Recent Courses (Course Manuscripts, Exams and Reports)

(textbook: A first course in modular forms by Diamond and Shurman)

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.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.*

(textbook - Modern Geometry vol.1 by Dubrovin--Fomenko--Novikov) - Manuscript, Mid-I, Mid-II, Final

*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)*

(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)*

Manuscripts on Minimal submanifolds, Chern classes, Local index theorem,

2012 Fall: Differential Geometry I -

Book

2012 Spring:

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

(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*

(math majored, textbook by Courant and John I) - Manuscript

2010 Spring: Complex Analysis

(textbook by Stein) - Mid, Final

2009 Fall: Geometry

(textbook by DoCarmo) - Mid, Final

(entry level course) - Mid, Final

Notes/Manuscripts on Short Courses

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