The main aim of this paper is to deal with a fuzzy version of Farkas lemma involving trapezoidal fuzzy numbers. In turns to that the fuzzy linear programming and duality theory on these problems can be used to provide a constructive proof for Farkas lemma. Keywords Farkas Lemma, Fuzzy Linear Programming, Duality, Ranking Functions.

In this paper, we establish weak and strong duality theorems for a pair of multiobjective symmetric dual problems. This removes several omissions in the paper "Symmetric and self duality in vector optimization problem, Applied Mathematics and Computation 183 (2006) 1121-1126".

We propose a suitable substitute for the classical Grothendieck ring of an algebraically closed field, in which any quasi-projective scheme is represented with its non-reduced structure. This yields a more subtle invariant, called the schemic Grothendieck ring. In order to include open subschemes and their complements, we introduce formal motives. Although originally cast in terms of definabili...

Based on the Cayley, Grothendieck, Knudsen Mumford theory of determinants we extend the CM polarization to the Hilbert scheme. We identify the weight of this refined line bundle with the generalized Futaki invariant of Donaldson. We are able to conclude that CM stability implies K-Stability. An application of the Grothendieck Riemann Roch Theorem shows that this refined sheaf is isomorphic to t...

Grothendieck introduced the notion of a “motif” in a letter to Serre in 1964. Later he wrote that, among the objects he had been privileged to discover, they were the most charged with mystery and formed perhaps the most powerful instrument of discovery.1 In this talk, I shall explain what motives are, and why Grothendieck valued them so highly. These are my notes for a “popular” talk in the ‘W...