#### 2020 SCMS Algebraic Geometry Summer School

Time: Jul 20 - Jul 31, 2020

Venue:via zoom

#### Speakers

**Series One:**Introduction to Curves and Surfaces

**Series Two:**Introduction to Algebraic Geometry

**Series Three:**The Geometric introduction to Algebraic Geometry

**Series Four:**Algebraic number theory

**Schedule**part below.

#### Schedule

**Series One: Introduction to Curves and Surfaces**

**Zhi Jiang (SCMS)****: Introduction to Algebraic Curves**

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

**Arnaud Beauville (Nice): Introduction to Algebraic Surfaces**

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**

**Chen Jiang (SCMS):****Commutative Algebra and 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

**Lawrence Ein (UIC):****Syzygies of Algebraic Varieties**

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**

**Jun Li (SCMS)****: Introduction to Algebraic Geometry I**

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

**Zhiyuan Li (SCMS)****: Introduction to Algebraic Geometry II**

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**

**Pramod N. Achar (Louisiana): Introduction to algebraic groups and their representations**

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

**Yifeng Liu (Yale)****: Introduction to Algebraic Number Theory**

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

#### Organizer(s)

Meng Chen (Fudan)

Chen Jiang (SCMS)

Zhi Jiang (SCMS)

Jun Li (SCMS)

Zhiyuan Li (SCMS)

#### Sponsor(s)

Shanghai Center For Mathematical Sciences

Fudan University

#### Contact us

Haoran Wu (wuhaoran@fudan.edu.cn)

#### Tips

The summer school will offer four two-week short courses covering topics in algebraic geometry and related subjects. The courses are designed for senior undergraduates with strong algebra and/or geometry back ground.The Summer School will offer Problem Sessions and other sessions; will provide opportunities for interaction among instructors, TAs of the courses, and the students in the Summer School. There will be group discussion sessions on problem solving, and session for students to present their solutions.

At the end of each week the Summer School will have examinations, on the three course-series students took. Students who demonstrate a decent grasp of the subjects will receive the certification of Summer School issued by the University. The Summer School will give recognition "with distinguished honor" to a selected few outstanding students; those who receive this honor will be invited to pay a short visit SCMS in 20-21 academic year, and eligible to receive top rank fellowship if they choose to enter SCMS-Fudan mathematics graduate program.