A set of density one satisfies the local-global conjecture for integral Apollonian packings

You need to know: Greatest common divisor (gcd) of a possibly infinite set of integers, circle in the plane, radius of a circle, curvature of a circle ( $1/r$ where r is the radius). Also, see this previous theorem description for the definition of bounded Apollonian circle packing (ACP).

Background: Let $b(C)$ denote the curvature of a circle C. The ACP is called integer if all circles in it has integer curvatures. For an integer ACP ${\cal P}$ , let ${\cal B}=\{b(C) : C \in {\cal P}\}$ be the set of all curvatures in ${\cal P}$ . The integer ACP ${\cal P}$ is called primitive if $\text{gcd}({\cal B})=1$ . A positive integer m is called admissible for ${\cal P}$ if for any integer $q\geq 1$ , there exists $k \in {\cal B}$ such that $k-m$ is divisible by q. Let $f_{\cal P}(N)$ be the number of admissible integers at most N which does not belong to ${\cal B}$ .

The Theorem: On 20th May 2012, Jean Bourgain and Alex Kontorovich submitted to arxiv a paper in which they proved that for any primitive integer ACP ${\cal P}$ there exist constants $\eta>0$ and $C<\infty$ such that $f_{\cal P}(N) \leq C N^{1-\eta}$ for all $N>0$ .

Short context: Apollonian circle packing is named after Apollonius of Perga, who lived more than 2000 years ago, and is studied by many researchers since that. Important research directions are counting circles with the curvature at most X (see here), as well as studying the set ${\cal B}$ of distinct integers occurring as curvatures. A central conjecture in the area is the the local-global conjecture stating that every sufficiently large admissible integer belongs to ${\cal B}$ . Previously, it was known that a positive percentage of integers satisfy this conjecture. The Theorem implies that the percentage of integers up to N that satisfy it approaches $100\%$ as N grows.

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

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The log-Brunn-Minkowski inequality holds for two convex bodies in the plane

You need to know: Euclidean plane ${\mathbb R}^2$ , origin $O=(0,0)$ in ${\mathbb R}^2$ , scalar product $(x,y)=x_1y_1+x_2y_2$ in ${\mathbb R}^2$ , unit circle $C=\{u \in {\mathbb R}^2: (u,u)=1\}$ , convex body $K \subset {\mathbb R}^2$ (compact convex set with non-empty interior), origin-symmetric convex body, area $S(K)$ of $K \subset {\mathbb R}^2$ .

Background: The support function $h_K:{\mathbb R}^2 \to {\mathbb R}$ of a convex body $K \subset {\mathbb R}^2$ is given by $h_K(x)=\max\limits_{y \in K} (x,y)$ , $\forall x \in {\mathbb R}^2$ . For convex bodies K and L, and $\lambda \in [0,1]$ , the log Minkowski combination, $(1-\lambda)K \oplus \lambda L$ , is the set of all points $x \in {\mathbb R}^2$ such that $(x,u) \leq h_K(u)^{1-\lambda}h_L(u)^\lambda$ for all $u \in C$ .

The Theorem: On 30th April 2012, Karoly Boroczky, Erwin Lutwak, Deane Yang, and Gaoyong Zhang submitted to Advances in Mathematics a paper in which they proved, among other results, that if K and L are origin-symmetric convex bodies in the plane, then inequality $S((1-\lambda)K \oplus \lambda L) \geq S(K)^{1-\lambda}S(L)^\lambda$ holds for all real $\lambda \in [0,1]$ .

Short context: One of several equivalent formulations of the (plane version) of famous Brunn–Minkowski inequality states that $S((1-\lambda)K + \lambda L) \geq S(K)^{1-\lambda}S(L)^\lambda$ , where $(1-\lambda)K + \lambda L=\{x\in {\mathbb R}^2: x=(1-\lambda)a + \lambda b, \, a\in K, \, b\in L\}$ . Because it is known that $(1-\lambda)K \oplus \lambda L \subseteq (1-\lambda)K + \lambda L$ , the Theorem strengthen this inequality. This is the first step towards proving the conjecture that the strengthened inequality (called the log-Brunn-Minkowski inequality) holds in all dimensions. In a later work (in 2019) Yang and Zhang proved it in ${\mathbb R}^3$ .

Links: The original paper is available here.

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The list chromatic number of a graph with average degree d is at least (1+o(1))log_2 d

You need to know: Graph, simple graph (one with no loops and multiple edges), degree of a vertex in the graph, average degree of a graph, small o notation. You also need to know chromatic number of graph and complete bipartite graph $K_{d,d}$ for the context section.

Background: A graph G is called k-choosable if, whenever for each vertex v of G we assign a list $L_v$ of k colours to v, then it is possible to choose a colour for v from the list $L_v$ , so that no two adjacent vertices receive the same colour. The list chromatic number $\chi_l(G)$ is the smallest k such that G is k-choosable.

The Theorem: On 30th April 2012, David Saxton and Andrew Thomason submitted to arxiv a paper in which they proved, among other results, that if G is a simple graph with average degree d, then $\chi_l(G) \geq (1+o(1))\log_2 d$ as $d\to \infty$ .

Short context: List chromatic number $\chi_l(G)$ is a natural version of the ordinary chromatic number $\chi(G)$ of G (which corresponds to the case when all $L_v$ are the same). It is known that $\chi_l(K_{d,d})=(1+o(1))\log_2 d$ , in sharp contrast with $\chi(K_{d,d})=2$ . In fact, Alon proved that $\chi_l(G) \geq (1/2+o(1))\log_2 d$ for graphs G of minimum degree d. The Theorem improves this inequality by a factor of 2 and is the best possible, as example $G=K_{d,d}$ shows. In fact, the Theorem is just one out of many corollaries of the general theorem proved by the authors, see the original paper for details.

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

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Sparse analog of Szemerédi’s theorem is true

You need to know: A (nontrivial) k-term arithmetic progression, notation $[n]$ for $\{1,2,\dots,n\}$ .

Background: For integer $m>0$ and real number $x\geq m$ , denote ${{x}\choose{m}}=\frac{1}{m!}\prod\limits_{i=0}^{m-1}(x-i)$ . By convention, we define ${{x}\choose{m}}=0$ if $x<m$ . Also, by m-subset of $[n]$ we mean a subset which has m elements.

The Theorem: On 29th April 2012, József Balogh, Robert Morris, and Wojciech Samotij submitted to arxiv a paper in which they proved, among other results, that for every real $\beta>0$ and every integer $k\geq 3$ , there exist constants C and $n_0>0$ such that for every integers $n \geq n_0$ and $m\geq Cn^{1-1/(k-1)}$ , there are at most ${{\beta n}\choose{m}}$ m-subsets of $[n]$ that contain no k-term arithmetic progression.

Short context: Famous Szemerédi’s theorem, proved in 1975, states that, for every $\beta>0$ , every subset of $[n]$ of size $m\geq \beta n$ must contain a k-term arithmetic progression, provided that n is sufficiently large. The authors view the Theorem as a sparse analog of this statement. It easily implies the original Szemerédi’s theorem, as well as its sparse random analog described here. In turn, the Theorem itself is just one out of many corollaries of the general powerful theorem in graph theory proved by the authors, see the original paper for details.

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

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There exist finitely generated simple groups that are infinite and amenable

You need to know: Group, finite and infinite groups, isomorphic and nonisomorphic groups, simple group, finitely generated group.

Background: A group G is called amenable if there is a map $\mu$ from subsets of G to $[0,1]$ such that (i) $\mu(G)=1$ , (ii) $\mu(A\cup B)=\mu(A)+\mu(B)$ whenever $A\cap B=\emptyset$ , and (iii) $\mu(gA)=\mu(A)$ for all $g\in G$ , where $gA=\{h\in G \,|\, h=ga, \, a\in A\}$ .

The Theorem: On 10th April 2012, Kate Juschenko and Nicolas Monod submitted to arxiv a paper in which they proved the existence of finitely generated simple groups that are infinite and amenable.

Short context: Amenable groups were introduced by von Neumann in 1929 in relation to Banach–Tarski paradox, see here, and are extensively studied since that. However, before 2012, it was an open question if there exists any finitely generated simple group that is infinite and amenable. The Theorem proves that such groups exist. Moreover, Juschenko and Monod proved that there are infinitely many (in fact uncountably many) nonisomorphic such groups. See here for even more general result proved in a later work.

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

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Every embedded minimal torus in S^3 is congruent to the Clifford torus

You need to know: Euclidean space ${\mathbb R}^4$ , unit sphere ${\mathbb S}^3 \subset {\mathbb R}^4$ , surface embedded in ${\mathbb S}^3$ , congruent surfaces, mean curvature of a surface (see here), torus.

Background: Let $\Sigma$ be a two-dimensional surface embedded in ${\mathbb S}^3$ . We say that $\Sigma$ is a minimal surface if the mean curvature of $\Sigma$ vanishes identically. The Clifford torus is defined by $\{(x_1, x_2, x_3, x_4)\in {\mathbb S}^3 : x_1^2+x_2^2 = x_3^2+x_4^2 = \frac{1}{2}\}$ .

The Theorem: On 29th March 2012, Simon Brendle submitted to arxiv a paper in which he proved that every embedded minimal torus in ${\mathbb S}^3$ is congruent to the Clifford torus.

Short context: Minimal surfaces are among the most studied objects in geometry and topology. Of particular interest are minimal surfaces embedded in Euclidean space ${\mathbb R}^3$ or in 3-sphere ${\mathbb S}^3$ . The simplest examples of minimal surfaces in ${\mathbb S}^3$ are equator and the Clifford torus. In 1970, Lawson conjectured that the Clifford torus is in fact the only embedded minimal torus in ${\mathbb S}^3$ . The Theorem confirms this conjecture.

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

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Integral of the square of the mean curvature of a torus in R^3 is at least 2 pi^2

You need to know: Curvature of a curve, closed surface S immersed in ${\mathbb R}^3$ , integration over S, torus.

Background: Let $S$ be a smooth immersed surface in ${\mathbb R}^3$ . For any point $x\in S$ , we can build a vector u perpendicular to S, choose any plain P containing u (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 $H(x)=\frac{1}{2}(k_1(x)+k_2(x))$ . The Willmore energy of S is $W(S)=\int_S H^2(x) dx$ .

The Theorem: On 27th February 2012, Fernando Marques and André Neves submitted to arxiv a paper in which they proved that $W(T)\geq 2\pi^2$ for every torus T immersed in ${\mathbb R}^3$ .

Short context: The Willmore energy is a way to measure the “total curvature” of a surface. It has nice mathematical properties and appears naturally in some physical contexts. It is known that $W(S)\geq 4\pi$ for every closed surface S, with equality if S is a sphere. In 1965, Willmore conjectured that stronger lower bound $W(T)\geq 2\pi^2$ holds for every torus T. The Theorem confirms this conjecture. The bound is the best possible, because there is a torus for which the equality holds.

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

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Every position of Rubik’s Cube can be solved in 20 moves or less

You need to know: Rubik’s Cube

Background: A move in the Rubik’s Cube is defined as any twist of any face. By solution of a position of Rubik’s Cube we mean the sequence of moves after which each face of Rubik’s Cube has only one colour.

The Theorem: On 24th February 2012, Tomas Rokicki, Herbert Kociemba, Morley Davidson, and John Dethridge submitted to the SIAM Journal on Discrete Mathematics a paper in which they gave an exposition of their computational proof that every starting position of Rubik’s Cube can be solved in 20 moves or less.

Short context: Rubik’s Cube, invented by Ernő Rubik in 1974, is widely considered to be the world’s best-selling toy. The least integer n such that every position can be solved in at most n moves is known as God’s number. The problem of determining the God’s number (let us denote it $g$ ) has been of interest ever since the Rubik’s Cube is invented. A simple counting argument proves that $g\geq 18$ . The first upper bound $g\leq 52$ was proved by Thistlethwaite in 1981. Then many researches proved better and better lower and upper bounds. The Theorem culminates this line of research and determines the God’s number exactly: $g=20$ .

Links: The original paper is available here.

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Every measure preserving word is primitive

You need to know: Group, finite group, free group, free generating set in a free group.

Background: Let $F_k$ be the free group with k generators $x_1, \dots x_k$ . Any element $w\in F_k$ (which we call word) can be written as $w=\prod\limits_{j=1}^r x_{i_j}^{\epsilon_j}$ for some $1 \leq i_1 < \dots < i_r \leq k$ and $\epsilon_j = \pm 1$ , $j=1,\dots,r$ . For every group G, w induces a word map from $w:G^k \to G$ given by $w(g_1, \dots, g_k)=\prod\limits_{j=1}^r g_{i_j}^{\epsilon_j}$ . If G is a finite group and $g\in G$ , let $N_G^w(g)$ be the number of $(g_1, \dots, g_k)\in G^k$ such that $w(g_1, \dots, g_k)=g$ . The word w is called measure preserving with respect to G if $N_G^w(g)=N_G^w(h)$ for every $g,h \in G$ . We say that w is measure preserving if it is measure preserving with respect to any every finite group G. The word w is called primitive if it belongs to some basis (free generating set) of $F_k$ .

The Theorem: On 15th February 2012, Doron Puder and Ori Parzanchevski submitted to arxiv a paper in which they proved that every measure preserving word is primitive.

Short context: Word maps in groups are the subject of intensive study with lots of nice recent results, see, for example, here. It is not difficult to check that every primitive word is measure preserving, and several authors have conjectured that the converse is also true. In a paper submitted in 2011, Puder proved this conjecture for words in $F_2$ . The Theorem confirms the conjecture in full.

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

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The clique density conjecture of Lovász and Simonovits is true

You need to know: Graph, clique in a graph, size of a clique, factorial $n!=1\cdot 2 \cdot \dots \cdot n$ with convention that $0!=1$ , notation ${n}\choose{k}$ for $\frac{n!}{k!(n-k)!}$ , with convention that ${{n}\choose{k}}=0$ if $n<k$ .

Background: For $\gamma\in[0,1/2)$ , let $s_{\gamma}$ be the unique positive integer such that $\gamma \in \left[\frac{s_{\gamma}-1}{2s_{\gamma}}, \frac{s_{\gamma}}{2(s_{\gamma}+1)}\right)$ , and let $\alpha_{\gamma}\in(0,1/\gamma]$ be the real number such that $\gamma=\frac{s_{\gamma}}{2(s_{\gamma}+1)}(1-\alpha_{\gamma}^2)$ .

The Theorem: On 19th January 2012, Christian Reiher submitted to arxiv and the Annals of Mathematics a paper in which he proved that, for every integer $r\geq 3$ and real $\gamma\in[0,1/2)$ , every graph on n vertices with at least $\gamma n^2$ edges contains at least $\frac{1}{(s_\gamma+1)^r}{{s_\gamma+1}\choose{r}}(1+\alpha_\gamma)^{r-1}(1-(r-1)\alpha_\gamma)\cdot n^r$ cliques of size r.

Short context: Famous Turán’s theorem, proved in 1941, states that for any integer $r\geq 3$ , every graph on n vertices with more than $\frac{r-2}{2(r-1)}n^2$ edges contains a clique of size r. A natural question arises to estimate the minimum number $g_r(n,e)$ of r-cliques a graph with n vertices and e edges must contain. The Theorem proves a lower bound for $g_r(n,e)$ , predicted by Lovász and Simonovits in 1983, which was known as the clique density conjecture. Together with known upper bound, the Theorem allows to exactly determine, for any $\lambda\in[0,1]$ , the limit $g_r(\lambda)=\lim\limits_{n\to\infty}\frac{g_r(n,e(n))}{{{n}\choose{r}}}$ , whenever $\lim\limits_{n\to\infty} \frac{e(n)}{{{n}\choose{2}}}=\lambda$ . See here for further progress in computing $g_3(n,e)$ exactly.