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:

  “Splitting of Vector Bundles via Fourier–Mukai Theory”;

• a presentation, titled:

  “Fourier–Mukai Theory in Commutative Algebra”;

• 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.