SISSA Geometry Seminars

[Ugo Bruzzo]

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SISSA GEOMETRY SEMINARS
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Professor Ettore Aldrovandi, Florida State University, Tallahassee

Postnikov invariants and morphisms of monoidal and bimonoidal stacks

Monday, July 8th, 2:30 pm in Room 136, SISSA Building A

Abstract: Certain classical cohomology theories, such as group or (variants of) Hochschild cohomology, classify homotopy types in low degrees. Furthermore, morphisms between homotopy types in low degrees can be  efficiently computed by way of special diagrams called Butterflies, owing to their shape.  In a geometric context, instead of  homotopy types we consider stacks equipped with monoidal or, in more recent developments, bimonoidal structures. I will survey the main points and some of the applications of the theory in the former case first, and then to discuss the latter case of stacks which are categorical rings (ring-like, for want of a better name). Ring-like stacks are related to truncated cotangent complexes, and connections to Shukla, MacLane cohomology of rings, and more generally functor cohomology, can be found.  These  cohomology objects provide one of the main motivations for looking into this subject and they will be the starting point for our discussion.

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Dr. Sikimeti Ma'u, ICTP

Holomorphic quilts and symplectic Fourier-Mukai functors

Tuesday, July 9th, 2:30 pm in Room 136, SISSA Building A

Abstract: I will introduce Wehrheim and Woodward's theory of pseudoholomorphic 
quilts in symplectic topology, which are pseudoholomorphic curves that use 
Lagrangian correspondences as boundary conditions.  I will then explain how 
quilts give rise to various (A-infinity) algebraic relationships for Fukaya categories 
of different symplectic manifolds, reporting on joint work with Wehrheim 
and Woodward.  For example, a Lagrangian correspondence betweeen 
symplectic manifolds determines a bimodule (alternatively, a functor) between 
Fukaya categories, so can be thought of as a symplectic analogue of Fourier-Mukai 
functor.  

U. Bruzzo