Georgia Institute of Technology College of Sciences

Quantum graphs are metric graphs with differential equations defined on the edges. Recent interest in control and inverse problems for quantum graphs
is motivated by applications to important problems of classical and quantum physics, chemistry, biology, and engineering.

In this talk we describe some new controllability and identifability results for partial differential equations on compact graphs. In particular, we consider graph-like networks of inhomogeneous strings with masses attached at the interior vertices. We show that the wave transmitted through a mass is more
regular than the incoming wave. Therefore, the regularity of the solution to the initial boundary value problem on an edge depends on the combinatorial distance of this edge from the source, that makes control and inverse problems
for such systems more diffcult.

We prove the exact controllability of the systems with the optimal number of controls and propose an algorithm recovering the unknown densities of thestrings, lengths of the edges, attached masses, and the topology of the graph. The proofs are based on the boundary control and leaf peeling methods developed in our previous papers. The boundary control method is a powerful
method in inverse theory which uses deep connections between controllability and identifability of distributed parameter systems and lends itself to straight-forward algorithmic implementations.

We give a complete description of the coefficients of the characteristic polynomial $\chi_H(\lambda)$ of a ($k$-uniform) hypergraph $H$, defined by the hyperdeterminant $\det(\mathcal{A} - \lambda \mathcal{I})$, where $\mathcal{A}$ is of the adjacency tensor/hypermatrix of $H$, and the hyperdeterminant is defined in terms of resultants of homogeneous systems associated to its argument. The co-degree $k$ coefficients can be obtained by an explicit formula yielding a linear combination of subgraph counts in $H$ of certain ``Veblen hypergraphs''. This generalizes the Harary-Sachs Theorem for graphs, provides hints of a Leibniz-type formula for symmetric hyperdeterminants, and can be used in concert with computational algebraic methods to obtain the full characteristic polynomial of many new hypergraphs, even when the degrees of these polynomials is enormous. Joint work with Greg Clark of USC.

One way to analyze a (finitely generated) module over a ring is to consider its minimal free resolution and look at its Betti table. The Betti table would be obtained by algebraic computations in general, but in case of the ideal (consists of relations) is generated by monomial quadratics, we can obtain Betti numbers (which are entries of the Betti table) by looking at the corresponding graphs and its associated simplicial complex. In this talk, we will introduce the Stanley-Reisner ideal which is the ideal generated by monomial quadratics and Hochster’s formula. Also, we will deal with some theorems and corollaries which are related to our topic.

We will go through the proof of the hypergraph container result of Balogh, Morris, and Samotij. We will also discuss some applications of this container result.