Description
Title: Four Flavours of Combinatorics: (Enumerative, Probabilistic,
Extremal, and Geometric)
Speaker: Kunal Dutta (INRIA, France)
Date-Time: Thursday, July 19, 2018, 4 PM.
Venue: 216, Dept. of Mathematics, IITB
Abstract: 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.
First, 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 generalization 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).
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.
The last 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).