## Preliminary Oral Examination

I passed my preliminary oral examination on **June 16th, 2021**.
My examination committee members were
Christine Berkesch (chair),
Gennady Lyubeznik,
Vic Reiner, and
Peter Webb.

The exam involved four parts:

- an expository paper, titled:
- a presentation, titled:
- questioning on the major topic (everything above);
- questioning on the minor topic.

### Minor Area Examination

Peter Webb was my minor area advisor, and asked me questions on

“Representation Theory of Finite Dimensional Algebras”,

which roughly meant studying the following:

- Gabriel’s Theorem:
- Describe the indecomposable representations of quivers of finite representation type with at most 3 (or maybe 4) vertices and without oriented cycles.
- Know what these quivers are. Construct the indecomposable representations by the mechanism of the proof of Gabriel’s theorem, and/or by Auslander-Reiten theory.
- Have some familiarity with the associated root system.
- Have some means to describe these representations (dimension vectors and/or diagrams, for instance).
- Be able to do it explicitly in these small cases.
- Identify the simple representations, the indecomposable projective and the indecomposable injective representations from among all indecomposable representations.
- Be able to describe explicitly the projective covers and injective hulls of all these representations.

- The radical:
- Identify the radical and socle series of all these representations.
- What is the radical of an algebra?

- Category Algebras:
- The path algebra of a quiver and the category algebra of a category.
- Be able to say what it means when one says that representations of a quiver are the same thing as modules for the path algebra,
- or that representations of a category are the same thing as modules for the category algebra (where a representation of the category is a functor from it to a module category over the ground ring).
- Explain how module homomorphisms correspond to natural transformations of the functors.

- Auslander-Reiten theory:
- Be able to construct the AR quiver of path algebras of quivers of finite representation type with at most 3 (or maybe 4) vertices and without oriented cycles.
- The same for representations of very small posets, such as ( a < {b,c} < d ), or the poset ( a < {b,c} < e with also c < d < e ).
- Identify all irreducible morphisms in these cases.
- Maybe the same for an algebra $Q[x]/ (f(x))^n$ where $f$ is a polynomial.

- Maybe something about functorial methods and Auslander algebras.
- The classification of simple functors from A-mod to K-mod, where A is a finite dimensional algebra over a field K.
- The projective dimension of the simple functors.
- Construction of a minimal projective resolution of a simple functor, doing it explicitly in the case of path algebras of small quivers considered already.