13 problems

Let $X(\Delta_{n-1})$ be the toric variety associated with a reflexive toric diagram $\Delta_{n-1}$, and let $\widetilde{X(\Delta_{n-1})}$ denote the relevant resolution. Let…

Let $V$ be a smooth Fano manifold of complex dimension $n-1$, and let $I(V)\in\mathbb{N}$ be its Fano index. GMSY's Fano inequality conjecture. … with equality if and only if…

GMSY's volume conjecture. For every regular Reeb vector field $\xi$, the Bishop obstruction never occurs. This means that the above volume inequality does not hold.

Let $X$ be an isolated conical Calabi–Yau $3$-fold singularity with link $L$, and consider the four-dimensional $\mathcal{N}=1$ superconformal field theory obtained in the low-ener…

A Sasaki–Einstein five-manifold $X_5$ is an Einstein manifold whose metric cone is Ricci-flat and Kähler, hence a Calabi–Yau threefold. The product $AdS_5\times X_5$, equipped with…

Let $\mathcal{X}$ be a toric Calabi–Yau $3$-fold with no factors of $\mathbb{C}$, let $\zeta_R$ be the Reeb symmetry, and let $\eta_h$ be the test symmetry associated with a genera…

AdS/CFT correspondence. Type IIB string theory on $\mathrm{AdS}_5\times\mathrm{SE}^5$ is equivalent to a four-dimensional $N=1$ superconformal quiver gauge theory. The statement is…

Let $L_f$ be a rational homology $7$-sphere arising as the link of a hypersurface singularity, and suppose the defining weighted homogeneous polynomial has degree…

Sasaki–Einstein existence conjecture. For each integer $k\leq 8$ and $n\geq 2$, the Smale $5$-manifold

Main conjecture. (i) The one-point correlation measure converges weakly as $N\to\infty$ to the unit-volume form of a Sasaki–Einstein metric $g_M$ on $(M,\xi)$:

Let ${\cal X}$ be a reflexive toric Calabi–Yau $n$-fold with toric diagram $\Delta_{n-1}$, let $Y$ be its Sasaki–Einstein base, let $V(b_i^{*};Y)$ be the minimum volume, and let…

Let ${\cal X}$ be a toric Calabi–Yau $n$-fold with toric diagram $\Delta_{n-1}$, let $Y$ be its Sasaki–Einstein base, and let $V(b_i^{*};Y)$ denote the minimum volume. Let…

Sasaki–Einstein circle-bundle conjecture. If there exists a Kähler–Ricci soliton on $M$, then the $S^1$-bundle associated to $K_M$ admits a Sasaki–Einstein metric with a suitable c…