Title: Closed Mean Curvature Flows with Prescribed Tangent Flows

Abstract: We show that every nonflat smooth embedded shrinker that is closed, smoothly asymptotically conical, or a product of such a shrinker with a Euclidean factor can arise as a multiplicity-one tangent flow at the first singularity of a mean curvature flow starting from a closed embedded hypersurface. We also discuss how certain first-order asymptotics near the singularity can be prescribed. This is joint work with Jingwen Chen, Tang-Kai Lee, and Ao Sun.

Title: Isoperimetry by stretching 

Abstract: Isoperimetric boundaries minimise area for fixed enclosed volumes, with sharp regularity theory ensuring they are smooth away from a closed singular set of codimension seven. I will discuss recent work, with G. Niu, which constructs isoperimetric regions from hypersurfaces in closed manifolds. As a direct application, we show that a wide variety of topological types and singular sets arise in isoperimetric boundaries.

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Title: On a construction of hermitian metrics on holomorphic vector bundles

Abstract: With any holomorphic vector bundle E over a compact base one can associate a certain line bundle L (often denoted O_{PE}(1)). According to a conjecture of Griffiths from the 1960s, if L admits a positively curved hermitian metric k, then E also admits a positively curved hermitian metric, h. In the talk I will discuss recent developments on this, in particular, a refutation of the conjecture.

Title: Topics in complex geometry

Abstract: We will discuss some recent advances in the complex algebraic geometry.

Title: Rigidity of infinite-dimensional geometries

Abstract: This talk explores recent research on rigidity phenomena in non-linear infinite-dimensional geometries, focusing on two key examples: Ebin's space of Riemannian metrics and the Wasserstein space of probability measures. Both are infinite-dimensional spaces built over a finite-dimensional base. I will show that these spaces do not gain exotic new symmetries; nearly every isometry arises directly from symmetries of the base, with a single exception for the Wasserstein space over the real line. Furthermore, these spaces faithfully preserve their structural origins: Ebin's space determines the underlying smooth manifold up to diffeomorphism, and the Wasserstein space determines the Riemannian manifold up to isometry.

Title: Equivariant Minimal Surfaces and The Allen-Cahn Equation

Abstract: For a closed manifold (M,g) with isometry group, G, we will discuss the construction of G-invariant minimal hypersurfaces. Pitts-Rubinstein initiated the program to construct such hypersurfaces via min-max methods. The community has made significant progress when the cohomogeneity of the manifold is at least 3 (i.e. the quotient M/G "looks" 3 dimensional). However, serious complications arise when the cohomogeneity is 2, as min-max on closed surfaces produce geodesic networks. Using the Allen-Cahn regularization of the area functional, we provide a general method for constructing minimal hypersurfaces in cohomogeneity 2 when there are no exceptional orbits of the group action, and any cohomogeneity greater than 2 for any action. This is joint work with Pedro Gaspar and Rayssa Caju.

Title: Metric invariants and scalar curvature

Abstract: We will discuss scalar curvature from the point of view of metric invariants. In particular, we will describe a new estimate for the codimension two width of circle bundles, and explain how it leads to counterexamples to the macroscopic version of Gromov’s Urysohn width conjecture, in dimensions four and higher. Based on joint work with B. Sen.

Title: Exploring the Chebyshev potential

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Title: Conformal invariants, nodal sets, negative eigenvalues and curvature prescription. 

Abstract: We shall discuss conformal invariants that arise from nodal sets of certain eigenfunctions of conformally covariant operators, as well as applications to scalar curvature prescription problems.   If time permits, we shall discuss related results in graph theory.  

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