The Ore conjecture is true: every element of every finite non-abelian simple group is a commutator

You need to know: Group, finite group, identity element e of a group, notation $g^{-1}$ for the inverse of group element g, subgroup.

Background: A subgroup 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. A group G is called abelian if $g\cdot h = h \cdot g$ for every $g \in G$ and $h \in G$ , and non-abelian otherwise. An element $a \in G$ is called a commutator if $a=g^{-1}h^{-1}gh$ for some $g \in G$ and $h \in G$ .

The Theorem: On 18th June 2009, Martin Liebeck, Eamonn O’Brien, Aner Shalev, and Pham Huu Tiep submitted to the Journal of the European Mathematical Society a paper in which they proved that every element of every finite non-abelian simple group is a commutator.

Short context: In any abelian group, $g^{-1}h^{-1}gh = g^{-1}(h^{-1}h)g = g^{-1}eg=e$ , hence only identity element is a commutator. In general, the size of set of commutators can be viewed as an indicator “how far” the group is from being abelian. In 1951, Ore made a conjecture that in every finite non-abelian simple group the set of commutators is the whole group, which makes such groups as “far” from being abelian as they possibly can. Despite substantial efforts, the conjecture remained open for 58 years. The Theorem proves this conjecture.

Links: The original paper is available here.

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Any semigroup in Z^n is asymptotically approximated by its regularisation

You need to know: Euclidean space ${\mathbb R}^n$ , subspace of ${\mathbb R}^n$ , subspace spanned by a set, cone, convex cone, closed cone, apex of a cone, groups, group ${\mathbb Z}^n$ of vectors with integer coordinates, subgroup, subgroup generated by a set, semigroup.

Background: Let $S \subset {\mathbb Z}^n$ be a semigroup. Let G be the subgroup of ${\mathbb Z}^n$ generated by S, L the subspace of ${\mathbb R}^n$ spanned by S, and C the smallest closed convex cone (with apex at the origin) containing S.

The Theorem: On 21st April 2009, Kiumars Kaveh and Askold Khovanskii submitted to arxiv a paper in which they proved the following result. Let $C' \subset C$ be a closed strongly convex cone that intersects the boundary (in the topology of the linear space L) of the cone C only at the origin. Then there exists a constant $N>0$ (depending on $C'$ ) such that any point in the group G that lies in $C'$ and whose distance from the origin is bigger than N belongs to S.

Short context: Semigroup $S' = C \cap G$ is called regularization of S. From the definition, S’ contains S. The Theorem states that the regularization S’ asymptotically approximates the semigroup S. The authors call this the approximation theorem. It is not very difficult to prove, but the authors showed that it has many applications in convex and algebraic geometry, intersection theory, and other areas of mathematics.

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

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Every tensor has a tensor-train decomposition

You need to know: Matrix, rank $\text{rank}(M)$ of a matrix M.

Background: Let $n_1, n_2, \dots, n_d$ be positive integers. Let ${\cal I}$ be the set of vectors $i=(i_1, i_2, \dots, i_d)$ such that all $i_j$ are integers and $1 \leq i_j \leq n_j$ for $j=1,2,\dots, d$ . A function $A:{\cal I} \to {\mathbb R}$ is called a d-dimensional $n_1 \times n_2 \times \dots \times n_d$ tensor. Let $r_k, k=0,1,2,\dots,d$ be positive integers such that $r_0=r_d=1$ . We say that A has tensor-train decomposition (TT decomposition in short) with TT-ranks $r_k$ if $A(i_1, i_2, \dots, i_d) = G_1(i_1)G_2(i_2) \dots G_d(i_d),$ where $G_k(i_k)$ is an $r_{k-1} \times r_k$ matrix. For $k=0,1,\dots,d$ , the unfolding matrix $A_k$ of A is an $(\prod\limits_{s=1}^k n_s) \times (\prod\limits_{s=k+1}^d n_s)$ matrix with rows indexed by $(i_1, \dots, i_k)$ and columns indexed by $(i_{k+1}, \dots, i_d)$ such that $A_k( (i_1, \dots, i_k), (i_{k+1}, \dots, i_d)) = A(i_1, \dots, i_d)$ .

The Theorem: On 10th March 2009, Ivan Oseledets submitted to the SIAM Journal on Scientific Computing a paper in which he proved that for each d-dimensional tensor A there exists a TT decomposition with TT-ranks $r_k \leq \text{rank}(A_k)$ , $k=0,1,\dots, d$ .

Short context: Tensors are natural generalizations of matrices and are central in many applications. However, storing a d-dimensional tensor requires memory which grows exponential in d, and performing any operations requires exponential time. TT decomposition with low TT-ranks allows to store tensors and do computations on them with significantly less time and memory resources.

Links: The original paper is available here.

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If a finite subset A of a free group has non-commuting elements, then |A^3|>|A|^2/(log|A|)^O(1)

You need to know: Group, free group, notation $|A|$ for the number of elements in a finite set A, $O(1)$ and $o(1)$ notations.

Background: For subsets A and B of group G denote $A \cdot B=\{g\in G \,|\,g=ab, \, a\in A, \, b \in B\}$ . Denote $A^3 = A \cdot A \cdot A$ .

The Theorem: On 18th June 2007, Alexander Razborov submitted to the Annals of Mathematics a paper in which he proved that, if A is a finite subset of a free group $F_m$ with at least two noncommuting elements, then $|A^3| \geq \frac{|A|^2}{(\log|A|)^{O(1)}}$ .

Short context: For sets of integers A,B, let $A+B=\{a+b \,|\, a\in A, \, b \in B\}$ . If A is an arithmetic progression, then $|A+A|<2|A|$ . A deep 1973 theorem of Freiman describes all possible examples of sets A such that $|A+A|\leq k|A|$ for fixed k. For many applications, it is important to derive the structure of A from a weaker estimate of the form $|A+A|\leq |A|^{1+o(1)}$ , but this is open. For subsets A of arbitrary group G, it is impossible to deduce the structure of A is $|A^2|$ is small, but sometimes possible if $|A^3|$ is small. The Theorem implies that in free groups $|A^3|$ is small only if $ab=ba$ for all $a,b \in A$ .

Links: The original paper is available here.

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w_1(A_n)w_2(A_n)=A_n for every non-trivial group words w_1,w_2 and every large alternating group A_n

You need to know: Group, identity element, finite group, simple group, free group, alternating group $A_n$ of degree n.

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 $A\cdot B=\{g \in G: g=a\cdot b, \, a\in A, \, b\in B\}$ .

The Theorem: On 11th January 2007, Michael Larsen and Aner Shalev submitted to arxiv a paper in which they proved that (i) for each pair of non-trivial words $w_1$ , $w_2$ there exists $N =N(w_1, w_2)$ such that for all integers $n \geq N$ we have $w_1(A_n)w_2(A_n) = A_n$ , and (ii) for every triple of non-trivial group words $w_1, w_2, w_3$ , there exists $N =N(w_1, w_2,w_3)$ such that $w_1(G)w_2(G)w_3(G) = G$ for every finite simple group with at least N elements.

Short context: In an earlier paper, Shalev proved that $w^3(G)=G$ for every non-trivial group word w, and every sufficiently large finite simple group $G$ . Part (ii) of the Theorem generalises this result (and implies it with $w_1=w_2=w_3=w$ ). Moreover, the authors conjectured that the same is true for just 2 group words! Part (i) of the Theorem proves this conjecture for the alternating groups.

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

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There exists infinitely many groups G with no product-free subset of size 2|G|^(8/9)

You need to know: Group, finite group, notation $|S|$ for the number of elements in a finite set S.

Background: A subset S of a finite group G is called product-free if it does not contain three elements x, y and z with $x y = z$ .

The Theorem: On 5th July 2005, Timothy Gowers submitted to Combinatorics, Probability and Computing a paper in which he proved (by providing explicit examples) the existence of infinitely many groups G which have no product-free subsets of cardinality greater than $2|G|^{8/9}$ .

Short context: A well-known theorem of Erdős states that for every n-element set X of integers there is a subset $S \subset X$ of size at least n/3 that is sum-free, that is, S does not contain integers x, y and z such that $x+y=z$ . In 1985, Babai and Sós asked for a generalisation: does every finite group G contain a product-free subset of size $c|G|$ for some $c>0$ ? As a partial result, they managed to prove the existence of product-free subsets of size $c|G|^{4/7}$ , which was improved to $c|G|^{11/14}$ by Kedlaya in 1997. The Theorem, however, shows that the answer to the Babai and Sós question is no.

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

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w(G)^3=G for every non-trivial group word w and every large finite simple group G

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

The Theorem: On 24th January 2006, Aner Shalev submitted to the Annals of Mathematics a paper in which he proved that for any non-trivial group word w there exists a positive integer $N=N(w)$ such that for every finite simple group G with $|G|\geq N(w)$ we have $w(G)^3=G$ .

Short context: Given a 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? This was proved for the commutator word $w=x_1^{-1}x_2^{-1}x_1x_2$ by Wilson in 1994, and for the power word $w=x_1^k$ by Martinez and Zelmanov in 1996. In 2001, Liebeck and Shalev deduced from this theorem that this is true for all non-trivial group words. However, the exact value of constant $c=c(w)$ was unknown. The Theorem proves that one can take $c=3$ , independently of w.

Links: The original paper is available here. See also Section 9.14 of this book for an accessible description of the Theorem.

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The B. and M. Shapiro conjecture in real algebraic geometry is true

You need to know: Set ${\mathbb C}$ of complex numbers, polynomials in complex variable, root of a polynomial, derivative, notation $P^{(j)}(z)$ for j-th derivative of polynomial $P(z)$ , matrix, determinant of a matrix, vector space, basis of a vector space.

Background: Let ${\cal P}=\{P_1(z), P_2(z), \dots, P_k(z)\}$ be a finite set of polynomials in complex variable z with complex coefficients. The complex span S of ${\cal P}$ is the set of polynomials $P(z)$ of the form $P(z)=\sum\limits_{i=1}^k \lambda_i P_i(z)$ for some $\lambda_i \in {\mathbb C}, \, i=1,\dots,k$ . The Wronskian $W(z)$ of ${\cal P}$ is the determinant of the $k \times k$ matrix with entries $P^{(j-1)}_i(z), \, i=1,\dots,k, \, j=1,\dots,k$ .

The Theorem: On 14th December 2005, Evgeny Mukhin, Vitaly Tarasov, and Alexander Varchenko submitted to arxiv a paper in which they proved that if all roots of the Wronskian of a set ${\cal P}$ of polynomials are real, then the complex span of ${\cal P}$ has a basis consisting of polynomials with real coefficients.

Short context: The Theorem confirms the 1995 conjecture known as the B. and M. Shapiro conjecture in real algebraic geometry. Earlier, it was known only in the case $k=2$ .

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 9.10 of this book for an accessible description of the Theorem.

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The Cayley graphs of SL_2(F_p) with random generators form a family expanders

You need to know: Group $\text{SL}_2({\mathbb F}_p)$ (group of $2 \times 2$ matrices over ${\mathbb F}_p$ with determinant 1, see here) , set of generators of a group, graph, edge expansion of a graph, family of expanders.

Background: For a finite group G with set of generators A, the (undirected) Cayley graph $\Gamma =\Gamma (G,A)$

The Theorem: On 3rd November 2005, Jean Bourgain and Alex Gamburd submitted to the Annals of Mathematics a paper in which they proved the following result. Fix $k \geq 2$ . For any prime $p$ , let $S_p$ be a $k$ -element subset of $SL_2({\mathbb F}_p)$ whose elements are chosen independently at random, and let $\Gamma_p=\Gamma (SL_2({\mathbb F}_p), S_p)$ be the corresponding Cayley graph. Then the sequence $\Gamma_2, \Gamma_3, \Gamma_5, \Gamma_7, \Gamma_{11}, \dots$ forms a family of expanders with probability $1$ .

Short context: Expanders are highly-connected sparse graphs widely used in both pure mathematics and computer science. A basic problem, posed by Lubotzky in 1990-s, is whether Cayley graphs of $SL_2({\mathbb F}_p)$ are expanders, for various choices of generating sets. The Theorem proves that they are expanders if the generating sets are chosen at random.

Links: The original paper is available here. See also Section 8.5 of this book for an accessible description of the Theorem.

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Elements of SL_2(F_p) have representations of length O((log p)^c)

You need to know: Arithmetic modulo p, field, field ${\mathbb F}_p$ (field with elements $0,1,\dots,p-1$ and operations modulo p, p prime), matrices, matrix multiplication, determinant of a $2 \times 2$ matrix, group, finite group, group $\text{SL}_2({\mathbb F}_p)$ (group of $2 \times 2$ matrices over ${\mathbb F}_p$ with determinant 1) , set of generators of a group. Also, non-abelian simple groups and their normal subsets (see here) are needed for the “context” section.

Background: For a finite group G with set of generators A, let $d(G,A)$ be the minimal integer m such that every $g\in G$ can be expressed as a product of at most m elements of A and their inverses.

The Theorem: On 1st September 2005, Harald Helfgott submitted to arxiv a paper in which he proved the existence of constants M and c such that $d(\text{SL}_2({\mathbb Z}/p{\mathbb Z}), A) \leq M (\log p)^c$ for every prime p and every set A of generators of $\text{SL}_2({\mathbb F}_p)$ .

Short context: Famous conjecture of Babai from 1992 states that $d(G,A) \leq M(\log |G|)^c$ for every set A of generators of a non-abelian finite simple group G with $|G|$ elements. The conjecture is open. See here for the solution of the special case when A is a normal set. The Theorem proves it for $G=\text{SL}_2({\mathbb F}_p)$ . The crucial contribution is that constants M and c in the Theorem does not depend neither on A nor on p.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 8.4 of this book for an accessible description of the Theorem.

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