## 20 Jan Winter School and Conference on Algebra and Number Theory

The department of Mathematical Sciences, Tezpur University organized a Winter School and Conference on Algebra and Number Theory during December 23-29, 2011 which was mainly sponsored by the National Board for Higher Mathematics (NBHM), Department of Atomic Energy, Government of India. Department of Science and Technology (DST), Government of India funded the travel of a few young Indian mathematicians. A grant from the UGC-SAP (DRS-I) was also utilized. Sixty five students from different parts of the country participated in the workshop. The students were final year BSc, MSc, and beginning PhD scholars. Apart from the participants from the colleges and universities of North East India, there were twenty participants from institutes outside the North East, namely, TIFR Mumbai, University of Hyderabad, UMDE Mumbai, CMI Chennai, IIT Bombay, IISER Mohali, IISER Kolkata, Burdwan University, and Sambalpur University. All the participants were provided with TA/DA, local hospitality and two books, namely, Algebra by Dummit & Foote and Rational Points on Elliptic Curves by Silverman & Tate.

In the Winter School (Dec 23-27), there were three mini courses, namely Mini Course-I: Structure of Groups, Mini Course-II: Polynomial Rings and Algebraic Sets, and Mini Course-III: Elliptic Curves and L-functions. Dr. K. V. Krishna of IIT Guwahati delivered three lectures in the Mini Course-I. He discussed free groups, finitely generated Abelian groups, and Sylow theorems. Professor Dilip P. Patil of IISc Bangalore gave four lectures in the Mini Course-II. He mainly discussed basic properties of polynomial rings, proof of Hilbert Basis Theorem, Hilbert's Nullstellensatz, and Algebraic sets.

Professor Eknath Ghate of TIFR Mumbai and Dr. Anupam Saikia of IIT Guwahati delivered four lectures in the Mini Course-III. Dr. Saikia described the Mordell Weil group laws on Elliptic curves, Elliptic curves over finite fields, L-functions, and Birch and Swinnerton-Dyer Conjecture. Professor Ghate gave a proof of the Modell Weil Theorem which states that the Mordell Weil group is a finitely generated Abelian group.

Apart from the talks, there were three rigorous problem solving sessions, one each for the three Mini Courses. There were many informal interactions and discussions among the participants and the experts including a local trip to Tezpur town and its vicinity.

The Winter School was followed by a Conference during Dec 28-29, 2011. Recent developments in Algebra and Number Theory were presented by a selected group of experts in a way accessible to B. Sc, M. Sc., and beginning Ph.D. students. The first talk was delivered by Professor R. Sujatha of University of British Columbia, Canada. In her first talk, she introduced Iwasawa Theory for elliptic curves. In her second talk, she talked about Congruent Number Problem and its connections to elliptic curves.

Professor Eknath Ghate of TIFR Mumbai delivered a talk on recent works of Professor Manjul Bhargava on the Birch & Swinnerton-Dyer Conjecture. The Birch & Swinnerton-Dyer Conjecture which is commonly known as BSD Conjecture is the 7^{th} Millennium Prize Problems of Clay Mathematics Institute. BSD Conjecture relates two objects coming from two complteley different areas of mathematics: one is algebraic rank of the Mordell Weil group of elliptic curves and the other is analytic rank of L-function of elliptic curves. Professor Kapil H. Paranjape delivered two talks in the Conference. In his talks, he introduced projective geometry.

Mr. Prem Prakash Pandey, research scholar at Institute for Mathematical Sciences (IMSc), Chennai delivered a talk on degrees of certain algebraic extension of the field of rational numbers and discussed certain results related to density of prime numbers.

At the end of the conference, feedback from the participants were taken. According to those feedbacks, the participants were highly benefited from the Winter School and Conference. They strongly suggested the organizers to hold such programmes regularly in future.

Author:- Dr. Rupam Barman, Associate Professor,

Department of Mathematical Sciences, Tezpur University.

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