We study n-dimensional totally real minimal submanifolds M immersed in complex space forms ${\tilde{M}}^n(c)$ for $c\le 0$. We prove that M is totally geodesic if the $L^{2p}$-norm of the second fundamental form is finite for some $p\in \mathbb{R}$. We also derive vanishing results for the space of harmonic forms on such submanifolds.

R. Sharifi formulated remarkable conjectures which relate the arithmetic of cyclotomic fields to modular curves. We give partial solutions to his conjectures.

We study the equivalence between the infinitesimal Torelli theorem for smooth hypersurfaces in rational homogeneous varieties with Picard number 1 and the theory of generalized Massey products. This equivalence shows that the differential of the period map vanishes on an infinitesimal deformation if and only if certain twisted differential forms are elements of the Jacobian ideal of the hypersurface. We also prove an infinitesimal Torelli theorem result for smooth hypersurfaces in log parallelizable varieties.

We give a new proof of the following theorem: moduli spaces of stable complexes on a complex projective K3 surface, with primitive Mukai vector and with respect to a generic Bridgeland stability condition, are hyper-Kähler varieties of ${\mathrm{K}\mathrm{3}}^{[n]}$-type of expected dimension. We use derived equivalences, deformations, and wall-crossing for Bridgeland stability to reduce to the case of the Hilbert scheme of points.

We discuss noninjectivity of mixed q-deformed Araki–Woods von Neumann algebras, extending the results in our previous work. These von Neumann algebras generalize those that were constructed by Hiai.

Let $d\in \{3,4,5,\dots \}$. Consider $L=-\frac{1}{w}\mathrm{div}(A\nabla u)+\mathrm{\mu }$ over its maximal domain in $L_w^2({ℝ}^d)$. Under certain conditions on the weight w, the coefficient matrix A, and the positive Radon measure μ, we characterize the Hardy space $H_L^1({ℝ}^d)$ and BMO space ${\mathrm{BMO}}_L({ℝ}^d)$ associated with L using square functions. Other results include $H_L^1({ℝ}^d)\equiv H_{L,max}^1({ℝ}^d)$ and ${(H_L^1({ℝ}^d))}^{\ast }\equiv {\mathrm{BMO}}_L({ℝ}^d)$ as norm spaces, where $H_{L,max}^1({ℝ}^d)$ denotes the maximal Hardy space associated with L.

We introduce a conjecture on homological mirror symmetry relating the symplectic topology of the complement of a smooth ample divisor in a K3 surface to the algebraic geometry of type III degenerations. We prove it when the degree of the divisor is either 2 or 4.