The seminar usually takes place in 515, Cosmology Building at 4-5PM every Tuesday, unless noted otherwise.
Organizers: Nicolau Sarquis Aiex 艾尼克 (NTNU), Siao-Hao Guo 郭孝豪 (NTU), Chao-Ming Lin 林朝明 (NTU), Wei-Bo Su 蘇瑋栢 (NCU), Chung-Jun Tsai 蔡忠潤 (NTU)
Shih-Kai Chiu 邱詩凱 (University of California, Irvine): Nowhere-vanishing harmonic 1-forms on real loci of K3-fibered Calabi-Yau 3-folds
Let $X$ be a Calabi-Yau 3-fold equipped with a real structure and a compatible Lefschetz K3 fibration. Assuming that the real locus of $X$ is nonempty and contains no critical points of the fibration, we show that each connected component admits a nowhere-vanishing harmonic 1-form with respect to the metric induced by the collapsing Ricci-flat Kähler metrics on $X$. We then apply this to the Joyce-Karigiannis construction to produce compact 7-manifolds with full G2 holonomy, and also to construct families of pairwise disjoint special Lagrangian submanifolds. This is joint work with Daniel Platt and Calum Spicer.
Tzu-Mo Kuo 郭子模 (National Center for Theoretical Sciences): Extrinsic Conformally Covariant Boundary Operators for Submanifolds
Gover and Peterson constructed a family of conformally covariant differential operators on conformal manifolds with range restricted to hypersurfaces. In this talk, I will describe an analogous family of operators defined on submanifolds, with range restricted to their boundaries. These operators depend on the embedding of the submanifold. This is joint work with Jeffrey Case, Jarosław Kopiński, Aaron Tyrrell, and Andrew Waldron.
Yuan Shyong Ooi 黄垣熊 (National Center for Theoretical Sciences): Recent Progress in Rigidity of Free Boundary Minimal Hypersurfaces
Mazet and Mendes recently proved that stable free boundary minimal hypersurfaces exterior to the unit ball in with parallel regular ends are necessarily catenoidal i.e. -symmetry. This talk will explore the core mathematical machinery driving this rigidity result. Specifically, we will detail how they analyze the asymptotic behavior of positive Jacobi functions on the regular ends. I will then conclude by discussing ongoing joint work with Keomkyo Seo, where we aim to relax this regular ends assumption.
Lucas Ambrozio (IMPA-Instituto de Matemática Pura e Aplicada): Zoll-like metrics and rigidity theorems for the area spectrum
Zoll metrics are Riemannian metrics all of whose geodesics are closed, simple and have the same length. Lusternik-Schnirelmann min-max theory for closed geodesics can be used to characterise Zoll metrics on the two-dimensional sphere. We discuss analogues of this phenomenon in higher dimensions, in the context of the min-max theory for minimal two-dimensional spheres in a three-dimensional sphere.
Guillermo Sánchez Arellano (Universidad Complutense de Madrid): h-principles in smooth and complex geometry
The goal of the h-principle theory is to understand when a geometric problem is governed by the laws of differential topology. When topology (more flexible) overrides geometry (more rigid), we say that the h-principle holds in that context.
The h-principle theory has been especially fruitful in scenarios that exhibit both rigid and flexible properties, such as in the symplectic and contact settings.
The identity principle and the resulting lack of partitions of unity endow complex geometry with great rigidity. However, there is a class of complex manifolds, Stein manifolds, in which the Oka principle, a type of h-principle for holomorphic functions, holds. The flexible properties of Stein manifolds have been exploited by F. Forstneric and M. Slapar to establish h-principles for holomorphic immersions and submersions, and by F. Forstnerič for complex contact forms.
In this talk, we will review some of the most recent h-principle results in the smooth category, and explore how to abstract Forstneric and Slapar's techniques to obtain more general holomorphic h-principles in Stein manifolds.
Joshua Daniels-Holgate (Queen Mary University of London): Mean curvature flow from conical singularities
We discuss some regularity results for mean curvature flow from smooth hypersurfaces with conical singularities. We then discuss how to use these results to tackle the conical singularity resolution conjecture of Ilmanen, demonstrating a non-uniqueness dichotomy: a closed flow encountering a conical singularity ‘fattens’ if and only if the asymptotic cone also fattens. This is joint work with Otis Chodosh and Felix Schulze.
Jiuzhou Huang (Nanjing University of Science and Technology): Mean curvature flow converging to a strictly stable minimizing cone and its foliation
In 1994, Velázquez constructed a family of mean curvature flows converging to Simons' cones (which have O(n)*O(n) symmetry) after a type I rescaling and its Hardt-Simon foliation after a type II rescaling. Later on, Guo-Sesum simplied Velázquez's proof, and Liu generalized Velázquez's construction to quadratic cones (which have O(p)*O(q) symmetry). In this talk, I will discuss how to construct such a mean curvature flow for a general strictly stable minimizing cone without symmetry assumptions.
Spandan Ghosh (University of Oxford): Lagrangian mean curvature flow out of conical singularities
Lagrangian mean curvature flow (LMCF) is a way to deform a Lagrangian submanifold inside a Calabi--Yau manifold according to the negative gradient of the area functional. There are influential conjectures about LMCF due to Thomas--Yau and Joyce, describing the long-time behaviour of the flow, singularity formation, and how one may flow past singularities. In this talk, we will show how one may flow out of a conically singular Lagrangian by gluing in expanders asymptotic to the cone, generalizing an earlier result by Begley--Moore. We solve the problem by a direct P.D.E.-based parabolic gluing method, along the lines of recent work by Lira--Mazzeo--Pluda--Saez on the network flow. The main technical ingredient we use is the notion of manifolds with corners and a-corners as introduced by Joyce following earlier work of Melrose.
Nicolaos Kapouleas (Brown University): Minimal surface and hypersurface doublings
After a brief introduction recalling classical and semi-classical results and conjectures, I will discuss the current status of minimal surface and hypersurface doublings and their geometry, including recent results with J. Zou on hypersurface doublings (arXiv:2405.18283), the index of some minimal surface doublings (arXiv:2512.07734), and if time allows ongoing work and open problems.
Sourav Ghosh (University of Notre Dame): Decay order bound for mean curvature flow near compact singularities
We consider the rescaled flow associated with a mean curvature flow that develops a compact singularity of multiplicity one. We prove that the "decay order'' of such a rescaled flow is uniformly bounded. As a consequence, we prove a unique continuation result.
Kai Wen Caleb Suan (Chinese University of Hong Kong): Hull--Strominger Systems and Geometric Flows
The Hull--Strominger system is a system of partial differential equations stemming from heterotic string theory in physics. Mathematically, these equations lead us to consider special structures with torsion and have been proposed as a natural generalization of the Ricci-flat condition to non-Kähler Calabi--Yau threefolds. In this talk, we discuss a geometric flow approach to the system, known as the anomaly flow. We shall also look at 7-dimensional analogues of the system and flow.
Shubham Dwivedi (University of Hamburg): Ricci-harmonic flow of $G_2$-structures
We will talk about a new general flow of $G_2$-structures which we call the Ricci-harmonic flow. The flow is a natural coupling of the Ricci flow of underlying metrics and the isometric flow of $G_2$-structures but with explicit lower order terms. We will explain how to obtain the lower order terms and how the flow can be interpreted as an analog of the Ricci flow of metrics for $G_2$-structures. We will discuss various analytic and geometric results for the flow including short-time existence and uniqueness of solutions, regularity theory and long time existence results. Time permitting, we will also talk about explicit solutions and singularities of the flow. The talk is based on arXiv:2601.05210 and arXiv:2601.16832 (joint with Ragini Singhal).
Shouhei Honda 本多正平 (University of Tokyo): Gel’fand’s inverse problem under Ricci curvature bounds
The classical Gel’fand’s inverse problem asks whether a Riemannian manifold is uniquely determined by the knowledge of the heat kernel on any open subset of the manifold. We study this inverse problem in the non-smooth setting in the framework of RCD(K,N) spaces, namely, metric-measure spaces with syntheticRiemannian Ricci curvature bounded below by K and dimension bounded above by N. We establish the unique solvability of Gel’fand’s inverse problem for the class of compact RCD(K,N) spaces whose regular set admits C^1-Riemannian structure. As an application, we obtain the stability of Gel’fand’s inverse problem in the class of closed Riemannian manifolds with bounded Ricci curvature, diameter and volume bounded from below. We note that the results are new even for Einstein orbifolds and (weighted) Riemannian manifolds with non-smooth boundary. This is a joint work with Jinpeng Lu (University of Helsinki), based on arXiv:2602.14527.
Poom Lertpinyowong (University of California, Irvine): Mapping Cone Cohomology and Fibrations
In 2016, Tsai, Tseng, and Yau utilized the Lefchetz decomposition to define a family of \(A_\infty\)-algebras \(F^p(M,\omega)\) for a symplectic manifold \((M^{2n},\omega)\) and \(p=0,1,\ldots,n-1\). In 2018, Tanaka and Tseng showed that for each \(p\), the \(A_\infty\)-algebra \(F^p(M,\omega)\) is quasi-isomorphic to a mapping cone cdga associated to the chain map \[ \omega^{p+1}\wedge : \Omega^\bullet(M) \longrightarrow \Omega^{\bullet+2p+2}(M). \] In general, given a closed form \(\psi\) on \(M\) one may define a mapping cone complex \(C(M,\psi)\), which, in addition, has a product structure when the degree \(|\psi|\) is even. Its cohomology \(H_C(M,\psi)\) is invariant under diffeomorphisms preserving the de Rham class \([\psi]_{dR}\). This cohomology possesses certain properties analogous to those of de Rham cohomology.
By proving mapping cone versions of the Leray--Serre spectral sequence, we establish a theory for computing the cone cohomology of fibrations equipped with compatible forms. In the symplectic context, this applies to symplectic products and symplectic fibrations.
