Pair correlation densities of squared distances sequences

You need to know: Rational and irrational numbers, limits, constant $\pi=3.1415...$ . In addition, you need to know Poisson process and almost sure convergence to understand the context.

Background: For fixed real numbers $\alpha,\beta\in[0,1]$ , consider an infinite sequence $0 \leq \lambda_1 \leq \lambda_2 \leq \dots$ formed by the numbers $(m-\alpha)^2 + (n-\beta)^2$ for integer $m,n$ , arranged in the non-decreasing order. For interval $[a,b] \subset {\mathbb R}$ , the pair correlation function is $R_2[a,b](\lambda):=\frac{1}{\pi\lambda}|\{j \neq k\,|\,\lambda_j\leq \lambda, \, \lambda_k \leq \lambda, a \leq \lambda_j-\lambda_k\leq b\}|,$ where $|S|$ denotes the number of elements in set S.

Real numbers $\alpha_1, \alpha_2, \dots \alpha_k$ are linearly independent over ${\mathbb Q}$ if there are no rational numbers $r_1, r_2, \dots r_k$ , not all $0$ , such that $\sum_{i=1}^k r_i \alpha_i = 0$ . An irrational number $\alpha$ is called diophantine if there exist constants $k, C > 0$ such that $\left|\alpha - \frac{p}{q}\right| > \frac{C}{q^k}$ for every pair of integers p and $q\neq 0$ .

The Theorem: On 10th May 2000 Jens Marklof submitted to Annals of Mathematics a paper in which he proved that, if $\alpha, \beta, 1$ are linearly independent over ${\mathbb Q}$ , $\alpha$ is diophantine, and $a<b$ , then $\lim\limits_{\lambda \to \infty} R_2[a,b](\lambda) = \pi(b-a).$

Short context: In 1991, Cheng and Lebowitz observed numerically that, somewhat surprisingly, non-random sequence $0 \leq \lambda_1 \leq \lambda_2 \leq \dots$ as defined above share many properties with random sequence of event times in the Poisson process with mean $\pi$ . Because it is known that $\lim\limits_{\lambda \to \infty} R_2[a,b](\lambda) = \pi(b-a)$ almost surely for Poisson process, the Theorem confirms this numerical observation.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 3.6 of this book for an accessible description of the Theorem.

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For all large x, there is a prime between x – x^0.525 and x

You need to know: Prime numbers, fractional and decimal exponents.

Background: Not needed.

The Theorem: On 3rd May 2000 Roger Baker, Glyn Harman, and János Pintz submitted to the Proceedings of the London Mathematical Society a paper in which they proved the existence of constant $x_0$ such that for all $x>x_0$ the interval $[x - x^{0.525},x]$ contains at least one prime number.

Short context: An old conjecture of Legendre (who lived in 1752-1833) states that, for every positive integer n, there is a prime number between $n^2$ and $(n+1)^2$ . This is problem 3 from famous list of Landau’s problems and remains open. It would follow from existence of primes between $x - x^\gamma$ and $x$ for $\gamma=0.5$ . However, this statement was known only for $\gamma=0.535$ . The Theorem proves it for $\gamma=0.525$ and $x>x_0$ . With enough effort, the explicit value of $x_0$ can be extracted from the proof, but this was not done.

Links: The original paper is available here.

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The only Banach space isomorphic to each of its subspaces is l^2

You need to know: Vector space, infinite dimensional vector space, Banach space, isomorphic Banach spaces, subspace, closed subspace.

Background: $l^2$ -space is the space of infinite sequences $x=(x_1, x_2, \dots, x_n, \dots)$ equipped with coordinate-wise addition and scalar multiplication and norm $|x| := \sqrt{\sum\limits_{i=1}^\infty x_i^2} < \infty$ .

The Theorem: On 30th March 2000 Timothy Gowers submitted to Annals of Mathematics a paper in which he proved, among other results, that $l^2$ -space is (up to isomorphism) the only infinite-dimensional Banach space which is isomorphic to every infinite-dimensional closed subspace of itself.

Short context: It is easy to see that $l^2$ -space is isomorphic to every infinite-dimensional closed subspace of itself. In his famous 1932 book, Banach asked whether this is the only such example. The Theorem gives a positive answer to this question.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 2.11 of this book for an accessible description of the Theorem.

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The intersection exponent for two simple random walks on the plane is 5/8

You need to know: Integer lattice ${\mathbb Z}^2$ , vertices of ${\mathbb Z}^2$ , basic probability theory, simple random walk on ${\mathbb Z}^2$ .

Background: For a simple random walk S on ${\mathbb Z}^2$ and integer $k\geq 0$ , denote $S[0,k]$ to be the (random) set of vertices S visited after k steps.

The Theorem: On 27th March 2000, Gregory Lawler, Oded Schramm and Wendelin Werner submitted to Acta Mathematica a paper in which they proved, among other results, that there exists a constant $c>0$ such that inequality $c^{-1}k^{-5/8} \leq P[S[0,k]\cap S'[0,k] = \emptyset] \leq ck^{-5/8}$ holds for all $k\geq 1$ , where S and S’ are two independent simple random walks that start from neighbouring vertices in ${\mathbb Z}^2$ .

Short context: If $P[S[0,k]\cap S'[0,k] = \emptyset]$ decays proportional to $k^{-\gamma}$ for some constant $\gamma>0$ , then $\gamma$ is called the intersection exponent. In 1988 Duplantier and Kwon used ideas from theoretical physics to predicts that, for two simple random walks on the plane, $\gamma=5/8$ . The Theorem provides a rigorous mathematical proof of this prediction.

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

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Least-area way to enclose and separate two regions of given volumes in R^3

You need to know: Sphere, spherical cap, angle at which spherical caps intersect, volume, surface area.

Background: Standard double bubble in ${\mathbb R}^3$ is a construction which consists on three spherical caps meeting along a common circle at 120-degree angles (If 2 of caps has equal radii, the third one becomes a ﬂat disc). It divides the space ${\mathbb R}^3$ into 3 regions: infinite one and 2 finite ones with some volumes $V_1$ and $V_2$ .

The Theorem: On 22nd March 2000 Michael Hutchings, Frank Morgan, Manuel Ritoré, and Antonio Ros submitted to Annals of Mathematics a paper in which they proved that the standard double bubble provides the way to enclose and separate two regions of prescribed volumes $V_1>0$ and $V_2>0$ in ${\mathbb R}^3$ with the least possible total surface area of the boundary.

Short context: The problem of finding area-minimising way to enclose and separate 2 given volumes was studied by Plateau in the 19th century, and the conjecture that the standard double bubble provides the solution was known as the double bubble conjecture. The 2-dimensional version of the conjecture was proved in 1991 by the team of undergraduate students. Later, Hass and Schlafly proved the $V_1=V_2$ case in ${\mathbb R}^3$ . The Theorem confirms the conjecture in general.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 2.5 of this book for an accessible description of the Theorem.

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All braid groups are linear

You need to know: Matrix multiplication, invertible matrix, group, subgroup, group isomorphism, field, braid group.

Background: General linear group of degree n over field F, denoted as GL(n, F), is the set of $n \times n$ invertible matrices with entries from F and with matrix multiplication as the group operation. A group is said to be linear if it is isomorphic to a subgroup of GL(n, F) for some natural number n and some field F.

The Theorem: On 11th March 2000 Daan Krammer submitted to Annals of Mathematics a paper in which he proved that all braid groups are linear.

Short context: In 1935, Burau suggested a way to represent elements of any braid group (braids) as matrices over some field. However, he did not prove whether it is faithful (that is, whether different braids corresponds to different matrices). In 1991, Moody showed that this not always the case. The Theorem proves that a different representation, suggested by Krammer in 1999, is faithful, thus establishing that all braid groups are linear. In May 2000, the same result was proved independently by Bigelow.

Links: Free arxiv version of the Bigelow’s paper is here, journal version of Krammer’s paper is here. See also Section 2.2 of this book for an accessible description of the Theorem.

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Every elliptic curve over Q is modular

You need to know: Basic complex analysis, infinite series, irreducible polynomial, and either “rational map” or function field.

Background: Let ${\mathbb H} = \{z \in {\mathbb C}, \text{Im}(z) > 0\}$ , $j:{\mathbb H}\to{\mathbb C}$ be given by $j(z) = 1728\frac{20 G_4(z)^3}{20 G_4(z)^3-49G_6(z)^2}$ , where $G_k(z)=\sum\limits_{(m,n)\neq (0,0)}(m+nz)^{-k}$ . For every positive integer n, there exists a non-zero irreducible polynomial $P_n(x,y)$ with integer coefficients such that $P_n(j(nz),j(z))=0, \, z\in {\mathbb H}$ . The set $X_0(n)$ of pairs of complex numbers $(x,y)$ such that $P_n(x,y)=0$ is called the classical modular curve.

Elliptic curve E over ${\mathbb Q}$ is the set of solutions to the equation $y^2=x^3+ax+b$ , where $a,b \in {\mathbb Q}$ are such that $4a^3+27b^2 \neq 0$ . It is called modular if it can be obtained via a rational map with integer coefficients from $X_0(n)$ for some positive integer n. Equivalently, E is modular if the field of functions on E (given by $Q(x)[y]/(y^2-(x^3+ax+b))$ ) is contained in the field of functions on $X_0(n)$ for some n (given by given by $Q(x)[y]/P_n(x,y)$ ).

The Theorem: On 28th February 2000 Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor submitted to the Journal of the AMS a paper in which they proved that every Elliptic curve over ${\mathbb Q}$ is modular.

Short context: The Theorem confirms conjecture of Taniyama and Shimura from 1961. In 1995, Wiles proved a special case of this conjecture and deduced Fermat’s Last Theorem, which was probably the most famous open problem in the whole mathematics for over 300 years. The Theorem confirms Taniyama-Shimura conjecture in full, has the name “the modularity theorem”, and is considered by many as one of the greatest achievements in the modern mathematics.

Links: The original paper is available here.

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Small set of initial points for Newton’s method to find all roots of polynomials

You need to know: Set ${\mathbb C}$ of complex numbers, absolute value of a complex number, polynomials in complex variable, convergence, derivative, natural logarithm $\ln$ .

Background: Let $P_d$ be the set of polynomials of degree d in one complex variable, such that all their roots have absolute values less than 1. For $P \in P_d$ and $z_0 \in {\mathbb C}$ , consider sequence $z_0, z_1=f_P(z_0), z_2=f_P(z_1), \dots$ , where $f_P(z) = z - \frac{P(z)}{P'(z)}$ . If this sequence converges to a root $z^*$ of P, we say that $z_0$ is in the basin of $z^*$ . This is called the Newton’s method for finding roots.

The Theorem: On 24th February 2000 John Hubbard, Dierk Schleicher, and Scott Sutherland submitted to Inventiones mathematicae a paper in which they proved that, for every $d \geq 2$ , there is a set $S_d$ consisting of at most $1.11 d \cdot \ln^2 d$ points in ${\mathbb C}$ with the property that for every polynomial $P \in P_d$ and every root $z^*$ of P, there is a point $s \in S_d$ in the basin of $z^*$ .

Short context: Finding roots of polynomials is one of the basic problems in mathematics, with Newton’s method being one of the most popular methods for its numerical solution. However, its convergence depends on the choice of initial point $z_0$ . The Theorem guarantees that, if we start Newton’s method from all points of $S_d$ , we are guaranteed to find all the roots of any polynomial $P \in P_d$ .

Links: The original paper is available here.

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Polynomial minimisation can be reduced to LMI problem

You need to know: Multivariate polynomial, degree of a polynomial, positive semidefinite matrix, NP-hard problems.

Background: By LMI (linear matrix inequality) problem we mean the problem of minimising $\sum\limits_{i=1}^m c_i y_i$ subject to the constraint that matrix $A_0+\sum\limits_{i=1}^m A_i y_i$ is positive semidefinite, where $y_i$ are real variables, $c_i$ are given real numbers, and $A_i$ are given square matrices.

The Theorem: On 28th January 2000 Jean Lasserre submitted to SIAM Journal on Optimization a paper in which he proved, among other results, the following theorem: Let $p:{\mathbb R}^n \to {\mathbb R}$ be a polynomial of even degree with (unknown) global minimum $p^*=p(x^*)$ . If the nonnegative polynomial $p(x) - p^*$ is a sum of squares of polynomials, then $x^*$ can be found from LMI problem as above, with $c_i$ and $A_i$ being efficiently computable from the coefficients of p.

Short context: The problem of finding the global minimum of a real multivariate polynomial is an important problem with numerous applications, which is, however, NP-hard even for polynomials of degree 4. Because LMI problem can be solved efficiently, the Theorem provides an efficient method to find $x^*$ if $p(x) - p^*$ is a sum of squares. Using the Theorem, Jean Lasserre also showed how to find approximate solution for general p by solving a ﬁnite sequence of LMI problems.

Links: The original paper is available here.

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All critical points of a rational function f are real only if f is equivalent to real

You need to know: Complex numbers, functions in complex variable, complex differentiation, polynomials.

Background: A (complex) rational function f is a ratio of two polynomials with complex coefficients, $f(z)=\frac{P(z)}{Q(z)}$ . If all coefficients of P(z) and Q(z) are real, we say that f is a real rational function. A complex number $z_0$ is a critical point of f if $f'(z_0)=0$ . We say that two rational functions f and g are equivalent if $g(z)=\frac{a f(z)+b}{c f(z)+d}$ for some complex numbers a,b,c,d such that $ad-bc\neq 0$ .

The Theorem: On 25th January 2000 Alexandre Eremenko and Andrei Gabrielov submitted to Annals of Mathematics a paper in which they proved, among other results, that if all critical points of a rational function f are real, then f is equivalent to a real rational function.

Short context: The Theorem equivalently states that, if for polynomials $P(z),Q(z)$ all the solutions of the equation $P(z)Q'(z)-P'(z)Q(z)=0$ are real, then there exist complex numbers $a,b,c,d$ such that $ad-bc\neq 0$ and $aP(z)+bQ(z)$ and $cP(z)+dQ(z)$ are real polynomials. This is a special case of a well-known conjecture of B. and M. Shapiro, made around 1993, which predicts a similar result for any number of polynomials. In a later work, Mukhin, Tarasov, and Varchenko proved this conjecture in general.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 2.1 of this book for an accessible description of the Theorem.

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