19 problems

Let $A$ be an abelian surface over $\mathbb{C}$, let $A^{[\ell]}$ be its Hilbert scheme of length-$\ell$ subschemes, and let $A^{\langle\ell-1\rangle}$ be the fiber over $0$ of the…

The chamber formula conjecture. In the fundamental chamber,

Mocanu's conjecture. For every $k \geq 2$, there are Hecke-equivalent isomorphisms

Let $\mathfrak Y$ be a compactification with higher supersymmetry, let $\gamma=(p^0,p^A,q_A,q_0)$ be an electromagnetic charge vector, and let … After restricting to the sublattice…

Let $\boldsymbol\gamma=(\gamma_1,\ldots,\gamma_n)$ be an ordered collection of charges, let $\hat{\boldsymbol\gamma}$ denote the corresponding hatted charge data, and let…

Let $J^{cusp}_{k,1}(N)$ denote the space of Jacobi cusp forms of index $1$ and level $N$, and let $\phi\in J^{cusp}_{k,1}(N)$ be an $L^2$-normalized newform. Write…

Let $J^{cusp}_{k,1}(N)$ denote the space of Jacobi cusp forms of index $1$ and level $N$, and let $\sup(J^{cusp}_{k,1}(N))$ denote its associated supremum quantity. Jacobi index-on…

Let $\mathfrak{g}_I$ be an intermediate Lie algebra between $\mathfrak{g}_1$ and $\mathfrak{g}_2$, and let the associated theta block be constructed from the positive roots and wei…

For each intermediate Lie algebra in the intermediate exceptional series, form the theta block from its associated positive roots and weights, using the construction described in t…

Let $k>4$ with $k\equiv 2$ modulo $4$. For every reduced matrix $T\in\halfint^+_2$ with bottom-right entry $m$, let $S_m(n,T)$ be the set of integers $r$ such that…

Let $\cone_k$ denote the modular cone associated with weight $k$. Polyhedrality conjecture. Suppose that $k\equiv 2$ modulo $4$ and $k>4$. The cone $\cone_k$ is polyhedral. The pap…

Let $m$ be an index and let $k$ be the weight of a $W(E_8)$-invariant weak Jacobi form. Sun and Wang's lower-bound conjecture. Every non-zero $W(E_8)$-invariant weak Jacobi form of…

Let $\Theta_f$ be a theta block of $q$-order one that is a holomorphic Jacobi form of integral weight $k$ and integral index $N$ for $A_1$. Let $\operatorname{Grit}(\Theta_f)$ deno…

Let $L$, $K$, $v$, $w$, and $\iota$ be as in Theorem $\operatorname{MFprodJac}$. Write $D_L$ and $D_K$ for the discriminant groups of $L$ and $K$, and let $D_{L+_{\iota}K}$ be the…

Double ramification cycle conjecture. There exist quasi-Jacobi forms $\varphi_m(z,\tau)$ and $\varphi_{m,n}(z,\tau)$ such that

Let the pure theta block $\Theta_f$ be a holomorphic Jacobi form of weight $k$ and index $m$ with vanishing order $1$ in $q$, where $k,m\in\mathbb Z_{>0}$. Define the nearly holomo…

Let $\Theta\in J_{k,t}$ be a theta block with trivial character and with order of vanishing $1$ in $q$. Here $J_{k,t}$ denotes the space of Jacobi forms of weight $k$ and index $t$…

Let $f_5(\boldsymbol{z};\tau)$ be the generating function defined by … where the parameters $\alpha_k$, for $k\in\{1,2,3,4\}$, satisfy…

Let $L$ be the lattice indexing the Jacobi-form space $J_{k,L,1}$, and let $\phi\in J_{k,L,1}$. Let $T_{-}(2)$ denote the index-raising Hecke operator, and let…