The seminar usually takes place in 515, Cosmology Building at 4-5PM every Thursday, unless noted otherwise.
Organizers: Nicolau Sarquis Aiex 艾尼克 (NTNU), Siao-Hao Guo 郭孝豪 (NTU), Chung-Jun Tsai 蔡忠潤 (NTU)
Kai-Hsiang Wang 望開翔 (Northwestern University): Introduction to Optimal Transport with Application to Geometric Inequalities
In this talk, I will first give an introduction to optimal transport (OT) theory and recall some fundamental results. This will then be illustrated by the common scenario where the cost function is given by distance squared and the measures are absolutely continuous. Under these assumptions, I will show the classical theorems of Brenier and McCann about OT maps for Euclidean spaces and manifolds, respectively, as well as the connection to the Monge-Amp\'ere equation and some geometric inequalities. After that, I will mention my previous work generalizing McCann's theorem to a submanifold setting and its application to the Michael-Simon inequality.
Wei-Bo Su 蘇瑋栢 (NCTS): Ancient solutions to curve shortening flow with finite entropy
Recently, asymptotic analysis on ancient solutions to mean curvature flow under certain convexity and low-entropy conditions has led to significant progress in the regularity theory in mean curvature flow in low dimensions. In the 1-dimensional case, convex ancient solutions are classified by Daskalopoulos–Hamilton–Sesum and Bourni–Langford–Tinaglia. In this talk, I will explain our recent progress towards the classification problem of ancient solutions to curve shortening flow under a much weaker assumption—finite entropy, which only places constraints on curves near space-time infinity. Specifically, in a joint work with Kyeongsu Choi, Donghwi Seo, and Kai-Wei Zhao, we show that ancient solutions with entropy less than 3 must be convex; hence they are completely classified by applying the convex result.
Moreover, we demonstrate that ancient solutions with finite entropy admit unique tangent flows at infinity given by lines with multiplicity. Additionally, there are finitely many ’tip points’ near which the curves resemble the translating ’Grim Reaper’. I will also show that the ancient solutions to curve shortening flow has finite entropy if and only if it has finite total curvature, based on a joint work with Kai-Wei Zhao.
Stephen Shing-Toung Yau 丘成棟 (Beijing Institute of Mathematical Sciences and Applications): Explore Geometry of Genome Space via Grassmann manifolds
We propose a novel method to represent genome data in a Grassmann manifold, enabling more accurate and efficient sequence comparison. By transforming genome sequences into frequency matrices, we employ the generalized Grassmannian distance to measure similarity. Our method outperforms existing techniques, as demonstrated by phylogenetic tree construction and comparative analysis. Additionally, low-dimensional visualization revealed intriguing geometric patterns in SARS-CoV-2 genome data. Our findings offer a new geometric framework for molecular biology research.
Shota Hamanaka 濱中翔太 (Osaka University): Limit theorems for the total scalar curvature
Gromov proved the following ''Limit theorem'': Let g be a C^2 Riemannian metric on a smooth manifold M (without boundary). If a sequence of C^2 Riemannian metrics on M converges to g in the C^0 sense, and each scalar curvature is bounded from below by k. Then the scalar curvature of the limiting metric g is also bounded from below by k. In this talk, I'd like to talk about some total-scalar-curvature-version theorems of this ''Limit theorem''. I also consider the limit theorem for certain weighted total scalar curvature and as its corollary, I give a definition of scalar curvature lower bound in a weak sense. To prove these, we use the Ricci flow and harmonic map heat flow with Ricci flow background. If I have time, I also would like to talk about limit theorems for the UPPER bound of total scalar curvatures. Compared to the above results, we use a different type of geometric flow: Yamabe flow.
Siqi He 何思奇 (Morningside Center of Mathematics): Z/2 harmonic forms, harmonic maps into R-trees, and compactifications of character variety
In this talk, we will explore the connection between the analytic compactification of the moduli space of flat SL2(C) connections on closed, oriented 3-manifolds defined by Taubes, and the Morgan–Shalen compactification of the SL2(C) character variety. We will discuss how these two compactifications are related through harmonic maps to R-trees. Additionally, we will discuss several applications of this construction in the analytic aspects of gauge theory. This is joint work with R. Wentworth and B. Zhang.
Maxwell Stolarski (Warwick): Closed Ricci Flows with Singularities Modeled on Asymptotically Conical Shrinkers
Shrinking Ricci solitons are Ricci flow solutions that self-similarly shrink under the flow. Their significance comes from the fact that finite-time Ricci flow singularities are typically modeled on gradient shrinking Ricci solitons. Here, we shall address a certain converse question, namely, “Given a complete, noncompact gradient shrinking Ricci soliton, does there exist a Ricci flow on a closed manifold that forms a finite-time singularity modeled on the given soliton?” We’ll discuss work that shows the answer is yes when the soliton is asymptotically conical. These Ricci flows can be shown to exist on a broad class of topologies. In particular, no symmetry or Kahler assumption is assumed, and so the proof involves an analysis of the Ricci flow as a nonlinear degenerate parabolic PDE system in its full complexity.
Jun Sun 孙俊 (Wuhan University): Translating solitons to a Lagrangian mean curvature flow with zero Maslov class
In this talk, we study translating solitons to the Lagrangian mean curvature flow with zero Maslov class, which are important models of Type II singularities. A necessary condition for a blow-up limit arising at a Type II singularity of a Lagrangian mean curvature flow with zero Maslov class is provided. As an application, we prove that the Lagrangian translating solitons constructed by Joyce-Lee-Tsui cannot be a blow-up limit for a Lagrangian mean curvature flow with zero Maslov class, answering an open question proposed by Joyce-Lee-Tsui and Neves-Tian, as well as we also prove that the Grim Reaper cannot be a blow-up limit for a Lagrangian mean curvature flow with zero Maslov class.
Yihan Wang 王一涵 (Peking University): Area-minimizing Hypersurfaces in Singular Ambient Manifolds
We are trying to generalize classical regularity theory for minimizing hypersurfaces to the case of singular ambient spaces. And our first step is to consider isolated singularities. We will prove a non-existence of 2-dimensional minimizing hypercones in 3-dimensional ambient cones with isolated singularities and nonnegative curvatures. Based on this model, a codimension 3 regularity result is expected using a standard blow-up argument. We will also talk about the new challenges we have to face when the ambient spaces become singular, such as the compactness, the monotonicity formula, and the density estimates of minimizing hypersurfaces.
Salvatore Stuvard (Universita degli studi di Milano): Dynamical instability of branched singularities of minimal hypersurfaces
The study of singularities of minimal surfaces is a fundamental problem in Geometric Analysis, which, for decades, has been fuelling some beautiful research in Differential Geometry, Geometric Measure Theory, Calculus of Variations, and PDE. It is widely known that a class of singularities which is particularly hard to treat is that of “branched” or “flat” singularities: these cannot be detected at the classical blow-up / linearization level, because their tangent cones are smooth planes with multiplicity larger than one. Yet, flat singularities are unavoidable even in case of hypersurfaces, and even if stability is assumed. In this talk, I will prove, for minimal hypersurfaces, that if blow-ups at a flat singularity converge to the tangent plane at a suitable (mild) rate then the singularity is dynamically unstable, in the sense that it can be perturbed away with a non-trivial, area reducing mean curvature flow (in the sense of Brakke). The rate of convergence we need to assume is satisfied in all known examples, and it can be proved to hold for large classes of solutions. I will then argue that the mean curvature flow may be used as a selection principle for “well-behaved” solutions to Plateau’s problem, even possibly in the case of very wild boundaries. This is joint work with Yoshihiro Tonegawa (Institute of Science Tokyo).
Beomjun Choi (POSTECH): Classification of ancient bubble-sheet ovals
Significant advances in geometric flows, such as the mean curvature flow and Ricci flow, rely on analyzing their singularity formations. The classification of possible singularity models has recently become feasible through refined asymptotic analysis. In the case of mean curvature flow in $\mathbb{R}^3$, B. White’s oval is the only non-self-similar ancient solution that can potentially arise as a singularity model. In this talk, we will discuss the recent classification of ancient ovals in $\mathbb{R}^4$, based on joint work with Daskalopoulos, Du, Haslhofer, and Sesum.
Ting-Jung Kuo 郭庭榕 (NTNU): A discussion on isomonodromic deformation and Painlevé property
The isomonodromic deformation plays a universal role in connecting various mathematics and physics research areas. In this talk, I will discuss the isomonodromy theory for a new class of Fuchsian-type elliptic second-order equations defined on the moduli space of elliptic curves with the parameter τ ∈ H. We observe that the associated isomonodromic deformation equation is governed by a new second-order nonlinear equation with a deep connection to the Painlevé VI equation but essentially admits different properties from PVI. Unlike the Painlevé VI equation, the new isomonodromic deformation equation admits so-called movable branch points that can be explicitly determined by their monodromy data.
