Abstract: The problem of interpolation is well known. But happens when we move to a higher dimension space over any arbitrary Field? While investigating such problem, Gregory Volfovich Chudnovsky gave a conjectural lower bound for the minimal degree of a homogeneous polynomial in the Polynomial ring that vanishes on a given finite set of points with a given set of multiplicities. Later Waldschmidth Constant came into the picture, but how this famous geometrical problem boils down to the Containment Problem of Symbolic Powers and Ordinary Powers is the primary objective of my talk. I will also talk on some recent developments in this area.

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Abstract: Quadratic Gauss Sums, more known as Arithmetic Gauss Sum, first made appearance on 'Disquitiones Arithmaticae' in the 4th and 6th proof of 'Quadratic Reciprocity' back in 1797. Thereafter it became a powerful tool in Number Theory, and around 1870 Camille Jordan(probably) came up with a new Algebraic Structure with finite number of elements, what we now know as Finite Fields (appeared in 'Théorie de Galois'). And more than 100 years later people (namely, Prof. Ronald J. Evans, Prof. Bruce C. Berndt) tried to unify these two notions. In this talk, we will try to get the glimpse of this unified concept and explore some of its applications.

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