In this talk we shall see three very different areas of applications of combinatorics in mathematics and computer science, illustrating four different flavours of combinatorial reasoning.
The first problem is on the decomposition, into irreducible representations, of the Weil representation of the full symplectic group associated to a finite module of odd order over a Dedekind domain. We shall discuss how a poset structure defined on the orbits of finite abelian p-groups under automorphisms can be used to show the decomposition of the Weil representation is multiplicity-free, as well as parametrize the irreducible subrepresentations, compute their dimensions in terms of p, etc. Joint works with Amritanshu Prasad (IMSc, Chennai).
Next, we consider lower bounds on the maximum size of an independent set, as well as the number of independent sets, in k-uniform hypergraphs, together with an extension to the maximum size of a subgraph of bounded degeneracy in a hypergraph. Joint works with C. R. Subramanian (IMSc, Chennai), Dhruv Mubayi (UIC, Chicago) and Jeff Cooper (UIC, Chicago) and Arijit Ghosh.
Finally, we shall look at Haussler’s Packing Lemma from Computational Geometry and Machine Learning, for set systems of bounded VC dimension. We shall go through its generalisation to the Shallow Packing Lemma for systems of shallow cell complexity, and see how it can be used to prove the existence of small representations of set systems, such as epsilon nets, M-nets, etc. Joint works with Arijit Ghosh (IMSc, Chennai), Nabil Mustafa (ESIEE Paris), Bruno Jartoux (ESIEE Paris) and Esther Ezra (Georgia Inst. Tech., Atlanta).