The u-invariant of any field of transcendence degree 2 over R is at most 4

You need to know: Field, subfield, field ${\mathbb R}$ of real numbers, matrix, symmetric matrix, matrix mupliplication, trasnpose $A^T$ of matrix A, determinant $\text{det}(A)$ of $n\times n$ matrix A.

Background: A subset S of a field K is algebraically independent over a subfield F if the elements of S do not satisfy any non-trivial polynomial equation with coefficients in F. The largest cardinality of such S is called the transcendence degree of K over F, which we denote $d_F(K)$ . If $K \neq F$ but $d_F(K)=0$ , K is called proper algebraic extension of F. A field F is called formally real if there is a total order $<$ on F such that, for all $a,b,c \in F$ , (i) if $a<b$ then $a+c < b+c$ and (ii) if $0<a$ and $0<b$ , then $0<a\cdot b$ . A formally real field is real-closed if it admits no proper formally real algebraic extensions. Any formally real field F with given ordering $<$ has a unique real-closed algebraic extension K preserving $<,$ which we denote $\text{cl}(F,<)$ .

Every symmetric $n\times n$ matrix A with entries $a_{ij}\in F$ generates a quadratic form $f=\sum\limits_{i=1}^n \sum\limits_{j=1}^n a_{ij} x_i x_j$ of dimension n over F. This form is called nondegenerate if $\text{det}(A)\neq 0$ . It is called anisotropic if there is no non-zero vector $(x_1,\dots,x_n)$ on which the form evaluates to zero. Let $K(F)$ be the set of all nondegenerate anisotropic forms over F. Quadratic forms generated by matrices A and B are equivalent if $A=MBM^T$ for some $n\times n$ matrix M over F with $\text{det}(M)\neq 0$ . If F is real-closed, any $f\in K(F)$ is equivalent to a form $\sum\limits_{i=1}^r x_i^2 - \sum\limits_{i=r+1}^{r+s} x_i^2$ for some positive integers $r,s$ . The difference $r-s$ is called the signature of f. If F is formally real, the signature of $f\in K(F)$ with respect to ordering $<$ is the signature of f as a form over $\text{cl}(F,<)$ . Let $K_0(F)$ be the set of all $f\in K(F)$ whose signature with respect to any ordering $<$ of F is $0$ . If F is not formally real, define $K_0(F)=K(F)$ . The general u-invariant $u(F)$ of field F is the maximal dimension of any form $f\in K_0(F)$ .

The Theorem: On 10th April 2018, Olivier Benoist submitted to arxiv a paper in which he proved that if F is any field of transcendence degree 2 over ${\mathbb R}$ , then $u(F)\leq 4$ .

Short context: The u-invariant of a field, usually defined as the maximal dimension of any form $f\in K(F)$ , is an important and well-studied concept, see here. However, with this definition it is infinite for any formally real field. The general u-invariant, introduced by Elman and Lam in 1973, coincides with the “usual” one when it is finite, but makes sense for formally real fields as well. In 1981, Pfister conjectured that if a field F has transcendence degree d over ${\mathbb R}$ , then $u(F)\leq 2^d$ . The Theorem verifies the $d=2$ case of this conjecture.

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

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The volume exponent of the first Grigorchuk group is equal to 0.7674…

You need to know: Group, generating set of a group, finitely generated group, subgroup generated by a set of elements of a group.

Background: Let $T$ be the set of all finite strings of $0$ s and $1$ s together with the the empty string $\emptyset$ . For $x,y \in T$ , we write $x<y$ if x is an initial segment of y. Let $\text{Aut}(T)$ be the set of all length-preserving permutations $\sigma: T \to T$ such that $\sigma(x) < \sigma(y)$ whenever $x<y$ . With composition operation $(\sigma_1 \cdot \sigma_2)(x)=\sigma_1(\sigma_2(x))$ , $\text{Aut}(T)$ forms a group. Let $a,b,c,d$ be specific elements of $\text{Aut}(T)$ defined recursively by the following relations, which hold for all strings x: $a(0)=1$ , $a(1)=0$ , $b(0)=c(0)=d(0)=0$ , $b(1)=c(1)=d(1)=1$ , $a(0x)=1x$ , $a(1x)=0x$ , $b(0x)=0a(x)$ , $b(1x)=1c(x)$ , $c(0x)=0a(x)$ , $c(1x)=1d(x)$ , $d(0x)=0x$ , $d(1x)=1b(x)$ . Let G be a subgroup of $\text{Aut}(T)$ generated by $a,b,c,d$ . For any generating set S of G, let $v_{G,S}(n)$ be the number of elements of G which are the product of at most n elements in $S \cup S^{-1}$ .

The Theorem: On 25th February 2018, Anna Erschler and Tianyi Zheng submitted to arxiv a paper in which they proved that for any generating set S of G, we have $\lim\limits_{n\to\infty}\frac{\log \log v_{G,S}(n)}{\log n}=\alpha_0$ , where $\alpha_0=0.7674...$ is the constant defined by $\alpha_0=\frac{\log 2}{\log \lambda_0}$ , where $\lambda_0$ is the positive solution to the equation $x^3-x^2-2x-4=0$ .

Short context: The group G defined above is called the first Grigorchuk group. In 1984, Grigorchuk proved that $\exp(cn^{1/2})\leq v_{G,S}(n) \leq \exp(Cn^\beta)$ , where $c,C$ are positive constants, and $\beta=\log_{32}31 < 1$ . This implies that $v_{G,S}(n)$ grows faster than any polynomial but slower than exponential function $\exp(C'n)$ . Such groups are called groups of intermediate growth, and their existence was a major open question asked by Milnor in 1968 and answered by Grigorchuk by constructing the group G. Grigorchuk, and later other authors, also constructed some other examples of groups of intermediate growth (see here and here), but the group G is the simplest and most studied example. From Grigorchuk’s bound it is clear that the limit $\lim\limits_{n\to\infty}\frac{\log \log v_{G,S}(n)}{\log n}$ (which is called the volume exponent of G), if exists, is between $1/2$ and $\log_{32}31$ . The Theorem proves that this limit exists and computes it exactly.

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

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The Strassen’s direct sum conjecture is false: the rank of tensors is not additive with respect to the direct sum

You need to know: Only basic math for the Theorem. However, you need to know matrices, linear and bilinear systems of equations, and the concept of infinite field to fully understand the context section.

Background: Let $I,J,K$ be finite sets. Let $I \times J \times K$ be the set of triples $(i,j,k)$ with $i\in I$ , $j\in J$ , and $k\in K$ . A 3-dimensional tensor with real entries (or just “tensor” for short) is a function $T: I \times J \times K \to {\mathbb R}$ . A tensor T is called decomposable, if there exist real numbers $a_i, \, i\in I$ , and $b_j, \, j\in J$ , and $c_k, \, k\in K$ , such that $T(i,j,k)=a_ib_jc_k$ for all $(i,j,k)\in I \times J \times K$ . The rank $\text{rank}(T)$ of a tensor T is the smallest r for which T can be written as a sum of r decomposable tensors. The direct sum $T=T_1 \oplus T_2$ of tensors $T_1: I_1 \times J_1 \times K_1 \to {\mathbb R}$ and $T_2: I_2 \times J_2 \times K_2 \to {\mathbb R}$ (with disjoint indexing sets $I_1, I_2, J_1, J_2, K_1, K_2$ ) is the tensor $T: I \times J \times K \to {\mathbb R}$ , where $I=I_1 \cup I_2$ , $J=J_1 \cup J_2$ , $K=K_1 \cup K_2$ , such that (i) $T(i,j,k) = T_1(i,j,k)$ if $i\in I_1, j\in J_1, k\in K_1$ , (ii) $T(i,j,k) = T_2(i,j,k)$ if $i\in I_2, j\in J_2, k\in K_2$ , and (iii) $T(i,j,k) = 0$ for all other $(i,j,k) \in I \times J \times K$ .

The Theorem: On 22nd December 2017, Yaroslav Shitov submitted to arxiv a paper in which he proved the existence of 3-dimensional tensor $T_1$ and $T_2$ with real entries such that $\text{rank}(T_1 \oplus T_2) < \text{rank}(T_1) + \text{rank}(T_2)$ .

Short context: Tensors are natural generalizations of matrices and are central in many applications. In particular, 3-dimensional tensors serve as natural representation of bilinear systems of equations in the same way as matrices are used to represent linear systems. In 1973, Strassen conjectured that the complexity (that is, number of multiplicative operations needed for solution) of the union of two bilinear systems depending on different variables is equal to the sum of the complexities of both systems. In the language of tensors, the conjecture states that $\text{rank}(T_1 \oplus T_2) = \text{rank}(T_1) + \text{rank}(T_2)$ for any tensors $T_1, T_2$ , and it became known as the Strassen’s direct sum conjecture. A similar statement is true for matrices, and was widely believed to be true for tensors. The Theorem, however, provides a counterexample. In fact, Shitov also disproved this conjecture for tensors whose entries are in an arbitrary infinite field.

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

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There exist finitely generated simple groups of intermediate growth

You need to know: Group, simple group.

Background: A generating set of a group G is a subset $S \subset G$ such that every $g\in G$ can be written as a finite product of elements of S and their inverses. Group G is called finitely generated if it has a finite generating set S. Let $\gamma_S(n)$ be the number of elements of G which are the product of at most n elements in $S \cup S^{-1}$ . We say that group G (i) has a polynomial growth rate if $\gamma_S(n) \leq C(n^k+1)$ for some constants $C,k < \infty$ ; (ii) has an exponential growth rate if $\gamma_S(n) \geq a^n$ for some $a>1$ , and (iii) is of intermediate growth if neither (i) nor (ii) is true. Properties (i)-(iii) does not depend on S.

The Theorem: On 6th January 2016, Volodymyr Nekrashevych submitted to arxiv a paper in which he, among other results, proved the existence of finitely generated simple groups of intermediate growth.

Short context: The rate of growth of $\gamma_S(n)$ can tell a lot about structure of the underlying group G, and is one of the central research directions in group theory. The existence of groups of intermediate growth was an open question, until the first examples were constructed by Grigorchuk in 1980. Since then, other constructions have been discovered, but all known groups of intermediate growth were not simple. The Theorem provides the first examples of simple groups of intermediate growth, affirnatively answering the 1984 question of Grigorchuk. In particular, the Theorem generalises (and implies) this previous result about existence of finitely generated simple amenable groups (because every group of intermediate growth is amenable).

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

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All Robbins conjectures on the number of ASMs with symmetries are true

You need to know: Notation $n!=1\cdot 2 \cdot \dots \cdot n$ , product notation $\prod\limits_{i=1}^n$ , matrices.

Background: An $n \times n$ matrix with elements $a_{ij}$ is called an alternating-sign matrix (ASM) if (i) all $a_{ij}$ are equal to -1, 0, or 1, (ii) the sum of elements in each row and column is 1, and (iii) the non-zero elements in each row and column alternate in sign. An ASM is called diagonally and antidiagonally symmetric (DASASM) if $a_{ij} = a_{ji} = a_{n+1-j,n+1-i}$ for all $1\leq i,j \leq n$ .

The Theorem: On 18th December 2015, Roger Behrend, Ilse Fischer, and Matjaž Konvalinka submitted to arxiv a paper in which they proved that, for any positive integer n, the number of $(2n+1)\times(2n+1)$ DASASMs is given by $\prod\limits_{i=0}^{n} \frac{(3i)!}{(n+i)!}$ .

Short context: Alternating-sign matrices are important both in pure mathematics (determinant computation) and applications (so-called “six-vertex model” for ice modelling in statistical mechanics). The formula for the total number of ASMs was conjectured by Robbins and Rumsey in 1986 and proved by Zeilberger in 1996. In 1991, Robbins conjectured formulas for the number of ASMs with various symmetries. For many years, these conjecture served as a roadmap, stimulating progress in the area. Starting from work of Kuperberg, see here, several groups of researchers resolved by 2006 all these conjectures except one: the formula for the number of $(2n+1)\times(2n+1)$ DASASMs. The Theorem resolves this remaining conjecture.

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

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If N is a product of two prime powers, map (x,y)->x^Ny^N is surjective on every finite non-abelian simple group

You need to know: Prime number, group, identity element, finite 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.

The Theorem: On 1st May 2015, Robert Guralnick, Martin Liebeck, Eamon O’Brien, Aner Shalev, and Pham Tiep submitted to Inventiones Mathematicae a paper in which they proved the following result. Let p, q be prime numbers, let a, b be non-negative integers, and let $N=p^aq^b$ . Then every element g of every finite non-abelian simple group G can be written as $g=x^Ny^N$ for some $x,y \in G$ .

Short context: Given group G, let w be a map mapping pair $x,y \in G$ into $w(x,y)=x^Ny^N$ . Let $w(G)$ be the set of all $g\in G$ such that $g=w(x,y)$ for some $x,y\in G$ . Map w is called surjective on G if $w(G)=G$ . In this language, the Theorem states that w is surjective on every finite non-abelian simple group G. Earlier, this result was proved for commutator map $w(x,y)=x^{-1}y^{-1}xy$ , see here. Also, it is known that $w(G)\cdot w(G)=G$ for much broader class of maps w, see here.

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

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No algorithm can decide if a finitely presented group has a proper finite-index subgroup

You need to know: Group, subgroup, proper subgroup, presentation of a group in the form $<S|R>$ , where S is a set of generators and R is a set of relations.

Background: Let G be a group and H be a subgroup. A (left) coset of H in G with respect to $g\in G$ is the set $\{gh: \, h\in H\}$ . We say that subgroup H is of a finite index is there are only finitely many different cosets of H in G. A group G is called finitely presented if it has a presentation $<S|R>$ with finite number of generators and finite set of relations.

The Theorem: On 1st November 2013, Martin Bridson and Henry Wilton submitted to Inventiones Mathematicae a paper in which they proved that there is no algorithm that can determine whether or not a finitely presented group has a proper subgroup of finite index.

Short context: A word in group G is any product of generators and their inverse elements. The word problem in G is the problem to decide whether a given word represents an identity element. In 1955, Novikov proved that there is no algorithm that can solve the word problem in all finitely presented groups. The problems that no algorithm can solve are called undecidable. After Novikov’s result, many other basic problems for finitely presented groups have been proved to be undecidable. However, the status of the problem to decide if such a group has a proper subgroup of finite index remained open. The Theorem proves that this problem is undecidable as well.

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

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Most odd degree hyperelliptic curves have no finite rational points

You need to know: Polynomial, degree of a polynomial.

Background: Let ${\cal P}$ be the set of odd-degree polynomials $P(x)=\sum\limits_{i=0}^{2g+1} a_i x^{2g+1-i}$ with rational coefficients $a_i$ and leading coefficient $a_0=1$ . With change of variables $x'=u^2x$ , $y'=u^{2g+1}y$ , we can have new coefficients $a'_i=u^{2i}a_i$ , $i=1,2,\dots,2g+1$ , and, by selecting $u$ to be the common denominator of $a_1, a_2, \dots, a_{2g+1}$ , we can make all coefficients integers. After this, define height of $P\in{\cal P}$ by $H(P) = \max\{|a_1|, |a_2|^{1/2}, \dots, |a_{2g+1}|^{1/(2g+1)}\}$ . For fixed integer $g>0$ and real $X>0$ , let $\mu(X,g)$ be a fraction of the polynomials $P\in {\cal P}$ of degree $2g+1$ and height less than $X$ for which the equation $y^2=P(x)$ has no rational solutions.

The Theorem: On 1st February 2013, Bjorn Poonen and Michael Stoll submitted to arxiv a paper in which they proved that $\lim\limits_{g\to\infty}(\lim\limits_{X\to\infty}\mu(X,g)) = 1$ .

Short context: Set of real solutions to $y^2=P(x)$ for $P \in {\cal P}$ is known as odd degree hyperelliptic curve, and rational solutions are called finite rational points on this curve. In this terminology, the Theorem states that most odd degree hyperelliptic curves have no finite rational points. Moreover, Poonen and Stoll also proved for “almost all” polynomials $P\in{\cal P}$ in the same sense as in the Theorem, there is an explicit algorithm, with polynomial P as an input, and output certifying that there are indeed no rational solutions to $y^2=P(x)$ . In other words, there exists a universal method able to solve almost all equations in the form $y^2=P(x), \, P \in {\cal P}$ at once!

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

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The Waring rank of any monomial x_1^(a_1)…x_n^(a_n) with a_1<=…<=a_n is (a_2+1)…(a_n+1)

You need to know: Complex numbers, polynomials in n variables and their multiplication.

Background: The linear form in n variables in an expression in the form $L=c_1x_1+\dots+c_nx_n$ , where $c_1,c_2, \dots, c_n$ are come complex coefficients. The Waring rank $\text{rk}(M)$ of monomial $M=x_1^{a_1}x_2^{a_2}\dots x_n^{a_n}$ of degree $d=\sum\limits_{i=1}^n a_i$ is the least value of s for which there exist linear forms $L_1,\dots,L_s$ such that $M=\sum\limits_{i=1}^s L_i^d$ .

The Theorem: On 2nd July 2012, Enrico Carlini, Maria Catalisano, and Anthony Geramita submitted to the Journal of Algebra a paper in which they proved the following result. Let $M=x_1^{a_1}x_2^{a_2}\dots x_n^{a_n}$ be a monomial in $n\geq 2$ variables, and let $1 \leq a_1 \leq \dots \leq a_n$ . Then $\text{rk}(M)=\prod\limits_{i=2}^n(a_i+1)$ .

Short context: Every natural number is the sum of at most 4 squares, or 9 cubes, or 19 fourth powers, etc. Waring’s problem asks whether for each positive integer k there exists positive integer $g(k)$ such that every natural number is the sum of at most $g(k)$ k-th powers. The Hilbert–Waring theorem, proved by Hilbert in 1909, gives a positive answer, and the exact formula for $g(k)$ has been derived in later works. The Theorem solves a similar problem for monomials.

Links: The original paper is available here.

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There exist finitely generated simple groups that are infinite and amenable

You need to know: Group, finite and infinite groups, isomorphic and nonisomorphic groups, simple group, finitely generated group.

Background: A group G is called amenable if there is a map $\mu$ 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 10th April 2012, Kate Juschenko and Nicolas Monod submitted to arxiv a paper in which they proved the existence of finitely generated simple groups that are infinite and amenable.

Short context: Amenable groups were introduced by von Neumann in 1929 in relation to Banach–Tarski paradox, see here, and are extensively studied since that. However, before 2012, it was an open question if there exists any finitely generated simple group that is infinite and amenable. The Theorem proves that such groups exist. Moreover, Juschenko and Monod proved that there are infinitely many (in fact uncountably many) nonisomorphic such groups. See here for even more general result proved in a later work.

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

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