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Purusottam Rath
Research Summary:
Zero-sum Ramsey theory is a rather recently developed and developing area in Combinatorics which has brought in algebraic tools and algebraic flavour to Ramsey theory. The paradigm of zero-sum problems can be formulated as understanding the following question: If elements of some combinatorial structure is mapped into a finite group, whether there exists a substructure sum of whose images in the group is the identity. The area is full of questions and conjectures and we have been endeavouring to answer and solve some of those.
The other question we are interested in constitutes one of the central questions in the area of Additive Number Theory, namely estimating the size of sum and product sets. Erdos and Szemeredi made the beautiful conjecture that a finite set of positive integers cannot have simultaneously few sums and few products. Freiman's path breaking work in multidimensional arithmetic progressions has made important inroads to this deep and difficult question and the conjecture has been proved in affirmative in some conditional cases. But the main conjecture remains unsolved.
Publications:
- Remarks on some zero-sum problems (Jointly with S.D.Adhikari) Expo. Math. 21, no. 2, 185 -- 191 (2003).
- Monochromatic configurations for finite colourings of the plane (Jointly with S.D.Adhikari) Note di Matematica, to appear.
Preprints:
- Method of approximation in Transcendental Number Theory
Conference/Workshops Attended:
- Attended the Conference " Young Researchers - Modular Forms and Transcendental Number Theory " held at CIRM, Marseilles, France in May 2003.
- Attended the "International Conference on Commutative Algebra and Combinatorics" held in H.R.I in December 2003.
- Attended the satelite conference in Number Theory held in Bangalore in December 2003.
- Attended the AMS-INDIA conference held in Bangalore in December.
Invited Lectures/Seminars:
- Gave a talk in the "International Conference on Commutative Algebra and Combinatorics" held in H.R.I in December 2003.
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