Extremal properties of simple geometric objects
What's the minimum number of distinct distances between $n$ points on a plane?
Erdős 1946 · Moser 1952 · Chung 1984
Chung, Szemerédi & Trotter 1992 · Székely 1993
Solymosi & Tóth 2001 · Tardos 2003 · Katz & Tardos 2004
Guth & Katz 2015
Geometry of ruled surfaces, polynomial method.
Given $n$ points not all collinear in $\mathbb{R}^2$,
what's the minimum number of ordinary lines ?
Sylvester 1893 · Gallai 1944
Dirac & Motzkin 1951 · Kelly & Moser 1958
Csima & Sawyer 1993 · Green & Tao 2012
Menelaus's theorem, Euler formula
Cayley–Bacharach theorem, sum-product estimate
What's the densest way to arrange
non-overlapping spheres in $\mathbb{R}^n$?
Kepler 1611 · Gauss 1831 · Thue 1890
Fejes Tóth 1940 · Hales 1998
Cohn, Kumar, Miller, Radchenko, Viazovska 2016
Poisson summation formula, modular forms, automated proof checking
How high do $n$ satellites need to fly?
Each satellite sees a zone of the sphere
What's the minimum width of $n$ equal zones
covering the unit sphere?
Is it $\pi / 3$ for $n = 3$?
Distribute $n$ great circles to minimize the greatest distance between a point and the nearest great circle.
What's the minimum width of $n$ equal zones
covering the unit sphere?
Fejes Tóth's zone conjecture
The width of $n$ equal zones covering the
unit sphere is at least $\pi / n$.
Research Problems: Exploring a Planet, 1973
Rosta 1972: 3 equal zones
Linhart 1974: 4 equal zones
Fodor, Vígh and Zarnócz 2016: a lower bound
for example, $\pi/6.83$ for 5 equal zones
Fejes Tóth's zone conjecture 1973 · J.–Polyanskii 2017
The total
width of $n$ equal zones covering
the unit sphere is at least $\pi$ in any dimension; Characterization of equality cases.
A plank in $\mathbb{R}^d$ is
Tarski's plank problem
What's the minimum total width of planks covering a given convex body?
What's the minimum total width of planks covering a unit disk?
Plank of width $w$
Planks cover disk
$\sum w_i \ge 2$
Archimedes: Arch of area $\pi w$
Arches cover hemisphere
$\sum \pi w_i \ge 2\pi$
What's the minimum total width of planks
covering a given convex body $C$?
Tarski's plank conjecture 1932 · Bang 1951
If convex body $C$ is covered by planks, then their total width is at least the width of $C$.
A direction of a plank is
Bang's lemma
Can choose directions such that $v_1 + \dots + v_n$ is not covered
Fejes Tóth's zone conjecture 1973 · J.–Polyanskii 2017
The total width of $n$ zones covering the unit sphere is at least $\pi$.
Each zone is a plank $\cap$
the unit sphere
and it has two directions
Bang's lemma: $v := v_1 + \dots + v_n$ is not covered
If $\lVert v \rVert \le 1$, then $\hat{v}$ is not covered.
Otherwise, $v$ is large in magnitude
We merge some zones
Erdős' circle covering problem · Goodman–Goodman 1945
A non-separable family of balls of radii $r_1, \dots, r_n$ can be
covered by a ball of radius $r_1 + \dots + r_n$
Bang's plank conjecture 1951
Total relative width of planks covering
a convex body is at least $1$.
Bezdek's annulus conjecture 2003
Total width of planks covering an annulus with a small hole at center is at least the diameter.
Ball 1992 using Bang's lemma
A lower bound on sphere packing density.
Equiangular lines in $\mathbb{R}^n$ are lines through origin pairwise separated by the same angle
What's the maximum number of
equiangular lines in $\mathbb{R}^n$?
| $n$ | 2 | 3-4 | 5 | 6 | 7-13 | 14 | ... | 23-41 | 42 | 43 |
| max | 3 | 6 | 10 | 16 | 28 | 28-29 | ... | 276 | 276-288 | 344 |
What's the maximum number of equiangular lines in $\mathbb{R}^n$?
Gerzon 1973
At most $\frac{1}{2}n(n+1)$
de Caen 2000
At least $cn^2$
Angles $\to$ 90° as $n\to\infty$
What happens if the angles are held fixed?
What's the maximum number $E_\alpha(n)$ of
equiangular lines in $\mathbb{R}^n$ with a fixed angle $\arccos\alpha$?
| Lemmens, Seidel 1973 | $E_{1/3}(n) = 2(n-1)$ for $n \ge 15$ |
| Neumann 1973 | $E_{\alpha}(n) \le 2n$, unless $1/\alpha$ is odd |
| Neumaier 1989 | $E_{1/5}(n) = \lfloor \frac{3}{2}(n-1) \rfloor$ for $n \ge n_0$ |
| Bukh 2016 | $E_{\alpha}(n) \le c_\alpha n$ |
| Balla, Dräxler, Sudakov, Keevash 2018 | $E_{\alpha}(n) \le 1.93n$ for $n \ge n_0(\alpha)$ if $\alpha \neq 1/3$ |
Bukh's conjecture on equiangular lines with fixed angle
$E_{1/(2k-1)}(n) \approx \frac{kn}{k-1}$. $E_{1/7}(n) \approx \frac{4}{3}n$.
What's the maximum number $E_\alpha(n)$ of
equiangular lines in $\mathbb{R}^n$ with a fixed angle $\arccos\alpha$?
Conjecture J.–Polyanskii
$E_{\alpha}(n) \approx \frac{kn}{k-1}$, where $k = k(\lambda)$, $\lambda = \frac{1-\alpha}{2\alpha}$.
Spectral radius order $k(\lambda) := $ smallest $k$ such that
$\exists$ $k$-vertex graph $G$ whose adjacency matrix has spectral radius $= \lambda$
| $\alpha$ | $\lambda$ | $G$ | $k$ | $E_\alpha(n)$ |
| $\tfrac{1}{3}$ | $1$ | $2$ | $2n$ | |
| $\tfrac{1}{5}$ | $2$ | $3$ | $\tfrac{3n}{2}$ | |
| $\frac{1}{7}$ | $3$ | $4$ | $\tfrac{4n}{3}$ |
J.–Polyanskii 2018
True for all $\lambda \le \sqrt{2 + \sqrt{5}}$.
What's the maximum number $E_\alpha(n)$ of
equiangular lines in $\mathbb{R}^n$ with a fixed angle $\arccos\alpha$?
J., Tidor, Yao, Zhang, Zhao 2019
$E_\alpha(n) = \lfloor \frac{k}{k-1}(n-1) \rfloor$ for $n \ge n_0(\alpha)$ if $k(\lambda) < \infty$;
$E_\alpha(n) = n+o(n)$ otherwise.
J., Tidor, Yao, Zhang, Zhao 2019
Multiplicity of second largest eigenvalue of an $n$-vertex connected graph of bounded degree is at most $Cn/\log\log n$.
Cayley graph of $\mathrm{PSL}(2,p)$ has
second eigenvalue multiplicity at least $\sqrt[3]{n}$
What is the maximum second eigenvalue multiplicity of
an $n$-vertex connected graph of bounded degree?
What is the maximum second eigenvalue multiplicity of
an $n$-vertex Cayley graph of bounded degree?
Gromov, Colding–Minicozzi, Kleiner
Lee–Makarychev: Constant for abelian groups
What is the maximum number of unit vectors in $\mathbb{R}^n$ such that pairwise inner products are either $\alpha$ or $\beta$?
Zauner's conjecture 1999
Maximum number of equiangular lines in $\mathbb{C}^n$ is $n^2$.