A. V. Jayanthan, IIT Madras.

Binomial Edge Ideals Associated to Finite Simple Graphs and their Algebraic Invariants

In 2009-2010, Herzog et al. and independently Ohtani introduced the concept of binomial edge ideal associated to finite simple graphs. They studied the properties of these ideals with a combinatorial point of view. Since then, there have been a lot of research on the connection between the algebraic properties of the ideals and the combinatorial properties of the graph. In this series of talks, I will introduce the ideal and discuss some of the important connection between the associated algebraic and combinatorial invariants. Towards the end I will discuss about some open problems in this direction and some other directions for future research.

Atul Dixit, IIT Gandhinagar.

Combinatorics of Partitions

The fundamental theorems in $q$-series such as the $q$-bionomial theorem, Jacobi triple product identity and Euler’s pentagonal number theorem will be discussed followed by applications of the elementary series-product identities to partitions. In this part, the combinatorial techniques in partition theory involving generating functions as well as bijective proofs will be introduced. For example, Legendre's combinatorial interpretation of Euler's pentagonal theorem and its bijective proof due to Franklin will be discussed. Properties of Gaussian polynomials will also be considered.

We will then focus on Ramanujan's well-known congruence for the partition function $p(n)$ modulo $5$, $7$, and $11$ and some of the partition statistics associated with them such as ranks and cranks. We will also introduce recently studied restricted partition functions which give simple combinatorial interpretations of some of the third order mock theta functions of Ramanujan.

Finally, analytic and combinatorial versions of Rogers-Ramanujan identities will be discussed along with the applications of the latter. If time permits, we will also delve into miscellaneous topics in partition theory, for example, MacMahon's partition analysis and plane partitions.