15 problems

Large Sets Conjecture. If $S$ is 2-large, then $S$ is large.

Let $\mathcal P$ be an Apollonian circle packing with symmetry group $\Gamma<\operatorname{PSL}_2(\mathbb C)$. For any circle $C\in \mathcal C_\Lambda^*$, consider its orbit…

Let $g\geq 1$, let $\alpha=(\alpha_1,\dots,\alpha_n)$ be a partition of $2g-2$, let $\mathcal H_1(\alpha)$ be the space of unit-area surfaces in the corresponding stratum of abelia…

Zariski dense orbit conjecture. Either there exists $\alpha\in X(K)$ whose orbit under $\Phi$ is well-defined and Zariski dense in $X$, or there exists a non-constant rational func…

Let $\mathcal{M}$ be an orbit closure of surfaces of genus $g$. Its rank is denoted by $\operatorname{rank}(\mathcal{M})$. Almost High Rank Conjecture. If … then $\mathcal{M}$ is a…

The Siegel–Veech and Lyapunov intersection formulas. The area Siegel–Veech constant and the sum of Lyapunov exponents are

Uniform boundedness conjecture. For each degree $d\geq 2$, there exists a constant $B=B(d)$ such that either

Let $C$ be a compact Riemann surface, and let the $n$-th Hitchin component be the connected component of … containing symmetric powers of Fuchsian representations. For a representa…

Let $G$ be a semisimple Lie group with a $\Theta$-positive structure, let $G/P_\Theta$ be its associated flag variety, and let $G_\Theta^{>0}$ denote the $\Theta$-positive semigrou…

Let $Q(v_1,v_2,v_3):=2v_1v_3-v_2^2$ and … Let $\mathcal{C}\subset\operatorname{Gr}_2(\mathbb{R}^3)$ be the circle of isotropic subspaces, and let…

Let $L$ and $S$ be as in Corollary, and let … Let $\mathfrak{D}_S(L)$ denote the corresponding set of conditioned directional lattices, and let…

Let $\mathcal{M}$ be an affine manifold in a stratum of translation surfaces. Its rank is the dimension of its tangent space projected to absolute cohomology divided by $2$, and…

Let $\mathcal{N}$ be a higher-rank affine invariant submanifold of a stratum of translation surfaces. A translation surface is not primitive if it covers a quadratic differential o…

Exceptional-value conjecture. Unless either $a=1$ and $\sqrt{2}\notin\mathbb{F}_q$, or the triple $(q,a,n)$ occurs in one of the explicitly listed exceptional cases, the map…

Let $F_n$ be the free group of rank $n$, let ${\mathcal X}(F_n)$ be its ${\operatorname{PSL}_2(\mathbb C)}$ character variety, and let ${\mathcal{PS}}(F_n)$ denote the primitive-st…