Mathematical Research Seminar - Archive

In this talk I will describe some progress in analytic number theory, from the pioneering work by Dirichlet and Riemann to the Selberg class of L-functions. 

Girth is a fascinating parameter in the study of symmetric graphs, since small girth often allows complete classification of the corresponding objects. Today, we focus on the existence of symmetric graphs of prescribed girth. In the cubic case, vertex-transitive graphs fall into three families according to the number of edge-orbits under their automorphism group. The existence problem has been settled for graphs with one and three edge-orbits, but remains open for the case of two. In this talk, I will sketch a proof establishing the existence of cubic vertex-transitive graphs with two edge-orbits (the flexible case) and even girth, by reducing the problem to one in geometric group theory.

Homophily is a sociological born principle. It asserts that individuals in any 'social' population preferably interact with like individuals. Vertex-colored graphs model finite populations of interacting individuals, where like individuals have the same color. The population exhibits homophily when the coloring correlates with
the graph. What does this mean? One natural answer could be (and has often been given in the literature): Subgraphs induced by the color classes are denser. So, the next question is: Denser than what? In this talk, we will attempt to answer this question by taking a combinatorial statistical approach: the Random Coloring Model. We will corroborate this by making comparisons with the state of the art, some asymptotics and a few new, concrete findings.

This talk is also based on joint work with other colleagues, especially Paolo Giulio Franciosa and Daniele Santoni.

Rotary maps are highly symmetric graph embeddings on orientable surfaces, and their classification is a central problem in topological graph theory. In this talk, we solve this problem for the family of Praeger-Xu graphs  C(p,r,s). We provide a complete classification of all such embeddings for any odd prime p that does not divide  r. 

Everyone is welcome and encouraged to attend.