Events

Table of contents

Upcoming events

TBA

Time:  

Room:  

Title:  

Speaker:  

Abstract:  


Links: announcement 


Past events

A Minkowski inequality on complete manifolds  (2024-6-18, 邱維毅)

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


Geometry and topology in statistical mechanics  (2024-6-13, 顏浩洋)

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


Path cohomology of digraphs as a Brown functor (2024-6-6, 阮登科)

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


Floer homology (2024-5-31, 劉筱玟)

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


A very brief introduction to Kodaira-Spencer map (2024-5-9, 黃筱涵)

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


On integral images of Curtis homomorphisms (2024-5-2, 李自然)

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


Ancient solutions to curve shortening flow with finite entropy (2024-4-18, 蘇瑋栢)

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


Introduction to Quantum Groups (2024-4-11, 賴俊儒)

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


Introduction to Langlands correspondence (2024-3-28, 陳昰宇)

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


Introduction to Gromov-Witten theory (2024-3-14, 周祐正)

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


Introduction to the upper shriek (2023-12-21, 邱聖夫)

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


Morse Theory (2023-12-13, 蔡宗霖)

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


Neighborhoods of leaves of a singular foliation (2023-11-23, Camille Laurent-Gengoux)

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


What is a K3 category? (2023-11-16, 賴冠文)

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

Category of fibrant objects. (2023-10-26, 阮登科)

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


Classification of effective Hamiltonian S^1-actions with finite fixed points on CP^2 (2023-10-19, 胡喬晏)

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


Introduction to the Maslov index (2023-10-12, 劉筱玟)

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


Formal transformation between geodesic coordinate systems (2023-9-21, 張華炘)

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


Introduction to categorical/geometric representation theory (2022-12-9, 許佑鴻)

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


Harmonic functions on complete Riemannian manifolds (2022-12-2, 蔡一豪)

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


DGLA & Maurer-Cartan element (2022-11-25, 呂建德)

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


Holomorphic vector bundles, Hermitian metrics, and positivity notions (2022-11-18, 吳侊儒)

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


A brief introduction to Riemann-Roch theorem (II) (2022-11-4, 張華炘)

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


A brief introduction to Riemann-Roch theorem (I) (2022-10-28, 梁孟豪)

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


Kontsevich deformation quantization in R^n and computation of weighted graphs (2022-10-21, 劉思承)

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


Hopf algebra and universal enveloping algebra (2022-10-14, 許智祐)

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


A Brief Introduction to Intersection Homology (2022-10-7, 陳俊碩)

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


A Brief Introduction to Simplicial Homology (2022-9-30, 陳俊碩)

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

Fedosov's quantization method on symplectic manifolds (2022-6-17, 張華炘)

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


Differential calculus on Lie algebroids (2022-6-17, 廖軒毅)

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.

Links: notes, video


Berezin-Toeplitz Quantization and Star Products (2022-6-10, 張晉嘉)

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


Principal bundles and invariant connections (2022-5-27, 徐靖家)

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


Definition and examples of toric varieties (2022-5-20, 蔡宗霖)

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


Graphs in deformation quantization (2022-5-13, 廖軒毅)

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