The main goal is to study some open questions of  Leavitt path algebras and graph C*-algebras, by methods from commutative algebras and groups. The main problems to be considered are:
Problem 1: Classify Leavitt path algebras and C*-algebras via K-theory and KK-theory.
Problem 2: Find criteria for two Leavitt path algebras to be isomorphic.
Problem 3: Find criteria for irreducible representations of Leavitt path algebras via their diagonal subalgebras. 

- Theorerical results: For Problem 1, we give a survey about the classification of Leavitt path algebras and C*-algebras via K-theory and KK-theory.
   For Problem 2, we give a survey about the criteria for two Leavitt path algebras to be isomorphic.
   For Problem 3, we give many ways to construct irreducible representions of Leavitt path algebras associted with vertices via Cohn Leavitt path algebras, and give a universal method to construct irreducible representions of Leavitt path algebras associted pairs (c, f) consisting of simple closed paths c together with irreducible polynomials f. We compute directly the kernel of these irreducible representations without using the theory of graded ideals of Tomforde. In particular, we give a new method to construct irreducible representions of Leavitt path algebras via their diagonal subalgebras.
- Applied results: Two PhD students were supported by this project to make two preprints.

We give many ways to construct irreducible representions of Leavitt path algebras associted with vertices via Cohn Leavitt path algebras, and give a universal method to construct irreducible representions of Leavitt path algebras associted pairs (c, f) consisting of simple closed paths c together with irreducible polynomials f. In particular, we give a new method to construct irreducible representions of Leavitt path algebras via their diagonal subalgebras.

- Scientific papers in referred journals (list):
   1. P. N. Ánh and T. G. Nam, Special irreducible representations of Leavitt path algebras, Advances in Mathematics 377 (2021) 107483.
- Other products (if applicable): A summary of the goals and results of the project.