Mathematical Research Seminar - Archive

We formulate a massively general framework for optimal dynamic stochastic control problems which allows for a control-dependent informational structure. The issue of informational consistency is brought to light and investigated. Bellman's principle is formulated and proved. In a series of related results, we expound on the informational structure in the context of (completed) natural filtrations of (stochastic) processes.

The topic of our talk is a refinement of the classical result of Frucht, who has shown that for any finite group G, there exists a graph whose full automorphism group is isomorphic to G. Instead of seeking a graph whose automorphism group is isomorphic to G, we choose a specific permutation representation of G on a set V, and ask about the existence of a combinatorial structure on V whose full automorphism group is equal to G in its particular representation. If the permutation representation considered is the regular permutation representation and the combinatorial structures we consider are graphs, the question becomes the well-known questions of which finite groups are the full automorphism groups of some Cayley graph (in case of existence, such Cayley graph is called a Graphical Regular Representation for G). In our talk, we extend the question of regular representations to other classes of combinatorial structures such as incidence structures, directed graphs, designs, maps, or hypergraphs. Our presentation is based on a collaboration with Tatiana Jajcayova.

A Hamiltonian cycle system of odd (even) order v, briefly a HCS(v), is a decomposition of the complete graph Kv (the complete graph Kv minus a 1-factor) into Hamiltonian cycles. I will survey known results and open problems about the automorphism groups of a HCS(v).

We will discuss the problem of describing the general form of adjacency and coherency preservers on 2 by 2 hermitian matrices. This is an elementary problem that can be easily explained to the first year undergraduate students of mathematics. I hope I will succeed to show that such a simple question can generate some interesting mathematics connecting linear algebra, functional analysis, geometry, algebraic topology, and mathematical physics.