Recent Courses (Course Manuscripts, Exams and Reports)


2024 Spring: Introduction to Classifications of Manifolds
(textbook - Milnor's Lectures on h-cobordism Theorem, Wall's Differential Topology and Moore's Lecture Notes on Seiberg-Witten Invariants)
Final Reports: Scalar Curvature under Surgery after Schoen-Yau (Chiang), Brieskorn-Hirzebruch Exotic Spheres (Hsin), Heat Kernels and Local Index Theorem (Chen), L^p Theory on Domains and Manifolds (Kuo), Banach Calculus and Sard-Samle Theory (Wang), Kervaire-Milnor Theory on Exotic Spheres (Chong), Unique Continuation of Elliptic Equations (Huang), Deformations of G2 Structures with Boundary after Donaldson (Chang), Scalar Curvature under Surgery after Gromov-Lawson (Wu), Seiberg-Witten Wall Crossing and Symplectic Ruled Surfaces after Aiko Liu (Lee).

2023 Fall: Geometry I (Honors Course)
(textbook - Modern Geometry vol.1 by Dubrovin--Fomenko--Novikov) Mid-I, Mid-II, Final

2023 Spring: Introduction to Modular Forms
(textbook: A first course in modular forms by Diamond and Shurman)
Notes taken by Shuang-Yen Lee, Mid. Final reports: Reductions of curves (Chen), Weil's converse theorem (Hsiao), Eichler--Shimura relation (Lee), Modularity for CM elliptic curves (Chong), Eichler--Shimura isomorphism (Yao), Witten genus and exotic spheres (Kuo), Ochanine's theorem on spin manifolds (Chang), Modular forms for even integral quadratic forms (Zhang), A newform with dihedral monodromy D_3 (Tsai).   

2022 Fall: Lie Groups and Lie Algebras
(textbooks: Introduction to Lie algebras and their representations - Humphreys, Compact Lie groups - Sepanski)
Lecture notes taken by Shuang-Yen Lee (course TA), Mid, Final.

2022 Spring: Geometry and Topological Field Theory II
(An introduction to quantum cohomology) Intro to BCOV Lect.1, Intro to BCOV Lect.2. Midterm reports: Chuang - Mirror principles, Lee - Compatibility of perfect obstruction theories, Wu - Generators of Picard groups of stable map moduli to projective spaces. Final reports: Chuang - Quantum singularity theory (FJRW), Lee - Quantum RR, Lefschetz and Serre (Coates--Givental), Wu - Classification of 2D topological field theories (Teleman). 

2021 Fall: Geometry and Topological Field Theory I
(textbook - Mirror Symmetry by Hori et. al., Clay-AMS 2003)
Solutions to Exercises: Chen, Chuang, Lee, Tsai, Wu.
Final reports: Chuang - critical dimension for bosonic strings, Lee- critical dimensions for superstrings, Tsai - toric construction of mirror CY, Chen - Morse--Witten complex with product, Wu - T^1 lifting and deformations of CY

2021 Spring: Differential Geometry II
Ch.6 Minimal Submanifolds, Ch.7 Characteristic Classes and Cobordism, Ch.8 Atiyah--Singer Index Theorem, Ch.9 Intro to Geometric Analysis, Ch.10 Intro to Geometric Flows - Mid, Final reports: combined reports in pdf, links to oral reports - Morse theory and Bott periodicity (Chang), AS Index theorem via twisted signature formula (Chuang), Brieskorn's construction of Milnor's 28 exotic 7-spheres (Huang), Kazhdan--Warner problem for surfaces (Wu), Witten's proof of the positive mass theorem (Yang), Solution to the Yamabe problem (Chen), Yau's proof of Calabi's conjecture (Lee, Tsai), Ricci flow with surgery for manifolds with PIC (Kuo, Lee)  

2020 Fall: Differential Geometry I
Ch.1 Manifolds, Ch.2 Tensors, Ch.3 Riemannian Geometry, Ch.4 Hodge Theory of Harmonic Forms, Ch.5 Lie Groups and Symmetric Spaces, Mid, Final

2020 Spring: Algebraic Geometry II
(textbooks - Algebraic Geometry by Hartshorne, Complex Algebraic Surfaces by Beauville) partial solutions to Exercises: Ch.4, Ch.5, Appendix, Beauville - Mid, Final reports: Smooth morphisms/Bertini's theorem, Theorem on formal functions, Semi-continuity/Cohomology and base change, Appendix-A: Kahler identity and Hodge decomposition, Lefschetz decomposition and basics on Line bundles, GAGA, Kodaira vanishing, Kodaira embedding, Appendix-B: Chow ring, Chern classes, Chow moving Lemma, GRR formalism, Proorf of GRR - projections, Proof of GRR - embeddings, Appendix -C: Etale cohomology, Descent data, Cohomology of curves, Base change theorems, Poincare duality, Lefschetz fixed point formula and rationality of zeta functions  

2019 Fall: Algebraic Geometry I
(textbook - Algebraic Geometry Ch.1-C.3 by Hartshorne) - Varieties, Schemes, Cohomology. Partial solutions to Exercises: Ch.1, Ch.2, Ch.3, Mid, Final  

2019 Spring: Algebra II (Honors Course)
(textbook - Basic Algebra II by Jacobson)- Modules, Rings, Representations of finite groups - Quiz-I, Mid, Quiz-II, Final

2018 Fall: Algebra I (Honors Course)
(textbook - Basic Algebra I by Jacobson) - Galois Theory, Quiz-I, Mid, Quiz-II, Final

2018 Spring:

• Topics on Frobenius Manifolds - Lecture Notes, taken by Ho, Special lectures on Witten's conjecture, Partial solutions to the exercises. Final reports: Isomonodromic deformations of linear ODE, Integrable deformations, Quantum cohomology, Semi-universal unfordings   
  
• Calculus for Life Science II - Mid, Final
  

2017: Fall: Calculus for Life Science I - MId-1, Mid-2, Final

2017 Spring:

• D-Modules - Introduction (revised), Basics, Kashiwara equivalence, Charactersitic varieties, Dualities, Holonomic D-modules, Finiteness and analytic theory, Constructibility, Meromorphic connections, Regular holonomic modules, Classical R--H regular case, Good formal structure, Stokes structure, Riemann--Hilbert--Birkhoff. Final reports: RH irregular case, Kazhdan--Lusztig conjecture, Decomposition theorem.
  
• Geometry II - Final reports: Topology of fiber spaces, Exotic spheres, Bott periodicity, GTR, C1 isometric imbeddings.
  

2016 Fall: Geometry I (Honors Course)
(textbook - Modern Geometry vol.1 by Dubrovin--Fomenko--Novikov) - Manuscript, Mid-I, Mid-II, Final

2015 Spring:

• Complex Analysis II (textbooks by Ahlfors, Whittaker and Watson, and Weyl), Dirichlet problem and Riemann mappings, Green functions and uniformizations, Modular functions, Global functions, Jacobi theta functions, Solving polynomials, Geometry of surfaces, Differentials and Riemann--Roch, Abel--Jacobi, Riemann theta functions, Asymptotics and irregular ODE. Final Reports: Week I, Week II, Week III.  
  
• Seminar in Algebraic Geometry (references: Fulton's Intersection Theory, Cox and Katz' Mirror Symmetry and Algebraic Geometry)
  

2014 Fall: Complex Analysis
(textbook by Stein) - Mid, Final.v1, Final.v2, manuscript on Elliptic and theta function.

2013 Fall:

• Geometry (textbook by DoCarmo) - Manuscript on Global surface theory, Mid, Final
• TIMS Undergraduate Seminar (towards the basic theory of algebraic D modules)

2013 Spring: Differential Geometry II -
Manuscripts on Minimal submanifolds, Chern classes, Local index theorem, Exotic 7 spheres (LaTeXed pdf), ASD moduli and exotic R4, or in one file combined pdf - Homework, Mid. Final reports: problem list, combined pdf

2012 Fall: Differential Geometry I -
Book (in preparation) - Appendix: Chinese translation of Riemann's lecture  - 1st Quiz, Mid, Final

2012 Spring:

• Advanced Calculus II (Honors Course) (textbooks by Apostol, Rudin and Royden) - Manuscipt on Lebesgue theory - Banach calculus, Mid, Final
• Calabi--Yau Geometry

2011 Fall: Advanced Calculus I (Honors Course)
(textbook by Apostol) - Manuscript, Mid, Final

2011 Spring:

• Calculus II (math majored, textbook by Courant and John II) - Notes by Hsu, Notes by Lee, Quiz, Mid, Final
• Geometry and Topological Field Theory (towards the Hori-Vafa mirror symmetry and introduction to GW theory) - Manuscript

2010 Fall: Calculus I
(math majored, textbook by Courant and John I) - Manuscript, Quiz, Mid, Mid-Sol, Final

2010 Spring: Complex Analysis
(textbook by Stein) - Mid, Final, manuscript on special topics

2009 Fall: Geometry
(textbook by DoCarmo) - Mid, Final (Notes by Chen-Yun Lin in 2003 at NTHU)

2009 Spring: Multivariable Calculus
(entry level course) - Mid, Final

2008 Fall: Complex Geometry -
(A survey of L2 theory developped by Hormander, Ohsawa-Takegoshi extension, and Siu's theory on pluricanonical forms)


Manuscripts of Selected Earlier Courses

1999 Spring at NTU - Complex Geometry II: Algebraic Surfaces - manuscript, 77 pages. (Based on Beauville's and Barth--Peters--Van de Ven's books.)

2000 Fall at CTS/NTHU - Lectures on Abelian Varieties - manuscript, 48 pages. (Based on Griffiths--Harris' and Mumford's books.)


Notes/Manuscripts on Short Courses

1998 August at Academia Sinica - Lectures on Kaehler Geometry - manuscript (12 hours lectures, 38 pages).

1998 Summer at NTU - Introduction to Modern Mathematics: Groups and Representations (5 lectures) - Manuscripts on Ch.1-2, Ch.3-5

1999 Spring at Academia Sinica - Introduction to Calabi--Yau Manifolds (one lecture) - pdf

1999 Summer at NTU - Introduction to Algebraic Geometry for Undergraduates (3 lectures) - pdf

2000 Summer at NCTS - Motivic Integations - Introduction and Applications (2 lectures) - pdf, notes on CVF

2011 Lectures on Gromov--Witten Theory at TIMS (3 lectures, based on lectures in NCTS 2004 and NTU 2009) - combined pdf