Introduction to Atiyah classes (2025-7-7~10, Panagiotis Batakidis)
Introduction to the mean curvature flow through singularities (2025-5-13, 郭孝豪)
Introduction to densities and integration on manifolds (2025-4-29, 劉筱玟)
Introduction to the Geometry of Conformal Manifolds (2025-4-22, 郭子模)
Moduli spaces and the Kuga-Satake construction (2025-4-8, Flora Poon)
Flat or Not? Rigidity of Graphical Translating Solitons in Mean Curvature Flow (2025-3-18, 黄垣熊)
An introduction to a theorem of Tian--Catlin--Zelditch and geometric quantization (2025-3-11, 吳侊儒)
The nonlocal ABP estimate on Riemmanian manifold (2024-11-22, Jongmyeong Kim)
Introduction to Ricci flow and Ricci soliton (2024-11-8, 陳柏揚)
Heat kernel asymptotics and Demailly‘s Morse inequalities (2024-11-1, 森田展弘)
Introduction to the Legendre transform and its characterization (2024-10-25, 邱聖夫)
Residue functions, adjoint ideal sheaves and their applications (2024-9-25, Mario Chan)
A Minkowski inequality on complete manifolds (2024-6-18, 邱維毅)
Geometry and topology in statistical mechanics (2024-6-13, 顏浩洋)
Path cohomology of digraphs as a Brown functor (2024-6-6, 阮登科)
A very brief introduction to Kodaira-Spencer map (2024-5-9, 黃筱涵)
Ancient solutions to curve shortening flow with finite entropy (2024-4-18, 蘇瑋栢)
Neighborhoods of leaves of a singular foliation (2023-11-23, Camille Laurent-Gengoux)
Formal transformation between geodesic coordinate systems (2023-9-21, 張華炘)
Introduction to categorical/geometric representation theory (2022-12-9, 許佑鴻)
Harmonic functions on complete Riemannian manifolds (2022-12-2, 蔡一豪)
Holomorphic vector bundles, Hermitian metrics, and positivity notions (2022-11-18, 吳侊儒)
A brief introduction to Riemann-Roch theorem (II) (2022-11-4, 張華炘)
A brief introduction to Riemann-Roch theorem (I) (2022-10-28, 梁孟豪)
Kontsevich deformation quantization in R^n and computation of weighted graphs (2022-10-21, 劉思承)
Hopf algebra and universal enveloping algebra (2022-10-14, 許智祐)
A Brief Introduction to Intersection Homology (2022-10-7, 陳俊碩)
A Brief Introduction to Simplicial Homology (2022-9-30, 陳俊碩)
Fedosov's quantization method on symplectic manifolds (2022-6-17, 張華炘)
Berezin-Toeplitz Quantization and Star Products (2022-6-10, 張晉嘉)
Principal bundles and invariant connections (2022-5-27, 徐靖家)
Time: 2025.9.17 (Wed) 16:30-17:30 (17:30-18:00自由討論)
Room: 綜三 631
Title: Introduction to derived intersections
Speaker: 陳俊碩
Abstract: The intersection of two submanifolds can fail to produce a smooth manifold in a classical sense, especially with non-transversal intersections. To solve this, we can enrich our mathematical structure beyond the category of smooth manifolds. In this talk, I will briefly recall the concept of L-infinity bundles and then define the derived intersection within this new framework. This provides a precise method to handle non-transverse intersections.
Links: announcement, video
Time: 2025.10.1 (Wed) 16:30-17:30 (17:30-18:00自由討論)
Room: 綜三 631
Title: TBA
Speaker: 吳侊儒 (Erdős Center)
Abstract: TBA
Links: announcement, video
Time: 2025.10.2 (Thu) 11:30-12:30 (12:30-13:00自由討論)
Room: 綜三 631
Title: An introduction to general inverse $\sigma_k$ equations
Speaker: 林朝明 (NTU)
Abstract: General inverse $\sigma_k$ equations arise in many areas, including PDE, differential geometry, and complex geometry. In this talk, I will begin by introducing general inverse $\sigma_k$ equations in Kähler geometry, highlighting classical examples such as the complex Monge–Ampère equation, the J-equation, the complex Hessian equation, and the deformed Hermitian–Yang–Mills equation. Next, by applying new techniques from real algebraic geometry, we are able to consider a more general format of these equations. Finally, from an analytic perspective, we will examine the solvability of these more complicated equations.
Links: announcement, video
Time: 2025.10.8 (Wed) 16:30-17:30 (17:30-18:00自由討論)
Room: 綜三 631
Title: Moduli Spaces of Instantons on Asymptotically Conical Spin(7)-Manifolds
Speaker: Tathagata Ghosh (NCTS)
Abstract: In this talk we discuss instantons on asymptotically conical Spin(7)-manifolds where the instanton is asymptotic to a fixed nearly G2-instanton at infinity. After discussing the preliminary notions of holonomy groups, G2 & Spin(7)-manifolds, asymptotically conical manifolds, and Yang-Mills equations & instantons in 4-dimensions, we mainly focus on the deformation theory of instantons on 8-dimensional the asymptotically conical Spin(7)-manifold.
As examples, we consider two important Spin(7) manifolds: $\mathbb{R}^8$, where $\mathbb{R}^8$ is considered to be an asymptotically conical manifold asymptotic to the cone over the round 7-sphere, and Bryant-Salamon manifold - the negative spinor bundle over 4-sphere, asymptotic to the cone over the squashed 7-sphere. We apply the deformation theory to describe deformations of Fairlie-Nuyts-Fubini-Nicolai (FNFN) Spin(7)-instantons on $\mathbb{R}^8$, and the Clarke-Oliviera instanton on the negative spinor bundle over 4-sphere. We also calculate the virtual dimensions of the moduli spaces using Atiyah-Patodi-Singer index theorem and the spectrum of the twisted Dirac operators.
Links: announcement, video
Time: 2025.10.29 (Wed) 16:30-17:30 (17:30-18:00自由討論)
Room: 綜三 631
Title: TBA
Speaker: Nicolau Aiex (NTNU)
Abstract: TBA
Links: announcement, video
Time: 2025.11.7 (Fri) 16:30-17:30 (17:30-18:00自由討論)
Room: 綜三 631
Title: TBA
Speaker: 黃彥彰 (NYCU)
Abstract: TBA
Links: announcement, video
Time: 2025.11.19 (Wed) 16:30-17:30 (17:30-18:00自由討論)
Room: 綜三 631
Title: TBA
Speaker: 阮志豪 (NYCU)
Abstract: TBA
Links: announcement, video
Time: 2025.11.26 (Thu) 11:30-12:30 (12:30-13:00自由討論)
Room: 綜三 631
Title: TBA
Speaker: Ronan Conlon (University of Texas at Dallas)
Abstract: TBC
Links: announcement, video
Time: 2025.9.10 (Wed) 16:30-17:30 (17:30-18:00自由討論)
Room: 綜三 631
Title: Introduction to L-infinity bundles
Speaker: 洪呈毅
Abstract: In this talk, I will start by briefly reviewing the idea of the category of fibrant objects (CFO) and then move on to introduce L-infinity bundles. After that, I will define weak equivalences and fibrations in this setting and explain how these make the category of L-infinity bundles into a CFO. If time permits, I will also describe the construction of derived path spaces in this category, which in turn shows that it satisfies the factorization lemma, a key lemma that ensures the structure truly fits within the framework of a CFO.
Links: announcement, video
Time: 2025.7.7 (Mon) ~ 2025.7.10 (Thu)
Room: 綜三 631
Title: Introduction to Atiyah classes
Speaker: Panagiotis Batakidis (Aristotle University of Thessaloniki, Greece)
Abstract: This is a mini course on vector bundles, connections and Atiyah classes.
Lecture 1: 2025.7.7 (Mon) 14:30 ~ 16:30, 2025.7.8 (Tue) 14:30 ~ 16:30, Introduction to vector bundles and Lie algebroids
Lecture 2: 2025.7.8 (Tue) 14:30 ~ 16:30, 2025.7.9 (Wed) 10:00 ~ 12:00, Connections on vector bundles and Lie algebroids
Lecture 3: 2025.7.9 (Wed) 14:30 ~ 16:30, Introduction to Atiyah classes
Lecture 4: 2025.7.10 (Thu) 10:00 ~ 12:00, Atiyah classes of homotopy modules
The lectures will be offered in English.
Note: the schedule was changed due to a typhoon.
Links: announcement; video: lecture 1, lecture 2, lecture 3, lecture 4
Time: 2025.5.27 (Tue) 16:30-17:30 (17:30-18:00自由討論)
Room: 綜三 631
Title: Brown functors of directed graphs
Speaker: 廖軒毅
Abstract: In this talk, we will consider Brown functors of directed graphs --- that is, contravariant functors from the homotopy category of finite directed graphs to the category of abelian groups, satisfying the triviality axiom, the additivity axiom, and the Mayer-Vietoris axiom. Our main theorem states that such a functor is representable. In addition, I will show that the first path cohomology of directed graphs provides a nontrivial example of a Brown functor. This talk is based on joint work with Zachary McGuirk, Dang Khoa Nguyen, and Byungdo Park.
Links: announcement, video
Time: 2025.5.20 (Tue) 16:30-17:30 (17:30-18:00自由討論)
Room: 綜三 631
Title: Introduction to algebraic stacks
Speaker: 周祐正 (中研院)
Abstract: In this talk, I will outline the ideas of the definition of algebraic stacks but without running into technical details. I will then focus on examples and motivations, including equivariant cohomology and the moduli problem.
Links: announcement, video
Time: 2025.5.13 (Tue) 16:30-17:30 (17:30-18:00自由討論)
Room: 綜三 631
Title: Introduction to the mean curvature flow through singularities
Speaker: 郭孝豪 (NTU)
Abstract: In this talk we will consider the evolution of a closed hypersurface moving by its mean curvature vector. In particular, we will introduce a natural continuation of the evolution once it becomes singular. Then the discussion will be focused on the special case where the motion is monotonically shrinking.
Links: announcement
Time: 2025.5.6 (Tue) 16:30-17:30 (17:30-18:00自由討論)
Room: 綜三 631
Title: Coadjoint orbits
Speaker: 洪呈毅
Abstract: It is well-known that a Lie group acts on the dual of its Lie algebra via the coadjoint action. In this talk, we will consider the orbit space of a coadjoint action and the symplectic form it carries. If time permits, we will also discuss the relationship between coadjoint orbits and symplectic reduction.
Links: announcement, video
Time: 2025.4.29 (Tue) 16:30-17:30 (17:30-18:00自由討論)
Room: 綜三 631
Title: Introduction to densities and integration on manifolds
Speaker: 劉筱玟
Abstract: In this talk, I will define densities on real vector spaces and smooth vector bundles, which generalize volume forms and enable coordinate-free integration, even in the absence of orientation. This provides a flexible framework for integration on manifolds, where traditional tools like orientation may not apply. The main focus will be on the properties of densities and their role in integration, with a discussion on the natural pairing between densities and generalized sections, and how they relate to differential operators and adjointness.
Links: announcement, video
Time: 2025.4.22 (Tue) 16:30-17:30 (17:30-18:00自由討論)
Room: 綜三 631
Title: Introduction to the Geometry of Conformal Manifolds
Speaker: 郭子模 (NCTS)
Abstract: The geometry of conformal manifolds is similar to that of Riemannian manifolds. In this talk, I will introduce three topics: conformal geodesics; GJMS operators, which are conformally covariant differential operators; and the renormalized volume, which was introduced by Graham and Witten.
Toward the end of the talk, I will focus on the geometry of conformal submanifolds. This includes recent work on extrinsic GJMS operators by Case, Graham, and Kuo, as well as work on tractor bundles by Curry, Gover, and Snell.
Links: announcement, video
Time: 2025.4.8 (Tue) 16:30-17:30 (17:30-18:00自由討論)
Room: 綜三 631
Title: Moduli spaces and the Kuga-Satake construction
Speaker: Flora Poon (NCTS)
Abstract: A coarse moduli space is a parametrisation space of isomorphism classes of structured algebra-geometric objects. We will look at the coarse moduli spaces for two types of complex projective varieties: abelian varieties and lattice polarized K3 surfaces. By the Global Torelli Theorem, the parameters in these moduli spaces can be given by the Hodge structures on the cohomology groups of the varieties. By modifying the classical Kuga-Satake construction which takes a K3 surface to an abelian variety, there is a particular coincidence of the parametrisation space of some families lattice polarized K3 surfaces and that of some families of abelian varieties.
Links: announcement, video
Time: 2025.3.18 (Tue) 16:45-17:45 (17:45-18:15自由討論)
Room: 綜三 631
Title: Flat or Not? Rigidity of Graphical Translating Solitons in Mean Curvature Flow
Speaker: 黄垣熊 (NCTS)
Abstract: Translating solitons play a fundamental role in mean curvature flow (MCF) as models for singularity formation. In this talk, we explore rigidity results for graphical translators. In joint work with Pyo and Ma, we show that any entire mean convex graphical translator must be a plane. Similar rigidity holds for translators with small entropy or controlled growth. This talk provides an accessible introduction to these results and their implications in geometric analysis.
Links: announcement, video
Time: 2025.3.11 (Tue) 16:45-17:45 (17:45-18:15自由討論)
Room: 綜三 631
Title: An introduction to a theorem of Tian--Catlin--Zelditch and geometric quantization
Speaker: 吳侊儒
Abstract: Geometric quantization is a vast subject. In one of its instances, the aim is to approximate the space of Kahler potentials by a sequence of finite dimensional spaces. Such an approximation was proved by Tian, and later refined by Catlin, Zelditch, and many other people. We will first go through some basics in complex geometry and then prove the Tian--Catlin--Zelditch theorem.
Links: announcement, video
Time: 2025.2.25 (Tue) 16:45-17:45 (17:45-18:15自由討論)
Room: 綜三 631
Title: Exploring Symmetries in Dynamical Systems
Speaker: 戴佳原
Abstract: Symmetries arise in many phenomena and enable dimensionality reduction and problem simplification in mathematical analysis. In this introductory talk, I will first review key concepts of equivariant dynamical systems, where a symmetry group acts on the phase space $X$ in a way that commutes with the underlying vector field. However, not all dynamical systems exhibiting symmetric phenomena possess group equivariance, prompting the need for weaker notions. One such approach (Ref. [1]) is to adopt a groupoid perspective, in which commutativity is only required within certain linear invariant subspace $X_j \subset X$. I will outline the framework and invite discussion on how it can broaden our understanding of symmetries in dynamical systems.
Reference: [1] Isabelle Schneider. Symmetry Groupoids in Dynamical Systems: Spatio-temporal Patterns and a Generalized Equivariant Bifurcation Theory, Freie Universität Berlin (2022).
Links: announcement, video
Time: 2024.12.13 (Fri) 16:45 – 17:45
Room: 綜三 631
Title: Introduction to A-infinity algebras
Speaker: 劉筱玟
Abstract: A-infinity algebras generalize associative algebras by incorporating higher homotopies through a sequence of multilinear operations. Motivated by the desire to reconstruct a multiplication on the cochain complex from the induced operation on its cohomology, A-infinity algebras provide a natural framework for this purpose. In this talk, I will introduce the basic definition, examples and properties of A-infinity algebras. If time permits, I will also discuss a coalgebraic approach to A-infinity algebras.
Links: announcement, video
Time: 2024.11.22 (Fri) 16:45 – 17:45
Room: 綜三 631
Title: The nonlocal ABP estimate on Riemmanian manifold
Speaker: Jongmyeong Kim (中研院)
Abstract: The Aleksandrov-Bakelman-Pucci (ABP) Maximum Principle is a pointwise estimate in elliptic PDE theory. From this estimate, One can achieve Hölder regularity of solution. In this talk I will talk about nonlocal version of ABP estimate on some manifolds. We will briefly review classical PDE regularity theory on both divergent and non-divergent form. After that, I will introduce nonlocal elliptic ABP estimate on Euclidean space and on some manifolds. Also, I would like to introduce nonlocal divergent (Dirichlet form) case and some geometric condition.
Links: announcement, video
Time: 2024.11.13 (Wed) 16:30 – 17:30
Room: 綜三 201
Title: An introduction to mirror symmetry
Speaker: Marco Suen 孫逸軒 (成功大學)
Abstract: Mirror symmetry is a duality between complex and symplectic geometry. In 1994, Kontsevich proposed a mathematical definition for mirror symmetry which is now known as homological mirror symmetry (HMS). HMS predicts that the Fukaya category of a symplectic manifold is quasi-equivalent to the derived category of its mirror complex manifold. Despite HMS has been proven in many interesting cases, it's usually hard to give an exact geometric correspondence between objects due to its homological nature. Two years after Kontsevich's proposal, Strominger-Yau-Zaslow introduced an entirely geometric approach to mirror symmetry, which is now known as the SYZ proposal. SYZ suggested that mirror pairs can be obtained by taking dual torus fibration and the mirror functor in HMS can be obtained by a Fourier-Mukai-type transform. In this talk, I would like to introduce mirror symmetry from the SYZ perspective. If time permits, I will talk about realization problems in tropical geometry.
Links: announcement
Time: 2024.11.8 (Fri) 16:45 – 17:45
Room: 綜三 631
Title: Introduction to Ricci flow and Ricci soliton
Speaker: 陳柏揚
Abstract: Ricci flow was introduced by Hamilton in 1982 to study the topology of 3-manifolds. It can be viewed as a heat flow of the Riemannian metric which evolves in the direction of its Ricci curvature. Ricci soliton is a self similar solution to the Ricci flow and often models the singularity of the flow. In this talk, we will discuss some basic concepts on the Ricci flow and Ricci soliton. Some of the results are based on my joint works with Zilu Ma and Yongjia Zhang, also with Luke Peachey and Man-Chun Lee.
Links: announcement
Time: 2024.11.1 (Fri) 16:45 – 17:45
Room: 綜三 631
Title: Heat kernel asymptotics and Demailly‘s Morse inequalities
Speaker: 森田展弘
Abstract: Demailly’s Morse inequalities plays an important role in complex geometry. So this talk will review the Demailly's Morse inequalities and give a heat kernel proof. The proof treats asymptotic problem in heat kernel induced by Kodaira Laplacian, then we apply the asymptotic heat kernel to prove the Demailly's Morse inequalities.
Links: announcement, video
Time: 2024.10.25 (Fri) 16:45 – 17:45
Room: 綜三 631
Title: Introduction to the Legendre transform and its characterization
Speaker: 邱聖夫
Abstract: The Legendre transform is a widely used tool in mechanics, thermodynamics, and convex geometry. Like its complex sibling, the Fourier transform, the importancy of the Legendre transform can never be underestimated. In this talk we will introduce the notion within the framework of mathematical duality. We will also go over the proof of an elegant characterization theorem of the transform under very mild assumptions on convex analysis.
Links: announcement
Time: 2024.10.18 (Fri) 16:45 – 17:45
Room: 綜三 631
Title: Modular classes of Lie algebroids
Speaker: 陳俊碩
Abstract: Two weeks ago, we discussed the modular class of a Poisson manifold, which characterizes the obstruction to the existence of a volume form invariant under Hamiltonian vector fields. Since a Poisson manifold naturally gives rise to a Lie algebroid, we may wonder whether the modular class can be studied within the framework of Lie algebroids. In this talk, I will begin with a fundamental introduction to Lie algebroids, followed by the definition of their modular classes. After that, we will explore some examples and revisit the case of Poisson manifolds.
Links: announcement, video
Time: 2024.10.04 (Fri) 16:30 – 17:30
Room: 綜三 631
Title: Modular classes of Poisson manifolds
Speaker: 陳俊碩
Abstract: In symplectic geometry, the Liouville theorem says that the symplectic volume form is preserved by Hamiltonian flows. For a Poisson manifold, the modular class is the topological obstruction to the existence of a volume form invariant under all Hamiltonian vector fields. In this talk, I will begin with a foundational introduction to Poisson geometry, followed by a detailed derivation of the modular class.
Links: announcement, video
Time: 2024.09.25 (Wed) 16:30 – 17:30
Room: 綜三 201
Title: Residue functions, adjoint ideal sheaves and their applications
Speaker: Mario Chan (釜山大學)
Abstract: We introduce in this talk a class of ``residue functions'', each of which ``deforms'' holomorphically certain weighted $L^2$ norm on the ambient complex manifold $X$ to an $L^2$ norm on the union of certain log-canonical (lc) centres of a given lc pair $(X,D)$. The properties of such residue functions can be encoded into a sequence of analytic adjoint ideal sheaves, which fit into various residue short exact sequences and are useful in facilitating induction on (co)dimension of lc centres in geometric problems involving lc singularities. As an illustration, we will see their use in a solution to Fujino's conjecture, that is, the injectivity theorem for lc pairs on compact Kähler manifolds. The content of this talk is based on the joint work with Young-Jun Choi and Shin-ichi Matsumura.
Links: announcement
Time: 2024.06.18 (Tue) 16:30 – 17:30
Room: 綜三 631
Title: A Minkowski inequality on complete manifolds (完備流形上的閔可夫斯基不等式)
Speaker: 邱維毅
Abstract: The classical Minkowski inequality implies the volume of a bounded convex domain in the Euclidean space is bounded by an integral of the mean curvature of its boundary. In this thesis, we obtain a version of such inequality without convexity assumptions for complete manifolds satisfying a weighted Poincare inequality. Additionally, we show that there are no embedded compact minimal surfaces on such manifolds.
在完備黎曼流形上，我們將討論閔可夫斯基不等式。假設該流形滿足加權龐加萊不等式，且里奇曲率有負值之下界。利用權重函數的增長，我們證明了閔可夫斯基不等式，而無需凸性條件。此外，我們還證明了該流形上不存在嵌入式緊緻極小曲面。
Links: announcement, video
Time: 2024.06.13 (Thu) 16:30 – 17:30
Room: 綜三 201
Title: Geometry and topology in statistical mechanics
Speaker: 顏浩洋
Abstract: Geometry and topology rank among the most valuable mathematical disciplines for theoretical physicists. In this presentation, I will introduce the foundational concepts of statistical mechanics, encompassing both quantum and classical forms, and elucidate their connections with geometry and topology. The main topics of discussion will include canonical transformations, Liville's theorem, the Landau-Ginzburg theory, and quantum topological phases.
Links: announcement
Time: 2024.06.06 (Thu) 16:30 – 17:30
Room: 綜三 201
Title: Path cohomology of digraphs as a Brown functor
Speaker: 阮登科
Abstract: In this talk, we will consider path cohomology groups of digraphs (i.e. directed graphs) as Brown functors. We will start with a homology theory for digraphs, and we will show that this homology is a homotopy invariant. The cohomology groups for digraphs can be obtained by taking the dual of the path chain complexes. We then show that the zeroth and the first path cohomology groups for finite directed graphs are the Brown functors, that is, they satisfy the Additivity axiom and the Mayer-Vietoris axiom.
Links: announcement
Time: 2024.05.31 (Fri) 11:00 – 12:30
Room: 綜三 631
Title: Floer homology
Speaker: 劉筱玟
Abstract: In this talk, we will discuss Floer homology, which is defined for a symplectic manifold with a time-dependent Hamiltonian. Floer homology is the homology of the Floer chain complex which is, as a vector space, a Z/2Z-vector space freely generated by the period solutions of the given Hamiltonian system. To define the boundary operator of the Floer complex, I will also need an action functional and the Floer equation. Finally I will compare Morse complexes and Floer complexes.
Links: announcement, video
Time: 2024.05.09 (Thu) 16:30 – 17:30
Room: 綜三 201
Title: A very brief introduction to Kodaira-Spencer map
Speaker: 黃筱涵
Abstract: The main goal of this talk is to briefly introduce the definition of the Kodaira-Spencer map. Before that, I will try to shortly introduce the concept of the deformation of complex structures on a manifold and the infinitesimal deformation.
If time allows, I might try to introduce theorems related to the Kodaira-Spencer map and the definition of versal deformation (also called complete) and semiuniversal deformation.
Links: announcement
Time: 2024.05.02 (Thu) 16:30 – 17:30
Room: 綜三 201
Title: On integral images of Curtis homomorphisms
Speaker: 李自然 (中央研究院)
Abstract: After introducing the Curtis homomorphism associated to a connected reductive group defined over a finite field, I shall talk about a characterization of integral images of Curtis homomorphisms by C. Bonnafé and R. Kessar, and then give a partial refinement of this characterization.
Links: announcement
Time: 2024.04.18 (Thu) 16:30 – 17:30
Room: 綜三 201
Title: Ancient solutions to curve shortening flow with finite entropy
Speaker: 蘇瑋栢 (理論中心)
Abstract: 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, 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'. This talk is based on joint work with Kyeongsu Choi, Donghwi Seo, and Kai-Wei Zhao.
Links: announcement
Time: 2024.04.11 (Thu) 16:30 – 17:30
Room: 綜三 201
Title: Introduction to Quantum Groups
Speaker: 賴俊儒 (中央研究院)
Abstract: The quantum groups, introduced in Drinfeld's 1986 ICM talk, have found numerous applications to diverse areas including mathematical physics, representation theory, algebraic combinatorics, and low dimensional topology.
I will give a crash course that includes the following features of the quantum groups:
(1) The origin of the terminology as solutions to the Yang-Baxter equations,
(2) The definition and its relation to Lie theory,
(3) The canonical basis and its connection to geometric representation theory,
(4) The quantum Schur-Weyl duality
Links: announcement
Time: 2024.03.28 (Thu) 15:30 – 16:30
Room: 綜三 201
Title: Introduction to Langlands correspondence
Speaker: 陳昰宇
Abstract: In the 1970s, R. Langlands proposed a conjectural correspondence between Galois representations and automorphic representations. Instances of the correspondence were established by consider étale cohomology of algebraic varieties and singular cohomology of locally symmetric spaces. In this talk, we will introduce Langlands correspondence based on examples.
Links: announcement
Time: 2024.03.14 (Thu) 17:30 – 18:30
Room: 綜三 201
Title: Introduction to Gromov-Witten theory
Speaker: 周祐正 (中央研究院)
Abstract: How do you count curves in a smooth variety? One approach is to study Gromov-Witten theory. Gromov-Witten invariants give a virtual count of the number of curves on a smooth projective variety with given conditions.
In this talk, I will introduce the moduli space of stable maps and the Gromov-Witten invariants. Then I will sketch a genus zero computation on P^2.
Links: announcement
Time: 2023.12.21 (Thu) 17:30 – 18:30
Room: 綜三 201
Title: Introduction to the upper shriek
Speaker: 邱聖夫
Abstract: The development of the upper shriek functors has occupied a central role in the formalism of Grothendieck duality. Nowadays, the functor has become an inevitable gadget in the categorical approach to algebraic geometry, symplectic geometry, and the geometric Langlands program. In this talk we will introduce the definition and emphasize its appearance in the homological algebra of triangulated categories. If time permits, we wish to go over Neeman’s ingenious proof of the general existence of the upper shriek functors.
Links: announcement
Time: 2023.12.13 (Wed) 17:30 – 18:30
Room: 綜三 201
Title: Morse Theory
Speaker: 蔡宗霖
Abstract: Morse theory gives us a way of analyzing the topology of a manifold by studying the differential functions on that manifold. In this talk, we consider a Morse function on a compact manifold, and use the Morse lemma to prove the Morse inequalities. Moreover, the Morse inequalities can be generalized to the Morse-Bott inequalities and the equivariant Morse inequalities.
Links: announcement
相關活動: NCTS lectures by Brett Parker from Australian National University.
Time: 11/27 - 12/8 at NCTS, Taipei (lecture 1 hour + discussion 1 hour, both weeks). Monday 10-12, Thursday 4-6, Friday 10-12
Title: Degenerations of holomorphic curves, tropical geometry, gluing theorems, and exploded manifolds
Abstract: Holomorphic curves play a central role in symplectic topology. They can be regarded as 2-dimensional analogues of a geodesics within a symplectic manifold, or as trajectories traced out by interacting strings in string theory, and provide a rich geometric framework for understanding symplectic topology. In many situations, holomorphic curves can be studied using 1-dimensional piecewise-linear objects called tropical curves. In the first lecture, I will explain the geometry behind the appearance of tropical curves, and explain why it is useful to employ a category blending tropical geometry with usual differential or algebraic geometry. In the remaining lectures, I will introduce the category of exploded manifolds, and explain how using such a category provides a guiding framework for proving gluing formulae and understanding holomorphic curves under a wide class of degenerations including normal crossing degenerations. Importantly, the transversality and intersection theory required for gluing theorems takes place within the category of exploded manifolds, so I will spend some time on transversality, intersection theory, and the implicit function theorem within the category of exploded manifolds.
Link: announcement
Time: 2023.11.23 (Thu) 17:30 – 18:30
Room: 綜三 201
Title: Neighborhoods of leaves of a singular foliation
Speaker: Camille Laurent-Gengoux (Université de Lorraine)
Abstract: For regular foliations, neighborhoods of leaves are classified by a group morphism from the fundamental group of the leaf to the group of local diffeomorphisms of the transversal. For singular foliations and singular leaves, it is much more involved. Still, there are many cases where there are not that many possible neighborhoods, even wide classes of singular leaves for which the classification is a finite dimensional problem. We will present a classification of formal neighborhoods of singular leaves in full generality, and address the question: how wide a class is it? Joint works with Leonid Ryvkin Simon Raphaël Fischer.
Links: announcement
Time: 2023.11.16 (Thu) 17:30 – 18:30
Room: 綜三 201
Title: What is a K3 category?
Speaker: 賴冠文 (中央研究院)
Abstract: A K3 category can be considered as an invariant attached to a complex cubic hypersurface of dimension four. They were introduced as an attempt to understand the birational geometry of such hypersurfaces. This talk is intended to be a gentle introduction to this topic.
Links: announcement
Time: 2023.10.26 (Thu) 17:30 – 18:30
Room: 綜三 201
Title: Category of fibrant objects
Speaker: 阮登科
Abstract: A category of fibrant object is a category equipped with two classes of maps called weak equivalences and fibrations, which satisfy certain axioms. In this talk, we will introduce the definition and show that the category of classical topological spaces and the category of (co)chain complexes are categories of fibrant objects.
Links: announcement
Time: 2023.10.19 (Thu) 17:30 – 18:30
Room: 綜三 201
Title: Classification of effective Hamiltonian S^1-actions with finite fixed points on CP^2
Speaker: 胡喬晏 (理論中心)
Abstract: In this talk, I would like to share the result of my master thesis under the supervision of Prof. Mei-Lin Yau. We classify the effective Hamiltonian S^1-actions with finite fixed points on CP^2. Building upon Professor Y. Karshon’s foundational work on ‘Periodic Hamiltonian Flows on Four Dimensional Manifolds’, our investigation leads to a significant conclusion: every such action can be symplectomorphically mapped to a standard linear case.
Links: announcement
Time: 2023.10.12 (Thu) 17:30 – 18:30
Room: 綜三 201
Title: Introduction to the Maslov index
Speaker: 劉筱玟
Abstract: The Maslov index is an integer associated with a loop in the Lagrangian Grassmannian manifold. In this talk, we will introduce Lagrangian subspaces and show bijection between the quotient group U(m)/O(m) and the Lagrangian Grassmannian manifold. Finally, we will give the definition of the Maslov index and compute it for some examples.
Links: announcement
Time: 2023.09.21 (Thu) 17:30 – 18:30
Room: 綜三 201
Title: Formal transformation between geodesic coordinate systems
Speaker: 張華炘
Abstract: Given an affine connection on a manifold, one has a geodesic exponential map which induces a family of geodesic coordinate systems. In this talk, we will consider a transformation between geodesic coordinate systems associated with two different choices of affine connections. We prove that the infinity jet of such a transformation is uniquely determined by an iteration equation following Fedosov’s iteration techniques. The talk is based on a joint work (arXiv:2309.02826) with Hsuan-Yi Liao.
Links: announcement, slides
Time: 2022.12.09 (Fri) 13:00 – 14:00
Room: 綜三 631
Title: Introduction to categorical/geometric representation theory
Speaker: 許佑鴻 (中央研究院)
Abstract: Representation theory is about the study of symmetry. In general, construction and classification of representations are the main problems in this field. On the other hand, categorification is the process of finding hidden higher level structure. In this talk, we will give an introduction to the combination of the two notions, which is called the categorical action. Usually the categories comes from geometric spaces that naturally arise in representation theory. Finally, if time permits we will discuss our current work where the motivation comes from the categorification of quantum groups and their actions.
Links: announcement, video
Time: 2022.12.02 (Fri) 13:00 – 14:00
Room: 綜三 631
Title: Harmonic functions on complete Riemannian manifolds
Speaker: 蔡一豪
Abstract: In this talk, we first introduce Bochner formula, Laplacian comparison theorem, and gradient estimate, which play fundamental roles in studying harmonic functions. After that, we show that any positive harmonic function on manifolds with non-negative Ricci curvature must be identically constant. Finally, we introduce some results about the dimension estimate of the space of polynomial growth harmonic functions.
Links: announcement
Time: 2022.11.25 (Fri) 13:00 – 14:00
Room: 綜三 631
Title: DGLA & Maurer-Cartan element
Speaker: 呂建德
Abstract: Stemming from Lie group and Lie Algebra, Maurer-Cartan form has played a role in modern physics such as gauge theory and quantization. We'll introduce the essential algebraic structure DGLA (no need for geometry background), and see how it generalizes concepts in some classical topics.
Links: announcement, video
Time: 2022.11.18 (Fri) 14:30 – 15:30
Room: 綜三 631
Title: Holomorphic vector bundles, Hermitian metrics, and positivity notions
Speaker: 吳侊儒 (中央研究院)
Abstract: We will introduce holomorphic vector bundles, Hermitian metrics, and Chern curvature. There are various positivity notions about the curvature, and we will focus on Griffiths positivity. Some basic properties will be given and proved. We will also mention some problems that seem still open.
Links: announcement, video
Time: 2022.11.04 (Fri) 13:00 – 14:00
Room: 綜三 631
Title: A brief introduction to Riemann-Roch theorem (II)
Speaker: 張華炘
Abstract: This is the second part of the talk on the Riemann-Roch theorem on compact Riemann surfaces. In this talk, I will introduce some basic topics about divisors on compact Riemann surfaces and then state the Riemann-Roch theorem and Serre duality. After that, I will give two applications of the Riemann-Roch theorem, which are the classification of compact Riemann surfaces and the computation of the dimension of Moduli space of pseudo-holomorphic curves.
Links: announcement, video
Time: 2022.10.28 (Fri) 13:00 – 14:00
Room: 綜三 631
Title: A brief introduction to Riemann-Roch theorem (I)
Speaker: 梁孟豪
Abstract: This is part (1) of a talk on Riemann-Roch theorem. We start with stating the Riemann-Roch theorem, which tells us how many "good meromorphic functions" are on a compact Riemann surface. After stating the formula, we explain the objects that occur in the formula, compact Riemann surface, genus, holomorphic functions, meromorphic functions, meromorphic 1-forms, etc. The other objects (like divisors), applications, etc., will be explained in the part (2). If the time permits, we will connect the relation between meromorphic functions and meromorphic 1-forms. Then, I will give a functional analysis proof of Runge approximation theorem on compact Riemann surfaces to finish part (1) of the talk.
Links: announcement, video
Time: 2022.10.21 (Fri) 12:00 – 13:00
Room: 綜三 631
Title: Kontsevich deformation quantization in R^n and computation of weighted graphs
Speaker: 劉思承
Abstract: The fundamental problem of deformation quantization is to find a star product for a given Poisson structure on a manifold. Among various methods, Kontsevich's application of the formality theorem solves the problem for general Poisson manifolds, which has an explicit formula on Euclidean spaces with a graphic representation. Though, the graphs become complicated in order-3 terms or higher. Buring and Kiselev hence classified the graphs to simplify the computation. In this talk I will present Kontsevich’s formula on Euclidean spaces and introduce the method, in which Buring and Kiselev generate and operate on weighted graphs to construct different star products.
Links: announcement, video
Time: 2022.10.14 (Fri) 13:00 – 14:00
Room: 綜三 631
Title: Hopf algebra and universal enveloping algebra
Speaker: 許智祐
Abstract: I will introduce the definition of Hopf algebra, and define the universal enveloping algebra U(g) of a Lie algebra g. Finally, I will show that U(g) is an example of Hopf algebra.
Links: announcement
Time: 2022.10.07 (Fri) 13:00 – 14:00
Room: 綜三 631
Title: A Brief Introduction to Intersection Homology
Speaker: 陳俊碩
Abstract: Poincare duality is a famous and fundamental theorem in geometry and topology. However, it is no longer true for singular variety. Mark Goresky and Robert MacPherson introduced the intersection homology in 1974, successfully reconstructed the duality property by restricting chains only interacting singular parts in controlled dimensions. In this talk, I will give the basic idea of intersection homology and some elementary calculations.
Links: announcement, video
Time: 2022.09.30 (Fri) 13:00 – 14:00
Room: online (https://nthu-meeting.webex.com/meet/hyliao)
Title: A Brief Introduction to Simplicial Homology
Speaker: 陳俊碩
Abstract: Classifying two topological spaces up to homeomorphism is one of the main topics in topology. To prove two topological spaces are not homeomorphic, one possible method is to show that some properties which are invariant under homeomorphisms are distinct. Homology group is a topologically invariant algebraic object and hence stands at the center of algebraic topology. In this talk, I will give a brief introduction of simplicial homology and the original idea of Poincare duality.
備註：前幾分鐘會用來說明研討會的運作方式
Links: announcement, video
Summer break (mid June -- mid September)
A relevant summer school: https://sites.google.com/view/usrp2022-dq/home
Time: 2022.06.17 (Fri) 10:00 – 11:00
Room: online (https://nthu-meeting.webex.com/join/hyliao)
Title: Fedosov's quantization method on symplectic manifolds
Speaker: 張華炘
Abstract: Deformation quantization is a mathematical formulation of quantization of a classical mechanical system, which is characterized by a Poisson bivector, to a quantum mechanical system, which is characterized by a “star product”. In 1997, Kontsevich solved this problem by proving the formality theorem. However, his method is quite complicated. A much easier and natural construction is feasible on a symplectic manifold, which is a special case of Poisson manifold. In this talk, I will provide some needed background knowledge and then introduce the idea of Fedosov's quantization method on symplectic manifolds.
Links: announcement, notes, video
Time: 2022.06.17 (Fri) 11:10 – 12:30
Room: online (https://nthu-meeting.webex.com/join/hyliao)
Title: Differential calculus on Lie algebroids
Speaker: 廖軒毅
Abstract: We discussed Section 2 in the paper
MR1726784
Evens, Sam (1-AZ); Lu, Jiang-Hua (1-AZ); Weinstein, Alan (1-CA)
Transverse measures, the modular class and a cohomology pairing for Lie algebroids.
Quart. J. Math. Oxford Ser. (2) 50 (1999), no. 200, 417–436.
Time: 2022.06.10 (Fri) 10:00 – 11:00
Room: online (https://nthu-meeting.webex.com/join/hyliao)
Title: Berezin-Toeplitz Quantization and Star Products
Speaker: 張晉嘉(Chin-Chia Chang)
Abstract: In this talk, I will introduce a method to construct star products on compact Kahler manifolds called Berezin-Toeplitz quantization. Using Toeplitz operators, one can construct a star product of the compactly supported smooth functions on C^{n}. On compact Kahler manifolds, one needs stationary phase formula for complex phase to obtain the asymptotic expansion of the Bergman kernel and the kernel function of Toeplitz operators, I will briefly introduce how to get the first term of the Berezin-Toeplitz star product.
Links: announcement, notes, video
Time: 2022.05.27 (Fri) 10:00 – 11:00
Room: online (https://nthu-meeting.webex.com/join/hyliao)
Title: Principal bundles and invariant connections
Speaker: 徐靖家
Abstract: In this talk, I would like to briefly introduce the concept of principal bundles and connections. With these terminologies, when there is an extra Lie group G acting on the principal bundle P by associating an automorphism of P to each element in G, is there a way to characterize the connections that are invariant under G-action. Under some conditions, the theorem by Wang (1958) gives a description of such connections.
Links: announcement, notes, video
Time: 2022.05.20 (Fri) 10:00 – 11:00
Room: online (https://nthu-meeting.webex.com/join/hyliao)
Title: Definition and examples of toric varieties
Speaker: 蔡宗霖
Abstract: Affine toric varieties is a kind of affine varieties obtained from rational convex polyhedral cones. In this talk, I will illustrate what is affine varieties, the maximal spectrum and rational convex polyhedral cone. Moreover, I will also explain how to construct affine varieties and demonstrate some examples.
Links: announcement, notes, video
Time: 2022.05.13 (Fri) 10:00 – 11:00
Room: online (https://nthu-meeting.webex.com/join/hyliao)
Title: Graphs in deformation quantization
Speaker: 廖軒毅
Abstract: Deformation quantization is a mathematical formulation of the general idea of quantization of a classical mechanical system to a quantum mechanical system. In this formulation, a classical mechanical system is characterized by a Poisson bivector, and a quantum mechanical system is characterized by a “star product” which is a formal deformation of the multiplication of smooth functions. In this talk, I will explain how Kontsevich applied graphs to the construction of star products.
備註：
前⼗分鐘會⽤來說明研討會的運作⽅式
Links: announcement, board, notes, video