The Second Memoir of Évariste Galois (Prof. Chandan Singh Dalawat)

Abstract: Galois was interested in polynomial equations solvable by radicals. In his First Memoir, he characterised equations of prime degree which are solvable. In the Second Memoir, he introduced the concept of primitive equations and showed that if a primitive equation is solvable by radicals, then its degree has to be the power of a prime. These days, instead of working with solvable primitive equations, one works with solvable primitive extensions of a field, and Galois's theorem says that the degree of such an extension is the power of a prime. But not every extension of degree equal to the power of a prime need be solvable or primitive. We will show how to parametrise the set of solvable primitive extensions of a given field. The presentation will be entirely elementary and all concepts will be explained, so the talk should be accessible to a wide audience.

Continued fraction expansions and their applications (Prof. S. G. Dani)

Abstract: Continued fraction expansions of numbers provide an intrinsic way of expressing and understanding real numbers in terms of integers. The underlying idea may be linked to the Euclidean algorithm, and is also involved in some of the work in medieval India, especially that of Narayana Pandit. Following the work of Lagrange leading to a systematic formulation of the concept, it has been applied very fruitfully to various problems in algebra, number theory, analysis and other areas. In recent years some of the theory has also been extended to complex numbers. In this talk we give a brief introduction to the topic, with an exposition of some major results, and some unsolved questions.

Envelopes, dual curves and the geometry of equations (Prof. Kaushal Verma)

Abstract: The aim of this elementary talk will be to define all the terms in the title and to understand the insights that they provide in understanding the geometry of equations.

An introduction to adaptive finite element methods (Prof. Neela Nataraj)

Abstract: In this talk, an introduction to the topic of a posteriori error estimation and adaptive finite element approximations of elliptic partial differential equations is presented. The residual method for a model Poisson problem is explained with details. The strategy of making the computational method adaptive with feedback from computational process is explained. We conclude with some applications.

Group Enumerations: History and Beyond (Prof. Geetha Venkataraman)

Abstract: A basic question that one can ask about groups is `How many groups are there of order n up to isomorphism?’ Group enumerations attempts to answer this and related questions. Early results can be traced back to Hölder in 1895 through a formula for counting isomorphism classes groups of square-free order. Much of the research in this area however dates from the 1960s. A brief history, survey of techniques and open problems will be covered in the lecture.

Eigenvalue problems: An evergreen topic with new dimensions (Prof. Shreemayee Bora)

Abstract: It is well known that eigenvalue problems find wide application in science and engineering. Different aspects of eigenvalue problems that arise in applications will be presented in this talk with pointers to some challenging open problems.