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Beyond Abel and Jacobi

Mathematics Colloquium | Kapil Hari Paranjape | Apr 11th, 2023

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The classical theorem of Abel and Jacobi characterises the zeroes and poles of a meromorphic function on a compact Riemann surface (or smooth projective variety) using a map which is nowadays called the Abel-Jacobi homomorphism.

Following the work of many algebraic geometers over the past 150 years, this result has been successively refined and studied for higher-dimensional varieties. The most sophisticated version of this refinement is via the theory of “motives” as first proposed by Grothendieck. The most elaborate conjectural refinements of the Abel-Jacobi theorem are those of Bloch and Beilinson.

As with most conjectures, there have been many attempts to construct counter-examples. One method is to construct “minimal” instances of these conjectures for “well-understood” varieties where the conjectures do not (as yet) follow from known results. Such examples were constructed by Griffiths, Mumford-Roitman and many others which led to the more precise formulations of the conjectures that we see today.

About the Speaker:

He joined Indian Institute of Technology Kanpur where he pursued a five-years integrated Master’s programme in Mathematics and graduated in 1982. He was awarded the General Proficiency Prize for Mathematics from IIT Kanpur (1982).

He joined School of Mathematics, Tata Institute of Fundamental Research and was awarded his PhD in Mathematics in 1992.

He worked as a Reader at TIFR from 1993-1998. During this he also held various visiting positions at University of Chicago, University of Paris-Sud and University of Warwick. He was appointed as Professor at the Theoretical Statistics and Mathematics Unit of the Indian Statistical Institute, Bangalore. He moved to Institute of Mathematical Sciences, Chennai in 1996. Between 2001 and 2009 he has held visiting positions at California Institute of Technology. Since 2009 he is a professor of Mathematics at Indian Institute of Science Education and Research, Mohali.


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