## 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.
• 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.