18 problems

Let $T_r=A_{2r}/\mathbb{Z}_2$ be the tadpole Dynkin diagram, and let $\mathbf{T}_r$ be its Cartan matrix. For $\mathbf{x}_r=(x_1,\ldots,x_r)$, define the tadpole Nahm sum … The sum…

Let $L$ be a quadratic lattice and let $\nu\in\mathbb{Q}L$. Define the proper spinor-genus theta series by … where the sum is over representatives of the proper classes in the prop…

Let $\Theta_{\Lambda}(x)$ be the theta series of a $d$-dimensional lattice, and let $\mathcal{P}_n$ denote the class of generating functions whose $n$th roots have integral coeffic…

Partial theta series conjecture. The partial theta series associated with the tensor products satisfy

Let $R$ and $S$ be positive integers with $1\leq S<R$, let $k>\ell\geq1$, and assume $k\geq4$. There exists an integer $N$ such that the theta series … has nonnegative coefficients…

Andrews–Merca–Guo–Zeng conjecture. This coefficient is nonnegative. This conjecture is solved: it was proved analytically by Mao and combinatorially by Yee in 2015, and later repro…

Let $n\geq 1$ and $n\geq m\geq 1$. For a triple $(\mathcal{G},h,\mathcal{E})$ in the moduli stack of skew-Hermitian bundles with a Lagrangian sub-bundle, let…

Global-minimum conjecture. The theta series ratio achieves its global minimum at $\tau=1$:

Let $\ell$, $R$ and $S$ be positive integers with $1\leq S<R/2$, and let $(a;q)_\infty$ denote the $q$-Pochhammer symbol. Andrews–Merca and Guo–Zeng's conjecture. The coefficient o…

Algebraicity conjecture. If all Fourier coefficients of the holomorphic part of $\tilde f$ are algebraic, then $\xi(\tilde f)$ is either a unary theta series with…

Let $\Gamma$ be a formally unimodular packing, and let $\Xi_{\Gamma}(\tau)$ denote its secrecy function and $\xi_{\Gamma}$ its maximum value. Formally unimodular packing secrecy fu…

Let $\Lambda$ be a formally unimodular lattice, and let $\Xi_{\Lambda}(\tau)$ denote its secrecy function and $\xi_{\Lambda}$ its maximum value. Belfiore–Solé secrecy function conj…

Let $G_p$ be the genus of lattices of rank $6$ and discriminant $D=p$. Kernel-dimension conjecture. The kernel of $\theta^{(2)}$ on $G_p$ has dimension equal to the number of class…

Let $G_p$ be the genus of lattices of rank $6$ and discriminant $p$, and let $k_p$ be the number of isomorphism classes in $G_p$ of lattices with no automorphism of determinant…

Let $\mathcal{E}$ be a rank $m$ vector bundle on $X'$, let $\mathcal{E}'=\sigma^{*}\underline{\operatorname{Hom}}(\mathcal{E},\nu^{*}\mathfrak{L})$, and suppose the maximal slope o…

Let $\mathcal{E}$ be a rank $m$ vector bundle on $X'$, let $\mathcal{E}'=\sigma^{*}\underline{\operatorname{Hom}}(\mathcal{E},\nu^{*}\mathfrak{L})$, and let…

Let $X'/X$ be the fixed double cover, let $U(2m)$ be the associated quasi-split unitary group, and let $\widetilde{Z}^{r}_{m}$ be the generating series taking values in…

Let $\mathcal{G}$ be the instanton generating function, expanded in terms of the completed generating functions $\widehat h_{p_i,\mu_i}$ and theta series…