The Kadison-Singer problem has a positive solution

You need to know: Set ${\mathbb C}$ of complex numbers, complex conjugate $\bar{z}$ and absolute value $|z|$ of $z\in {\mathbb C}$ .

Background: Let ${\mathbb C}^d$ be the set of vectors $x=(x_1,\dots,x_d)$ with complex components $x_i$ . Denote $\langle x,y \rangle = \sum\limits_{i=1}^d x_i \bar{y_i}$ the inner product in ${\mathbb C}^d$ . Let $||x||=\sqrt{\langle x,x \rangle}$ be the norm in ${\mathbb C}^d$ . We say that $u\in {\mathbb C}^d$ is a unit vector if $||u||=1$ .

The Theorem: On 17th June 2013, Adam Marcus, Daniel Spielman, and Nikhil Srivastava submitted to arxiv a paper in which they proved the following result. There exist universal constants $\eta\geq 2$ and $\theta>0$ so that the following holds. Let $w_1,\dots,w_m \in {\mathbb C}^d$ satisfy $||w_i|| \leq 1$ for all i and suppose $\sum\limits_{i=1}^m |\langle u,w_i \rangle|^2=\eta$ for every unit vector $u\in {\mathbb C}^d$ . Then there exists a partition $S_1, S_2$ of $\{1,\dots,m\}$ so that $\sum\limits_{i\in S_j} |\langle u,w_i \rangle|^2 \leq \eta - \theta$ , for every unit vector $u\in {\mathbb C}^d$ and each $j\in\{1,2\}$ .

Short context: The statement of the Theorem is one of many equivalent formulations of famous Kadison-Singer problem. In was posed in 1959 in the language of functional analysis. Later, it was discovered that numerous open problems in pure mathematics, applied mathematics, engineering and computer science are all equivalent to this problem. Hence, it was sufficient to solve one of these problems to solve them all. This is what the Theorem achieves!

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

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For any finite subset S of R^n, there exists a quasiregular map from R^n onto R^n\S

You need to know: Euclidean space ${\mathbb R}^n$ , norm $||x|| = \sqrt{\sum_{i=1}^n x_i^2}$ in ${\mathbb R}^n$ , linear map $D: {\mathbb R}^n \to {\mathbb R}^n$ , matrix, determinant $\text{det}(A)$ of $n\times n$ matrix A, map from set $S_1$ onto set $S_2$ , notation $S_1 \setminus S_2$ for set difference.

Background: A function $f:{\mathbb R}^n \to {\mathbb R}^n$ is called differentiable at $x\in{\mathbb R}^n$ if there exists a linear map $D_{f.x}: {\mathbb R}^n \to {\mathbb R}^n$ such that $\lim\limits_{h\to 0}\frac{||f(x+h)-f(x)-D_{f,x}(h)||}{||h||}=0$ . Let $||D_{f.x}||=\max\limits_{y\in{\mathbb R}^n}\frac{||D_{f.x}(y)||}{||y||}$ be the norm of $D_{f.x}$ . Function f can be written as $f=(f_1,\dots,f_n)$ with $f_i:{\mathbb R}^n \to {\mathbb R}$ , $1\leq i \leq n$ . If f is differntiable at $x=(x_1,\dots, x_n)$ then all partial derivatives $\frac{\partial f_i}{\partial x_j}$ exist for $1\leq i,j \leq n$ . Let $J_{f,x}$ be the $n\times n$ matrix with entries $\frac{\partial f_i}{\partial x_j}$ . A function $f:{\mathbb R}^n \to {\mathbb R}^n$ , differentiable at every $x\in{\mathbb R}^n$ , is called K-quasiregular if $||D_{f.x}||^n \leq K|\text{det}(J_{f,x})|$ for all $x\in {\mathbb R}^n$ . f is called quasiregular if it is K-quasiregular for some $K\geq 1$ .

The Theorem: On 25th April 2013, David Drasin and Pekka Pankka submitted to arxiv a paper in which they proved that, given integer $n\geq 3$ , and any finite set $S\subset {\mathbb R}^n$ , there exists a quasiregular map from ${\mathbb R}^n$ onto ${\mathbb R}^n \setminus S$ .

Short context: Quasiregular mappings are natural higher dimensional analogues for
holomorphic functions (differentiable functions in complex variables). In 1980, Rickman proved that given any $K>1$ and $n\geq 2$ there exists q depending only on K and n so that a non-constant K-quasiregular mapping $f:{\mathbb R}^n \to {\mathbb R}^n$ omits at most q points. The Theorem states that this result is sharp in all dimensions $n\geq 3$ . Earlier, the Theorem was known to hold only for $n=3$ , and also for all $n$ if S is a one-element set.

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

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The converse to the Rademacher theorem holds if and only if m>=n

You need to know: Euclidean space ${\mathbb R}^n$ , norm $||x||_n = \sqrt{\sum_{i=1}^n x_i^2}$ in ${\mathbb R}^n$ , linear map $D: {\mathbb R}^n \to {\mathbb R}^m$ , Lebesgue null set (set of Lebesgue measure $0$ ) in ${\mathbb R}^n$ .

Background: Let n and m be positive integers. A function of several real variables $f:{\mathbb R}^n \to {\mathbb R}^m$ is called differentiable at a point $x_0\in{\mathbb R}^n$ if there exists a linear map $D: {\mathbb R}^n \to {\mathbb R}^m$ such that $\lim\limits_{h\to 0}\frac{||f(x_0+h)-f(x_0)-D(h)||_m}{||h||_n}=0$ . A function $f:{\mathbb R}^n \to {\mathbb R}^m$ is called Lipschitz if there exists a constant $K\geq 0$ such that $||f(x)-f(y)||_m \leq K ||x-y||_n$ for all $x,y \in {\mathbb R}^n$ .

The Theorem: On 25th April 2013, David Preiss and Gareth Speight submitted to arxiv and Inventiones Mathematicae a paper in which they proved that, for every integer $n>1$ , there exists a Lebesgue null set $N \subseteq {\mathbb R}^n$ such that every Lipschitz function $f:{\mathbb R}^n \to {\mathbb R}^{n-1}$ is differentiable at some point $x_0\in N$ .

Short context: The classical Rademacher theorem states that if a Lipschitz function $f:{\mathbb R}^n \to {\mathbb R}^m$ is differentiable at no point of a set $A \subset {\mathbb R}^n$ , then A must be Lebesgue null. The natural converse of this statement asks: given a Lebesgue null set $A \subset {\mathbb R}^n$ , does there exist a Lipschitz function $f:{\mathbb R}^n \to {\mathbb R}^m$ which is differentiable at no point of A? After efforts of many researchers, it was known by 2013 that the answer is “Yes” if $m\geq n$ and “No” if $m\leq n-2$ , leaving open only the case $m=n-1$ . The Theorem solves this remaining case: the answer is “No”. Hence, the the answer is “Yes” if and only if $m\geq n$ .

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

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Any finite union of intervals supports a Riesz basis of exponentials

You need to know: Notations ${\mathbb Z}$ , ${\mathbb R}$ , and ${\mathbb C}$ for sets of integer, real, and complex numbers, respectively. Notation $i$ for complex number $\sqrt{-1}$ , absolute value $|z|$ and conjugate $\bar{z}$ of $z\in{\mathbb C}$ , exponent of a complex number, integral, sum of infinite series.

Background: Let $S\subseteq {\mathbb R}$ be a set (for this Theorem, we only need the case when S in a finite union of intervals). Let $L^2(S)$ be the set of functions $f:S \to {\mathbb C}$ for which the integral $\int_S |f(x)|^2dx$ exists and finite. For $f,g \in L^2(S)$ , denote $(f,g)=\int_S f(x)\bar{g}(x)dx$ and $\|f\|^2=(f,f)$ . A sequence $\{f_n\}_{n=1}^\infty$ of functions in $L^2(S)$ is called complete if the only $f \in L^2(S)$ satisfying $(f,f_n)=0$ for all n is $f=0$ . A complete sequence $\{f_n\}_{n=1}^\infty$ is called a Riesz basis if there exist positive constants c and C such that $c\sum\limits_{n=1}^\infty |a_n|^2 \leq \|\sum\limits_{n=1}^\infty a_n f_n\| \leq C\sum\limits_{n=1}^\infty |a_n|^2$ for every sequence $\{a_n\}_{n=1}^\infty$ of complex numbers such that $\sum\limits_{n=1}^\infty |a_n|^2 < \infty$ .

The Theorem: On 23th October 2012, Gady Kozma and Shahaf Nitzan submitted to arxiv a paper in which they proved that, whenever $S\subseteq {\mathbb R}$ is a finite union of intervals, there exists a set $\Lambda\subset{\mathbb R}$ such that the functions $\left\{e^{i \lambda t}\right\}_{\lambda \in \Lambda}$ form a Riesz basis in $L_2(S)$ . Moreover, if $S \subseteq [0, 2\pi]$ then $\Lambda$ may be chosen to satisfy $\Lambda \subseteq {\mathbb Z}$ .

Short context: A Riesz basis of exponential functions as in the Theorem gives each function $f \in L^2(S)$ a unique representation $f (t) = \sum c_{\lambda} e^{i\lambda t}$ , which is useful for many applications. However, there are relatively few examples of sets S for which such basis is known to exists. For example, it was known that it exists if S is an interval. The Theorem extends this to the important case when S is a finite union of intervals.

Links: Free arxiv version of the original paper is here, journal version is 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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The BMV (Bessis-Moussa-Villani) conjecture is true

You need to know: Complex number, conjugate $\bar{z}$ of complex number z, matrix with complex entries, matrix multiplication, eigenvalues of an $n\times n$ matrix, infinite series, positive measure, integration.

Background: Let A be $n\times n$ matrix with complex entries $a_{ij}$ . The trace of A is $\text{tr}[A]=\sum\limits_{i=1}^n a_{ii}$ . An exponent $\exp(A)$ of matrix A is $n\times n$ matrix given by $\exp(A)=\sum\limits_{k=0}^\infty \frac{1}{k!}A^k$ . Matrix A is called Hermitian if $a_{ij}=\bar{a}_{ji}$ for all $i,j$ . A Hermitian matrix is called positive semidefinite if all its eigenvalues are non-negative real numbers.

The Theorem: On 25th July 2011, Herbert Stahl submitted to arxiv a paper in which he proved the following result. Let A and B be two $n\times n$ Hermitian matrices and let B be positive semidefinite. For $t\geq 0$ , let $f(t) = \text{Tr}[\exp(A-tB)]$ . Then there exists a positive measure $\mu$ on $[0,\infty)$ (which may depend on A and B), such that $f(t)=\int_0^\infty e^{-ts}d\mu(s)$ for all t.

Short context: The Theorem confirms a conjecture of Bessis, Moussa, and Villani made in 1975, which was known as the BMV conjecture. The conjecture was extensively studied and had a number of equivalent formulations.

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

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A fixed point theorem holds for alpha–psi-contractive type mappings

You need to know: Metric space, complete metric space, notation $\psi^n$ for the n-th iterate of $\psi$ .

Background: Let $\Psi$ be the set of all non-decreasing functions $\psi:[0,+\infty)\to[0,+\infty)$ such that $\sum\limits_{n=1}^{\infty}\psi^n(t)<+\infty$ for all $t>0$ . Let $(X,d)$ be a complete metric space. A mapping $T:X \to X$ is called an $\alpha$ – $\psi$ -contractive if there exist two functions $\alpha: X \times X \to [0,+\infty)$ and $\psi\in \Psi$ such that $\alpha(x,y)d(Tx,Ty)\leq \psi(d(x,y))$ for all $x,y \in X$ . We say that T is $\alpha$ -admissible if for $x,y \in X$ , $\alpha(x,y) \geq 1$ implies that $\alpha(Tx,Ty)\geq 1$ .

The Theorem: On 17th April 2011, Bessem Samet, Calogero Vetro, and Pasquale Vetro submitted to Nonlinear Analysis: Theory, Methods & Applications a paper in which they proved the following result. Let $(X,d)$ be a complete metric space and $T:X \to X$ be an $\alpha$ – $\psi$ -contractive mapping satisfying the following conditions: (i) T is $\alpha$ -admissible; (ii) there exists $x_0 \in X$ such that $\alpha(x_0,Tx_0)\geq 1$ ; and (iii) T is continuous. Then, T has a fixed point, that is, there exists $x^* \in X$ such that $Tx^*=x^*$ .

Short context: The 1922 Banach’s Fixed Point Theorem states that, if $(X,d)$ is a non-empty complete metric space and $f:X\to X$ is such that (*) $d(f(x),f(y))\leq c\cdot d(x,y), \, \forall x,y$ for some $c\in(0,1)$ , then f has a fixed point $x^*$ . It is a fundamental result in mathematics with countless applications. However, in some other applications condition (*) does not hold. The Theorem replaces (*) with a weaker condition which reduces to (*) in the special case $\alpha(x,y)=1$ and $\psi(t)=ct$ . As an example, the authors provide applications to the theory of differential equations when the Theorem is applicable while Banach’s result is not. See here and here for other versions of fixed point theorems.

Links: The original paper is available here.

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For any bounded subset A of L^1 space, exists x fixed by every isometry of L^1 preserving A

You need to know: Banach space B with norm $||.||$ , linear map $f:B\to B$ , bounded subset of B, $L^1$ space.

Background: Let B be a Banach space. A linear map $f:B\to B$ is called an isometry if $||f(x)||=||x||$ for all $x \in B$ . For a subset $A \subset B$ , we say that f preserves A if $f(x) \in A$ for all $x \in A$ . A point $x\in B$ is called a fixed point of $f$ if $f(x)=x$ .

The Theorem: On 7th December 2010, Uri Bader, Tsachik Gelander, and Nicolas Monod submitted to arxiv and Inventiones Mathematicae a paper in which they proved the following result. Let A be a non-empty bounded subset of an $L^1$ space B. Then there is a point in B fixed by every isometry of B preserving A. Moreover, one can choose a fixed point which minimises $\sup\limits_{a\in A}||v-a||$ over all $v \in B$ .

Short context: Starting with famous Banach’s Fixed Point Theorem, mathematicians developed fixed point theorems in various contexts, useful in different applications, see here for an example. The Theorem is a version of fixed point theorem for L1 spaces. The authors demonstrated that it has many applications. For example, it can be used to derive the optimal solution to so-called “derivation problem”, see the original paper for details.

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

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There is a HAPpy Banach space, other than l_2, which has a symmetric basis

You need to know: Hilbert space $l_2$ , Banach space B, notation $\|.\|_B$ for norm in B, convergence in this norm, isomorphic Banach spaces, dimension of a Banach space, finite and infinite dimensional Banach spaces, subspace of a Banach space, bounded linear operator K between Banach spaces, range of K, compact set in a Banach space.

Background: A sequence $\{x_n\}$ of a Banach space B is called basis of B if every $x\in B$ has a unique representation of the form $x=\sum\limits_{n=1}^\infty a_nx_n$ for some real numbers $a_n$ . Two bases $\{x_n\}$ and $\{y_n\}$ of B are called equivalent if series $x=\sum\limits_{n=1}^\infty a_nx_n$ converges in B if and only if $x=\sum\limits_{n=1}^\infty a_ny_n$ converges. A basis $\{x_n\}$ of B is called symmetric if every permutation of $\{x_n\}$ is a basis of B equivalent to $\{x_n\}$ .

A Banach space B is said to have the approximation property (AP) if for every compact set K in B and for every $\epsilon>0$ , there is a bounded linear operator $T:B \to B$ , whose range is finite-dimensional, and such that $\|Tx-x\|_B \leq \epsilon$ for all $x \in K$ . We say that Banach space B has the hereditary AP (or is a HAPpy space) if all of its subspaces have the AP.

The Theorem: On 7th November 2010, William Johnson and Andrzej Szankowski submitted to the Annals of Mathematics a paper in which they proved the existence of a HAPpy Banach space, not isomorphic to the Hilbert space $l_2$ , which has a symmetric basis.

Short context: The first examples of HAPpy Banach spaces not isomorphic to a Hilbert space was constructed by Johnson in 1980. Later, Pisier constructed more such examples, but none of them have a symmetric basis. In fact, Johnson asked in 1980 whether there exists any HAPpy space, other than $l_2$ , that has a symmetric basis. The Theorem answers this question affirmatively.

Links: The original paper is available here.

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For each N>c_d t^d, there exists an N-point spherical t-design in the sphere S^d

You need to know: Notation ${\mathbb N}$ for the set of positive integers, Euclidean space ${\mathbb R}^n$ , sphere in ${\mathbb R}^n$ , d-dimensional Lebesgue measure $\mu_d$ , integration over $\mu_d$ .

Background: Let $S^d$ be a sphere in ${\mathbb R}^{d+1}$ normalised such that $\mu_d(S^d) = 1$ . A set of points $x_1, \dots, x_N \in S^d$ is called a spherical t-design if equality $\int_{S^d} P(x) d \mu_d(x) = \frac{1}{N}\sum\limits_{i=1}^N P(x_i)$ holds for all polynomials P in $d+1$ variables, of total degree at most t.

The Theorem: On 22nd September 2010, Andriy Bondarenko, Danylo Radchenko, and Maryna Viazovska submitted to arxiv a paper in which they proved that for every $d\in {\mathbb N}$ there exist a constant $c_d>0$ such that for each $t\in {\mathbb N}$ and each $N\geq c_d t^d$ , there exists a spherical t-design in $S^d$ consisting of N points.

Short context: The concept of a spherical design was introduced by Delsarte, Goethals, and Seidel in 1977, and, as follows from the definition, it is useful for evaluating integrals. Of course, the smaller N, the easier to compute $\frac{1}{N}\sum\limits_{i=1}^N P(x_i)$ . In 1993, Korevaar and Meyers conjectured the existence of spherical t-designs in $S^d$ with as little as $c_d t^d$ points, which is optimal up to the constant factor. The Theorem confirms this conjecture.

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

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