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- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Kelly Shane
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- Mon5-6(H104)
- Group
- -
- Course number
- MTH.A503
- Credits
- 1
- Academic year
- 2018
- Offered quarter
- 3Q
- Syllabus updated
- 2018/9/17
- Lecture notes updated
- -
- Language used
- English
- Access Index
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Course description and aims
Algebraic cycles are a central theme in algebraic geometry, appearing in places such as Abel’s Theorem, The Riemann-Roch Theorem, enumerative geometry, higher K-theory, motivic cohomology, and the Hodge conjecture. In this course we develop some basic ideas, and review some of these applications. For more information see: http://www.math.titech.ac.jp/~shanekelly/Cycles2018-19WS.html
Student learning outcomes
(1) Obtain overall knowledge on basics of algebraic cycle theories, such as Chow groups
(2) Understand the relationship between Chow groups and other theories, such as de Rham cohomology
(3) Attain basic understanding of motivic cohomology
Keywords
Algebraic cycles, Chow groups, motivic cohomology
Competencies that will be developed
| ✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Class flow
Standard lecture course
Course schedule/Required learning
| Course schedule | Required learning | |
|---|---|---|
| Class 1 | Introduction | Details will be provided during each class session. |
| Class 2 | Cohomology | Details will be provided during each class session. |
| Class 3 | Algebraic cycles | Details will be provided during each class session. |
| Class 4 | Equivalence | Details will be provided during each class session. |
| Class 5 | Cycle maps | Details will be provided during each class session. |
| Class 6 | Comparison | Details will be provided during each class session. |
| Class 7 | Albanese | Details will be provided during each class session. |
| Class 8 | Milnor Conjecture | Details will be provided during each class session. |
Textbook(s)
None required
Reference books, course materials, etc.
Murre, Lectures on algebraic cycles and Chow groups
Mazza, Carlo, Vladimir Voevodsky, and Charles A. Weibel. Lecture notes on motivic cohomology. Vol. 2. American Mathematical Soc., 2011.
Assessment criteria and methods
Course scores are evaluated by homework assignments. Details will be announced during the course.
Related courses
- MTH.A301 ： Algebra I
- MTH.A302 ： Algebra II
- MTH.A331 ： Algebra III
- MTH.A504 ： Advanced topics in Algebra H
Prerequisites (i.e., required knowledge, skills, courses, etc.)
Basic knowledge of scheme theory (e.g., Hartshorne)
