Georgia Institute of Technology College of Sciences

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.

This is the second lecture of the series on h-principle. We will introduce jet bundle and it's various properties. This played a big role in the devloping modern geometry and topology. And using this we will prove Whitney embedding theorem. Only basic knowledge of calculus is required.