3 X}$ .

The Theorem: On 16th December 2014, Kevin Ford, Ben Green, Sergei Konyagin, James Maynard, and Terence Tao submitted to arxiv a paper in which they proved the existence of constant $c>0$ such that $G(X) \geq c F(X)$ for all X.

Short context: How fast does the largest gap $G(X)$ between consecutive primes below X grow with X? It is conjectured that $G(X)$ grows like a constant times $\log^2 X$ , but this is wide open. See here for the best upper bound on $G(x)$ . For the lower bound, the authors proved earlier in 2014 that G(X) grows faster than $\frac{\log X \log_2 X \log_4 X}{(\log_3 X)^2}$ , giving the first substantial improvement since 1938. The Theorem improves this bound by a $\log_3 X$ factor. The authors argue that this bound in the limit of the current techniques, and a radically new approach will be needed to improve beyond this.

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

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A compact, embedded self-similar shrinker in R^3 of genus 0 is a round sphere

You need to know: Euclidean space ${\mathbb R}^3$ , origin $O=(0,0,0)$ , inner product $\langle x,y \rangle$ in ${\mathbb R}^3$ , unit vector in ${\mathbb R}^3$ (the one with $\langle u,u \rangle = 1$ ), round sphere in ${\mathbb R}^3$ of radius r centred at $A \in {\mathbb R}^3$ (set of points on distance r from $A$ ), surface in ${\mathbb R}^3$ , connected surface, compact surface, curvature of a curve.

Background: Let $S$ be a surface embedded into ${\mathbb R}^3$ . For any point $x\in S$ , we can build a unit vector $u(x)$ perpendicular to S, choose any plain P containing $u(x)$ (called a normal plain), and measure the curvature $\kappa(x,P)$ at x of a curve which is the intersection of S and P. Let $k_1(x)$ and $k_2(x)$ be the minimum and maximum values of $\kappa(x,P)$ over all choices of normal plain P. The mean curvature of S at x is $H(x)=\frac{1}{2}(k_1(x)+k_2(x))$ . Surface S is called a self-similar shrinker if it satisfies the equation $H(x) = \frac{1}{2}\langle x,u(x) \rangle$ for every $x\in S$ . We say that surface S has genus $0$ if it is connected but cutting it along any closed curve makes it disconnected.

The Theorem: On 17th November 2014, Simon Brendle submitted to arxiv a paper in which he proved that the only compact, embedded self-similar shrinker in ${\mathbb R}^3$ of genus $0$ is the round sphere of radius 2 centred at the origin.

Short context: The classication of self-similar shrinkers is an important problem with implications for the analysis of singularities. The simplest example of a compact self-similar shrinker in ${\mathbb R}^3$ is the round sphere of radius 2 centred at the origin. The Theorem confirms a well-known conjecture that this is in fact the only example of a compact, embedded self-similar shrinker in ${\mathbb R}^3$ of genus $0$ . It is known that the assumptions that the surface is embedded and has genus $0$ are both necessary: if we remove any of them, we get more examples.

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

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The largest gap between consecutive primes below X grows faster than log X log_2 X log_4 X/(log_3 X)^2

You need to know: Prime numbers.

Background: Let $p_n$ denotes the n-th prime. For $X>2$ , let $G(X)=\sup\limits_{p_n \leq X}(p_{n+1}-p_n)$ be the largest gap between consecutive primes below X. For $X>16$ , let $f(X)=\frac{\log X \log \log X \log \log \log \log X}{(\log \log \log X)^2}$ . With notation $\log_r (X)$ for the r-th fold logarithm, this simplifies to $f(X)=\frac{\log X \log_2 X \log_4 X}{(\log_3 X)^2}$ .

The Theorem: On 20th August 2014, Kevin Ford, Ben Green, Sergei Konyagin, and Terence Tao submitted to arxiv a paper in which they proved that $\lim\limits_{X\to\infty} \frac{G(X)}{f(X)}=\infty$ .

Short context: How fast does the largest gap $G(X)$ between consecutive primes below X grow with X? It is conjectured that $G(X)$ grows like a constant times $\log^2 X$ , but this is wide open. In 1931, Westzynthius proved that $G(X) \geq c\frac{\log X \log_3 X}{\log_4 X}$ for some constant $c>0$ . This was improved to $G(X) \geq c\frac{\log X \log_2 X}{(\log_3 X)^2}$ by Erdős in 1935, and to $G(X) \geq c f(X)$ by Rankin in 1938. This strange-looking bound, surprisingly, standed for almost 80 years, except of improvements by a constant factor. At last, the Theorem proves that $G(X)$ grows faster than $f(X)$ . The same result was proved independently by Maynard. See here for even better lower bound proved in a later work, and here for an upper bound on $G(x)$ .

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

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The Gaussian correlation conjecture is true

You need to know: Euclidean space ${\mathbb R}^n$ , origin in ${\mathbb R}^n$ , convex set in ${\mathbb R}^n$ , centrally symmetric set in ${\mathbb R}^n$ (set symmetric with respect to the origin), integration in ${\mathbb R}^n$ , $n \times n$ matrix, matrix multiplication, determinant $\text{det}(A)$ of a matrix A, inverse $A^{-1}$ of a matrix A with $\text{det}(A)\neq 0$ , symmetric matrix, transpose $A^T$ of matrix A.

Background: A symmetric $n\times n$ matrix $\Sigma$ is called positive definite, if $x^T\Sigma x > 0$ for all non-zero $x \in {\mathbb R}^n$ . Given a positive definite $n\times n$ matrix $\Sigma$ , let $f:{\mathbb R}^n \to {\mathbb R}$ be a function given by $f(x)=(2\pi)^{-n/2}\text{det}(\Sigma)^{-1/2}\exp(-\frac{1}{2}x^T\Sigma^{-1} x)$ , $x\in {\mathbb R}^n$ . For a subset $A\subset{\mathbb R}^n$ , let $\mu(A)=\int_A f(x)dx$ . $\mu$ is called an n-dimensional Gaussian probability measure on ${\mathbb R}^n$ , centred at the origin.

The Theorem: On 5th August 2014, Thomas Royen submitted to arxiv and Far East Journal of Theoretical Statistics a paper in which he proved that inequality $\mu(A \cap B) \geq \mu(A) \cdot \mu(B)$ holds for all convex and centrally symmetric sets $A,B \subset {\mathbb R}^n$ .

Short context: The statement of the Theorem was known as Gaussian correlation conjecture, and was an important conjecture in probability theory. A special case of the conjecture was formulated by Dunnett and Sobel in 1955, the general case was stated in 1972. Despite simple-looking formulation and efforts of many researchers, the conjecture remained open until 2014. The Theorem proves the conjecture, and it is now called Gaussian correlation inequality.

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

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The set of points in a rational polygon P not illuminated by any x in P is finite

You need to know: Polygon, angles measured in radians, angle of incidence, angle of reflection.

Background: Let P be a rational polygon, that is, polygon whose angles measured in radians are rational multiples of $\pi$ . For any point $x \in P$ (called a light source), consider a light ray starting from x and moving inside P, with the usual rule that the angle of incidence equals the angle of reflection. A point $x \in P$ is called illuminated if there is a light ray starting from x which passes through y.

The Theorem: On 10th July 2014, Samuel Lelievre, Thierry Monteil, and Barak Weiss
submitted to arxiv a paper in which they proved that in any rational polygon P, and for any point $x\in P$ there are at most finitely many points $y\in P$ which are not illuminated from light source x.

Short context: In 1969, Klee asked whether any point x in any polygon P illuminates the whole P. In 1995, Tokarsky answered this question negatively by constructing an example in which one point y is not illuminated. The Theorem states that, at least for rational polygons, only finitely many points can remain unilluminated. It is just one out of many corollaries of deep “Magic Wand Theorem” proved in 2013 by Eskin and Mirzakhani.

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

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Fast (1-1/e-epsilon)-approximation for the infuence maximization problem

You need to know: Directed graph, algorithm, approximation algorithm, polynomial-time algorithm, NP-hard problem.

Background: Let $G$ be a directed graph, in which each directed edge $w\to v$ has a weight $b_{vw}\in[0,1]$ . Each vertex v has a threshold $\theta_v$ , and the thresholds $\theta_v$ are chosen independently and uniformly at random from the interval $[0,1]$ . At time $0$ , we select some set A of vertices at mark them as “active”. At times $t=1,2,\dots$ , vertex v become active if $\sum\limits_{w\to v, \, w \text{ active}} b_{vw} \geq \theta_v$ , and if vertex become active at stays active forever. This process ends if no activations occur from round $t$ to $t+1$ . The influence $\sigma(A)$ of a set A of vertices is the expected number of active nodes at the end of the process. The influence maximization problem asks, given graph G, weights $b_{vw}$ , and parameter k, to find a k-vertex set A with maximal $\sigma(A)$ .

The Theorem: On 17th April 2014, David Kempe, Jon Kleinberg, and Éva Tardos submitted to arxiv a paper in which they proved that for any $\epsilon>0$ , there is a polynomial-time algorithm approximating the maximum influence to within a factor of $1-1/e-\epsilon$ , where $e=2.718...$ is the base of the natural logarithm.

Short context: The influence maximization problem has important practical applications. For example, G may model a social network, the “activation process” is when people start to use some product or service because their friends do, and the influence maximization problem asks which set A of individuals should we initially convince to use it, to maximize the “cascade effect”. The problem is NP-hard to solve exactly, but the Theorem provides an efficient approximation algorithm.

Links: The original paper is available here.

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