Abstract: Classical Interpolation problem of estimating new data from a set of known data is well understood under one variable situation. There are modified numerical methods of estimation, even for multiple (but small) variables. In higher dimensional spaces (namely, Projective Space) the main problem is of finding the lowest possible degree of the hyper-surface passing through a given set of points with prescribed multiplicity. To tackle such problems there are some famous conjectures: Chudnovsky’s Conjecture (provides the lower bound for the degree of the lowest possible hyper-surface that passes through the given set of points at least once), Demailly’s Conjecture (similar to the Chudnovsky’s Conjecture, but it captures the multiplicities also), and etc. I will be discussing those conjectures and some recent developments in this area, also some connected well-famous problems associated to interpolation problem.

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Abstract: The classical Interpolation problem is a numerical analysis problem that estimates new data, from a known data set. There have been different approaches to provide more refined numerical results on Interpolation, like Polynomial Interpolation, Spline Interpolation, Mimetic Interpolation, etc. But in higher dimensions (specifically in Projective N-Space), the main problem is to find the minimal degree homogeneous polynomial that vanishes on a finite set of points with a given set of multiplicities. To deal with such a problem G.V. Chudnovsky and J.P. Demailly provided some conjectural bounds to the minimal degree, which I will discuss in my talk. I will also talk about some recent developments on this topic. 

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Abstract: The problem of interpolation is well known. But what happens when we leave planer space and move to some general space? While investigating such a problem, G. V. 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 Waldschmidt Constant came into the picture, but how this famous geometrical problem boils down to the Containment Problem of Symbolic Powers and Ordinary Powers and how other associated problems developed is the primary objective of my talk. I will start with the basic overview of Algebraic Geometry and connect it to my research via Commutative Algebra.

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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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