I believe in Federico Ardila-Mantilla’s axioms:
Axiom 1. Mathematical talent is distributed equally among different groups, irrespective of geographic, demographic, and economic boundaries.
Axiom 2. Everyone can have joyful, meaningful, and empowering mathematical experiences.
Axiom 3. Mathematics is a powerful, malleable tool that can be shaped and used differently by various communities to serve their needs.
Axiom 4. Every student deserves to be treated with dignity and respect.
Graph symmetries:
Our aim is to develop a theory for transitive graphs that can be seen as a graph-theoretical analogue of the Bass-Serre-theory for groups.
Stallings' splitting theorem is one of the crucial theorems in geometric\combinatorial group theory. It says a finitely generated group with more than one end is either an amalgamated free product, or a HNN extension over a finite subgroup. Hamann, Lehner, Rühmann, and I recently proved an analogous result for infinite graphs and I am currently working on its applications.
One of the applications of Stallings' theorem is accessibility. A finitely generated group G is said to be accessible if the process of iterated nontrivial splitting of G by Stalllings' theorem always terminates in a finite number of steps. Recently Hamann and I proved an analogous result for accessible graphs.
Spanning subgraphs in Cayley (di)graphs:
This project splits into two parts. In the first part, we study the decomposition of the Cartesian product of two directed cycles into two arc-disjoint hamiltonian dipaths. This can be considered as the decomposition of the Cayley digraph of two cyclic groups with the standard generating set. I am interested in the general case of whether any Cayley digraph of two cyclic groups can be decomposed into two arc-disjoint hamiltonian dipaths.
The next part is about hamiltonicity of infinite Cayley graphs. The notion of topological circles of a graph has made it possible to extend theorems about cycles in finite graphs to locally finite graphs with ends. In particular, the existence of a hamiltonian circle, a topological circle in |Cay(G,S)| that passes through every vertex and every end is an open question. I am interested in Cayley graphs with hamiltonian circles.
Median-decompositions:
Treewidth can be seen as a measure of how “treelike” a graph is. The usefulness of tree decompositions as a decomposition tool, especially in the theory of Graph Minors, is highlighted by various structural theorems and several NP-hard decision and optimization problems are fixed-parameter tractable when parameterized by treewidth.
Since graphs of bounded treewidth inherit several advantages of trees, it is very natural to investigate how to go beyond tree-decompositions and try to model a graph on graphs other than trees. One of these approaches is median-decompositions which are originally introduced for graphs by Konstantinos Stavropoulos as a generalization of tree-decompositions.
In this project, we investigate different aspects of median-decompositions including algorithmic aspects.
Email: bobby.miraftab@gmail.com
Follow me here:
Man muß jederzeit an Stelle von „Punkte, Geraden, Ebenen“ „Tische, Stühle, Bierseidel“ sagen können. [One must always be able to say “tables, chairs, beer mugs” in place of “points, lines, planes”.]
— David Hilbert to Otto Blumenthal, on the axiomatic method in mathematics
