In the continuation of the lecture on 5th April, we complete the discussion on semigroup theory. Using this, we study the controllability and stabilizability of linear partial differential equations. In details, the control of transport, heat and wave equations will be discussed.

Problem at hand: Suppose you have the choice of taking one of the "n" decisions every day which will consequently put you to certain loss which is not known apriori What shall you do to "*minimize*" the loss? Let's meet and concrete out a strategy this Wednesday at Popular Talk.

We will give a rough idea of Voisin's proof of the Green's Conjecture in the even genus case.

First, we recall semigroup theory. Using this, we study the controllability and stabilizability of linear partial differential equations. In details, th

A living cell responds in sophisticated ways to its environment. Such behavior is all the more remarkable when one considers that a cell is a bag of molecules. A detailed algorithmic explanation is required for how a network of chemical reactions can produce sophisticated behavior. Several previous works have shown that reaction networks are computationally universal and can, in principle, implement any algorithm. The problem is that these constructions have not mapped well onto biological reality, have made wasteful use of the computational potential of the native dynamics of reaction networks, and have not made any contact with statistical mechanics. We seek to address these problems. We find that the mathematical structure of reaction networks is particularly well suited to implementing modern machine learning algorithms. We describe a new reaction network scheme for solving a large class of statistical problems including the problem of how a cell would infer its environment from receptor-ligand bindings. Specificially we show

A polynomial ring S over a field k has a basis consisting of monomials. Given a quotient ring R of S, one would like to identify the monomials whose images form a k-basis of R. We will discuss the ideas of a monomial order, and initial ideals, leading to Groebner basis, which help answer this question.

Over the last decade, a Brill-Noether theory for tropical curves analogous to the corresponding theory for algebraic curves has taken shape. This was discussed by Ashwin Deopurkar in the first few CACAAG talks. I will sketch the proof of the nonexistence part of the Brill-Noether theorem for tropical curves. This is based on the paper ``A tropical proof of the Brill-Noether Theorem'' by Cools,Draisma,Payne and Robeva.

This will be a motivational seminar about the importance of line bundles( ie, vector bundles of rank 1). Giving a vector bundle of rank k over some space is equivalent to give some data, such as an open cover for the space, and a collection of maps from the intersections of these open sets to Gl(k,C). We will first construct CP^n by gluing (n+1) copies of C^n. We will then construct a special line bundle, called O(1), over CP^n , which possesses global sections. The main goal will be to prove that giving a morphism from X to CP^n is equivalent to give a line bundle over X and (n+1) sections which do not vanish simultaneously. Time and Venue: Friday, 30th March, 4pm, Room 216

Attached.

Nori proved that a vector bundle on a projective variety satisfies a polynomial equation with integral coefficients if and only if its pullback to a finite cover is trivial. I shall talk about an extension of this result to Gauduchon astheno-Kahler manifolds. This is joint work with Indranil Biswas.