10 problems

Apparent-singularity conjecture. The point $0$ is an apparent singularity if and only if $B=D_0=D=0$; equivalently, this polynomial system has only the trivial solution $B=D_0=D=0$…

Non-degeneracy conjecture. Except for finitely many $\tau$ modulo the $SL(2,\mathbb{Z})$ action, every solution of the Toda system is non-degenerate. If true, this would support th…

Solution-count conjecture. Except for finitely many $\tau$ modulo the $SL(2,\mathbb{Z})$ action, the Toda system has exactly $N(n_1,n_2)$ solutions. The conjecture extends the esta…

The Leray–Schauder degree conjecture. Under these conditions, the topological Leray–Schauder degree for the Toda system equals $N(\\{n_{1,k}\\}_{k},\\{n_{2,k}\\}_{k})$. This conjec…

Bruhat trajectory conjecture. There exist trajectories connecting $w$ and $w'$ if and only if the corresponding elements are comparable in Bruhat order. In that case, the space of…

Let $u^k$ be a sequence of blowup solutions of a rank $n$ Toda system, with $n\geq 3$. A component $u_i^k$ is not concentrating if it does not satisfy $u_i^k(x)\to -\infty$ away fr…

Phase-portrait conjecture. The phase picture of the Toda system coincides with the Bruhat order on the corresponding Weyl group: stationary points correspond bijectively to Weyl-gr…

Let $L$ be the $B_2$ Toda Lax matrix, and consider homogeneous quadratic Poisson brackets on $B_2$ that are compatible with the $B_2$ Lie-Poisson bracket, for which the functions…

Consider the general $SU(3)$ Toda system studied in the paper, with the parameter values $\mu_n$ defined in the preceding discussion. Lemma 6 describes bifurcation from the radial…

Fluxbrane polynomial conjecture. The functions $H_s(z)$ solving these equations and boundary conditions are polynomials of the form