Preliminary syllabus, Fall 2020-21

**Instructor:** Dan Abramovich

**Class meeting:** Mondays, Wednesdays and Fridays at 10:00-10:50am. For the beginning we will meet online on Zoom. We will use Foxboro auditorium when meeting in person. See statement on remote learning below.

**Office:** Kassar 118

**Office Telephone:** (401) 863 7968

**E-mail:** `Dan_Abramovich+2510@brown.edu`

**Web site:**
`http://www.math.brown.edu/~abrmovic/MA/f2021/index.html`

**Preliminary Office hours:** Monday, 2:00-3:00, Wednesday and Friday 11:00-11:50 (TBC)

Canvas: direct link.

There is a brief course qestionnaire on Canvas, also available here.

**Text:** ALGEBRA by Serge Lang, 3rd rev. ed. 2002. Corr. 4th printing, 2005.
ISBN: 0-387-95385-X

**Topics:** Groups and group actions; Categories and functors; Rings
and modules; Fields and Galois theory.

More detail below.

Note - the book is available through the bookstore and on other platforms for sale
(The price is more reasonable than some undergraduate textbooks.)
It is also available electronically through the library with Brown credentials (e.g. reverse proxy) -
see this link.

Taking a page from my colleage Chan,
you will find that Chapters 1-3 of Riehl's book
Category Theory in Context
provide excellent background on category theory, quite a bit more than I will require.
(I'm also drawn by material from Chapter 4, but am not likely to get there.)

**Goals:** Students will develop as mathematicians, in several ways I find crucial for a successful career:

**Grading:**

**Exams:** I plan none.

My experience says that students who join the course with the appropriate background - including all math PhD students - do well.

One aspect where I am making a significant change is presentations, with the goal that everybody enjoys them and learns from them - watch for details.

I will also make changes in self-study projects with the aim of making some of these collaborative, as collaboration has become a dominant mode of a mathematician's work.

**Collaboration policy:**
I will be delighted if you work on all assignments in collaboration, **between students in this class only**,
except that

You will be required to coordinate various presentations,
and you will be required to collaborate on some self study projects.
You will be required to provide peer feedback on these too.

The use of internet resources, other than general reference, is prohibited.

** It is the student's responsibility to know which rules
govern each assignment and to adhere to the university's academic conduct code. **

**Topics in more detail:**

Groups: quick review and self study. Highlights: group actions, Sylow's theorem, Jordan Holder theorem,
first examples of simple groups.

Categories and functors: definitions, examples, and universal properties.

Rings, algebras, and modules: quick review. Self study of modules over a PID.
Illustrations of universal properties through free objects,
tensor products and localization.
Very rudimentary homological algebra.

Fields and Galois: quick review of fields. The Galois correspondence in finite and infinite setups.
Selected topics in Galois theory.

I'm known to have covered representation theory of finite groups. We'll see.

** Credit hours and estimate of work load.**

Expect 15 weekly hours over 13 weeks of class + reading + assignments + presentation preparations + self study. 195 hours in total.

** Accommodations for students with disabilities.**
Any student with a documented disability is welcome to contact me
as early in the semester as possible so that we may arrange reasonable accommodations.
As part of this process, please be in touch with Student and Employee Accessibility Services by calling 401-863-9588 or online at
http://brown.edu/Student_Services/Office_of_Student_Life/seas/index.html

** Remote and hybrid learning.** Several members of the class are not going to be on campus.
I am confident everybody will have a full opportunity to learn the material.
Beyond that, I will make every effort to make everybody's experience as fulfilling as possible,
especially in creating a community of mathematicians working together towards common goals,
whether or not you are taking the course remotely.

We will discuss plans before classes start and during the "cooling period", September 9-16.
I will respond to situation changes in line with university and state guidelines.

Currently (writing in early August) I expect to hold some regular meetings,
probably focused on discussion of problems,
ideas, and projects, in the classroom, with video streaming, lecture capture,
and opportunities for remote participants to contribute,
and adhering to physical distancing and safety guidelines.

** Other resources.** Especially during this period it is important to care for one's well-being.
The university has resources available, to all students,
in particular CAPS,
the college,
the graduate school,
and in our department the directors of undergraduate and graduate studies.