In this paper, we prove a general existence theorem for properly embedded minimal surfaces with free boundary in any compact Riemannian 3‐manifold M with boundary ∂M. These minimal surfaces are either disjoint from ∂M or meet ∂M orthogonally. The main feature of our result is that there is no assumptions on the curvature of M or convexity of ∂M. We prove the boundary regularity of the minimal surfaces at their free boundaries. Furthermore, we define a topological invariant, the filling genus, for compact 3‐manifolds with boundary and show that we can bound the genus of the minimal surface constructed above in terms of the filling genus of the ambient manifold M. Our proof employs a variant of the min‐max construction used by Colding and De Lellis on closed embedded minimal surfaces, which were first developed by Almgren and Pitts.

We prove a lower bound for the first Steklov eigenvalue of embedded minimal hypersurfaces with free boundary in a compact n-dimensional Riemannian manifold which has nonnegative Ricci curvature and strictly convex boundary. When n=3, this implies an apriori curvature estimate for these minimal surfaces in terms of the geometry of the ambient manifold and the topology of the minimal surface. An important consequence of the estimate is a smooth compactness theorem for embedded minimal surfaces with free boundary when the topological type of these minimal surfaces is fixed.

We prove a version of equivariant split generation of Fukaya category when a symplectic manifold admits a free action of a finite group <i>G</i>. Combining this with some generalizations of Seidel's algebraic frameworks from [35], we obtain new cases of homological mirror symmetry for some symplectic tori with non-split symplectic forms, which we call <i>special isogenous tori</i>. This extends the work of AbouzaidSmith [2]. We also show that derived Fukaya categories are complete invariants of special isogenous tori.

In this paper, we prove the DonohoStark uncertainty principle for locally compact quantum groups and characterize the minimizer which are bi-shifts of group-like projections. We also prove the HirschmanBeckner uncertainty principle for compact quantum groups and discrete quantum groups. Furthermore, we show Hardy's uncertainty principle for locally compact quantum groups in terms of bi-shifts of group-like projections.

The representation category of a conformal net is a unitary modular tensor category. We investigate the reconstruction program: whether all unitary modular tensor categories are representation categories of conformal nets. We give positive evidence: the fruitful theory of multi-interval Jones-Wassermann subfactors on conformal nets is also true for modular tensor categories. We construct multi-interval Jones-Wassermann subfactors for unitary modular tensor categories. We prove that these subfactors are symmetrically self-dual. It generalizes and categorifies the self-duality of finite abelian groups. We call this duality the modular self-duality, because the modularity of the modular tensor category appears in a crucial way. For each unitary modular tensor category, we obtain a sequence of unitary fusion categories. The cyclic group case gives examples of Tambara-Yamagami categories.