MAT 544 Course Webpage Algebra III: Commutative Algebra, Homological Algebra, Representation Theory. Tu Th 2:30PM — 3:50PM, Library W4535 Fall 2016

Course Announcements Announcements about the course will be posted here. Please check the site regularly for announcements (which will also be given in lecture).

Course Description The course will have three (very related) components. First we will give an overview of commutative algebra through the basics of dimension theory. Next we will review homological algebra from Algebra II, and we will survey advanced topics through spectral sequences. Finally, we will outline the structure theory of finite dimensional complex representations of finite groups and of semisimple Lie algebras. The emphasis will be on statements of results and use of the results in applications.

Prerequisites Students should have passed the graduate algebra sequence. Any student who has not passed the graduate algebra sequence should speak with me at the beginning of the semester.

Text There is no required textbook. Lectures will draw on the following references.

• M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra, Westview Press, 1969.
• Hideyuki Matsumura, Commutative Ring Theory, Cambridge University Press, 1986.
• Charles Weibel, An Introduction to Homological Algebra, Cambridge University Press.
• W. Fulton and J. Harris, Representation Theory, A First Course, Springer-Verlag.

Course Learning Objectives

• Acclimate to New Mathematics. Gain familiarity with a new mathematical idea (be it a definition, a result, an algorithm, etc.) through examples, through basic results that involve that idea, and through deeper results that reflect the significance of the idea. Example. Mapping complexes of chain maps of complexes are the model for the notion of distinguished triangles in a triangulated category.
• Apply and Model. Understand how an abstract notion or result can lead to an algorithm or computation arising in a context different from the original notion or result. Understand the necessary hypotheses and limitations of that model. Example. For the associative (but noncommutative) group algebra of a group, the Ext and Tor functors define group cohomology and group homology. The standard resolution for computing Ext and Tor for a (flat) associative algebra leads to the standard description of group cohomology and group homology in terms of cocycles and coboundaries.
• Specialize. Pass from general theorems, definitions, and methods to specific examples. Be able to compute with those examples. Example. The general method for producing irreducible finite dimensional representations of a semisimple complex Lie algebra in the special case of the special linear algebra gives specific Schur functors applied to the "standard irreducible representation".
• Generalize. Understand examples of ideas, constructions and arguments originally developed in one context yet that extend to another context. Example. The Universal Coefficients Theorem generalizes to the two standard spectral sequences associated to the hyperderived functors.
• Prove. For a conjectured result, often expected from examples, heuristics and other indirect evidence, rigorously prove the result using techniques such as proof by induction, proof by contradiction, proof by cases, and more advanced proof techniques. Example. The Theorem of Appell-Humbert describing complex line bundles on a complex torus follows from basic algebra of the group cohomology of the fundamental group of the complex torus (and this makes precise the sketch in Griffiths and Harris, Principles of Algebraic Geometry).

Course Outcomes / Key Skills

• Understand the basic theorems of commutative algebra through basic results in dimension theory, as developed in the textbook of Atiyah and MacDonald.
• Understand the basic construction of derived functors following the strategy of Grothendieck in Tohoku.
• Understand spectral sequences and how to compute with them.
• Understand the basic notions of derived categories. Understand hypotheses under which additive functors extend to a derived functor. Understand applications, such as composition of three or more derived functors.
• Understand the representation theory of finite groups over a field of characteristic prime to the order of the group.
• Understand the specific representation theory of the symmetric groups and alternating groups, particularly the Schur functors.
• Understand the basic structure theory of complex Lie algebras.
• Understand the classification of semisimple Lie algebras over the complex numbers.
• Understand the general theory of finite representations of a semisimple Lie algebra over the complex numbers.
• Understand the classification of finite irreducible representations for each of the classical simple Lie algebras over the complex numbers.

Lectures The instructor for this course is Jason Starr. All instruction will occur in lectures. A tentative schedule will be posted in the syllabus.

Lecture is held Tuesdays and Thursdays, 2:30 PM — 3:50PM in Library W4535.

Office Hours Office hours for Jason Starr are scheduled as follows.

• Tuesdays 10 — 11AM, Math Tower P—143 (advising).
• Thursdays 10 — 11AM, Math Tower 4—108.
• Thursdays 1:15 — 2:15PM, Math Tower 4—108.

Grading System The letter grade for the course is based on class participation, weekly problem sets, and a final oral report by each student.

Oral presentation
Every student will give an oral presentation of approximately 20 minutes during office hours in the last few weeks of the semester. The topic of the presentation is to be determined by the student and instructor jointly. It may be on anything related to the course but not directly covered during lecture or on problem sets, e.g., a skipped section of the textbook, 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 15th.

Hand-backs Graded problem sets and exams will be handed back in lecture. If you cannot attend the lecture in which a problem set or exam is handed back, it is your responsibility to contact the instructor and arrange a time to pick up the work (typically in office hours).

Students are responsible for collecting any graded work by the end of the semester.

Required Syllabi Statements
The University Senate Undergraduate and Graduate Councils have authorized that the following required statements appear in all teaching syllabi (graduate and undergraduate courses) on the Stony Brook Campus.

Americans with Disabilities Act.
If you have a physical, psychological, medical or learning disability that may impact your course work, please contact Disability Support Services, ECC (Educational Communications Center) Building, Room 128, (631) 632-6748. They will determine with you what accommodations, if any, are necessary and appropriate. All information and documentation is confidential.

Students who require assistance during emergency evacuation are encouraged to discuss their needs with their professors and Disability Support Services. For procedures and information go to the following website: http://www.stonybrook.edu/ehs/fire/disabilities.

Academic Integrity Each student must pursue his or her academic goals honestly and be personally accountable for all submitted work. Representing another person's work as your own is always wrong. Faculty is required to report any suspected instances of academic dishonesty to the Academic Judiciary. Faculty in the Health Sciences Center (School of Health Technology Management, Nursing, Social Welfare, Dental Medicine) and School of Medicine are required to follow their school-specific procedures. For more comprehensive information on academic integrity, including categories of academic dishonesty please refer to the academic judiciary website at http://www.stonybrook.edu/commcms/academic_integrity/index.html.

Critical Incident Management Stony Brook University expects students to respect the rights, privileges, and property of other people. Faculty are required to report to the Office of University Community Standards any disruptive behavior that interrupts their ability to teach, compromises the safety of the learning environment, or inhibits students' ability to learn. Faculty in the HSC Schools and the School of Medicine are required to follow their school-specific procedures.

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