12 problems

Let ${\cal M}$ be Hitchin's moduli space of rank-$2$ Higgs bundles with fixed determinant of degree $1$ over a Riemann surface $\Sigma$ of genus $g>1$. It is a simply connected, no…

Let $\mathcal{M}$ be Hitchin's moduli space of Higgs bundles with fixed determinant of degree $1$ over a Riemann surface $\Sigma$ of genus $g>1$. It is a simply connected, non-comp…

Let a Riemannian metric on a contact manifold be blown up along the direction orthogonal to the contact distribution. Although the curvatures become unbounded, the metric converges…

Let $g$ be a metric on $T^3$ that is close to the flat metric. A harmonic $1$-form on $T^3$ is a $1$-form $\theta$ satisfying $\Delta\theta=0$, and it is nowhere vanishing when…

Nonexistence conjecture. There exist no $\mathbb Z_2$-harmonic $1$-forms on $Y$ with $\mathcal Z\neq\emptyset$ with respect to any metric. In particular, there exist no…

Crossing and smoothing deformation conjecture. There exist $\Z_2$ harmonic $1$-forms $(\om,\Sigma,\cI)$ on $\R^3$ such that $\Sigma$ is the union of two intersecting lines. Moreove…

Harmonic intersection-cohomology conjecture. The vector space of $\mathbb{V}$-valued $i$-forms that are harmonic and $L^2$ with respect to the Kähler metric is isomorphic to

Let $G$ be a pseudo-reflection group. Write $\mathcal{SH}_G^{i,k}$ for the component of $x$-degree $i$ and $\theta$-degree $k$, let $\Delta_G$ be the discriminant, and let…

Differential operator conjecture. If $G=G(m,1,n)$, then this identity holds. The conjecture extends the proved cases for rank at most two and the real pseudo-reflection groups; com…

Near-extremal intrinsic functional conjecture. There exists an $\epsilon>0$ such that

The conjecture. The theorem can still hold after replacing condition $(*)$ by condition $(*)'$: $X'$ admits a complete Riemannian metric satisfying $(*)'$, whereas $X$ admits no co…

Let $M_\beta$ be the family of Einstein manifolds parametrized by $\beta\in[0,2]$, and let $\mathcal{F}_3$ be the harmonic 2-form defined on $M_\beta$. Uniqueness conjecture. The f…