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INDIAN INSTITUTE OF SCIENCE EDUCATION AND RESEARCH (IISER) PUNE
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An Autonomous Institution, Ministry of Human Resource Development, Govt. of India
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Seminars and Colloquia

Mathematics

Four Flavours of Combinatorics: (Enumerative, Probabilistic, Extremal, and Geometric) 
 
Fri, Jul 27, 2018,   04:00 PM to 05:00 PM at Madhava Hall

Dr. Kunal Dutta
Associate Researcher in the DataShape group at INRIA Sophia-Antipolis, France

 

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 (IMSc Chennai).
 
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).

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