The main goal is to study some open questions of Leavitt path algebras and graph C*-algebras, aimed at the classification of these algebras in terms of easily computable invariants of graph theory. The main problems to be considered are:
Problem 1: Is there a set of graph transformations that classifies, up to Morita equivalence, Leavitt path algebras and graph C*-algebras? In particular, whether the on the graph transformations (R), (S), (O), (I) are enough to classify, up to Morita equivalence, Leavitt path algebras and graph C*-algebras.
Problem 2: Find the stable rank of ultragraph Leavitt path algebras and ultragraph C*-algebras.
Problem 3: Find graph-theoretic sufficient conditions for automorphisms of the Leavitt algebras L_{(1, n)}.

-    For Problem 1, we give criteria to determine which the commutator Lie algebras related to simple Steinberg algebras are simple. Then, we provide easily computable criteria for the commutator Lie algebra related to a simple Leavitt path algebra to be simple. Consequently, we obtain that two purely infinite simple Leavitt path algebras whose Grothendieck groups correspond appropriately, their associated commutator Lie algebras are either both simple or both non-simple. These results were published in the first paper of Item 11 below. 
     Moreover, we introduce and study Steinberg algebras over idempotent commutative semirings. We provide criteria for these algebras to be simple. Consequently, we construct an example of simple algebras over the Boolean semifield B = {0, 1} by using Steinberg algebras over B and graph gruopoids, which is not isomorphic to the corresponding Leavitt path algebras. These results were published in the second paper of Item 11 below.   
-    For Problem 2, we characterize graded simple ultragraph Leavitt path algebras. Consequently, we compute the stable rank of these ultragraph Leavitt path algebras and the corresponding C*-algebras. These results were published in the third paper of Item 11 below.
-    We have not obtained new results about Problem 3. However, by the study of Problem 3, Nguyen Duy Tan, a member of the project, receives some stronger forms of Koszulity for the Galois cohomology algebra (which is an associative noncommutative algebra) of the maximal pro-p quotient of an absolute Galois group.

- The project studies simple Steinberg algebras via their associated commutator Lie algebras. Then, we provide easily computable criteria for the commutator Lie algebra related to a simple Leavitt path algebra to be simple. 
- The project introduces and studies Steinberg algebras of Hausdorff ample groupoids over idempotent commutative semirings, and gives criteria for these algebras to be simple.
- The project classifies graded simple ultragraph Leavitt path algebras and computes the stable rank of these algebras and the corresponding C*-algebras.

- Scientific papers in referred journals (list):
1) T. G. Nam, Simple Lie algebras arising from Steinberg algebras of Hausdorff ample groupoids, J. Algebra 595 (2022), 194-215.             
2) T. G. Nam and J. Zumbragel, On Steinberg algebras of Hausdorff ample groupoids over commutative semirings, J. Pure Appl. Algebra 225 (2021) 106547.
3) T. G. Nam and N. D. Nam, Purely infinite simple ultragraph Leavitt path algebras, Mediterranean Journal of Mathematics 19 (2022), no. 1, PaperNo.7, 20pp.
4) J. Minác, M. Palaisti, F. W. Pasini and N. D. Tân, Enhanced Koszul properties in Galois cohomology, Research in the Mathematical Sciences, 7: 10 (2020). https://doi.org/10.1007/s40687-020-00208-5.    
- Other products (if applicable): 
+ Successfully organized one scientific conference (16/10/2021).
+ A summary of the goals and results of the project.