The cap set conjecture is true: the size of any cap set in F_3^n is o(2.756^n)

You need to know: Addition modulo 3, vectors, notation $|A|$ for the number of elements in finite set A, small o notation.

Background: Let ${\mathbb F}_3$ be the set $\{0,1,2\}$ with addition defined modulo 3. Let ${\mathbb F}_3^n$ be the set of n-component vectors $x=(x_1, \dots, x_n)$ with each $x_i \in {\mathbb F}_3$ , and with addition defined component-wise. We say that three different points $x,y,z\in {\mathbb F}_3^n$ form a line if $x+y=z+z$ . A set $A \subseteq {\mathbb F}_3^n$ is called a cap set if it contains no lines.

The Theorem: On 30th May 2016, Jordan Ellenberg and Dion Gijswijt submitted to arxiv a paper in which they proved that if $A \subseteq {\mathbb F}_3^N$ is a cap set, then $|A|=o(2.756^n)$ .

Short context: What can the maximal size of a cap set in ${\mathbb F}_3^n$ ? This problem is interesting in its own, but also studied because of hope that methods to solve it may be useful for the similar problem of finding dense sets of integers without 3-term arithmetic progressions, see here. A cap set conjecture predicted the existence of constant $c<3$ such that $|A|<c^n$ for every cap set $A \subseteq {\mathbb F}_3^n$ . The Theorem confirms this conjecture, building on an earlier similar result for subsets of ${\mathbb Z}_4^n$ . Before 2016, the best upper bound was $|A|\leq C\frac{3^n}{n^{1+\epsilon}}$ , see here.

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

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The Nadirashvili’s conjecture on the volume of the zero sets of harmonic functions is true

You need to know: Eucledean space ${\mathbb R}^n$ , origin $0$ in ${\mathbb R}^n$ , norm $||x||=\sqrt{\sum_{i=1}^nx_i^2}$ of $x=(x_1,\dots,x_n)\in{\mathbb R}^n$ , unit ball $B_n(0,1) =\{ x \in {\mathbb R}^n\,:\,||x||<1\}$ , open subset V of ${\mathbb R}^n$ , twice continuously differentiable function f on V, notation $\frac{\partial^2 f}{\partial x_i^2}$ for second-order partial derivatives.

Background: Let V be an open subset of ${\mathbb R}^n$ . A function $f: V \to {\mathbb R}$ is called harmonic if $\frac{\partial^2 f}{\partial x_1^2} + \dots + \frac{\partial^2 f}{\partial x_n^2} = 0$ for every $x\in V$ . A diameter of set $U\subset {\mathbb R}^n$ is $\text{diam} U=\sup\limits_{x,y \in U}||x-y||$ . For $\delta>0$ , a $\delta$ -cover of set $S \subset {\mathbb R}^n$ is a sequence of sets $U_1, U_2, \dots, U_i, \dots$ of diameters $\text{diam}U_i\leq \delta$ for all i such that $S \subset \bigcup\limits_{i=1}^\infty U_i$ . For $d\geq 0$ , let $H_\delta^d(S) = \inf \sum\limits_{i=1}^{\infty}(\text{diam}U_i)^d$ , where the infimum is taken over all $\delta$ -covers of S. The number $H^d(S)=\lim\limits_{\delta\to 0} H_\delta^d(S)$ is called the d-dimensional Hausdorff measure of S.

The Theorem: On 9th May 2016, Alexander Logunov submitted to arxiv a paper in which he proved that for every dimension $n\geq 3$ , there exists a constant $c=c(n)>0$ , depending only on n, such that inequality $H^{n-1}(\{f = 0\} \cap B(0,1)) \geq c$ holds for every harmonic function $f: B(0,1)\to {\mathbb R}$ such that $f(0)=0$ .

Short context: Harmonic functions are studied in many areas of mathematics, such as mathematical physics and the theory of stochastic processes. The Hausdorff measure $H^{n-1}$ is (up to a constant factor) just the usual $n-1$ -dimensional volume, e.g. $H^2$ is the area. The Theorem confirms a conjecture of Nadirashvili made in 1997. In particular, it implies that the zero set of any non-constant harmonic function $h:{\mathbb R}^3 \to {\mathbb R}$ has an infinite area Also, the Theorem is an important step towards proving a more general conjecture of Yau, which predicts a similar result on n-dimensional curved spaces (called “ $C^\infty$ -smooth Riemannian manifolds”) in place of ${\mathbb R}^n$ .

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

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A properly embedded genus-g minimal surface with finite topology have at most C_g ends

You need to know: Surface, orientable surface, connected surface, boundary of a surface, area of a surface, compact surface, homeomorphic surfaces.

Background: 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 $M \subset {\mathbb R}^3$ is called properly embedded in ${\mathbb R}^3$ , if it has no boundary, no self-intersections, and its intersection with any compact subset of ${\mathbb R}^3$ is compact. The genus of a connected orientable surface is the maximum number of cuttings along non-intersecting closed simple curves without making it disconnected. A surface is said to have finite topology if it is homeomorphic to a compact surface with a finite number of points removed. The number of points removed is called the number of ends of a surface.

The Theorem: On 9th May 2016, William Meeks III, Joaquin Perez, and Antonio Ros submitted to arxiv and Acta Mathematica a paper in which they proved that for every positive integer g, there exists a constant $C_g$ such that any properly embedded minimal surface in ${\mathbb R}^3$ with genus g and finite topology has at most $C_g$ ends.

Short context: Minimal surfaces remain an active area of research since 19th century, and this research area has a golden age at the beginning of the 21st century, with a large number of impressive results, see, for example, here, here, and here. One important conjecture of Hoffman and Meeks states that if a properly embedded connected minimal surface of genus g has a finite number k ends, then $k\leq g+2$ . However, before 2016, it was not known if there exists any upper bound on k depending only on g. This is what the Theorem establishes.

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

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Properly embedded minimal surfaces with one end and genus g exist for all g

You need to know: Surface, orientable surface, connected surface, boundary of a surface, area of a surface, compact surface, homeomorphic surfaces.

The Theorem: On 20th June 2013, David Hoffman, Martin Traizet, and Brian White submitted to Acta Mathematica a paper in which they proved that for every positive integer g, there exists a connected, properly embedded minimal surface in ${\mathbb R}^3$ with one end and genus g.

Short context: Meeks III and Rosenberg proved that the only examples of properly embedded simply connected minimal surfaces in ${\mathbb R}^3$ are plane and helicoid. Later, Weber, Hoffman, and Wolf constructed a properly embedded minimal surface in ${\mathbb R}^3$ with one end and genus $g=1$ . The Theorem proves that such surface exists for every genus g.

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

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Progression-free sets in Z_4^n are exponentially small

You need to know: Addition modulo 4, vectors, notation $|A|$ for the number of elements in finite set A, notation $\log_2 x$ for the base-2 logarithm of x.

Background: Let ${\mathbb Z}_4$ be the set $\{0,1,2,3\}$ with addition defined modulo 4. Let ${\mathbb Z}_4^N$ be the set of N-component vectors $x=(x_1, \dots, x_N)$ with each $x_i \in {\mathbb Z}_4$ , and with addition defined component-wise. We say that three different points $x,y,z\in {\mathbb Z}_4^N$ form a 3-term arithmetic progression if $x+y=z+z$ . A set $A \subseteq {\mathbb Z}_4^N$ is called progression-free if it contains no 3-term arithmetic progressions.

For $x\in(0,1)$ , let $H(x)=-x \log_2 x - (1-x) \log_2(1-x)$ . Let $\gamma = \max\limits_{0<\epsilon<0.25}\left(\frac{1}{2}(H(0.5-\epsilon)+H(2\epsilon))\right)$ . Note that $\gamma \approx 0.926$ . In particular, $\gamma<1$ .

The Theorem: On 5th May 2016, Ernie Croot, Vsevolod Lev, and Peter Pach submitted to arxiv and the Annals of Mathematics a paper in which they proved that if $n\geq 1$ and $A\subseteq {\mathbb Z}_4^n$ is progression-free, then $|A| \leq 4^{\gamma n}$ .

Short context: What can be the maximal size of a progression-free set in ${\mathbb Z}_4^N$ ? This problem is interesting in its own, and also closely related to the problems of bounding progression-free sets in ${\mathbb Z}_3^N$ (known as the cap set conjecture, see here) and in the sets of integers (see here). The Theorem proves, for the first time, that progression-free sets in ${\mathbb Z}_4^n$ are exponentially small comparing to $|{\mathbb Z}_4^n|=4^n$ . In just few weeks after the Theorem was published online, its proof was extended to solve the cap set conjecture.

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

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The family {x, xy, x+y} is Ramsey

You need to know: Set ${\mathbb N}$ of positive integers

Background: By finite colouring of ${\mathbb N}$ we mean partition ${\mathbb N}=C_1\cup\dots\cup C_r$ into a finite number r of disjoint subsets $C_i$ . We say that set $S \subset {\mathbb N}$ is monochromatic if $S \subset C_i$ for some i.

The Theorem: On 4th May 2016, Joel Moreira submitted to the Annals of Mathematics a paper in which he proved that for any finite colouring of ${\mathbb N}$ there exist (infinitely many) pairs $x,y \in {\mathbb N}$ such that the set $\{x, xy, x + y\}$ is monochromatic.

Short context: A set of k polynomials $P_1,\dots, P_k$ in s variables $x_1, \dots, x_s$ with integer coefficients is called a Ramsey family if for any finite colouring of ${\mathbb N}$ there exist $x_1, \dots, x_s \in {\mathbb N}$ such that the set $P_1(x_1, \dots, x_s),\dots, P_k(x_1, \dots, x_s)$ is monochromatic. A classical problem asks to develop necessary and sufficient conditions on the polynomials $P_1,\dots, P_k$ that guarantee that they form a Ramsey family. However, before 2016, it was not even known if a simple family $\{xy, x + y\}$ is Ramsey. The Theorem establishes this, even for a lager family $\{x, xy, x + y\}$ . In fact, the authors developed a general methodology and established that many other families are Ramsey as well. See here and here for related recent results.

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

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The boolean Pythagorean Triples problem has a positive answer

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

Background: A Pythagorean triple is a set of three positive integers a, b, and c, such that $a^2+b^2=c^2$ .

The Theorem: On 3rd May 2016, Marijn Heule, Oliver Kullmann, and Victor Marek submitted to arxiv a paper in which they proved the following result. The set $\{1,\dots,7824\}$ can be partitioned into two parts, such that no part contains a Pythagorean triple, while this is impossible for $\{1,\dots,7825\}$ .

Short context: The boolean Pythagorean Triples problem has been a longstanding open problem in Ramsey Theory. It asks whether in every partition of the set ${\mathbb N}$ of positive integers into two parts, one can always find a Pythagorean triple in one of the parts. There was a progress in the variants of the problem (see here and here), but the original problem remained open. The Theorem provides a positive answer. In fact, it guarantees that one can find a Pythagorean triple in every partition of the set of first 7825 positive integers, and that 7825 is the smallest integer with this property. The computer-assisted proof is almost 200 terabytes in size.

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

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Upper bound 2^(n+o(n)) holds in the Erdős-Szekeres “Happy Ending” conjecture

You need to know: Elementary geometry, convex n-gon, notation $\log_2 n$ for logarithm of n with base 2, $o(n)$ notation.

Background: We say that m points on the plane are in general position if no three of them are on a line. For every integer $n\geq 3$ , let $\text{ES}(n)$ be the minimal integer m, such that any set of m points in the plane in general position contains n points which are the vertices of a convex n-gon.

The Theorem: On 29th April 2016, Andrew Suk submitted to arxiv a paper in which he proved the existence of absolute constant $n_0$ such that $\text{ES}(n)\leq 2^{n+6n^{2/3}\log_2 n}$ for all $n\geq n_0$ .

Short context: In 1935, Erdős and Szekeres proved that $ES(n)$ is finite for $n\geq 3$ . In fact, they proved an explicit upper bound for $ES(n)$ growing as $4^{n+o(n)}$ . In 1960, they proved the lower bound $ES(n)>2^{n-2}$ and conjectured that in fact $ES(n)=2^{n-2}+1$ . This became known as the Erdős-Szekeres conjecture, or the “Happy Ending” conjecture. It is known to hold for $n\leq 6$ but is open for all $n>6$ . The Theorem dramatically improves the upper bound for $ES(n)$ from $4^{n+o(n)}$ to $2^{n+o(n)}$ , thus establishing the conjecture up to the $o(n)$ term in the exponent.

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

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There exists a discrete bounded envy-free cake cutting protocol for any number of agents

You need to know: Algorithm.

Background: Let us call interval $[0,1]$ a cake, and any finite union of subintervals of $[0,1]$ – a piece of cake. Let ${\cal X}$ be the set of all pieces of cake. Let $N=\{1,\dots,n\}$ be the set of agents, each agent i having its own valuation function $V_i:{\cal X}\to[0,\infty)$ such that (i) $V_i(X\cup X')=V_i(X)+V_i(X')$ for all disjoint $X, X' \in {\cal X}$ , and (ii) for every $X\in {\cal X}$ and $0\leq \lambda\leq 1$ , there exists $X'\subset X$ in ${\cal X}$ with $V_i(X') = \lambda V_i(X)$ . A cake allocation is a partition $[0,1]=\bigcup\limits_{i=1}^n X_i$ into disjoint pieces $X_i\in{\cal X}$ . We say that agent i is envious if $V_i(X_i)<V_i(X_j)$ for some j. A discrete protocol is an algorithm which returns an allocation after a sequence of queries which either (1) for given $x \in [0,1]$ and $r \geq 0$ , ask agent i to return a point $y\in[0,1]$ such that $V_i([x,y])=r$ (CUT query), or (2) for given $x,y\in[0,1]$ , ask agent i to return $V_i([x,y])$ (EVALUATE query). A protocol is called envy-free if no agent is envious if he/she follows the protocol truthfully. A protocol is bounded if the number of queries required to return a solution is bounded by a constant $C=C(n)$ depending only on n but not on $V_i$ .

The Theorem: On 13th April 2016, Haris Aziz and Simon Mackenzie submitted to arxiv a paper in which they proved the existence of a discrete bounded envy-free cake cutting protocol for any number of agents.

Short context: Cake cutting is a metaphor for the allocation of a heterogeneous divisible good among multiple agents with possibly different preferences over different parts of the cake. The existence of a discrete and bounded envy-free cake cutting protocol was the central open problem in this area. Such protocol is easy to $n\leq 3$ . In 2015, Aziz and Mackenzie solved the problem for $n=4$ . The Theorem solves it for all n.

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

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For any digit d, there are infinitely many primes which do not have d in their decimal expansion

You need to know: Prime numbers

Background: Any positive integer $n$ can be written in a unique way as $n=\sum\limits_{k=1}^{m}a_k10^{m-k}$ , where $m\geq 0$ is an integer, and $a_1, \dots, a_m$ are integers between $0$ and $9$ . This is called decimal expansion of n with digits $a_1, \dots, a_m$ , and this is the usual way we write integers.

The Theorem: On 4th April 2016, James Maynard submitted to arxiv a paper in which he proved that given any digit $d\in\{0,1,\dots,9\}$ , there are exist infinitely many primes p which do not have the digit d in their decimal expansion.

Short context: For $d\in\{0,1,\dots,9\}$ , let ${\cal A}_d$ be the set of positive integers which do not have the digit d in their decimal expansion. For $x \approx 10^k$ , there are about $9^k \approx x^{0.954}$ integers less than x in ${\cal A}_d$ . Sets of integers with at most $x^c$ integers up to x for some $c<1$ are called sparse. Usually, it is hard to prove that there are infinitely many primes in sparse sets. The Theorem achieves this for sets ${\cal A}_d$ , $d=0,1,\dots,9$ . See here for a result about infinitely many primes in (somewhat similar but not sparse) sets of integers with even/odd sum of digits.