The connective constant of the honeycomb lattice is equal to (2+2^0.5)^0.5

You need to know: Limits.

Background: The honeycomb lattice (or the hexagonal lattice) is the partition of the plane into same-size regular hexagons. Denote by $c_n$ the number of n-step self-avoiding (that is, visiting every vertex at most once) walks on the honeycomb lattice H started from some fixed vertex. It is known that there exists $\mu \in (0,+\infty)$ such that $\mu=\lim\limits_{n\to\infty} \sqrt[n]{c_n}$ . This constant $\mu$ is called the connective constant of the honeycomb lattice.

The Theorem: On 4th July 2010, Hugo Duminil-Copin and Stanislav Smirnov submitted to arxiv a paper in which they proved that the connective constant of the honeycomb lattice is $\mu=\sqrt{2+\sqrt{2}}$ .

Short context: Self-avoiding walks on a lattice were proposed by a famous chemist Flory in 1953 as a model for spatial position of polymer chains. In 1982, Nienhuis, using non-rigorous methods from theoretical physics, predicted that the connective constant of the honeycomb lattice is equal to $\sqrt{2+\sqrt{2}}$ . The Theorem confirms this prediction.

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

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A positive proportion of elliptic curves over Q ordered by height have rank 0

You need to know: For integers a,b, notation $a|b$ if b is divisible by a, and $a\nmid b$ if not, notation ${\mathbb Q}$ for the set of rational numbers, elliptic curve over ${\mathbb Q}$ , rank $\text{rank}(E)$ of elliptic curve E, limit superior $\limsup$ , limit inferior $\liminf$ , notation $|S|$ for the number of elements in any finite set S.

Background: Any elliptic curve E over ${\mathbb Q}$ can be written in the form $y^2 = x^3+ax+b$ , where $a,b$ are integers such that if $p^4|a$ for some prime p, then $p^6\nmid b$ . The height of E is $\max\{4|a^3|, 27b^2\}$ . For any h>0, let $S(h)$ be the set of elliptic curves of height at most h. Also, let $R_0$ be the set of all elliptic curves having rank $0$ .

The Theorem: On 1st July 2010 Manjul Bhargava and Arul Shankar submitted to arxiv a paper in which they proved that (i) $\limsup\limits_{h\to\infty}\left(\frac{1}{|S(h)|}\sum\limits_{E\in S(h)}\text{rank}(E)\right) \leq \frac{7}{6}$ , and (ii) $\liminf\limits_{h\to\infty}\frac{|S(h)\cap R_0|}{|S(h)|} >0$ .

Short context: Quantity $\frac{1}{|S(h)|}\sum\limits_{E\in S(h)}\text{rank}(E)$ is the average rank of elliptic curves of height at most h. In an earlier work, Bhargava and Shankar proved for the first time that the average rank is bounded from above by some constant C. Moreover, they show that one can take $C=1.5$ . Part (i) of the Theorem improves this to $C=\frac{7}{6}<1.17$ . Part (ii) of the Theorem states that a positive proportion of elliptic curves have rank $0$ .

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

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For most cubic forms F, the solution set of F(x,y)=z^l, gcd(x,y)=1, l>=4 is finite

You need to know: Notation $\text{gcd}(x,y)$ for the greatest common divisor of integers x and y.

Background: Let $F(x,y)=ax^3+bx^2y+cxy^2+dy^3$ be a binary cubic form with integer coefficients. F is called irreducible if it cannot be written as $(a'x+b'y)(c'x^2+d'xy+e'y^2)$ for some integers $a',b',c',d',e'$ . Number $\Delta_F=18abcd-4b^3d+b^2c^2-4ac^3-27a^2d^2$ is called the discriminant of F. Let $S_F$ be the set of primes dividing $2\Delta_F$ , and let $U_F$ be the set of nonzero integers u with the property that if u is divisible by some prime p then $p\in S_F$ . Let ${\cal F}$ be the set of binary cubic forms F such that equation $F(x,y) \in U_F$ has no solutions in integers x and y.

The Theorem: On 16th June 2010, Michael Bennett and Sander Dahmen submitted to the Annals of Mathematics a paper in which they proved that, for any $F \in {\cal F}$ , equation $F(x,y) = z^l$ has at most finitely many solutions in integers $(x,y,z,l)$ , such that $\text{gcd}(x,y)=1$ and $l\geq 4$ .

Short context: A special case of the 1995 theorem of Darmon and Granville states that, for any irreducible binary cubic form F and for any fixed $l\geq 4$ , equation $F(x,y) = z^l$ has at most finitely many solutions in co-prime integers $(x,y,z)$ . The Theorem states that, for forms $F\in{\cal F}$ , this equation has at most finitely many solutions even if $l$ is also a variable. For example, the authors show that $F(x,y)=3x^3-ax^2y-(a+9)xy^2-3y^3$ belong to ${\cal F}$ for infinitely many a, which provides an infinite family of forms for which the Theorem is applicable. Moreover, the authors provides a heuristic argument which suggests that “almost all” binary cubic forms belong to ${\cal F}$ .

Links: The original paper is available here.

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The Hirsch Conjecture is false

You need to know: d-dimensional polytope, its vertices, edges, and facets.

Background: A (combinatorial) diameter of a polytope is the maximum number of steps needed to go from one vertex to another, where a step consists in traversing an edge.

The Theorem: On 14th June 2010, Francisco Santos submitted to arxiv a paper in which he proved the existence of a 43-dimensional polytope with 86 facets and diameter at least 44.

Short context: In 1957, Hirsch conjectured that d-dimensional polytope with n facets cannot have diameter greater than $n-d$ . The conjecture has applications to the complexity of the simplex method for linear programming. It was well-believed and proved in some special cases. Because $44>86-43$ , the Theorem provides a counterexample to this conjecture.

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

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The average rank of elliptic curves over Q ordered by height is at most 1.5

You need to know: For integers a,b, notation $a|b$ if b is divisible by a, and $a\nmid b$ if not, notation ${\mathbb Q}$ for the set of rational numbers, elliptic curve over ${\mathbb Q}$ , rank $\text{rank}(E)$ of elliptic curve E, limit superior $\limsup$ , notation $|S|$ for the number of elements in any finite set S.

The Theorem: On 4th June 2010 Manjul Bhargava and Arul Shankar submitted to arxiv a paper in which they proved that $\limsup\limits_{h\to\infty}\left(\frac{1}{|S(h)|}\sum\limits_{E\in S(h)}\text{rank}(E)\right) \leq 1.5$ .

Short context: Quantity $\frac{1}{|S(h)|}\sum\limits_{E\in S(h)}\text{rank}(E)$ is the average rank of elliptic curves of height at most h, and the Theorem states that average rank of all elliptic curves over Q ordered by height is at most 1.5. It is an old conjecture that 50% of all elliptic curves over Q have rank $0$ and 50% have rank 1, which would imply that the average rank is 0.5. However, before 2010, no-one could prove that the average rank is bounded from above by any finite constant whatsoever.

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

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The Hoeffding inequality holds for semidefinitely upper bounded matrices

You need to know: Matrix, self-adjoint matrix, eigenvalues $\lambda_1, \dots, \lambda_d$ of $d \times d$ matrix, notation $||A||$ for the norm $||A||=\max\limits_{i}|\lambda_i|$ of self-adjoint matrix A, positive semidefinite matrix (one with all $\lambda_i \geq 0$ ), notation $A \preceq B$ if matrix $B-A$ is positive semidefinite, probability ${\mathbb P}$ , independence, expectation ${\mathbb E}$ .

Background: Consider a finite sequence $A_1, \dots, A_n$ of fixed self-adjoint $d \times d$ matrices. Denote $\sigma^2 = ||\sum\limits_{k=1}^n A_k^2||$ . Let $X_1, \dots, X_n$ be a sequence of independent, random, self-adjoint $d \times d$ matrices satisfying ${\mathbb E}[X_k]=0$ and $X_k \preceq A_k$ almost surely.

The Theorem: On 25th April 2010, Joel Tropp submitted to arxiv a paper in which he proved that, for $X_1, \dots, X_n$ as above, and for all $t \geq 0$ , ${\mathbb P}\left(\lambda_{\text{max}}\left(\sum\limits_{k=1}^n X_k\right)\geq t\right) \leq d \cdot e^{-t^2/8\sigma^2}$ .

Short context: There are many classical inequalities which allows to bound the probability that the sum of independent random variables exceeds some threshold. For many applications, it is important to have a similar result for sum of matrices, where we bound the size of maximal eigenvalue $\lambda_{\text{max}}$ of the sum. The Theorem provides a matrix version of the classical Hoeffding inequality. In the same paper, many other classical inequalities are extended to the matrix setting as well.

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

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For any connected graph, the blanket and cover times are within an O(1) factor

You need to know: Graph, finite graph, connected graph, notation $G=(V,E)$ for graph with vertex set V and edge set E, notation $|E|$ for the number of edges in G, degree $\text{deg}(v)$ of vertex $v\in V$ , probability, selection uniformly at random, expectation, $O(1)$ notation.

Background: Let $G=(V,E)$ be a connected graph. A simple random walk on G is the process which starts at some vertex v at time $t=0$ , and then at times $t=1,2,\dots$ moves to an adjacent vertex selected uniformly at random. Let $\tau_\text{cov}$ be the first time at which every vertex of G has been visited, and let ${\mathbb E}_v[\tau_\text{cov}]$ be the expectation of $\tau_\text{cov}$ (which depend on the initial vertex v). Number $t_\text{cov}(G) = \max\limits_{v \in V}{\mathbb E}_v[\tau_\text{cov}]$ is called the cover time of G.

For any $v\in V$ , let $N_v(t)$ be a (random) number of times the random walk has visited v up to time t. For $\delta\in(0,1)$ , let $\tau_{\text{bl}}(\delta)$ be the first time $t\geq 1$ at which $N_v(t) \geq \delta t \frac{\text{deg}(v)}{2|E|}$ holds for all $v\in V$ . Number $t_\text{bl}(G,\delta) = \max\limits_{v \in V}{\mathbb E}_v[\tau_{\text{bl}}(\delta)]$ is called the $\delta$ –blanket time of G.

The Theorem: On 25th April 2010, Jian Ding, James Lee, and Yuval Peres submitted to arxiv a paper in which they proved that for every $\delta\in(0,1)$ , there exists a $C=C(\delta)$ such that for every connected graph G, one has $t_\text{bl}(G,\delta) \leq C t_\text{cov}(G)$ .

Short context: It is known that as $t\to\infty$ , the proportion of time a random walk spend at any vertex v converges to $\frac{\text{deg}(v)}{2|E|}$ . Hence, $\tau_{\text{bl}}(\delta)$ is the first time at which all nodes have been visited at least a $\delta$ fraction as much as we expect. Because $t_\text{cov}(G) \leq t_\text{bl}(G,\delta)$ by definition, the Theorem states that the blanket and cover times are within an O(1) factor. It confirms conjecture of Winkler and Zuckerman made in 1996.

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

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w_1(G)w_2(G)=G for any nontrivial group words w_1,w_2 and any large finite non-abelian simple group G

You need to know: Group, identity element, finite group, notation $|G|$ for the number of elements in a finite group G, abelian and non-abelian groups, simple group, free group.

Background: Let $w = w(x_1,\dots,x_d)$ be a non-trivial group word, that is, a non-identity element of the free group $F_d$ on $x_1,\dots,x_d$ . For a group G, denote $w(G)$ the set of all elements $g \in G$ which can be obtained by substitution of some $g_1, g_2, \dots, g_d \in G$ into w instead of $x_1, x_2, \dots, x_d$ , respectively, and performing the group operation. For subsets $A,B$ of group G, let $A\cdot B=\{g \in G: g=a\cdot b, \, a\in A, \, b\in B\}$ . Denote $A^2=A \cdot A$ .

The Theorem: On 10th February 2010, Michael Larsen, Aner Shalev, and Pham Tiep submitted to the Annals of Mathematics a paper in which they proved that for each pair of non-trivial words $w_1$ , $w_2$ there exists $N =N(w_1, w_2)$ such that for every finite non-abelian simple group G with $|G|\geq N$ we have $w_1(G)\cdot w_2(G) = G$ .

Short context: In 2001, Liebeck and Shalev deduced from this theorem that for any non-trivial group word w, is there a constant $c=c(w)$ such that $w(G)^c=G$ for every large finite simple group G. In 2006, Shalev proved that this holds with $c=3$ , independently of w. In 2007, Larsen and Shalev proved that, for some groups (called alternating groups), one can even take $c=2$ . The Theorem states that one can in fact take $c=2$ for all w and all large G. This is clearly the best possible in general, but see here and here for the proof that $c=1$ works for some specific words w.

Links: The original paper is available here.

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The positive density conjecture for integer Apollonian circle packings is true

You need to know: Circles in the plane, radius of a circle, tangent circles, point of tangency, curvature of a circle ( $1/r$ where r is the radius).

Background: A set of four mutually tangent circles in the plane with distinct points of tangency is called a Descartes configuration. Given a Descartes configuration, one
can construct four new circles, each of which is tangent to three of the given ones.
Continuing to repeatedly fill the interstices between mutually tangent circles with
further tangent circles, we arrive at an infinite circle packing. It is called a bounded Apollonian circle packing (ACP). It is known that if the original four circles have integer curvatures, all of the circles in the packing will have integer curvatures as well. In this case, the ACP is called integer. For an integer ACP ${\cal P}$ , let $\kappa({\cal P},X)$ denote the number of distinct integers up to X occurring as curvatures in the packing. Let $X_{\cal P}$ be the curvature of the largest circle in ${\cal P}$ .

The Theorem: On 21st January 2010, Jean Bourgain and Elena Fuchs submitted to the Journal of the AMS a paper in which they proved that for any integer ACP ${\cal P}$
there exist a constant $c>0$ depending on ${\cal P}$ such that $\kappa({\cal P},X) \geq c X$ for all $X \geq X_{\cal P}$ .

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 the number $\kappa({\cal P},X)$ of distinct integers up to X occurring as curvatures. Graham et.al. proved in 2003 that $\kappa({\cal P},X) \geq c'\sqrt{X}$ for some constant $c'$ and conjectured that this can be improved to $\kappa({\cal P},X) \geq c X$ . The Theorem proves this conjecture. It also makes an important step towards even more general local-global conjecture, see here.

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

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The Schmidt conjecture in simultaneous Diophantine approximation is true

You need to know: Notations: ${\mathbb N}$ for the set of positive integers, $||x||$ for the distance from real number x to the nearest integer. In addition, you need to know the concept of Hausdorff dimension of set $A\subset {\mathbb R}^2$ to understand the context.

Background: Let S be the set of pairs of real numbers $(i,j)$ such that $0\leq i,j \leq 1$ and $i+j=1$ . For $(i; j)\in S$ , let $\text{Bad}(i; j)$ denote the set of points $(x; y)\in {\mathbb R}^2$ for which there exists a positive constant $c=c(x,y)$ such that $\max\{||qx||^{1/i}, ||qy||^{1/j}\} > c/q$ for all $q \in {\mathbb N}$ .

The Theorem: On 15th January 2010, Dzmitry Badziahin, Andrew Pollington, and Sanju Velani submitted to arxiv and the Annals of Mathematics a paper in which they proved that for any finite number $(i_1; j_1); \dots ; (i_d; j_d)$ of pairs from S, the set $\bigcap\limits_{t=1}^d \text{Bad}(i_t; j_t)$ is non-empty.

Short context: A real number x is said to be badly approximable if there exists a constant $c(x)>0$ such that $||qx|| > c(x)/q$ for all $q \in {\mathbb N}$ . Sets $\text{Bad}(i; j)$ are natural generalisations containing pairs $(x; y)$ of simultaneously badly approximable numbers. The Theorem confirms a 1983 conjecture of Schmidt. In fact, Schmidt conjectured it for specific values $d=2$ , $i_1=1/3$ , and $j_1=2/3$ , and even this remained open. Any counterexample to this conjecture would imply the famous Littlewood conjecture (see here), but the Theorem states that there is no counterexample. In fact, Badziahin, Pollington, and Velani proved a much stronger result that the set $\bigcap\limits_{t=1}^d \text{Bad}(i_t; j_t)$ is not only non-empty, but has Hausdorff dimension 2 – the same as the whole plane. In a later work, this result has been extended to higher dimensions.