For any r-colouring of integers, there is a monochromatic set S with sum of reciprocals 1

You need to know: Basic arithmetic, notation $b^r$ for b to the power r, notation $[x,y]$ for integers between x and y.

Background: By r-colouring of integers we mean assigning to each integer one of r colours, or, equivalently, partition all integers into r classes. Then set S of integers is called monochromatic if all integers $k\in S$ are coloured in the same colour, or, equivalently, belong to the same class in the partition.

The Theorem: On 16th May 2001, Ernie Croot submitted to the Annals of Mathematics a paper in which he proved the existence of a constant b such that for every partition of the integers in $[2, b^r]$ into r classes, there is always one class containing a subset S with the property $\sum\limits_{n\in S}\frac{1}{n}=1$ .

Short context: In 1980, Erdős and Graham asked if for any division of the integers $2, 3, 4, \dots$ into $r \geq 2$ classes, we can always represent 1 as a sum of distinct unit fractions using denominators from one class only. The Theorem gives a positive answer to this question. Moreover, it says we can do this using only denominators from 2 to $b^r$ . This bound is the best possible up to the value of b. Croot also proved that, for sufficiently large r, one may choose $b = 10^{72527}$ .

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 3.2 of this book for an accessible description of the Theorem.

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Any non-trivial knot has ropelength at least 21.45

You need to know: Curve in ${\mathbb R}^3$ , closed curves, self-intersecting and non-self-intersecting curves, length of the curve.

Background: A knot is a closed, non-self-intersecting curve in ${\mathbb R}^3$ . Two knots $K_1$ and $K_2$ are equivalent if $K_1$ can be transformed smoothly, without intersecting itself, to coincide with $K_2$ . A knot is called non-trivial if it is not equivalent to a circle. The thickness of knot K is $\tau(K)=\inf\limits_{x,y,z \in K} r(x,y,z)$ , where the infimum is taken over all triples of pairwise distinct points $x,y,z$ on K, and $r(x,y,z)$ is the radius of the circle passing through these points. The ropelength of a knot K is the quotient of its length by its thickness.

The Theorem: On 30th March 2001, Jason Cantarella, Robert Kusner, and John Sullivan submitted to arxiv a paper in which they proved, among other results, that any non-trivial knot has ropelength at least $2\pi(2+\sqrt{2}) \approx 21.45$ .

Short context: Given a family of equivalent knots, the knots with minimal ropelength in the family (such knots are also known as ideal knots) can serve as “nice-looking” representatives of the family. A basic open question is what the minimal possible ropelength of a non-trivial knot can be. Litherland et.al. proved in 1999 that it is at least $5\pi \approx 15.71$ . The Theorem improves this lower bound to $2\pi(2+\sqrt{2}) \approx 21.45$ . In a subsequent paper (in 2006), the lower bound was improved to $31.32$ . Computer experiments suggests that the correct answer may be around $32.66$ .

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

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A properly embedded simply connected minimal surface in R^3 is either a plane or a helicoid

You need to know: Euclidean space ${\mathbb R}^3$ , surface in ${\mathbb R}^3$ , surface area, neighbourhood of a point in the surface, boundary, path, endpoints of a path, surface with and without self-intersections, compact subset of ${\mathbb R}^3$ , compact subset of a surface.

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 simply connected if for every 2 points $x\in M$ and $y\in M$ there is a path in M with endpoints x and y, and every 2 such paths can be continuously transformed into each other within M while keeping both endpoints x and y fixed. 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. A helicoid is the surface defined by equations $x = s \cos (\alpha t), \,\, y = s \sin (\alpha t), \,\, z=t,$ where $\alpha$ is a constant, and $s,t$ are real parameters, ranging from $-\infty$ to $\infty$ .

The Theorem: On 13th March 2001, William Meeks III and Harold Rosenberg submitted to Annals of Mathematics a paper in which they proved that any properly embedded simply connected minimal surface in ${\mathbb R}^3$ is either a plane or a helicoid.

Short context: Minimal surfaces, ones that locally minimises their areas, are among the most studied objects in geometry and topology. A trivial example of minimal surface is the plane. One of the simplest non-trivial examples, described by Euler in 1774, is a helicoid. Since 1774, many other minimal surfaces has been discovered, but none of them are properly embedded and simply connected. The Theorem proves that in fact there are no other minimal surfaces with these natural properties.

Links: The original paper is available here. See also Section 5.3 of this book for an accessible description of the Theorem.

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There is a set S on the plane such that each isometric copy of Z^2 meets S in 1 point

You need to know: Euclidean plane ${\mathbb R}^2$ (set of points $x=(x_1,x_2)$ with real coordinates). Integer lattice ${\mathbb Z}^2 \subset {\mathbb R}^2$ (set of points $x=(x_1,x_2)$ with both coordinates being integers). Notation $\cap$ for set intersection. Notation $|A|$ for the number of elements in set A.

Background: Function $f:{\mathbb R}^2 \to {\mathbb R}^2$ is called isometry if $d(f(x),f(y)) = d(x,y)$ for all $x,y \in {\mathbb R}^2$ , where d is the usual Euclidean distance. For isomerty f, denote $f({\mathbb Z}^2) = \{y\in{\mathbb R}^2\,|\,y=f(x), \, x\in {\mathbb Z}^2\}$ .

The Theorem: On 14th February 2001, Steve Jackson and Daniel Mauldin submitted to the Journal of the AMS a paper in which they proved the existence of set $S \subset {\mathbb R}^2$ such that $|S \cap f({\mathbb Z}^2)|=1$ for every isometry f.

Short context: In the 1950-th, Steinhaus asked whether there exist two sets A and B in the plane such that every set congruent to A has exactly one point in common with B, except of the trivial cases $A={\mathbb R}^2, \, |B|=1$ and $B={\mathbb R}^2, \, |A|=1$ . He also asked if this is possible with $A={\mathbb Z}^2$ . The first question was answered affirmatively in 1958 by Sierpinski, while the second one remained open for decades until it was answered by this Theorem.

Links: The original paper is available here.

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In any infinite sequence of graphs, some graph is a minor of some other one

You need to know: (Undirected, finite) graph, isomorphic graphs, operations of edge contraction, edge deletion, and vertex deletion.

Background: A graph H is a minor of graph G if H can be obtained from G by a sequence of contractions of edges of G, and deletions of edges and vertices of G.

The Theorem: On 13th February 2001, Neil Robertson and Paul Seymour submitted to the Journal of Combinatorial Theory, Series B, a paper in which they proved that in any infinite sequence $G_1, G_2, \dots$ of graphs, one can select two graphs, $G_i$ and $G_j$ , such that one of them is isomorphic to a minor of another one.

Short context: A family ${\cal F}$ of graphs if called minor-closed if $G \in {\cal F}$ implies that $H \in {\cal F}$ for all minors H of G. For example, if $H_1, H_2, \dots, H_n$ is any finite set of graphs, then the family ${\cal F}$ of graphs G that do not have any of $H_i$ as a minor is obviously minor-closed. The Theorem implies that in fact any minor-closed family can be represented in this way. This statement was known as Wagner’s conjecture (although Wagner said he never conjectured it!) and is now known as the Robertson-Seymour theorem, or the graph minor theorem. The Theorem was proved in a series of 20 papers of total length over 500 pages (the date 13th February 2001 is the submission date of the last paper).

Links: The original paper is available here.

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A finitely presented non-amenable group without free non-cyclic subgroups

You need to know: Group, subgroup, presentation of a group in the form $<S|R>$ , where S is a set of generators and R is a set of relations, group $SO(3)$ of rotations in ${\mathbb R}^3$ .

Background: A group G is called finitely presented if it has a presentation $<S|R>$ with finite number of generators and finite set of relations. A group G is called free if it has presentation $<S|\emptyset>$ , that is, with no relations. A free group is called non-cyclic if it has at least 2 generators.

A group G is called amenable if it has a left invariant finitely additive probability measure, that is, a function $\mu$ 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 8th February 2001, Alexander Ol’shanskii and Mark Sapir submitted to Publications Mathématiques a paper in which they proved the existence of a finitely presented non-amenable group without free non-cyclic subgroups.

Short context: In 1924, Banach and Tarski proved that one can decompose a unit ball into a finite number of disjoint pieces, move and rotate the pieces, and construct two unit balls! This is known as Banach–Tarski paradox. In 1929, von Neumann introduced the notion of amenable groups, and proved that Banach–Tarski paradox arises because the group $SO(3)$ is not amenable. In fact, he proved that any group containing a free non-cyclic subgroup is not amenable, and asked if the converse is true. In 1980, Ol’shanskii proved the existence of counterexample to this von Neumann’s problem. The Theorem provides the first finitely presented counterexample.

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

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Ellipsoid rE in R^d, d>=5, contains Vol(rE)+O(r^(d-2)) lattice points

You need to know: Euclidean space ${\mathbb R}^d$ of vectors $x=(x_1, \dots, x_d)$ , (coordinate-wise) addition of vectors, their multiplication by constants, scalar product $(x,y)=\sum_{i=1}^d x_iy_i$ , matrices, matrix multiplication, invertible matrix, (Lebesgue) volume in ${\mathbb R}^d$ , big $O()$ notation.

Background: For a (measurable) set $S \subset {\mathbb R}^d$ , let $\text{Vol}(S)$ be its (Lebesgue) volume in ${\mathbb R}^d$ , and let $\text{Vol}_{\mathbb Z}(S)$ be the number of points $x \in S$ with all coordinates $x_i$ being integers. For $S \subset {\mathbb R}^d$ , $a\in {\mathbb R}^d$ , and $r>0$ , denote $rS + a$ the set of points $y \in {\mathbb R}^d$ , such that $y=rx+a$ for some $x \in S$ .

Let Q be an invertible $d \times d$ matrix with real entries such that (a) $(Qx,y)=(x,Qy)$ for all $x,y \in {\mathbb R}^d$ , and (b) $(Qx,x)>0$ for every non-zero $x \in {\mathbb R}^d$ . For $s>0$ , let $E_s$ be the set of points $x \in {\mathbb R}^d$ such that $(Qx,x) \leq s$ . Any set of the form $rE_1 + a$ (for $a\in {\mathbb R}^d$ , and $r>0$ ) is called ellipsoid.

The Theorem: On 12th December 2000, Friedrich Götze submitted to the Inventiones Mathematicae a paper in which he proved that for any $d\geq 5$ and any Q as above, $\sup\limits_{a\in {\mathbb R}^d}\left|\frac{\text{Vol}_{\mathbb Z}(rE_1 + a)-\text{Vol}(rE_1)}{\text{Vol}(rE_1)}\right|=O(r^{-2})$ as $r\to\infty$ .

Short context: Set $rE_1 + a$ is just the ellipsoid $E_1$ scaled by r and shifted by a. The Theorem states that the number of integer points (also called lattice points) in such an ellipsoid can be approximated by its volume up to the relative error, decreasing at rate $O(r^{-2})$ as $r\to\infty$ , whenever $d\geq 5$ . Previously, this result was known only for $d\geq 9$ . The condition $d\geq 5$ is the best possible, because, for $d\leq 4$ , the error rate is known to decrease slower than $O(r^{-2})$ , even if $E_1$ is the unit sphere.

Links: The original paper is available here.

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Complete graph of odd order can be decomposed into equal-length cycles

You need to know: Basic graph theory: graph, vertex, edge, order of graph (number of vertices), complete graph, cycle, length of the cycle, degree of a vertex, subgraph, isomorphic graphs, perfect maching.

Background: We denote $K_n$ the complete graph on n vertices. For even n, we denote $K'_n$ the complete graph on n vertices with a perfect maching removed. Also, let $C_m$ denote a cycle of length m. We say that graph G is decomposed into its subgraphs $H_1$ and $H_2$ , and write $G = H_1 \oplus H_2$ , if G is the edge disjoint union of $H_1$ and $H_2$ . We cay that G is H-decomposable, if $G = H_1 \oplus H_2 \oplus \dots \oplus H_k$ , where each $H_i$ is isomorphic to H.

The Theorem: On 28th November 2000, Mateja Šajna submitted to the Journal of Combinatorial Designs a paper in which she proved that (i) for every odd $n\geq 3$ and every m such that $3\leq m \leq n$ and m divides the number of edges in $K_n$ , graph $K_n$ is $C_m$ -decomposable; (ii) for every even $n\geq 4$ and every m such that $3\leq m \leq n$ and m divides the number of edges in $K'_n$ , graph $K'_n$ is $C_m$ -decomposable.

Short context: When can we decompose the complete graph $K_n$ into cycles of some fixed length m? Obviously, this can be possible only if $3\leq m \leq n$ and m divides the number of edges in $K_n$ . Also, the degrees of all vertices of $K_n$ should be even. Hence, n should be odd. The Theorem proves that these obvious necessary conditions are in fact sufficient, culminating a long series of papers with partial results. It also establishes a similar result for even n, in which case perfect matching is removed from $K_n$ to make the vertices degrees even.

Links: The original paper is available here.

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Irreducible form F(x,y,z) of degree d has at most O(B^(2/d+e)) simple B-bounded roots

You need to know: Greatest common divisor (gcd), multivariate polynomial, absolutely irreducible multivariate polynomial, homogeneous polynomial (or form), degree of a form.

Background: For an absolutely irreducible form $F(x_1, x_2, \dots, x_n)$ of degree d with integer coefficients, we call solution $\textbf{x}=(x_1, x_2, \dots, x_n)$ to the equation $F(x_1, x_2, \dots, x_n)=0$ (or $F(\textbf{x})=0$ in short) simple if $\text{gcd}(x_1, x_2, \dots, x_n)=1$ and the first non-zero component of $(x_1,x_2,\dots x_n)$ is positive. Let $N_F(B)$ be the number of simple solutions to $F(\textbf{x})=0$ such that $\max\limits_{1 \leq i \leq n}|x_i| \leq B$ .

The Theorem: On 17th November 2000, Roger Heath-Brown submitted to the Annals of Mathematics a paper in which he proved that, for any absolutely irreducible form $F(x,y,z)$ of degree d in $n=3$ variables, and any $\epsilon>0$ , we have $N_F(B) \leq C(d, \epsilon)B^{2/d+\epsilon}$ , where $C(d, \epsilon)$ is a constant depending only on d and $\epsilon$ .

Short context: Counting integer solutions to polynomial equations is one of the basic and most-studied problems in mathematics, which is easy for forms in $n=1$ and $n=2$ variables. For (irreducible) forms in $n=3$ variables, Pila showed in 1995 that $N_F(B) \leq C(d, \epsilon)B^{1+1/d+\epsilon}$ . The Theorem improves the exponent in this estimate from $1+1/d+\epsilon$ to $2/d+\epsilon$ , which is in fact optimal.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 2.6 of this book for an accessible description of the Theorem.

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Sinai conjecture on orbit recurrence of nonregular quadratic maps is true

You need to know: Limits, the concept of (Lebesgue) almost every.

Background: For a quadratic polynomial $f:{\mathbb R}\to {\mathbb R}$ and initial point $x_0 \in {\mathbb R}$ , consider infinite sequence defined by induction: $x_n=f(x_{n-1})$ , $n=1,2,\dots$ . Let I be the longest interval which f maps into itself. If, for almost every $x_0 \in I$ , the sequence above converges to a cycle, f is called regular.

The Theorem: On 6th October 2000, Artur Ávila and Carlos Moreira submitted to arxiv a paper in which they proved that, for almost every $a\in[-1/4, 2]$ such that quadratic polynomial $f_a(x)=a-x^2$ is not regular, and $x_0=0$ , the set of n such that $|x_n| < 1/n^\gamma$ is finite if $\gamma > 1$ and infinite if $\gamma < 1$ .

Short context: The sequence $x_n$ defined above is an example of (an orbit of) a dynamical system. If $x_n \approx 0 = x_0$ , then $x_{n+1} \approx x_1$ , $x_{n+2} \approx x_2$ , $\dots$ and the sequence is close to being periodic for some time. The Theorem provides an exact rate how close $x_n$ can be to $0$ , answering a long-standing conjecture of Sinai. This result helps to understand to what extend this simple dynamical system exhibits a periodic behaviour, and is the step towards understanding approximate periodicity in more complex dynamical systems.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 5.4 of this book for an accessible description of the Theorem.

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