The Grothendieck constant is strictly smaller than Krivine’s bound

You need to know: Euclidean space ${\mathbb R}^n$ , inner product $\langle x,y\rangle=\sum\limits_{i=1}^n x_i y_i$ in ${\mathbb R}^n$ , unit vector in ${\mathbb R}^n$ , matrix, big O notation.

Background: In 1953, Grothendieck proved the existence of a universal constant $K<\infty$ such that, given any $m\times n$ matrix with real entries $a_{ij}$ and arbitrary unit vectors $x_1,\dots, x_m,y_1,\dots, y_n$ in ${\mathbb R}^{m+n}$ , there are signs $\epsilon_1,\dots,\epsilon_m,\delta_1,\dots,\delta_n \in \{-1, +1\}$ such that inequality $\sum\limits_{i=1}^m \sum\limits_{j=1}^n a_{ij}\langle x_i, y_j \rangle \leq K \sum\limits_{i=1}^m \sum\limits_{j=1}^n a_{ij}\epsilon_i \delta_j$ holds. This is called the Grothendieck’s inequality. The minimal constant K for which it holds is called the Grothendieck constant, and is denoted $K_G$ . See here for a version of the Grothendieck constant of a graph.

The Theorem: On 31st March 2011, Mark Braverman, Konstantin Makarychev, Yury Makarychev, and Assaf Naor submitted to arxiv a paper in which they proved the existence of a constant $\epsilon_0>0$ such that $K_G < \frac{\pi}{2\log(1+\sqrt{2})}-\epsilon_0$ .

Short context: The Grothendieck’s inequality, despite being innocent-looking at the first glance, turned out to be important in many areas of mathematics and applications, such as functional analysis, harmonic analysis, operator theory, quantum mechanics, and computer science. For many applications, it is important to have it with the best possible constant. However, despite substantial efforts, even the second digit of $K_G$ remains unknown. The best lower bound is $K_G\geq 1.67696...$ proved by Davie in 1984. For the upper bound, Krivine proved in 1979 that $K_G \leq \frac{\pi}{2\log(1+\sqrt{2})}=1.7822...$ , and conjectured that $K_G$ is actually equal to this value. This conjecture was well-believed, and many researchers focused on proving the matching lower bound. The Theorem states that Krivine’s conjecture is false.

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

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If E is a compact set on the plane with dimension >5/4, then its distance set has positive Lebesgue measure

You need to know: Euclidean plane ${\mathbb R}^2$ , compact set on the plane, distance $|x-y|$ between points $x,y$ on the plane, Lebesgue measure, Hausdorff dimension.

Background: For a set $E \subset {\mathbb R}^2$ , define the distance set $\Delta(E) = \{|x-y|: x \in E, y \in E\}$ .

The Theorem: On 28th August 2018, Larry Guth, Alex Iosevich, Yumeng Ou, and Hong Wang submitted to arxiv a paper in which they proved that if $E \subset {\mathbb R}^2$ is a compact set with Hausdorff dimension greater than $5/4$ , then $\Delta(E)$ has positive Lebesgue measure.

Short context: In 1985, Falconer posed the following problem. What is the smallest constant $c(d)$ such that every compact set E in ${\mathbb R}^d$ with the Hausdorff dimension larger than $c(d)$ must have the distance set $\Delta(E)$ of positive Lebesgue measure? Falconer proved that $\frac{d}{2}\leq c(d) \leq \frac{d+1}{2}$ , and conjectured that in fact $c(d)=\frac{d}{2}$ . This is known as the Falconer Distance Conjecture. For the plane, it predicts that $c(2)=1$ . In 1999, Wolff proved that $c(2)\leq \frac{4}{3}$ , The Theorem improves this to $c(2)\leq \frac{5}{4}$ .

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

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There exist finitely generated infinite simple left orderable groups

You need to know: Group, finite and infinite group, simple group, total order $\leq$ on a set.

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. A group G with group operation + is called left orderable if there exists a total order $\leq$ on G such that $a\leq b$ implies $c+a \leq c+b$ for all $a,b,c \in G$ .

The Theorem: On 17th July 2018, James Hyde and Yash Lodha submitted to arxiv a paper in which they proved the existence of finitely generated infinite simple left orderable groups.

Short context: The Theorem answered a long-standing open question posed by Rhemtulla in 1980. In fact, the authors proved the existence of continuum many isomorphism types of such groups. The same is true for right orderable groups (groups G with a total order $\leq$ such that $a\leq b$ implies $a+c \leq b+c$ for all $a,b,c \in G$ ).

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

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The hot spots conjecture is true for all triangles in the plane

You need to know: Euclidean plane ${\mathbb R}^2$ , smooth (that is, differentiable) function in 2 variables, first and second order partial derivatives, notation $\Delta f = \frac{\partial^2 f}{\partial x_1^2}+\frac{\partial^2 f}{\partial x_2^2}$ , directional derivative $\frac{\partial f}{\partial n}$ for a vector n, normal vector.

Background: Let $\Omega \subset {\mathbb R}^2$ be a domain in ${\mathbb R}^2$ . In fact, we may assume that $\Omega$ is a triangle. Let $\partial\Omega$ denote the boundary of $\Omega$ . The second Neumann eigenvalue $\mu_2=\mu_2(\Omega)$ is the smallest positive real number such that there exists a not identically zero, smooth function $u : \Omega \to {\mathbb R}$ that satisfies the equation $\Delta u =-\mu_2 \cdot u$ on $\Omega$ and the boundary condition $\left.\frac{\partial u}{\partial n}\right|_{\partial \Omega} \equiv 0$ at the smooth points of $\partial\Omega$ , where n denotes the outward pointing normal vector (see here for an equivalent definition of $\mu_2$ ). A function u that satisfies these conditions is called the second Neumann eigenfunction for $\Omega$ .

The Theorem: On 6th February 2018, Chris Judge and Sugata Mondal submitted to arxiv a paper in which they proved that, for any triangle $\Omega$ , the second Neumann eigenfunction attains its extrema at the boundary of $\Omega$ .

Short context: The conjecture that the second Neumann eigenfunction attains its extrema at the boundary of $\Omega$ is known as the hot spot conjecture. In simple English, it states that if a flat piece of metal is given an initial heat distribution which then flows throughout the metal, then, after some time, the hottest point on the metal will lie on its boundary. The conjecture was proposed by Rauch in 1974. In 1999, Burdzy and Werner constructed a domain (with holes) for which the conjecture fails, but it still believed to be true for all convex domains. For triangles, the conjecture has been known to hold for obtuse triangles, right triangles, and acute triangles with at least one angle less than $\pi/6$ . The Theorem confirms the conjecture for all triangles. See here for another partial result towards the conjecture.

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

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The Pólya conjecture for the Neumann eigenvalues holds for the second eigenvalue

You need to know: Euclidean space ${\mathbb R}^n$ , open subsets of ${\mathbb R}^n$ , bounded subsets of ${\mathbb R}^n$ , notation $|\Omega|$ for the volume of $\Omega \subset {\mathbb R}^n$ , differentiable function on ${\mathbb R}^n$ , its gradient $\nabla u(x)=\left(\frac{\partial u}{\partial x_1}(x), \dots, \frac{\partial u}{\partial x_n}(x) \right)$ , integral over $\Omega \subset {\mathbb R}^n$ , normed space, subspace, dimension of a subspace, compactly embedded space.

Background: Let $n\geq 2$ , and let $\Omega \subset {\mathbb R}^n$ be a bounded open set. Let $L^2(\Omega)$ be the set of functions $u:\Omega\to{\mathbb R}$ with norm $||u||_2:=\left(\int_\Omega |u(x)|^2 dx\right)^{1/2}<\infty$ . Let $H^1(\Omega)$ be the set of differentiable functions $u:\Omega\to{\mathbb R}$ with norm $||u||_H := \left(\int_\Omega\left(|\nabla u(x)|^2 + |u(x)|^2\right)dx\right)^{1/2}<\infty$ . We say that domain $\Omega$ is regular if $H^1(\Omega)$ is compactly embedded in $L^2(\Omega)$ . For every integer $k\geq 1$ , let ${\cal S}_k$ be the family of all subspaces of dimension k in $\{u\in H^1(\Omega) :\int_\Omega u(x) dx = 0\}$ , and let $\mu_k(\Omega)=\min\limits_{S \in {\cal S}_k} \max\limits_{u\in S}\frac{\int_\Omega |\nabla u(x)|^2 dx}{\int_\Omega |u(x)|^2 dx}$ . If $B \subset {\mathbb R}^n$ is a ball, the quantity $\mu_2^*=2^{2/n}|B|^{2/n}\mu_1(B)$ is a constant which does not depend on B.

The Theorem: On 22nd January 2018, Dorin Bucur and Antoine Henrot submitted to Acta Mathematica a paper in which they proved that inequality $|\Omega|^{2/n}\mu_2(\Omega) \leq \mu_2^*$ holds for every regular set $\Omega \subset {\mathbb R}^n$ , with equality if $\Omega$ is the union of two disjoint, equal balls.

Short context: The sequence $\mu_k(\Omega)$ is known as “the spectrum of the Laplace operator with Neumann boundary conditions”, and is well-studied in mathematics with applications in physics. In 1950-th, Szegő and Weinberger proved that $|\Omega|^{2/n}\mu_1(\Omega)$ is maximised when $\Omega$ is the ball in ${\mathbb R}^n$ . The Theorem solves the maximization problem for $|\Omega|^{2/n}\mu_k(\Omega)$ for $k=2$ . As one of the applications, it immediately implies the $k=2$ case of important Polya conjecture for the Neumann eigenvalues, which states that $\mu_k(\Omega) \leq 4\pi^2\left(\frac{k}{w_n|\Omega|}\right)^{2/n}$ , where $w_n$ is the volume of the unit ball in ${\mathbb R}^n$ .

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

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A typical 1-Lipschitz image of a purely n-unrectiable subset of R^k has H^n dimension 0

You need to know: Euclidean space ${\mathbb R}^n$ , norm $||x||_n=\sqrt{\sum_{i=1}^nx_i^2}$ of $x=(x_1,\dots,x_n)\in {\mathbb R}^n$ , countable set, n-dimensional Hausdorff measure $H^n$ (see here), metric space, dense open set in a metric space.

Background: A function $f:{\mathbb R}^n \to {\mathbb R}^m$ is called K-Lipschitz if $||f(x)-f(y)||_m \leq K||x-y||_n$ for all $x,y\in {\mathbb R}^n$ . The set $\{y\in {\mathbb R}^m\,|\,y=f(x)\}$ for some K-Lipschitz $f:{\mathbb R}^n \to {\mathbb R}^m$ is called Lipschitz image of ${\mathbb R}^n$ . A subset $S\subset {\mathbb R}^m$ is called n-rectifiable if it can be covered by a set of $H^n$ measure zero and a countable number of Lipschitz images of ${\mathbb R}^n$ . A set S is purely n-unrectifiable if all of its n-rectifiable subsets have $H^n$ measure zero. Let $\text{Lip}_1(n.m)$ be the set of all bounded 1-Lipschitz functions $f:{\mathbb R}^n \to {\mathbb R}^m$ equipped with the norm $||f||=\sup\limits_{x \in {\mathbb R}^n}||f(x)||_m$ . A subset ${\cal F}\subseteq \text{Lip}_1(n.m)$ is called residual if it contains a countable intersection of dense open sets.

The Theorem: On 19th December 2017, David Bate submitted to arxiv a paper in which he proved that for any purely n-unrectifiable $S\subset {\mathbb R}^k$ with $H^n(S)<\infty$ , the set of all $f\in\text{Lip}_1(k.m)$ with $H^n(f(S))=0$ is residual in $\text{Lip}_1(k.m)$ . Conversely, if S is not purely n-unrectifiable, then the set of $f\in\text{Lip}_1(k.m)$ with $H^n(f(S))>0$ is residual.

Short context: Rectifiable and purely unrectifiable sets are a central object of study in geometric measure theory. The term “residual” is just a formalisation of intuitive concept of “almost all” Lipschitz functions. Hence, the Theorem states that for “almost every” function $f\in\text{Lip}_1(k.m)$ , we have $H^n(f(S))=0$ if and only if set S is purely n-unrectifiable. This gives a complete and useful characterisation of such sets. In fact, Bate derived a similar characterisation of purely unrectifiable sets is general metric spaces.

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

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There is no coarsely minimal infinite-dimensional Banach space

You need to know: Metric space, Banach space, infinite dimensional Banach space.

Background: We say that metric space X coarsely embeds into metric space Y if there is a function $f:X\to Y$ and non-decreasing functions $\rho_1, \rho_2 : [0,\infty)\to[0,\infty)$ such that (i) $\rho_1(d_X(x, y)) \leq d_Y (f(x), f(y)) \leq \rho_2(d_X(x, y))$ for all $x, y \in X$ , and $\lim\limits_{t\to\infty}\rho_1(t)=+\infty$ . A Banach space is called coarsely minimal if it coarsely embeds into every infinite-dimensional Banach space.

The Theorem: On 18th May 2017, Florent Baudier, Gilles Lancien, and Thomas Schlumprecht submitted to arxiv and the Journal of the AMS a paper in which they proved that there is no coarsely minimal infinite-dimensional Banach space

Short context: There was an important open problem asking weather the $l^2$ -space (the space of infinite sequences $x=(x_1, x_2, \dots, x_n, \dots)$ equipped with norm $|x| := \sqrt{\sum\limits_{i=1}^\infty x_i^2} < \infty$ ) is coarsely minimal. There were some evidence and partial results supporting that the answer may be positive. It is would be so, this would imply the solution of some other important problems in the field. The Theorem, however, states that the answer is negative in a strong sence: not only $l^2$ is not coarsely minimal, but, moreover, there are no coarsely minimal infinite-dimensional Banach spaces at all.

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

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There are CN^(2p-4)(1+o(1)) meanders with at most 2N crossings and p minimal arcs

You need to know: Simple closed curve in the plane, transversal intersection of a curve and a line, topological configuration, binomial coefficient ${n\choose k}=\frac{n!}{k!(n-k)!}$ , small o notation.

Background: A meander is a topological configuration of a line and a simple closed curve in the plane intersecting transversally. We called crossings the points at which curve and lines intersects. There is always an even number of crossings which we denote $2N$ . An arc is the part AB of the curve between crossings A and B which do not contain any other crossings. If the interval AB on the line also does not contain any other crossings, the arc AB is called minimal. Let $M_p(N)$ be the number of meanders with exactly p minimal arcs and with at most $2N$ crossings.

The Theorem: On 15th May 2017, Vincent Delecroix, Elise Goujard, Peter Zograf, and Anton Zorich submitted to arxiv a paper in which they proved that $M_p(N)=C_pN^{2p-4}+o(N^{2p-4})$ as $N\to\infty$ (with $p\geq 3$ fixed), where $C_p=\frac{2^{p-3}}{\pi^{2p-4}p!(p-2)!}{{2p-2}\choose {p-1}}^2$ .

Short context: Meanders were studied already by Poincaré over 100 year ago, and appear in various areas of mathematics, and in applications, for example, in physics. A long-standing open problem is to derive the precise asymptotic how the number of meanders with $2N$ crossings grows with N. The Theorem solves a version of this problem when we also assume that the number p of minimal arcs is fixed.

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

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Every Besicovitch set in R^3 has dimension at least 5/2+e for some e>0

You need to know: Euclidean space ${\mathbb R}^3$ , unit line segment in ${\mathbb R}^3$ , direction of a line segment, compact sets in ${\mathbb R}^3$ , norm $||x||=\sqrt{x_1^2+x_2^2+x_3^2}$ of $x=(x_1,x_2,x_3)\in {\mathbb R}^3$ , diameter $\text{diam}\,U=\sup\limits_{x,y \in U}||x-y||$ of set $U\subset {\mathbb R}^3$ .

Background: A Besicovitch set in ${\mathbb R}^3$ is a compact set $X\subset {\mathbb R}^3$ that contains a unit line segment pointing in every direction.

For $\delta>0$ , a $\delta$ -cover of set $S \subset {\mathbb R}^3$ is a sequence of sets $U_1, U_2, \dots, U_i, \dots$ of diameters $\text{diam}\,U_i\leq \delta$ for all i such that $S \subset \bigcup\limits_{i=1}^\infty U_i$ . For $d\geq 0$ , let $H_\delta^d(S) = \inf \sum\limits_{i=1}^{\infty}(\text{diam}\,U_i)^d$ , where the infimum is taken over all $\delta$ -covers of S. Let $H^d(S)=\lim\limits_{\delta\to 0} H_\delta^d(S)$ . The Hausdorff dimension of S is $\text{dim}(S)=\inf\{d\geq 0\,:\,H^d(S)=0\}$ .

The Theorem: On 24th April 2017, Nets Katz and Joshua Zahl submitted to arxiv a paper in which they proved the existence of constant $\epsilon_0>0$ , such that every Besicovitch set in ${\mathbb R}^3$ has Hausdorff dimension at least $\frac{5}{2}+\epsilon _0$ .

Short context: The ( ${\mathbb R}^3$ case of) famous Kakeya conjecture states that every Besicovitch set in ${\mathbb R}^3$ must have Hausdorff dimension equal to 3. In 1995, Wolff proved that every such set must have Hausdorff dimension at least $\frac{5}{2}$ . For about 22 years, noone was able to improve the Wolff’s lower bound. The Theorem provides a small improvement.

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

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Vertical versus horizontal isoperimetric inequality holds in Heisenberg groups

You need to know: Set ${\mathbb Z}$ of integers, set ${\mathbb N}$ of positive integers, notation $A\times B$ for the set of pairs $(x,y)$ with $x\in A$ and $y\in B$ , group G, notation $g^{-1}$ for an inverse element of $g \in G$ , notation $1$ for the identity element, notation $g^n$ for $g\cdot g \cdot \dots \cdot g$ ( $n$ times) and $g^{-n}$ for $(g^{-1})^n$ , commutator $[g,h]=ghg^{-1}h^{-1}$ for $g,h \in G$ , set of generators of a group, presentation of a group in the form $<S|R>$ , where S is a set of generators and R is a set of relations, notation $|A|$ for the cardinality (the number of elements) of finite set A.

Background: For every $k \in {\mathbb N}$ , the $k$ -th discrete Heisenberg group, denoted $H^{2k+1}_{\mathbb Z}$ , is the group with $2k+1$ generators $a_1, b_1, \dots, a_k, b_k, c$ and relations $[a_1,b_1]=\dots=[a_k,b_k]=c$ and $[a_i, a_j]=[b_i, b_j]=[a_i, b_j ]=[a_i, c]=[b_i, c] =1$ for every distinct $i,j \in \{1,\dots,k\}$ . The horizontal perimeter of a finite $\Omega \subseteq H^{2k+1}_{\mathbb Z}$ is $|\partial_h \Omega|$ , where $\partial_h \Omega = \{(x,y)\in \Omega \times (H^{2k+1}_{\mathbb Z} \setminus \Omega)\,:\,x^{-1}y \in S_k\}$ , where $S_k=\{a_1, b_1, a_1^{-1}, b_1^{-1}, \dots, a_k, b_k, a_k^{-1}, b_k^{-1}\}$ . The vertical perimeter of $\Omega$ is $|\partial_v \Omega| = \left(\sum\limits_{t=1}^\infty \frac{|\partial_v^t \Omega|^2}{t^2}\right)^{\frac{1}{2}}$ , where $\partial_v^t \Omega = \{(x,y)\in \Omega \times (H^{2k+1}_{\mathbb Z} \setminus \Omega)\,:\,x^{-1}y \in \{c^t, c^{-t}\}\}$ .

The Theorem: On 3rd January 2017, Assaf Naor and Robert Young submitted to arxiv a paper in which they proved the existence of a universal constant $C<\infty$ such that inequality $|\partial_v \Omega| \leq \frac{C}{k}|\partial_h \Omega|$ holds for every integer $k\geq 2$ and every finite subset $\Omega \subseteq H^{2k+1}_{\mathbb Z}$ .

Short context: The inequality proved in the Theorem is called vertical versus horizontal isoperimetric inequality and has applications in the embedding theory for metric spaces and in the analysis of complexity of algorithms for combinatorial problems, such as the Sparsest Cut Problem (see here).

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

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