There exist full-sized groups that are profinitely rigid in the absolute sense

You need to know: Group, finite and infinite group, abelian and non-abelian 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.

Background: A subgroup N of group G is called normal, if $g\cdot a\cdot g^{-1}\in N$ for any $a \in N$ and $g \in G$ . A (left) coset of N in G with respect to $g\in G$ , denoted $gN$ , is the set $\{g \cdot a: \, a\in N\}$ . If N is normal, cosets form a group with operation $(gN)\cdot (hN)=(g \cdot h)N$ , which is called the quotient group and denoted $G/N$ . If $G/N$ is a finite group, we say that N has a finite index. A group G is called residually finite if the intersection of all its normal subgroups of finite index is an identity element. Let $c(G)$ be the set of all finite quotients of G.

A group G is called finitely generated if it has a presentation $<S|R>$ with finite number of generators. A group G is free if R is an empty set. A group G is called full-sized if it has a non-abelian free subgroup. A finitely generated, residually finite group G is called profinitely rigid in the absolute sense, if, for any finitely generated residually finite group H, $c(G)=c(H)$ (up to isomorphism) implies that H is isomorphic to G.

The Theorem: On 11th November 2018, Martin Bridson, David McReynolds, Alan Reid and Ryan Spitler submitted to arxiv a paper in which they provided the first examples of full-sized groups that are profinitely rigid in the absolute sense.

Short context: One of the fundamental questions in the theory of finitely generated groups is to what extend such groups are determined by their set of finite quotients. To have a hope to recover group G from its finite quotients, we must assume that G is residually finite. It is conjectured that a broad class of residually finite groups are recoverable, but, before 2018, this was known only for some special groups like abelian. In particular, no examples of full-sized groups with this property was known. The Theorem provides first such examples.

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

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There exist finitely generated coarsely non-amenable groups that are coarsely embeddable into l^2

You need to know: Symmetric difference $A \Delta B = (A \setminus B) \cup (B \setminus A)$ of sets $A$ and $B$ , notation $|A|$ for the number of elements in finite set $A$ , notation $\times$ for set product, set ${\mathbb N}$ of natural numbers, metric space $(X; d)$ where $X$ is a set and $d$ is a metric, infinite dimensional Hilbert space $l^2$ , group, generating set of a group, finitely generated group, word metric of a group with respect to a generating set.

Background: We say that metric space $(X; d_X)$ is coarsely embeddable into metric space $(Y; d_Y)$ is there exists a map $f:(X; d_X)\to(Y; d_Y )$ such that $d_Y(f(x_n); f(y_n)) \to \infty$ if and only if $d_X(x_n; y_n) \to \infty$ for all sequences $(x_n)_{n\in {\mathbb N}}$ and $(y_n)_{n\in {\mathbb N}}$ in $X$ . We say that a finitely generated group $G$ is coarsely embeddable into $(Y; d_Y )$ if it is so for the word metric with respect to a finite generating set $S$ . This property does not depend on the choice of $S$ .

A metric space $(X; d)$ is called uniformly discrete if there exists a constant $r>0$ such that, for any $x,y \in X$ , we have either $x=y$ or $d(x,y)>r$ . A uniformly discrete metric space $(X; d)$ is coarsely amenable if for every $\epsilon>0$ and $R>0$ , there exist a constant $S>0$ and a collection of finite subsets $\{A_x\}_{x \in X}$ , $A_x \subseteq X \times {\mathbb N}$ for every $x\in X$ , such that (a) $|A_x \Delta A_y|/|A_x \cap A_y|\leq \epsilon$ when $d(x,y) \leq R$ , and (b) $A_x \subseteq B(x,S) \times N$ . A finitely generated group $G$ is called coarsely amenable if it is so for the word metric with respect to a finite generating set $S$ , and is called coarsely non-amenable otherwise.

The Theorem: On 19th June 2014, Damian Osajda submitted to arxiv a paper in which he proved the existence of finitely generated coarsely non-amenable groups that are coarsely embeddable into the infinite dimensional Hilbert space $l^2$ .

Short context: The notion of coarse amenability is a week version of amenability, and has many equivalent formulations and applications. In particular, it is known that any finitely generated coarsely amenable groups is coarsely embeddable into $l^2$ . The question whether the converse is true (Are groups coarsely embeddable into $l^2$ coarsely amenable?) is a natural question raised by a number of researchers. The Theorem provides a negative answer.

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

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The set of non-weakly mixing IETs has Hausdorff codimension at most 1/2

You need to know: Notation $f^k(x)=f(f(\dots f(x) \dots))$ for k-fold function composition, notation $f^{-k}(A)$ for set $\{x: f^k(x) \in A\}$ , Lebesgue measure, measurable sets, Lebesgue almost every, Hausdorff dimension of a set.

Background: Let $d \geq 2$ be an integer, and let $\Lambda_d \subset {\mathbb R}^{d-1}$ be the set of vectors $\lambda=(\lambda_1,\dots,\lambda_{d-1})$ such that $0<\lambda_1<\dots<\lambda_{d-1}<1$ . For $\lambda \in \Lambda_d$ , let $f_\lambda^*:[0,1)\to[0,1)$ be a function that divides $[0,1)$ into subintervals $[0,\lambda_1), [\lambda_1, \lambda_2), \dots, [\lambda_{d-1},1)$ and rearranges them in the opposite order. Function $f:[0,1)\to[0,1)$ is called weakly mixing if for every pair of measurable sets $A, B \subset [0,1)$ , $\lim\limits_{n\to\infty} \frac{1}{n}\sum\limits_{k=1}^{n-1} \left|m(f^{-k}(A) \cap B) -m(A)m(B)\right| = 0$ , where $m$ denotes the Lebesgue measure. Let $\text{dim}(S)$ and $\text{codim}(S)=d-1-\text{dim}(S)$ denote Hausdorff dimension and Hausdorff codimension of set $S \subset \Lambda_d$ , respectively.

The Theorem: On 2nd January 2018, Jon Chaika and Howard Masur submitted to arxiv a paper in which they proved that for every odd $d\geq 3$ , the set $S$ of all $\lambda$ such that $f_\lambda^*$ is not weakly mixing has $\text{codim}(S)=1/2$ .

Short context: Interval exchange transformations (IETs in short) are functions defined similarly to $f_\lambda^*$ but such that subintervals $[0,\lambda_1), [\lambda_1, \lambda_2), \dots, [\lambda_{d-1},1)$ are rearranged according to an arbitrary permutation $\pi$ , see here for details. In 2004, Ávila and Forni proved that almost all IETs are weakly mixing. In 2016, Avila and Leguil proved a stronger result that set S of non-weakly mixing IETs is not even full-dimensional, or, in other words, has a positive codimension. This raises the problem of determining this codimension. The Theorem solves this problem for permutation $\pi = (1,\dots,d)\to(d,\dots,1)$ with odd $d$ . It is a corollary of a more general result establishing inequality $\text{codim}(S) \leq 1/2$ for a broad class of permutations $\pi$ .

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

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The Szemerédi–Trotter estimate holds for hypersurfaces in R^d

You need to know: Euclidean space ${\mathbb R}^d$ , multivariate polynomial, degree of a polynomial (maximal degree of its monomials), notation $|S|$ for the number of elements in finite set S.

Background: We say that $V\subset {\mathbb R}^d$ is a hypersurface of degree $m$ , if there exists a polynomial $P(x_1, \dots, x_d)$ of degree $m$ such that $V$ is the solution set to the equation $P(x_1, \dots, x_d)=0$ . If ${\cal P}$ is a finite set of points in ${\mathbb R}^d$ , and ${\cal H}$ is a finite set of hypersurfaces, let $I({\cal P},{\cal H}) := \{(p,V)\in {\cal P}\times {\cal H} : p \in V\}$ be the set of incidences.

The Theorem: On 19th November 2018, Miguel Walsh submitted to arxiv a paper in which he proved the following result. Let $d \geq 2$ , $k, c \geq 1$ , and let ${\cal P}$ and ${\cal H}$ be finite sets of points and hypersurfaces in ${\mathbb R}^d$ satisfying the following conditions: (a) the degrees of the hypersurfaces in ${\cal H}$ are bounded by $c$ ; (b) the intersection of any family of $d$ distinct hypersurfaces in ${\cal H}$ is finite, and (c) for any subset of $k$ distinct points in ${\cal P}$ , the number of hypersurfaces in ${\cal H}$ containing them is bounded by $c$ . Then $|I({\cal P},{\cal H})| \leq C_{d,k,c}\left( |{\cal P}|^{\frac{k(d-1)}{dk-1}}|{\cal H}|^{\frac{d(k-1)}{dk-1}}+|{\cal P}|+|{\cal H}|\right)$ , where $C_{d,k,c}$ is a constant depending on $d,k,c$ .

Short context: Famous Szemerédi–Trotter Theorem, proved in 1983, states that if P is a finite set of distinct points in the plane and L is a finite set of distinct lines, then $|I(P,L)| \leq C(|P|^{2/3}|L|^{2/3}+|P|+|L|)$ for some absolute constant C. This bound is the best possible up to the constant factor. In 2016, Basu and Sombra conjectured that the same bound holds for hypersurfaces in ${\mathbb R}^d$ in place of lines on the plane. The Theorem confirms this conjecture. See here for a different generalization of the Szemerédi–Trotter Theorem.

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

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The Benjamini-Schramm conjecture is true for nonunimodular graphs

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

Background: Let $G=(V,E)$ be an infinite graph. Graph $G$ is called locally finite if every vertex $v\in V$ has finite degree. An automorphism of graph $G$ 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. For vertices $u,v \in V$ , let us write $u \sim v$ if there is an automorphism $\sigma$ such that $\sigma(u)=v$ . An (infinite) graph G is called quasi-transitive if its vertex set V can be partitioned into finitely many classes $V_1, \dots, V_k$ , such that $u \sim v$ if and only if vertices u and v belong to the same class $V_i$ . Let $S_u v$ be the set of vertices $w \in V$ such that there is an automorphism $\sigma$ such that $\sigma(u)=u$ and $\sigma(v)=w$ . A graph G is called unimodular if $|S_u v| = |S_v u|$ whenever $u \sim v$ , and nonunimodular otherwise.

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. For given G, let $p_c(G)$ be the infimum of all $p\in[0,1]$ such that the Bernoulli(p) bond 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 7th November 2017, Tom Hutchcroft submitted to the Journal of the AMS a paper in which he proved, among other results, that for any connected, locally finite, quasi-transitive, nonunimodular graph G, we have $p_c(G) < p_u(G)$ .

Short context: Inequality $p_c(G) < p_u(G)$ implies the existence of non-empty range of parameters $p\in(p_c, p_u)$ such that Bernoulli(p) bond percolation on G has many infinite connected components. Characterisation of graphs with this property is a well-known important open problem. In 1996, Benjamini and Schramm conjectured that a connected, locally finite, quasi-transitive graph $G$ has $p_c(G) < p_u(G)$ if and inly if it is nonamenable, see here for the definition. In fact, Gandolfi, Keane, and Newman proved the “only if” part in 1992, so only the “if” part remains open. The proof of this theorem can be extended to quasi-transitive graphs, and this implies the Benjamini-Schramm conjecture for planar graphs. The Theorem confirms the conjecture for nonunimodular graphs.

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

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For every Banach space, its Rademacher type and Enflo type coincide

You need to know: Metric space, Banach space, notation $\|.\|$ for the norm in the Banach space, independent random variables, expectation (denoted by E).

Background: Let $\epsilon_1, \epsilon_2, \dots$ be a sequence of indepedndent random variables, each equal to $+1$ or $-1$ with equal probabilities. We say that a Banach space $X$ has Rademacher type $p\in[1,2]$ if there exists a constant $C\in (0,\infty)$ such that inequality $E\left\|\sum\limits_{i=1}^n \epsilon_j x_j\right\|^p \leq C^p \sum\limits_{j=1}^n\|x_j\|^p$ holds for all $n\geq 1$ and all $x_1, \dots, x_n \in X$ . For $\epsilon=(\epsilon_1, \dots, \epsilon_n)$ , and index $j$ , denote $\epsilon^{j-}$ the vector $(\epsilon_1, \dots, -\epsilon_j, \dots, \epsilon_n)$ . We say that a Banach space $X$ has Enflo type $p$ if there exists a constant $C\in (0,\infty)$ such that inequality $E\|f(\epsilon)-f(-\epsilon)\|^p \leq C^p \sum\limits_{j=1}^n E\|f(\epsilon)-f(\epsilon^{j-})\|^p$ holds for all $n\geq 1$ and every function $f:\{-1,1\}^n \to X$ .

The Theorem: On 13th March 2020, Paata Ivanisvili, Ramon van Handel, and Alexander Volberg submitted to arxiv a paper in which they proved that a Banach space $X$ has Rademacher type $p\in[1,2]$ if and only if $X$ has Enflo type $p$ . In other words, Rademacher type and Enflo type coincide.

Short context: The notion of Rademacher type is one of the central and useful concepts in the theory of Banach spaces, and researchers have long tried to extend it to the general metric space. However, the original definition use addition in $X$ substantially, and is difficult to extend. In 1978, Enflo introduced a definition of type which is straightforward to extend to metric spaces, and conjectured that, for Banach spaces, the two notions of type coincide. If a Banach space $X$ has Enflo type $p$ , then, applying the definition to function $f(\epsilon)=\sum_{j=1}^n \epsilon_j x_j$ , it is easy to see that $X$ has Rademacher type $p$ as well. However, the converse direction of the conjecture remained open for decades, despite intensive study and a number of partial results. The Theorem proves the conjecture in full.

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

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Hyperbolic motions of any shape exist in the Newtonian N-body problem

You need to know: Euclidean space $E={\mathbb R}^d$ of dimension $d$ , Euclidean norm $||a||$ of vector $a$ , (second) derivative of a function $x:{\mathbb R}\to E$ , small o notation $o(.)$ .

Background: Let N point particles with masses $m_i > 0$ and positions $x_i(t) \in E$ at time $t$ are moving according to Newton’s laws of motion: $m_j \frac{d^2 x_j}{dt^2} = \sum\limits_{i \neq j} \frac{m_i m_j(x_i - x j)}{r_{ij}^3}, \, 1 \leq j \leq N$ , where $r_{ij}$ is the distance between $x_i$ and $x_j$ . The motion of particles is determined by the initial conditions: their masses and their positions and velocities at time $t=0$ . Let $\Omega$ be the set of configurations such that the motion has no collisions ( $r_{ij}(t)>0$ for all $i,j$ and $t$ ). A motion is called hyperbolic if each particle has a different limit velocity vector, that is, $\lim\limits_{t\to \infty} \frac{d x_j}{dt}=a_j \in E$ and $a_i \neq a_j$ whenever $i \neq j$ .

The Theorem: On 25th August 2019, Ezequiel Maderna and Andrea Venturelli submitted to arxiv a paper in which they proved that for the Newtonian N-body problem in a space E of dimension $d\geq 2$ , there are hyperbolic motions $x:[0;+\infty) \to E^N$ such that $x(t) = \sqrt{2h} t a + o(t)$ as $t \to \infty$ for any choice of $x_0 = x(0) \in E^N$ , for any $a=(a_1, \dots, a_N) \in \Omega$ normalized by $||a||=1$ , and for any constant $h>0$ .

Short context: The problem of describing motion of N bodies under gravitation (N-body problem) in Euclidean space is a fundamental problem in physics and mathematics, studied by many authors, see, for example, here and here. In general, the motion can be very complicated even for $N=3$ , but can we at least understand hyperbolic motions? The only explicitly known hyperbolic motions are such that the shape of the configuration does not change with time, but it is conjectured that there are only finitely many such motions for any fixed $N$ . In contrast, the Theorem states that hyperbolic motions exist for all initial configurations and all choices of the limited velocities.

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

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There are at most B_n X^(C log^3 n) degree n number fields with discriminant at most X

You need to know: Field, isomorphic and non-isomorphic fields. Also, see this previous theorem description for the concepts of number field, degree of a number field, and discriminant of a number field.

Background: For $X>0$ , let $N_n(X)$ denotes the number of non-isomorphic number fields of degree n with absolute value of the discriminant at most X.

The Theorem: On 31st July 2019, Jean-Marc Couveignes submitted to arxiv a paper in which he proved the existence of constant $C>0$ such that inequality $N_n(X) \leq n^{Cn\log^3 n} X^{C\log^3 n}$ holds for all integers $n\geq C$ and $X\geq 1$ .

Short context: Counting number fields up to isomorphism is a basic and important open problem in the area. There is a folklore conjecture that $N_n(X)$ grows as linear function of X for every fixed n, but this is known only for $n\leq 5$ , see here and here. For general n, the best upper bound was $N_n(X) \leq B_n X^{\exp(C\sqrt{\log n})}$ , see here. The Theorem significantly improves the last result and provides the first upper bound with exponent polynomial in $\log n$ . In a later work (submitted 28th May 2020), Robert Lemke Oliver and Frank Thorne improved the bound further to $N_n(X) \leq B_n X^{c \log^2 n}$ for $n\geq 6$ , where one can take $c=1.564$ .

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

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The Weyl-type upper bound holds for Dirichlet L-functions of cube-free conductor

You need to know: Set ${\mathbb Z}$ of integers, greatest common divisor (gcd) of 2 integers, coprime integers, cubefree integer (integer $n$ not divisible by $m^3$ for any integer $m\geq 2$ ), set ${\mathbb C}$ of complex numbers, notation $i$ for $\sqrt{-1}$ , absolute value $|z|$ and real part $\text{Re}(z)$ of complex number z, function of complex variable, meromorphic function, analytic continuation.

Background: Function $\chi:{\mathbb Z}\to{\mathbb C}$ is called a Dirichlet character modulo integer $q>0$ if (i) $\chi(n)=\chi(n+q)$ for all n, (ii) if $\text{gcd}(n,q) > 1$ then $\chi(n)=0$ ; if $\text{gcd}(n,q) = 1$ then $\chi(n) \neq 0$ , and (iii) $\chi(mn)=\chi(m)\chi(n)$ for all integers m and n. The conductor of $\chi$ is the least positive integer $q_1$ dividing $q$ for which $\chi(n+q_1)=\chi(n)$ for all $n$ coprime to $q$ . For any Dirichlet character $\chi$ , and complex number $z$ with $\text{Re}(z)>1$ , let $L(z,\chi)=\sum\limits_{n=1}^\infty \frac{\chi(n)}{n^z}$ . By analytic continuation, function $L(z,\chi)$ can be extended to a meromorphic function on the whole complex plane, and it is called a Dirichlet L-function.

The Theorem: On 6th November 2018, Ian Petrow and Matthew Young submitted to the Annals of Mathematics a paper in which they proved that for any $\epsilon>0$ there exist a constant $C_\epsilon$ such that inequality $|L(1/2+it,\chi)| \leq C_\epsilon q^{1/6 + \epsilon}(1+|t|)^{1/6 + \epsilon}$ holds for every Dirichlet character $\chi$ with a cubefree conductor $q$ , and for all real $t$ .

Short context: Dirichlet L-functions $L(z,\chi)$ are generalisations of famous Riemann zeta function (which corresponds to the case $\chi(n)=1$ for all $n$ ), and finding upper bounds for their values at line $z=1/2+it$ (which is called the critical line), is an important problem in number theory with many applications. See here for a related problem. For the general case, Burgess proved in 1963 the bound with exponent $3/16 + \epsilon$ , which remains unimproved for over 55 years. The Theorem improves the exponent to $1/6 + \epsilon$ in the case when the conductor $q$ is cubefree. Earlier, bound with this exponent was derived (by Weyl) only for the Riemann zeta function, and is known as the Weyl bound.

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

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The size of subset of {1,…,N} without 3-term arithmetic progressions is O(N/(log N)^(1+c)) for some c>0

You need to know: A (nontrivial) 3-term arithmetic progression, big O notation, small o notation.

Background: For integer $N>0$ , let $r_3(N)$ be the cardinality of the largest subset of $\{1, 2, \dots , N\}$ which contains no nontrivial 3-term arithmetic progressions.

The Theorem: On 7th July 2020, Thomas Bloom and Olof Sisask submitted to arxiv a paper in which they proved that $r_3(N) = O\left(\frac{N}{\log^{1+c} N}\right)$ for some absolute constant $c>0$ .

Short context: In 1936, Erdős and Turán conjectured that any set containing a positive proportion of integers must contain a 3-term arithmetic progression (3APs). This is equivalent to $\lim\limits_{N\to\infty}\frac{r_3(N)}{N}=0$ . In 1953, Roth confirmed this conjecture by proving that $r_3(N) = O\left(\frac{N}{\log\log N}\right)$ . Another famous conjecture of Erdős states that if A is a set of positive integers such that $\sum\limits_{n \in A}\frac{1}{n}$ diverges then A contains arithmetic progressions of length k for all k. The $k=3$ case of this conjecture was known to follow from $r_3(N) = o\left(\frac{N}{\log N}\right)$ . In 2011, Sanders came close to this by proving that $r_3(N) = O\left(\frac{N}{\log^{1-o(1)} N}\right)$ . The Theorem finally achieves the bound better than $\frac{N}{\log N}$ , and thus implies the $k=3$ case of the Erdős conjecture. Because there are $O\left(\frac{N}{\log N}\right)$ primes up to N, the Theorem also implies the 1939 Van der Corput theorem that the set of primes contains infinitely many 3APs, as well as this generalisation by Green.