SEAMS school “Arithmetic, Geometry and Model Theory”
SEAMS school “Arithmetic, Geometry and Model Theory”
Institute of Mathematics (VAST), Hanoi Vietnam
17 – 28 February 2020
Description
- Recent years have seen a flourishing interaction between model theory, arithmetic geometry and number theory. The IHM-SEAMS school aims to familiarize advanced undergraduate students, graduate students and young researchers to basic concepts and techniques of model theory and main notions of algebraic geometry and number theory. This school will be an activity aiming to boost the interaction and collaboration between Asian and European researchers in mathematics.
Program:
- The school is two week long from 17 February to 28 February 2020, with free Wednesday afternoons. There are three mini-courses, each consists of 6 x 120-minute lectures. There will be at least one tutorial session every day (11h00-12h00 and / or 16h00-17h00). Though the audience is expected to have a basic mathematical background, knowledge of technical terminology is not assumed.
- Introduction to Model Theory given by Pablo Cubides Kovacsics (TU Dresden) and Le Quy Thuong (VNU Hanoi). The aim of this course is to introduce the students to the basics of model theory with a particular emphasis towards potential algebraic and arithmetic applications. In particular, o-minimality will be defined and some of its main properties will be discussed
- Introduction to Algebraic Curves given by Nguyen Chu Gia Vuong and Ha Minh Lam (IMH Hanoi). Algebraic geometry is a highly developed field in mathematics. The aim of this course is to give an introduction to algebraic geometry via algebraic curves. We will introduce basic notions and theorems of algebraic geometry, especially those related to curves: affine varieties, projective varieties, affine plane curves, projective plane curves, local invariants, intersection numbers, Bezout’s theorem, Noether’s theorem, divisors and Riemann-Roch theorem. Note that, the materials covered here are closely related to Angles’ course. The students are supposed to have some background on commutative algebra: rings, algebras, ideals, Hilbert basis theorem, modules.
- Introduction to Algebraic Number Theory given by Bruno Angles (Univ. Caen) and Ta Thi Hoai An (IMH Hanoi). This course introduces the student to the main arithmetic objects of algebraic number theory: zeta values and ideal class groups. We will introduce basic notions of algebraic number theory: ring of integers, ideal class groups and ramification and study them in the case of quadratic fields and cyclotomic fields. Then we will present the basic properties of L-series attached to Dirichlet characters and see how the values at negatives integers of such Dirichlet L-series are connected to Bernoulli numbers. Finally, we introduce the basic properties of Gauss and Jacobi sums and prove the Stickelberger Theorem and the celebrated Herbrand-Ribet’s Theorem
Registration: 02-05/06/2019
- There are no registration fees and no additional fees
- If you are interested in participating in the school
Location: Institute of Mathematics
SEAMS school “Arithmetic, Geometry and Model Theory”
Institute of Mathematics (VAST), Hanoi Vietnam
17 – 28 February 2020
ORGANIZING COMMITTEE
- Bruno Anglès (Université Caen Normandie)
- Phung Ho Hai (Institute of Matheamtics, VAST)
- Nguyen Duy Tan (Institute of Matheamtics, VAST)
- Ngo Dac Tuan (CNRS and Université Claude Bernard Lyon 1)
- Nguyen Chu Gia Vuong (Institute of Mathematics, VAST)
SEAMS school “Arithmetic, Geometry and Model Theory”
Institute of Mathematics (VAST), Hanoi Vietnam
17 – 28 February 2020
FINANCIAL SUPPORT
- We can cover local expenses for selected participants.
- Travel support for up to 12 students from Asean and neighboring countries and 10 domestic students is available.
- If you apply for travel support, please fill out the corresponding section carefully.
- a copy of the academic transcript;
- a recommendation letter.
SEAMS school “Arithmetic, Geometry and Model Theory”
Institute of Mathematics (VAST), Hanoi Vietnam
17 – 28 February 2020
- There are no registration fees and no additional fees.
- If you are interested in participating in the school, please register online here.
SEAMS school “Arithmetic, Geometry and Model Theory”
Institute of Mathematics (VAST), Hanoi Vietnam
17 – 28 February 2020
Program
The school is two week long from 17 February to 28 February 2020, with free Wednesday afternoons. There are three mini-courses, each consists of 6 x 120-minute lectures. There will be at least one tutorial session every day (11h00-12h00 and / or 16h00-17h00). Though the audience is expected to have a basic mathematical background, knowledge of technical terminology is not assumed.
- LECTURE 1: Introduction to Model Theory
Lecturers: Pablo Cubides Kovacsics (TU Dresden) and Le Quy Thuong (VNU Hanoi).
Abstract: The aim of this course is to introduce the students to the basics of model theory with a particular emphasis towards potential algebraic and arithmetic applications. Basic notions such as first-order theories, completeness, compactness and quantifier elimination will be introduced with several examples. The end of the course will contain a brief introduction to o-minimality to let the participants have an idea of potential research directions in this area.
Detailed plan:
1) Background on mathematical logic (2-3 hours)
References: [1] Chapter 1, [3] Section 1.
Basic notions such as languages, structures and formulas will be introduced together with the main proof methods: induction on terms and formulas. Several examples will be discussed.
2) Basics of model theory (7-8 hours)
References: [1] Chapters 2, [3] Sections 2-5.
In the second part of the course we dive directly into basic model theory concepts. Among others, we will introduce the concepts of theory, logical consequence, elementary classes, completeness and categoricity, and prove classical theorems such as the compactness theorem and Löwenheim-Skolem theorems. Some applications will be given to algebraic structures, including groups and fields.
3) A brief introduction to o-minimality (2 hours)
References : [2] Chapter 3, [3] Section 6.
The final two hours of this course will be dedicated to provide the students with a brief introduction to o-minimality. After formally define o-minimal structures and explain some of its main results (without proofs), an overview of possible research directions in this area will be presented.
References:
[1] Marker, David Model theory. An introduction. Graduate Texts in Mathematics, 217. Springer-Verlag, New York, 2002
[2] Tent, Katrin; Ziegler, Martin A course in model theory. Lecture Notes in Logic, 40. Association for Symbolic Logic, La Jolla, CA; Cambridge University Press, Cambridge, 2012
[3] Pablo Cubides Kovacsics, Course notes. - LECTURE 2: Introduction to Algebraic Curves
Lecturers: Nguyen Chu Gia Vuong and Ha Minh Lam
Algebraic geometry is a highly developed field in mathematics. The aim of this course is to give an introduction to algebraic geometry via algebraic curves. We will introduce basic notions and theorems of algebraic geometry, especially those related to curves: affine varieties, projective varieties, affine plane curves, projective plane curves, local invariants, intersection numbers, Bezout’s theorem, Noether’s theorem, divisors and Riemann-Roch theorem. Note that, the materials covered here are closely related to Angles’ course. The students are supposed to have some background on commutative algebra: rings, algebras, ideals, Hilbert basis theorem, modules.
Abstract: Algebraic geometry is a highgly developed field in mathematics. The aim of this course is to give an introduction to algebraic geometry via algebraic curves. We will introduce basic notions and theorems of algebraic geometry, specially those related to curves: affine varieties, projective varieties, affine plane curves, projective plane curves, local invariants, intersection numbers, Bezout’s theorem, Max Noether’s theorem, divisors and Riemann-Roch theorem. Note that, the materials covered here are closely related to Angles’ course. The students are supposed to have some background on commutative algebra: rings, algebras, ideals, Hilbert basis theorem, modules.
Detailed plan:
1) Affine algebraic sets and affine varieties, local properties of plane curves (4h).
Ref: Chap. 1, 2 and 3 of [1].
2) Projective varieties, projective planec curves, Bezout and Max Noether theorems.
Ref: Chap 4, 5 and 6 of [1].
3) Varieties, morphisms, function fields, singularities, divisors, Riemann-Roch theorem.
Ref: Chap. 7, 8 of [1].
References: The main reference is Fulton’s book:
[1] Algebraic curves, W. Fulton. - LECTURE 3: Introduction to Algebraic Number Theory
Lecturers: Bruno Angles (Univ. Caen) and Ta Thi Hoai An (IMH Hanoi).
This course introduces the student to the main arithmetic objects of algebraic number theory: zeta values and ideal class groups. We will introduce basic notions of algebraic number theory: ring of integers, ideal class groups and ramification and study them in the case of quadratic fields and cyclotomic fields. Then we will present the basic properties of L-series attached to Dirichlet characters and see how the values at negatives integers of such Dirichlet L-series are connected to Bernoulli numbers. Finally, we introduce the basic properties of Gauss and Jacobi sums and prove the Stickelberger Theorem and the celebrated Herbrand-Ribet’s Theorem.
Abstract: This course introduces the student to the main arithmetic objects of algebraic number theory: zeta values and ideal class groups. We will introduce basic notions of algebraic number theory: ring of integers, ideal class groups and ramification and study them in the case of quadratic fields and cyclotomic fields. Then we will present the basic properties of L-series attached to Dirichlet characters and see how the values at negatives integers of such Dirichlet L-series are connected to Bernoulli numbers. Finally, we introduce the basic properties of Gauss and Jacobi sums and prove the Stickelberger Theorem and the celebrated Herbrand-Ribet’s Theorem.
Detailed plan:
1) Zeta Values (4 hours)
References: [1] chapters 15 and 16, [2] chapters 3 and 4
- Bernoulli numbers and present some of their basic properties.
- L-series attached to Dirichlet characters.
- The values at negatives integers of such Dirichlet L-series and how they are connected to Bernoulli numbers.
2) Basics of Algebraic Number Theory (4 hours)
Reference: [1] chapters 12 and 13
- The basics of algebraic number theory: ring of integers, ideal class groups and ramification
- Examples: quadratic fields and cyclotomic fields.
3) Bernoulli Numbers and Class Groups (4 hours)
References: [1] chapters 14 and 15, [2] chapters 6 and 15
- Basic properties of Gauss and Jacobi sums.
- The Stickelberger Theorem.
- Herbrand’s Theorem and its converse: Ribet’s Theorem.
References:
[1] K. Ireland & M. Rosen, A Classical Introduction to Modern Number Theory, Second Edition, Springer-Verlag, 1990
[2] L. C. Washington, Introduction to Cyclotomic Fields, Second Edition, Springer, 1997
Tentative Schedule
Date | Morning lecture 9h00-11h00 | Tutorial 11h-12h | Afternoon lecture 14h00-16h00 | Tutorial 16h-17h |
17/2 | Model theory Lecture 1 (Cubides) | Tutorial (Model theory) | Algebraic curves Lecture 1 (NCG Vuong) | |
18/2 | Algebraic number theory Lecture 1 (B Angles) | Tutorial (Algebraic number theory) | Algebraic curves Lecture 2 (NCG Vuong) | Tutorial (Algebraic curves) |
19/2 | Model theory Lecture 2 (P Cubides) | Tutorial (Model theory) | Free time | Free time |
20/2 | Algebraic number theory Lecture 2 (B Angles) | Tutorial (Algebraic number theory) | Algebraic curves Lecture 3 (NCG Vuong) | Tutorial (Algebraic curves) |
21/2 | Model theory Lecture 3 (P Cubides) | Algebraic number theory Lecture 3 (B Angles) | ||
24/2 | Algebraic number theory Lecture 4 (B Angles) | Tutorial (Algebraic number theory) | Algebraic curves Lecture 4 (NCG Vuong) | Tutorial (Algebraic curves) |
25/2 | Model theory Lecture 4 (P Cubides) | Tutorial (Model theory) | Algebraic curves Lecture 5 (NCG Vuong / HM Lam) | Tutorial (Algebraic curves) |
26/2 | Algebraic number theory Lecture 5 (B Angles) | Tutorial (Algebraic number theory) | Free time | Free time |
27/2 | Model theory Lecture 5 (P Cubides) | Tutorial (Model theory) | Algebraic curves Lecture 6 (NCG Vuong) | |
28/2 | Algebraic number theory Lecture 6 (B Angles) | Model theory Lecture 6 (P Cubides) |