Thursday

There have been proposals to study different classes of manifolds by counting calibrated submanifolds: for instance, holomorphic curves in symplectic geometry, special Lagrangians in Calabi-Yau 3-folds, and associatives in G2-manifolds. While the holomorphic curve case leads to the well-established Gromov-Witten theory, the remaining cases, despite strong geometric and physics-inspired motivations for these counts, face serious analytical difficulties. A central difficulty arises from the formation of singular calibrated submanifolds, which complicates the definition of an invariant count, and therefore any meaningful count depends on understanding the formation of singularities. Although a general theory remains out of reach, there has been significant recent progress. We review the background and describe the recent developments both in compactifications of moduli spaces and in methods to define invariant counts. The talk is based on a collection of joint works with Yang Li, work with Gora Bera, an ongoing project with Tobias Ekholm, Vivek Shende, and Luya Wang, and an in progress work with Cliff Taubes.

We resolve the Mean Convex Neighborhood Conjecture for mean curvature flows in all dimensions and for all types of cylindrical singularities. Our proof relies on a complete classification of ancient, asymptotically cylindrical flows. We prove that any such flow is non-collapsed, convex, rotationally symmetric, and belongs to one of three canonical families: ancient ovals, the bowl soliton, or the flying wing translating solitons. This is joint work with Richard Bamler.

In the round 3-sphere, there exists an family of minimal 2-spheres (the equatorial spheres) and an family of minimal tori (the Clifford tori). Almgren (1966) and Calabi (1967) proved that the equatorial spheres are the only minimal 2-spheres, and Brendle (2013) confirmed that the Clifford tori are the only minimal tori, thereby resolving the Lawson conjecture. The topology of these moduli spaces motivated Yau (1982) and White (1989) to conjecture the existence of at least four minimal spheres and five minimal tori, respectively, in any closed Riemannian 3-sphere. For other topological types, Lawson (1970) constructed minimal surfaces of arbitrary genus in , now known as Lawson surfaces. Inspired by Yau’s and White’s conjectures, it is further expected that for any genus g, there exist multiple genus-g minimal surfaces in any closed Riemannian 3-sphere, with the number related to the topology of the space of Lawson surfaces. In this talk, I will present the resolution of Yau’s conjecture by Wang–Zhou (2023) and of White’s conjecture by joint work of Adrian and myself (2024), in the setting of positive Ricci curvature. I will then discuss how the techniques we developed can be applied to construct multiple minimal surfaces of higher genus. This is based on joint work with Adrian Chu (Cornell University) and Zhihan Wang (Cornell University).

A Riemannian manifold (M,g) is said to be curvature-homogeneous if, for any two points x and y in M, there is an isometry between T_xM and T_yM that identifies the curvature tensors R_x and R_y. Of course, any locally homogeneous Riemannian manifold is curvature-homogeneous, and the converse is true in dimension 2 (where the condition is equivalent to the constancy of the Gauss curvature), but this is not so in higher dimensions. I will survey the history of this problem, starting with the work of I. Singer and culminating with very recent joint work of myself and Renato Bettiol in which we prove a number of new results, giving sufficient conditions for curvature-homogeneity to imply local homogeneity (for example, we prove that a curvature homogeneous 4-manifold that is either Einstein or conformally flat must be locally homogeneous) and also constructing new examples of curvature-homogeneous Riemannian 4-manifolds that are not locally homogeneous and not products of lower dimensional curvature-homogeneous manifolds.

In this talk, I will present several results about isotopy problems in dimension 4. First, we give a classification of the isotopy classes of embeddings of $\Sigma$ in $\Sigma\times S^2$ that are geometrically dual to $\{pt\}\times S^2$, where $\Sigma$ is a closed oriented surface with a positive genus, and show that there exist infinitely many such embeddings that are homotopic to each other but mutually non-isotopic. This answers a question of Gabai. Second, we show that the space of symplectic forms on an irrational ruled surface homologous to a fixed symplectic form has infinitely many connected components. This gives the first such example among closed 4-manifolds and answers a question of McDuff-Salamon. We also show that symplectic forms on a closed 4-manifold with a fixed cohomology class do not admit the h-principle, which answers a question of Cieliebak-Eliashberg-Mishachev. The proofs are based on a generalization of the Dax invariant to embedded closed surfaces. This is joint work with Jianfeng Lin, Weiwei Wu, and Yi Xie.