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Description of course: This course will develop some of the main tools of Galois cohomology and descent theory with an eye toward applications to rational points. Topics will include Galois descent and twisting, principal homogeneous spaces and torsors, the Brauer group, local-to-global phenomena, and obstructions to the existence of rational points. Expected background: Prior experience with algebra and Galois theory will be necessary. Some exposure to topology and algebraic geometry will be useful.
Grading:
Your grade will be based on class participation, collaborative group
work, and a (final) presentation. A passing grade requires at least
one group work presentation or write-up and a (final) presentation.
Work with anyone on solving your homework problems,Writing up the final draft is as important a process as figuring out the problems on scratch paper with your friends. If you work with people or other resources on a particular assignment, you must list your collaborators on the top of the first page. This makes the process fun, transparent, and honest. External resources: Mathematical writing is very idiosyncratic; if your proofs are copied, it is easy to tell. You will not learn (nor adhere to the Honor Principle) by copying solutions from others, or from external sources such as internet forums and generative artificial intelligence (GAI) output. Concerning internet forums (e.g., math.stackexchange), you are free to look at them and use any understanding you've gained from them in your course work, of course, subject to the above rules. Just be warned that these forums often contain incorrect or circuitous solutions, misleading discussions, use of techniques outside of the course material, and other material that may be detrimental to your learning process. Even the time that it takes to repeatedly search for solutions and read through dozens of forum posts could be better spent learning the material on your own or composing a question via email to the instructor. Concerning GAI (e.g., ChatGPT), you are free to experiment with asking questions, but be warned that these systems are currently still very bad at deductive reasoning, and that the output may contain a mix of correct, incorrect, and unverified statements. Ask them to prove something false, they will work hard to do so, often giving contradictory answers. Therefore, I would be very careful with using these tools as learning resources on your own. Attendance: You are expected to attend class, including required X-hour sessions, in person unless you have made alternative arrangements due to illness, medical reasons, or the need to isolate due to COVID-19. For the health and safety of our class community, please follow Dartmouth's health guidance. Accommodations: Students requesting disability-related accommodations and services for this course are required to register with Student Accessibility Services and to request that an accommodation email be sent to me in advance of the need for an accommodation. Then, students should follow-up with me to determine relevant details such as what role SAS or its Testing Center may play in accommodation implementation. This process works best for everyone when completed as early in the term as possible. If students have questions about whether they are eligible for accommodations or have concerns about the implementation of their accommodations, they should contact the SAS office. All inquiries and discussions will remain confidential. |
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