Undoubtedly one of the best economists his generation, Emmanuel Farhi transformed theories taxation, macroeconomics, and international finance. This essay describes itinerary research style attempts to pay tribute immense contribution economics.

We shall discuss generic extension monoids associated with finite dimensional (basic) hereditary algebras of finite or cyclic type and related applications to Ringel–Hall algebras, (and hence, to quantum groups). We shall briefly review the geometric setting of quantum gln by Beilinson, Lusztig and MacPherson and its connections to Ringel– Hall algebras and q-Schur algebras. In the second part ...

A method is suggested for construction of quadrangulations of the closed orientable surface with given genus g and either (1) with given chromatic number or (2) with given order allowed by the genus g. In particular, N. Hartsfield and G. Ringel’s results [J. Comb. Theory, Ser. B 46 (1989), 84-95] are generalized by way of generating minimal quadrangulations of infinitely many other genera.

By counting with triangles and the octohedral axiom, we find a direct way to construct the Ringel-Hall algebra associated to a triangulated category with (left) homological-finite condition. Our formula is a refinement of that in Peng-Xiao [5] and provides a modified version of Toën’s formula in [7]. 1. Calculation with triangles Given a finite field k with q elements, let C be a k-additive tri...