# The number of n-faces genus-g triangulations if g/n converges to a constant

You need to know: Basic topology, homeomorphism, topological surface, orientable surface, genus of a surface, $o(n)$ notation.

Background: A (two-dimensional, orientable) triangulation $T$ is a way to glue together a collection of oriented triangles, called the faces, along their edges in a connected way that matches the orientations. If the number of triangles is finite, then $T$ is homeomorphic to an orientable topological surface $S$. Let us define the genus of $T$ as the genus of $S$. The triangulation $T$ is called rooted if it has a distinguished oriented edge called the root edge. Let $\tau(n,g)$ be the number of rooted triangulations with $n$ faces and genus $g$.

Let $\lambda_c=\frac{1}{12\sqrt{3}}$. For any $\lambda\in(0,\lambda_c]$, let $h\in (0,\frac{1}{4}]$ be such that $\lambda = \frac{h}{(1+8h)^{3/2}}$, and let
$d(\lambda) = \frac{h\log\frac{1+\sqrt{1-4h}}{1-\sqrt{1-4h}}}{(1+8h)\sqrt{1-4h}}$. For any $\theta\in(0,\frac{1}{2})$, let $\lambda(\theta)$ be the unique solution to the equation $d(\lambda)=\frac{1-2\theta}{6}$, and let $f(\theta) = 2 \theta \log \frac{12\theta}{e} + \theta \int_2^{1/\theta} \log\frac{1}{\lambda(1/t)}dt, \, 0<\theta<\frac{1}{2}$. Also, let $f(0)=\log 12\sqrt{3}$ and $f(1/2)=\log\frac{6}{e}$.

The Theorem: On 1st February 2019, Thomas Budzinski and Baptiste Louf submitted to arxiv a paper in which that proved that for any sequence $g_n$ such that $0 \leq g_n \leq \frac{n+1}{2}$ for every $n$ and $\lim_{n\to\infty}\frac{g_n}{n}=\theta \in [0,\frac{1}{2}]$, one has $\tau(n,g_n) = n^{2 g_n} \exp(f(\theta)n + o(n)) \,\text{as}\, n\to\infty$.

Short context: Triangulations are important in many applications, for example, as finite representations of a surface to a computer, or in the theory of random surfaces. In the last application, it is important to understand how a uniformly random triangulation looks like, and, for this, we need to count triangulations with various properties. In 1986, Bender and Canfield established the asymptotic growth of $\tau(n,g)$ when $n\to\infty$ and $g$ is fixed. However, the important case when $n$ and $g$ both go to infinity remained open. The Theorem estimates $\tau(n,g)$ up to sub-exponential factors in the case when the ratio $g/n$ converges to a constant.

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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# Characterisation of polyhedral types inscribed in the hyperboloid or cylinder

You need to know: Euclidean space ${\mathbb R}^3$, convex polyhedron in ${\mathbb R}^3$, vertices of a polyhedron, graph, planar graph, closed path in a graph, 3-connected graph.

Background: Let $S \subset {\mathbb R}^3$ be either unit sphere (that is, set of all points $x=(x_1,x_2,x_3)\in {\mathbb R}^3$ such that $x_1^2+x_2^2+x_3^2=1$),  or hyperboloid (defined by equation $x_1^2+x_2^2-x_3^2=1$), or cylinder (defined by $x_1^2+x_2^2=1$ with $x_3$ free). We say that a convex polyhedron P is inscribed in S if $P \cap S$ is exactly the set of vertices of P. A 1-skeleton of a convex polyhedron P is the set of vertices and edges of P, considered as a graph. A Hamiltonian cycle in a graph is a closed path visiting each vertex exactly once.

The Theorem: On 13th October 2014, Jeffrey Danciger, Sara Maloni, and Jean-Marc Schlenker submitted to arxiv a paper in which they proved that, for a planar graph G, the following conditions are equivalent: (i) G is the 1-skeleton of some convex polyhedron inscribed in the cylinder; (ii) G is the 1-skeleton of some convex polyhedron inscribed in the hyperboloid; and (iii) G is the 1-skeleton of some convex polyhedron inscribed in the sphere and G admits a Hamiltonian cycle.

Short context: In 1832, Steiner asked which graphs are 1-skeletons of (a) an arbitrary convex polyhedron in ${\mathbb R}^3$, (b) a convex polyhedron inscribed in the sphere, (c) inscribed in the cylinder, and (d) inscribed in the hyperboloid. Question (a) was answered by a famous Theorem of Steinitz, which states that a graph G is the 1-skeleton of a convex polyhedron in ${\mathbb R}^3$ if and only if G is planar and 3-connected. In 1992, Hodgson et al. gave a full characterisation of possible 1-skeletons of convex polyhedra inscribed in the sphere, thus answering question (b). The Theorem solves the remaining parts (c) and (d) of the Steiner’s question by reducing them to the already solved part (b).

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

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# The Conway knot is not slice

You need to know: Euclidean space ${\mathbb R}^4$, unit ball $B^4=\{x\in {\mathbb R}^4 : |x|\leq 1\}$ in ${\mathbb R}^4$, unit sphere $S^3=\{x\in {\mathbb R}^4 : |x|=1\}$ in ${\mathbb R}^4$, closed curves in $S^3$, self-intersecting and non-self-intersecting curves, smoothly embedded 2-dimensional disk in $B^4$.

Background: A knot is a closed, non-self-intersecting curve in $S^3$. Two knots $K_1$ and $K_2$ are equivalent if $K_1$ can be transformed smoothly, without intersecting itself, to coincide with $K_2$. There are tables which lists some knots up to equivalence, which are called the Rolfsen tables. The Conway knot is the knot labelled 11n34 in these tables. A knot $K \subset S^3$ is called slice if it bounds a smoothly embedded 2-dimensional disk in $B^4$.

The Theorem: On 8th August 2018, Lisa Piccirillo submitted to arxiv a paper in which she proved that the Conway knot is not slice.

Short context: A knot diagram is a projection of a knot into a plane, which (i) is injective everywhere, except at a finite number of crossing points, which are the projections of only two points of the knot, and (ii) records over/under information at each crossing. We say that the knot K has n crossings if n is the minimal number of crossing a knot diagram of a knot equivalent to K may have. The notion of slice knot is a central notion in 4-dimensional theory of knots, originated by Fox in 1962. By 2005, all knots of under 13 crossings were classified whether they are slice knots or not, with the only exception of the Conway knot (which has 11 crossings). The Theorem completes this classification.

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

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# There exist lattices with exponentially large kissing numbers

You need to know: Set ${\mathbb Z}$ of integers, Euclidean space ${\mathbb R}^n$, basis for ${\mathbb R}^n$, norm $||x||=\sqrt{\sum_{i=1}^n x_i^2}$ of $x=(x_1, \dots, x_n) \in {\mathbb R}^n$.

Background: A lattice $L$ in ${\mathbb R}^n$ is a set of the form $L =\left\{\left.\sum\limits_{i=1}^n a_i v_i\,\right\vert\, a_i \in {\mathbb Z}\right\}$, where $v_1, \dots, v_n$ is a basis for ${\mathbb R}^n$. Let $\lambda_1(L)$ be the length of the shortest non-zero vector in L. The kissing number $\tau(L)$ of L is the number of vectors of length $\lambda_1(L)$ in L. The lattice kissing number $\tau_n^l$ in dimension n is the maximum value of $\tau(L)$ over all lattices $L$ in ${\mathbb R}^n$.

The Theorem: On 3rd February 2018, Serge Vlăduţ submitted to arxiv a paper in which he proved the existence of constant $c>0$ such that $\tau_n^l \geq e^{cn}$ for any $n\geq 1$.

Short context: The kissing number in ${\mathbb R}^n$ is the highest number of equal nonoverlapping spheres in ${\mathbb R}^n$ that can touch another sphere of the same size. Determining the kissing number in various dimensions is an active area of research, see here and here. The lattice kissing number corresponds to the case when spheres form a “regular pattern”. It was a long-standing open problem whether the lattice kissing number grows exponentially with dimension. The Theorem resolves this problem affirmatively.

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

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# There are fifteen types of convex pentagons that can tile the plane

You need to know: Convex set on the plane, convex polygon.

Background: We say that convex polygon P can tile the plane if the plane can be covered by identical copies of P (possibly translated and rotated) with no overlaps and no gaps. For a convex pentagon, denote $\alpha_1, \dots, \alpha_5$ the sizes of its angles (in radians) and $l_1, l_2, \dots, l_5$ the lengths of its sides, enumerated counterclockwise starting from a fixed vertex.

The Theorem: On 1st August 2017, Michael Rao submitted to arxiv a paper in which he proved that a convex pentagon can tile the whole plane by identical copies if and only if it belongs to one of the following fifteen types: (1) $\alpha_1+\alpha_2+\alpha_3 = 2\pi$; (2) $\alpha_1+\alpha_2+\alpha_4 = 2\pi$ and $l_1=l_4$; (3) $\alpha_1 = \alpha_3 = \alpha_4 = \frac{2}{3}\pi$, $l_1=l_2$, and $l_4=l_3+l_5$; (4) $\alpha_1 = \alpha_3 = \frac{1}{2}\pi$, $l_1=l_2$, and $l_3=l_4$; (5) $\alpha_1=\frac{1}{3}\pi$, $\alpha_3=\frac{2}{3}\pi$, $l_1=l_2$, and $l_3=l_4$; (6) $\alpha_1+\alpha_2+\alpha_4 = 2\pi$, $\alpha_1= 2\alpha_3$, $l_1=l_2=l_5$, and $l_3 = l_4$; (7) $2\alpha_2+\alpha_3 = 2\alpha_4+\alpha_1 = 2\pi$, and $l_1=l_2=l_3=l_4$; (8) $2\alpha_1+\alpha_2=2\alpha_4 + \alpha_3 = 2\pi$, and $l_1 = l_2 = l_3 = l_4$; (9) $\alpha_5 = \frac{\pi}{2}$, $\alpha_1+\alpha_4 = \pi$, $2\alpha_2-\alpha_4 = 2\alpha_3+\alpha_4 = \pi$, and $l_1 = l_2+l_4 = l_5$; (10) $\alpha_2 + 2\alpha_5 = 2\pi$, $\alpha_3 + 2\alpha_4 = 2\pi$, and $l_1=l_2=l_3=l_4$; (11) $\alpha_1 = \frac{\pi}{2}$, $\alpha_3+\alpha_5 = \pi$, $2\alpha_2+\alpha_3 = 2\pi$, and $2l_1 + l_3 = l_4 = l_5$; (12) $\alpha_1 = \frac{\pi}{2}$, $\alpha_3+\alpha_5 = \pi$, $2\alpha_2+\alpha_3 = 2\pi$, and $2l_1 = l_3+l_5 = l_4$; (13) $\alpha_1 = \alpha_3 = \frac{\pi}{2}$, $2\alpha_2 + \alpha_4 = 2\alpha_5 + \alpha_4 = 2\pi$, $l_3 = l_4$, and $2l_3 = l_5$; (14) $\alpha_1 = \frac{\pi}{2}$, $2\alpha_2 + \alpha_3 =2\pi$, $\alpha_3 + \alpha_5 = \pi$, and $2l_1 = 2l_3 = l_4 = l_5$; (15) $\alpha_1 = \frac{\pi}{3}$, $\alpha_2 = \frac{3\pi}{4}$, $\alpha_3 = \frac{7\pi}{12}$, $\alpha_4 = \frac{\pi}{2}$, $\alpha_5=\frac{5\pi}{6}$, and $l_1 = 2l_2 = 2l_4 = 2l_5$.

Short context: In the 1910s, Bieberbach suggested to determine all the convex domains which can tile the whole plane. It is easy to see that if convex set P tiles the plane that P must actually be a convex m-gon, $m\geq 3$. It is also not difficult to verify that any triangle and any convex quadrilateral can tile the plane. In 1918, Reinhardt proved that no convex m-gon with $m>6$ can tile the plane, and a convex hexagon can tile the plane if and only if it belongs to one of the following three types: (i) $\alpha_1+\alpha_2+\alpha_3=2\pi$, and $l_1 = l_4$; (ii) $\alpha_1 + \alpha_2 + \alpha_4 = 2\pi$, $l_1 = l_4$, and $l_3 = l_5$; (iii) $\alpha_1 = \alpha_3 = \alpha_5 = \frac{2}{3}\pi$, $l_1 = l_2$, $l_3 = l_4$, and $l_5 = l_6.$ After this, only the case of pentagons remained open. From 1918 to 2015, various researchers discovered the 15 types of pentagons that can tile the plane. The Theorem proves that this list is complete, and thus finishes the solution to the Bieberbach’s problem.

Links: Free arxiv version of the original paper 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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# For all n>n_a, there are at most 2n-2 lines in R^n with common angle a

You need to know: Euclidean space ${\mathbb R}^n$, origin in ${\mathbb R}^n$, lines in ${\mathbb R}^n$, angle between lines, the inverse trigonometric function of cosine $\arccos$.

Background: A set of lines through the origin in ${\mathbb R}^n$ is called equiangular if any pair of lines defines the same angle. Let $N(n)$ denotes the maximum cardinality of an equiangular set of lines in ${\mathbb R}^n$. Let $N_\theta(n)$ denotes the maximum number of equiangular lines in ${\mathbb R}^n$ with common angle $\theta$, where $\theta$ does not depend on dimension.

The Theorem: On 21st June 2016, Igor Balla, Felix Dräxler, Peter Keevash, and Benny Sudakov submitted to arxiv a paper in which they proved that for any angle $\theta \in (0,\pi/2)$, $\theta \neq \arccos \frac{1}{3}$, there is a constant $n_\theta$, such that $N_\theta(n)\leq 1.93n$ for all $n \geq n_\theta$.

Short context: Equiangular sets of lines appear naturally in many areas of mathematics, and the problem of estimating the maximum size of such sets has been studied starting from at least the work of Haantjes in 1948, who proved that $N(3)=N(4)=6$. In 1973, Lemmens and Seidel formulated a problem of estimating $N_\theta(n)$ for fixed $\theta$, and proved that $N_\theta(n) = 2n-2$ for $\theta=\arccos \frac{1}{3}$ and sufficiently large n. The Theorem proves a stronger upper bound for all $\theta \neq \arccos \frac{1}{3}$. This implies that, for all large n, $N_\theta(n)$ is maximised at $\theta=\arccos \frac{1}{3}$.

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

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# A properly embedded genus-g minimal surface with finite topology have at most C_g ends

You need to know: Surface, orientable surface, connected surface, boundary of a surface, area of a surface, compact surface, homeomorphic surfaces.

Background: A surface $M \subset {\mathbb R}^3$ is called a minimal surface if every point $p \in M$ has a neighbourhood S, with boundary B, such that S has a minimal area out of all surfaces S’ with the same boundary B. A surface $M \subset {\mathbb R}^3$ is called properly embedded in ${\mathbb R}^3$, if it has no boundary, no self-intersections, and its intersection with any compact subset of ${\mathbb R}^3$ is compact. The genus of a connected orientable surface is the maximum number of cuttings along non-intersecting closed simple curves without making it disconnected. A surface is said to have finite topology if it is homeomorphic to a compact surface with a finite number of points removed. The number of points removed is called the number of ends of a surface.

The Theorem: On 9th May 2016, William Meeks III, Joaquin Perez, and Antonio Ros submitted to arxiv and Acta Mathematica a paper in which they proved that for every positive integer g, there exists a constant $C_g$ such that any properly embedded minimal surface in ${\mathbb R}^3$ with genus g and finite topology has at most $C_g$ ends.

Short context: Minimal surfaces remain an active area of research since 19th century, and this research area has a golden age at the beginning of the 21st century, with a large number of impressive results, see, for example, here, here, and here. One important conjecture of Hoffman and Meeks states that if a properly embedded connected minimal surface of genus g has a finite number k ends, then $k\leq g+2$. However, before 2016, it was not known if there exists any upper bound on k depending only on g. This is what the Theorem establishes.

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

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# Properly embedded minimal surfaces with one end and genus g exist for all g

You need to know: Surface, orientable surface, connected surface, boundary of a surface, area of a surface, compact surface, homeomorphic surfaces.

Background: A surface $M \subset {\mathbb R}^3$ is called a minimal surface if every point $p \in M$ has a neighbourhood S, with boundary B, such that S has a minimal area out of all surfaces S’ with the same boundary B. A surface $M \subset {\mathbb R}^3$ is called properly embedded in ${\mathbb R}^3$, if it has no boundary, no self-intersections, and its intersection with any compact subset of ${\mathbb R}^3$ is compact. The genus of a connected orientable surface is the maximum number of cuttings along non-intersecting closed simple curves without making it disconnected. A surface is said to have finite topology if it is homeomorphic to a compact surface with a finite number of points removed. The number of points removed is called the number of ends of a surface.

The Theorem: On 20th June 2013, David Hoffman, Martin Traizet, and Brian White submitted to Acta Mathematica a paper in which they proved that for every positive integer g, there exists a connected, properly embedded minimal surface in ${\mathbb R}^3$ with one end and genus g.

Short context: Meeks III and Rosenberg proved that the only examples of properly embedded simply connected minimal surfaces in ${\mathbb R}^3$ are plane and helicoid. Later, Weber, Hoffman, and Wolf constructed a properly embedded minimal surface in ${\mathbb R}^3$ with one end and genus $g=1$. The Theorem proves that such surface exists for every genus g.

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

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