**Romie Banerjee**

TIFR

October 20, 2011

**Non-Abelian Grothendieck Duality and Stable Homotopy**:
I will give a formulation of Grothendieck duality for derived algebraic stacks. These are ordinary Artin/D-M stacks whose structure sheaf can be enriched to a sheaf of multiplicative cohomology theories. In our formulation, following Toen, Vessozi and Lurie, these objects are $(\infty,1)$-topoi equipped with a structure sheaf of $E_{\infty}$-rings.The category of quasi-coherent modules over such derived stacks form symmetric monoidal stable model categories, which are natural homotopical generalizations of abelian categories.
I will relate duality in this context to certain computational aspects of stable homotopy theory. It is common to have generalized cohomology theories arise as homotopy global sections of certain derived stacks. Examples of these are the theory of topological modular forms, real K-theory, real Morava E-theories and stable homotopy itself. Computing the coefficient rings of these theories involve computing Ext in a category of comodules over a Hopf algebroid, whic