14 problems

Let $H_{0;1,N-1}$ be the Hurwitz space of rational functions on the Riemann sphere with two poles, one of order $1$ and one of order $N-1$, where $N\geq 2$. Each such space carries…

Let ${\cal A}(P)$ be the algebra containing the elements $P_m$, for $m\in\mathbb{N}$, and let $\Phi$ be the map from ${\cal A}(P)$ to expressions for the noncommutative KP hierarch…

Novikov's conjecture. Just the first equation, with $N=1$, is sufficient for the characterization of the Jacobians.

Eigenvalue-support conjecture. The support of the distributions annihilating $z^n\psi_M$ is the set of eigenvalues of the matrices $Y$ and $Z$, with the highest derivative taken at…

Novikov's conjecture. The function $u(x,y,t)$ satisfies the KP equation if and only if $\Omega$ is the period matrix of some compact Riemann surface $\Sigma$.

Consider the hierarchy obtained from three copies of the dispersionless Toda lattice hierarchy by the cyclic coupled reduction … The Whitham correspondence conjecture. The resultin…

Let the topological vertex be regarded in the fermionic picture, and let the fermionic vacuum vector belong to the $3$-component fermionic Fock space. A Bogoliubov transform is the…

Let $\mathcal C=(\Sigma,x,y)$ be a spectral curve with boundaries ${\bf b}=\{b_1,\ldots,b_m\}$, and let $\Lambda=(\lambda_1,\ldots,\lambda_m)$ be local coordinates near the boundar…

The constrained KP hierarchy is an integrable hierarchy, and its dispersionless limit determines a Frobenius manifold. The Dubrovin–Zhang hierarchy is the hierarchy associated with…

Let $a,b\in\mathbb{C}$, let $\Theta$ be the Riemann theta function appearing in the construction, let $\mathbf{v}_{0}(a),\mathbf{v}_{1}(a),\mathbf{v}_{2}(a),\mathbf{v}_{0}(b)$ be t…

Let $\mathcal{L}$ be the Lax operator of the extended rationally constrained KP hierarchy, with dependent variables $a_1,\dots,a_N$, and let $\log \mathcal{Q}$ and $\tilde{B}_\ell$…

BKP identification conjecture. The BKP hierarchy can be identified with the restriction of the odd-times flows of the KP hierarchy to this locus.

Let $Z_{q,p}({\bf T})$ be the generating function of triple Hodge integrals … with … The parameters satisfy the Calabi–Yau condition, and $\mathbf T$ denotes the Hodge-integral var…

Let $\mathrm{QSym}$ be the algebra of quasi-symmetric functions, let $\mathrm{Sym}$ be the algebra of symmetric functions, and let $\bullet$ denote the product used to build identi…