The number of integer solutions to |F|<= m for decomposable form F

You need to know: Polynomial in n variables, degree of a polynomial, complex numbers.

Background: Decomposable form is a polynomial $F(x_1, \dots x_n)=\prod\limits_{i=1}^d (a_{i1}x_1+\dots+a_{in}x_n),$ where the coefficients $a_{ij}$ are non-zero complex numbers.

The Theorem: On 24th January 2000 Jeffrey Thunder submitted to Annals of Mathematics a paper in which he proved, among other results, that for every decomposable form F of degree d in n variables with integer coefficients, the number $N_F(m)$ of integer solutions to the inequality $|F(x_1, \dots, x_n)|\leq m$ is either infinite or at most $c(n,d)m^{n/d}$ , where $c(n,d)$ is an effectively computable constant depending only on n and d.

Short context: Counting integer solutions to equations and inequalities is an old theme in mathematics. Case d=2 of the Theorem was proved by Mahler in 1933, but progress in estimating $N_F(m)$ for d>2 was limited. The Theorem estimates $N_F(m)$ for all d, confirming 1989 conjecture of Schmidt.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 1.3 of this book for an accessible description of the Theorem.

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There exists a field with u-invariant 9

You need to know: The concept of a Field.

Background: A number u(F) is called u-invariant of a field F, if (a) equation $\sum_{i=1}^n \sum_{j=1}^n a_{ij} x_i x_j = 0$ (where $x_1, x_2, \dots, x_n$ are variables and $a_{ij}$ are some coefficients from F) in n=u(F) variables may have no solutions, except of $x_1 = x_2 = \dots = x_n = 0$ , and (b) the same equation in $n=u(F)+1$ variables always have a non-zero solution.

The Theorem: On 4th January 2000 Oleg Izhboldin submitted to Annals of Mathematics a paper in which he proved that there exists a field F with u-invariant 9.

Short context: In 1953, Kaplansky conjectured that u-invariant of any field is always a power of 2. In 1989, Merkurev disproved this, and showed that u-invariant can be any even number, but left open whether it can be any odd number grater than 1. It is known that u-invariant can never be 3, 5, or 7. However, the Theorem proves that it can be 9.

Links: The original paper is available here. See also Section 1.7 of this book for an accessible description of the Theorem.

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There are three nonempty phases for percolation on planar hyperbolic graph

You need to know: Graph, infinite graph, planar graph, connected components of a graph, basic probability theory, almost sure convergence, bijection, infimum, notation |A| for the number of elements in any finite set A.

Background: An automorphism of graph $G=(V,E)$ is a bijection $\sigma: V \to V$ , such that pair of vertices $(u,v)$ is an edge if and only if $(\sigma(u), \sigma(v))$ is an edge. A (finite or infinite) graph G is called transitive if, for every 2 vertices u and v of G, there is an automorphism $\sigma$ such that $\sigma(u)=v$ . An infinite transitive graph $G=(V,E)$ has one end if, after deletion of any finite set of vertices (and the corresponding edges), the remaining graph has exactly one infinite connected component. A graph $G=(V,E)$ is amenable if, for every $\epsilon>0$ , there is a finite set $A \subset V$ , such that $|\partial A| \leq \epsilon |A|$ , where $\partial A$ is the set of vertices outside A connected by edge to at least one vertex in A. Transitive, nonamenable, planar graph with one end are known as “planar hyperbolic graphs”.

Let $p\in[0,1]$ . The Bernoulli(p) bond percolation on $G=(V,E)$ is a subgraph of G to which each edge of G is included independently with probability p. The Bernoulli(p) site percolation on $G=(V,E)$ is a subgraph of G to which each vertex of G is included independently with probability p, and each edge $(u,v)$ is included if and only if both vertices u and v are. For given G, let $p_c(G)$ be the infimum of all $p\in[0,1]$ such that the Bernoulli(p) (bond or site) percolation on G has an infinite connected component almost surely, and let $p_u(G)$ be the infimum of all p for which this infinite connected component is unique almost surely.

The Theorem: On 30th December 1999, Itai Benjamini and Oded Schramm submitted to the Journal of the AMS a paper in which they proved, among other results, that for any planar hyperbolic graph G we have $0 < p_c(G) < p_u(G) < 1$ , for Bernoulli bond or site percolation on G.

Short context: The Theorem proves that, as p increases from 0 to 1, the Bernoulli(p) percolation on planar hyperbolic graphs passes through three distinct nonempty phases. In the first phase, $p\in(0,p_c]$ , there are no infinite connected components; in the second phase, $p\in(p_c, p_u)$ , there are infinitely many of them; and in the third phase, $p\in[p_u, 1)$ , there is a unique infinite connected component.

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

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For every n>30, n-th Lehmer number has a primitive divisor

You need to know: prime numbers, coprime integers, complex numbers, polynomials, root of a polynomial

Background: An algebraic integer is a complex number that is a root of some polynomial with integer coefficients and leading coefficient 1. A root of unity is a complex number z such that $z^n=1$ for some positive integer n. A pair of algebraic integers $(\alpha,\beta)$ is called Lehmer pair if $(\alpha+\beta)^2$ and $\alpha\beta$ are non-zero coprime integers and $\alpha/\beta$ is not a root of unity. For every Lehmer pair $(\alpha,\beta)$ , define sequence $u_n = u_n(\alpha,\beta)$ by formulas $u_n = \frac{\alpha^n - \beta^n}{\alpha-\beta}$ for odd n and $u_n = \frac{\alpha^n - \beta^n}{\alpha^2-\beta^2}$ for even n. Then $u_n, \, n=0,1,,2,\dots$ is a sequence of integers which is called the sequence of Lehmer numbers. A prime number p is called a primitive divisor of $u_n$ if p divides $u_n$ but does not divide $(\alpha^2-\beta^2)^2 u_1 \dots u_{n-1}$ .

The Theorem: On 9th February 1999, Yuri Bilu, Guillaume Hanrot and Paul Voutier submitted to Journal fur die Reine und Angewandte Mathematik a paper in which they proved that for every Lehmer pair $(\alpha,\beta)$ and every n>30, $u_n(\alpha,\beta)$ has a primitive divisor.

Short context: The condition n>30 in the Theorem is the best possible because, for example, $\alpha=(1-\sqrt{-7})/2$ and $\beta=(1+\sqrt{-7})/2$ is a Lehmer pair for which $u_{30}$ has no primitive divisors. In fact, an old and difficult problem, which goes back to at least the beginning of 20-th century, is to find all triples $(\alpha,\beta, n)$ such that $(\alpha,\beta)$ is a Lehmer pair and $u_n(\alpha,\beta)$ has no primitive divisors. With the help of computer, the authors found all such triples with $n\leq 30$ , while the Theorem states that there are no such triples with $n>30$ . This completely solves the problem. As a special case, the Theorem also solves the corresponding problem for so-called Lucas numbers.

Links: The original paper is here.

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Elements of finite simple groups have short normal subsets representations

You need to know: Group, finite group, subgroup, identity element e of a group, logarithm $\log$ .

Background: A subset 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. For any finite set T, denote $|T|$ the number of elements in it.

The Theorem: On 28th October 1999, Martin Liebeck and Aner Shalev submitted to Annals of Mathematics a paper in which they proved, among other results, that there exists a constant c such that if G is a finite simple group and $S\neq \{e\}$ is a normal subset of G, then, for any $m \geq c \log |G|/\log |S|$ , any element of G can be expressed as a product of m elements of S.

Short context: Because there are $|S|^m$ ways to write a product of m elements of S, the conclusion in the Theorem cannot hold if $|S|^m < |G|$ , or $m< \log |G|/\log |S|$ . Hence, the inequality $m \geq c \log |G|/\log |S|$ in the Theorem is the best possible up to constant factor. The Theorem has many applications, see here for an example.

Links: The original paper is here. See also Section 1.6 of this book for an accessible description of the Theorem.

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There exist two different finite groups with isomorphic integral group rings

You need to know: Groups, finite groups, rings, isomorphism for groups and rings.

Background: For any (finite) group G, an integral group ring is the ring whose elements are all possible sums $\sum\limits_{g\in G}a_gg$ , where $a_g$ are integers, addition is defined as $\sum_{g\in G}a_gg + \sum_{g\in G}b_gg = \sum_{g\in G}(a_g+b_g)g,$ and multiplication is given by $(\sum_{g\in G}a_gg)\cdot(\sum_{h\in G}b_hh) = \sum_{g\in G}\sum_{h\in G}a_gb_hgh$ .

The Theorem: On 15th September 1999, Martin Hertweck submitted to Annals of Mathematics a paper in which he proved that there exist two finite non-isomorphic groups G and H such that their integral group rings are isomorphic.

Short context: Obviously, if two groups G and H are isomorphic, then so are their integral group rings. In 1940, Higman asked if the converse is true, that is, whether different finite groups have different integral group rings. The question was answered positively in many special cases. The Theorem, however, implies that in general the answer is negative.

Links: The original paper is here. See also Section 1.5 of this book for an accessible description of the Theorem.

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The asymptotic mean of bounded muptiplicative function is at least -0.656999…

You need to know: limits, integrals, logarithm $\log$ , base of natural logarithms e.

Background: Let ${\mathbb N}$ be the set of positive integers and ${\mathbb R}$ be the real line. A function $f:{\mathbb N} \to {\mathbb R}$ is called completely multiplicative, if $f(x \cdot y) = f(x) \cdot f(y)$ for all positive integers x, y. We say that function $g:{\mathbb N} \to {\mathbb R}$ is $o(1)$ if $\lim\limits_{n\to\infty} g(n) =0$ .

The Theorem: On 8th September 1999, Andrew Granville and Kannan Soundararajan submitted to Annals of Mathematics a paper in which they proved that for every completely multiplicative function $f:{\mathbb N} \to {\mathbb R}$ , taking values in $[-1,1]$ , $\frac{1}{N}\sum\limits_{n=1}^N f(n) \geq \delta + o(1)$ , where $\delta = 1-2\log(1+\sqrt{e})+4\int_1^{\sqrt{e}} \frac{\log t}{t+1}dt = -0.656999...$ .

Short context: In 1994, Heath-Brown conjectured that $\frac{1}{N}\sum\limits_{n=1}^N f(n) \geq c + o(1)$ for some constant $c > -1$ . In 1996, Hall proved this conjecture, and in turn conjectured that one can take $c=\delta$ with $\delta$ defined above. The Theorem confirms this conjecture. As observed by Hall, this result is the best possible because there exists f for which the equality holds.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 1.2 of this book for an accessible description of the Theorem.

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The honeycomb conjecture is true

You need to know: Basic geometry (area, perimeter, curve, ball, etc.). In addition, concepts like “locally finite graph”, “bounded connected components” and $\limsup$ are needed for formal statement but not for intuitive understanding.

Background: Let C be a partition of the plane into regions of equal unit area (formally, let G be a locally finite graph in ${\mathbb R}^2$ , consisting of smooth curves, and such that ${\mathbb R}^2\setminus G$ has infinitely many bounded connected components, all of unit area, and let C be the union of these bounded components). Let B(0,r) be the ball with centre in the coordinate centre and radius r. Let $p(C) = \limsup\limits_{r \to \infty} \frac{\text{Perimeter}(C \cap B(0,r))}{\text{Area}(C \cap B(0,r))}$ be the average perimeter of C per unit area.

The Theorem: On 8th June 1999 Thomas Hales submitted to arxiv a paper in which he proved that, for any C as above, $p(C) \geq \sqrt[4]{12}$ .

Short context: Let C* be a partition of the plane into regular hexagons of unit area. C* also has names “regular hexagonal grid” and “honeycomb”. The perimeter of any one such hexagon is $2\sqrt[4]{12}$ , hence $p(C^*)= \sqrt[4]{12}$ . The statement that C* is the perimeter-optimal way to divide the plane into regions/cells of equal area was known as the honeycomb conjecture and is more than 2000 years old: Varro studied it around 36 B.C., and proved that C* is better than tilings by squares or regular triangles. In 1943, Toth proved the honeycomb conjecture under the hypothesis that the cells are convex. The Theorem proves it unconditionally.

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

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Moderate deviations probability for the Wiener Sausage volume decays fast

You need to know: d-dimensional Euclidean space ${\mathbb R}^d$ , volume in ${\mathbb R}^d$ , limits, basic probability theory, stochastic processes, Wiener process in ${\mathbb R}^d$

Background: Let $W(t)$ be standard Wiener process in ${\mathbb R}^d$ . For $a>0$ and $t \geq 0$ , let $W_a(t) = \{ x \in {\mathbb R}^d \,|\,\exists s \in [0,t] : \rho(x,W(s)) \leq a \}$ , where $\rho$ is the distance in ${\mathbb R}^d$ . $W_a(t)$ is known as Wiener sausage. Let $|W_a(t)|$ denote its (d-dimensional) volume. It is known that $E|W_a(t)| \approx 2\pi t/\ln t$ for $d=2$ , $E|W_a(t)| \approx 2\pi a t$ for $d=3$ , and $E|W_a(t)| \approx K_{a,d} t$ for $d \geq 3$ , where $K_{a,d}>0$ is a constant depending on a and d, $E$ denotes mathematical expectation, and by $f(t) \approx g(t)$ we mean $\lim\limits_{t \to \infty}\frac{f(t)}{g(t)}=1$ .

The Theorem: On 5th July 1999, Michiel van den Berg, Erwin Bolthausen and Frank den Hollander submitted to Annals of Mathematics a paper in which they proved that for every $a>0$ , $b \in (0,K_{a,d})$ , $\lim\limits_{t\to\infty}\frac{1}{t^{(d-2)/d}}\ln P(|W^a(t)|\leq bt) = -C_{a,d,b}, \, d \geq 3,$ where P denotes the probability, and $C_{a,d,b}>0$ is some constant depending on a, d, and b. Similarly, for every $a>0$ , $b \in (0,2\pi)$ , $\lim\limits_{t\to\infty}\frac{1}{\ln t}\ln P(|W^a(t)|\leq bt/\ln t) = -C_b, \, d=2.$

Short context: Let $a$ be fixed and denote $V_t=|W^a(t)|$ . The Theorem implies that, for large t, $P(V_t\leq bE[V_t])$ (which is called moderate deviation probability) decays as $t^{-C}$ for $d=2$ , and as $\text{exp}\left(-C t^{(d-2)/d}\right)$ for $d \geq 3$ . Earlier, similar fast decays were known only for probabilities of the form $P(V_t \leq f(t) E[V_t])$ for some $f(t)$ such that $\lim\limits_{t \to \infty} f(t) = 0$ . These are known as probabilities of large deviations.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 1.1 of this book for an accessible description of the Theorem.

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The Erdős-Taylor conjecture on the most visited point of simple plane random walk is true

You need to know: Coordinate plane, vectors, natural logarithm $\log$ , basic probability theory, almost sure convergence.

Background: Let $X_1, X_2, \dots, X_n, \dots$ be a sequence of independent vector-valued random variables, each taking values (1,0), (0,1), (-1,0) and (0,-1) with equal probabilities. Sequence $S_n = \sum\limits_{k=1}^n X_k$ , $n=1,2,\dots$ is called simple random walk in ${\mathbb Z}^2$ . For each pair of integers (x,y) and positive integer n, let $T(n,x,y)$ be the number of positive integers $k \leq n$ such that $S_k=(x,y)$ . Finally, let $T^*(n) = \max_{x,y} T(n,x,y)$ be the number of times that random walk visits the most visited point.

The Theorem: On 10th May 1999, Amir Dembo, Yuval Peres, Jay Rosen, and Ofer Zeitouni submitted to Acta Mathematica a paper in which they proved, among other results, that $\frac{T^*(n)}{(\log n)^2}$ converges to $\frac{1}{\pi}$ almost surely as $n \to \infty$ .

Short context: The question “How many times does the simple plane random walk revisit the most frequently visited point in the first n steps?” was asked by Erdős and Taylor in 1960. They proved that the limit $\lim\limits_{n\to\infty}\frac{T^*(n)}{(\log n)^2}$ , if it exists, is between $\frac{1}{4\pi}$ and $\frac{1}{\pi}$ . They further conjectured that the limit actually exists and is equal to $\frac{1}{\pi}$ almost surely. The Theorem confirms this conjecture.

Links: The original paper is here.

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