TuTh 10-11:20am Math 4-130

**Instructor:** Ljudmila Kamenova
**Office:** Math Tower 3-115

**Office hours:** Wednesdays 1-3pm, MLC on Fridays 1-2pm

The Instructor may be reached by e-mail at
kamenova@math.sunysb.edu.

Feel free to send me an e-mail or drop by outside of office hours.

The course will introduce three components. First we will give an overview of commutative algebra. Next we will review homological algebra from MAT 535, Algebra II, and we will survey more advanced topics. Finally, we will outline the structure theory of finite dimensional complex representations of finite groups and of semisimple Lie algebras.

**Syllabus:**

1. Modules, Nakayama's lemma, Noetherian rings.

2. The Nullstellensatz and Spec A.

3. Rings of fractions and localization.

4. Primary decomposition.

5. DVRs and normal integral domains.

6. Chain complexes.

7. Derived functors.

8. Spectral sequences.

9. The Derived category.

10. Representations of finite groups.

11. Representations of symmetric groups.

12. Classification of semisimple Lie algebras.

13. The Weyl Character Formula.

- M. Reid,
*Undergraduate Commutative Algebra*, LMS Student Texts 29, Cambridge University Press, 1995. - M. F. Atiyah, I. G. MacDonald,
*Introduction to Commutative Algebra*, Westview Press, 1969. - C. Weibel,
*Introduction to Homological Algebra*, Cambridge University Press, 1995. - S. Lang,
*Algebra*, Addison Wesley, 1999. - W. Fulton, J. Harris,
*Representation Theory, a First Course*, GTM 129, Springer-Verlag, 2004.

Every student will give an oral in-class presentation of approximately 30 minutes in the last few weeks of the semester. The topic of the presentation is to be determined by the student and the instructor jointly. It may be on anything related to the course but not directly covered during lectures or on the problem sets, e.g., a skipped section of the textbooks, an interesting sequence of (unassigned) textbook problems, related material from another algebra textbook, etc. The topic should be discussed with the instructor and approved no later than November 1st.

**HW 1** (due 9/6/18):
Click here for the problems.

**HW 2** (due 9/20/18):
Click here for the problems.

**HW 3** (due 10/4/18):
Click here for the problems.

**HW 4** (due 10/18/18):
Click here for the problems.

**HW 5** (due 11/1/18):
Click here for the problems.

**HW 6** (due 11/15/18):
Click here for the problems.

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