Series One: Introduction to Curves and Surfaces
Abstract: We will discuss the geometry of non-singular projective curves. We will cover classification of curves, Riemann-Roch Theorem, automorphisms of curves, special divisors, and canonical embeddings.
Prerequisites: commutative algebra, homological algebra, basic algebraic geometry
Course Notes: Notes 1, Notes 2, Notes 3, Notes 4, Notes 5
Lecture Videos: Video 1, Video 2, Video 3, Video 4, Video 5
Abstract: I will survey the basic theory of algebraic surfaces and their classification. The course will be illustrated by many examples: rational and ruled surfaces, K3 and abelian surfaces, surfaces of general type.
Background: It would help to have some familiarity with the basic objects of algebraic geometry: cohomology of coherent sheaves, divisors, line bundles, tangent and cotangent bundle. In fact the course will illustrate the remarkable efficiency of these notions in understanding algebraic varieties in general, and in particular surfaces.
Course Notes: Notes 1, Notes 2, Notes 3, Notes 4, Notes 5
Lecture Videos: Video 1, Video 2, Video 3, Video 4, Video 5
Series Two: Introduction to Algebraic Geometry
Abstract: I will introduce basic concepts and important theorems in commutative algebra, preparing for the study of geometry of syzygies.
I will cover interesting theorems with applications and examples in selected topics as Tensor and Tor, regular sequences, Koszul complexes, dimensions, depth, minimal resolutions, and Auslander--Buchbaum formula.
Prerequisites: Very basic knowledge of commutative algebra (as definitions of ring, ideal, modules)
Course Notes: Notes 1, Notes 2, Notes 3, Notes 4, Notes 5(all)
Lecture Videos：Video 1, Video 2, Video 3, Video 4, Video 5
Abstract: In these lectures, I'll give an introduction to syzygies of algebraic varieties. One of the best ways to input a variety or a module to computer is to input the equations of the variety or give an presentation of the module. It was the insight of Hilbert one may want to compute the whole free resolution of the module. I would discuss Castelnuovo-Mumford regularity of a module or an ideal. This turns out to be an excellent invariant to measure the complexity of the module. At the last lecturer, I would discuss syzygies of algebraic curves.
Prerequisites: Basic knowledge of commutative algebra and homological algebra ( depth of a module, associated prime ideals of a module, definition of Tor and Koszul complexes etc) In algebraic geometry, I assume the students are familiar with cohomologies of line bundles on a projective space. For the last lecture, I would assume that the students are familiar with basic facts about linear systems on a smooth projective curve.
Course Notes: Notes 1, Notes 2, Notes 3, Notes 4, Notes 5
Lecture Videos: Video 1, Video 2, Video 3, Video 4, Video 5
Series Three: The Geometric introduction to Algebraic Geometry
Abstract: After a brief introduction to the subject Algebraic Geometry (lecture one), we will see how the notion scheme is developed in the last fifty years, along with the subject algebraic geometry grow to becoming a core subject in mathematics.
Lecture Videos: Video 1, Video 2, Video 3, Video 4, Video 5
Abstract: In this lecture series, we will focus on the geometry of projective schemes. It includes projective schemes, Bezout's theorem, Bertini's theorem, Nakai-Moishezon criterion. We may also cover functor of points and the Hilbert schemes.
Prerequisites: commutative algebra, basic homological algebra.
Course Notes: Notes 1, Notes 2, Notes 3, Notes 4, Notes 5
Lecture Videos: Video 1, Video 2, Video 3, Video 4, Video 5
Series Four: Algebraic number theory
Abstract: This mini-course is an introduction to representations of algebraic groups, with a focus on examples. A rough outline of the five lectures in the mini-course is as follows:
1. Definition of algebraic groups; examples
2. Representations of SL_2(C)
3. Representations of tori
4. Character theory for semisimple groups
5. Introduction to representation theory in positive characteristic
Course Notes: Notes 1, Notes 2, Notes 3, Notes 4, Notes 5
Lecture Videos: Video 1, Video 2, Video 3, Video 4, Video 5
Abstract: In this lecture series, we will cover modular forms, modular curves, Hecke operators, the Eichler-Shimura relation, and possibly mention their modern generalization -- automorphic forms and Shimura varieties.
Prerequisites: abstract algebra, basic algebraic number theory, Riemann surfaces, basic algebraic geometry
Course Notes: Notes 1, Notes 2, Notes 3, Notes 4, Notes 5
Lecture Videos: Video 1, Video 2, Video 3, Video 4, Video 5
Exams and Solutions: Exams, Solutions
