# Homological Algebra Research Paper

### Homological Algebra

Homological algebra was invented to formalise aspects of algebraic topology, but has developed into something far more ambitious. With the advent of triangulated categories and, more generally, model categories, it can be seen as unifying many areas of mathematics, stretching from topology over algebra to analysis.

Triangulated categories and, more generally, model categories, have more structure than ordinary categories. For instance, there is a way to construct the mapping cone of a morphism; this formalises the eponymous construction from topology. Topological spaces form a modelcategory and, suitably modified, a triangulated category. But the point is that such categories abound. Complexes of modules over a ring, complexes of quasi-coherent sheaves on a scheme, and C*-algebras can all be turned into triangulated categories.

Each mathematical area is now able to feed its ideas back into the categories. From topological spaces come homotopy limits, from modules comes Auslander-Reiten theory, and from sheaves come t-structures, and these notions can all be made purely categorical and subsequently applied to triangulated categories from other areas.

From this philosophy has flowed a number of stunning developments. To mention but two, our understanding of Grothendieck duality, a main item of algebraic geometry, has been revolutionised by triangulated category methods imported from topology, and recent developments in Lie theory and combinatorics depend crucially on the introduction of cluster categories which are triangulated categories drawing on a range of inputs, not least Auslander-Reiten theory.

### Representation Theory

Representation theory is a study of symmetry. It is concerned with understanding all possible ways in which some algebraic structure (group, associative algebra, Lie algebra) can be represented as linear operators on some vector space (so, in a sense, it is linear algebra for groups).

### Including number theory, algebraic geometry, and combinatorics

We have large groups of researchers active in number theory and algebraic geometry, as well as many individuals who work in other areas of algebra: groups, noncommutative rings, Lie algebras and Lie super-algebras, representation theory, combinatorics, game theory, and coding. A number of members of the algebra group belong to the Research Training Group in Representation Theory, Geometry and Combinatorics, which runs activities and supports grad students and postdocs in its areas of interest.

## Courses

### Undergraduate upper division courses

**Math 110** (and honors version, **Math H110**). Linear algebra.**Math 113** (and honors version, **Math H113**). Introduction to abstract algebra.**Math 114.** Second course in abstract algebra. (Instructor's choice; usually Galois Theory)**Math 115.** Introduction to number theory.**Math 116.** Cryptography.**Math 143.** Elementary algebraic geometry.**Math 172.** Combinatorics.

The Mathematics Department also offers, at the undergraduate level, courses which may include algebraic topics along with others: Problem Solving (H90), Experimental Courses (191), a Special Topics course (195), and several courses of directed and independent individual and group work (196-199).

### Graduate courses

**Math 245A.** General theory of algebraic structures.**Math 249.** Algebraic combinatorics.**Math 250A.** Groups, rings and fields.**Math 250B.** Multilinear algebra and further topics.**Math 251.** Ring theory.**Math 252.** Representation theory.**Math 253.** Homological algebra.**Math 254A-254B.** Number theory.**Math 255.** Algebraic curves.**Math 256A-256B.** Algebraic geometry.**Math 257.** Group theory.

## Seminars

### 2008-2009

**Math 290**. Seminar - Commutative algebra and algebraic geometry, *David Eisenbud***Math 290**. Seminar - Number theory, *Kenneth Ribet*

*Spring 2009*

**Math 274**. Topics in Algebra - Tropical geometry, *Bernd Sturmfels***Math 274**. Topics in Algebra - Infinitesimal geometry, *Mariusz Wodzicki*

*Fall 2008*

**Math 290**. Seminar - Algebraic geometry, *David Eisenbud and Daniel Erman***Math 290**. Seminar - Discrete mathematics, *Bernd Sturmfels***Math 290**. Seminar - Representation theory, geometry and combinatorics, *Mark Haiman and Nicolai Reshetikhin***Math 290**. Seminar - Student arithmetic geometry seminar, *Martin Olsson***Student Seminar**. Student algebraic and arithmetic geometry seminar, *David Brown, Daniel Erman and Anthony Varilly*

### 2007-2008

**Math 270**. Hot Topics - Derived algebraic geometry and topology, *Peter Teichner***Math 290**. Seminar - Commutative algebra and algebraic geometry,*David Eisenbud***Math 290**. Seminar - Representation theory, geometry and combinatorics, *Mark Haiman and Nicolai Reshetikhin*

*Spring 2008*

**Math 274**. Topics in Algebra - Real p-adic analysis,*Robert Coleman*

*Fall 2007*

**Math 274**. Topics in Algebra - Locally finite lie algebras and their representations with a view toward open problems, *Ivan Penkov***Math 290**. Seminar - Student representation theory, geometry and combinatorics, *Vera Serganova and Peter Tingley***Math 290**. Seminar - Number theory, *Kenneth Ribet***Math 290**. Seminar - Perverse sheaves, *Joel Kamnitzer and Xinwen Zhu*

Earlier years, from 2006-2007

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