# Fall 2014 MATH 253 001 LEC

Section | Days/Time | Location | Instructor | CCN |
---|---|---|---|---|

001 LEC | TuTh 330-5P | 31 EVANS | WODZICKI, M | 54476 |

Units/Credit | Final Exam Group | Enrollment |
---|---|---|

4 | NONE | Limit:24 Enrolled:11 Waitlist:0 Avail Seats:13 [on 10/09/14] |

**Prerequisites:** 250A (or my consent). The student is expected to have some familiarity with the vocabulary of Category Theory and basic Algebra (rings, modules)

**Syllabus:** The course will offer an intensive introduction into modern Homological Algebra. The latter since its birth has been seen more and more as permeating all of the modern Mathematics. Today it is present on the fundamental level in nearly every area of Mathematics, and especially so in Algebra, Geometry, Topology, Representation Theory, Algebraic Geometry, Mathematical Physics, and Number Theory.

I will cover adjointness and exactness of functors. I will discuss additive, exact, abelian and triangulated categories, as well as the fundmental operations on categories: forming a quotient and localization. Using quotients I will discuss homotopy categories associated with an exact category. Localizations of categories will be used to describe a passage from an exact category to its derived category. Time permeating I will also give a rapid overview of model categories in the sense of Quillen. For each of these contexts I will describe the correspoding theory of derived functors.

I will also cover the "mechanics" of Homological Algebra: complexes, double complexes, spectral sequences, developing as illustrations a number of classical spectral sequences (the First and the Second spectral sequences of a double complex, the Hochschild-Serre spectral sequence, Künneth spectral sequence).

**Office:** 995 Evans Hall

**Office Hours:** TBA

**Required Text:** Cartan, Eilenberg. Homological Algebra. Princeton 1956 (still by far the best text written on the subject)

**Recommended Reading:** Freyd. Abelian categories. New York, Harper & Row 1964

Mac Lane. Category Theory for a Working Mathematician. Graduate Courses in Mathematics 5, Springer Verlag (either edition)

I will supply a number of suppemental texts and recommendations througout the semester

**Grading:** homework and the take home exam

**Homework:**

**Course Webpage:**