Speaker: Sujata Ghosh (ISI Chennai)
Title: Formal methods in mathematics
Abstract: In computer science, formal methods are used to develop, specify and verify software and hardware systems. Over the years, such methods have provided breakthroughs in mathematical discovery and verification of mathematics as well. This talk will provide a sketch of the underlying methodology with a focus on the developments in the last decade
Speaker: Apoorva Khare (Indian Institute of Science, Bangalore)
Title: Pólya frequency sequences
Abstract: This talk provides a gentle introduction to totally positive matrices and Polya frequency sequences (and sequences). We will see basic examples, history, and fundamental results on total positivity, variation diminution, and sign non-reversal – as well as a few proofs to illustrate how the main ingredients fit together. Several classical results (and one Hypothesis) from before 1955 feature in this journey. We will end by connecting Polya frequency sequences to the Laguerre–Polya class and hence to Polya–Schur multipliers, and mention 21st century incarnations of the latter.
Speaker: Jyotshana Prajapat (University of Mumbai)
Title: From Linear Algebra to Functional Analysis
Abstract: I will discuss the Riesz representation theorem and give a new proof using a variational method.
Speaker: Parthanil Roy (Indian Statistical Institute Bangalore)
Title: Weierstrass Approximation Through Bernstein Polynomials: When Probability Helps Analysis
Abstract: In this lecture, a probabilistic proof of Weierstrass approximation theorem will be presented. Basic probability theory needed for the proof will be developed in a self-contained manner. The statement and motivation of the approximation theorem will also be discussed from the viewpoint of real analysis. This lecture will explain, among other things, how inter-dependent the two subjects (probability theory and real analysis) are. Special care will be taken so that this lecture is accessible to everyone.
Speaker: Jyoti Singh (VNIT Nagpur)
Title: Quadratic and Cubic forms in Characteristic two
Abstract: In this talk, we provide the classification, up to isomorphism, of the set of all finitely generated graded algebras of fixed characteristic p that are non F-pure. In particular, we discuss when quadratic forms and cubic surfaces are Frobenius split or equivalently, when their homogeneous coordinate rings are F-pure. We show that the vast majority of cubic surfaces in characteristic two are Frobenius split.
Speaker: Kaneenika Sinha (IISER Pune)
Title: Reflections on the prime omega function
Abstract: The prime omega function counts the number of prime divisors of a number n. This beautiful function has several interesting arithmetic properties. A detailed study of this function led to the birth of a fascinating area in mathematics, namely probabilistic number theory (that is, mathematics at the interface of probability and number theory). We will discuss these themes in this talk.
Speaker: Prahlad Vaidyanathan (IISER Bhopal)
Title: Korovkin's Theorem and the Choquet Boundary
Abstract: We discuss an approximation theorem due to Korovkin, which provides a short proof of Weierstrass' approximation theorem. We then discuss how this relates to the Choquet boundary of the corresponding function system.