The "alpha" version of factorization homology pairs (framed) $n$-manifolds with $E_n$-algebras.  This construction generalizes classical homology of a manifold, yields novel results concerning configuration spaces of points in a manifold, and supplies a sort of state-sum model for sigma-models (ie, mapping spaces) to
$(n-1)$-connected targets.  This "alpha" version of factorization homology novelly extends Poincaré duality, shedding light on deformation theory and dualities among field theories.  Being defined  using homotopical mathematical foundations, "alpha" factorization homology is manifestly functorial and continuous in all arguments, notably in moduli of manifolds and embeddings between them, and it satisfies a local-to-global expression that is inherently homotopical in nature.  
Now, $E_n$-algebras can be characterized as $(\infty,n)$-categories equipped with an $(n-1)$-connected functor from a point.  The (full) "beta" version of factorization homology pairs (framed) $n$-manifolds with pointed $(\infty,n)$-categories (with adjoints).  Applying $0$th homology, or $\pi_0$, recovers a version of the String Net construction of surfaces, as well as of Skein modules of $3$-manifolds.  In some sense, the inherently homotopical nature of (full) "beta" factorization homology affords otherwise unforeseen continuity in all arguments, and local-to-global expressions.  
In this talk, I will outline a definition of "beta" factorization homology, focusing on low-dimensions and on suitably
reduced $(\infty,n)$-categories (specifically, braided monoidal categories).  I will outline some examples, and demonstrate some operational practice of factorization homology.  Some of this material is established in literature, some a work in progress, and some conjectural — the status of each assertion will be made clear.  I will be especially interested in targeting this talk wot those present, and so will welcome comments and questions.
All of this work is joint with John Francis.

Zoom uygulaması Bilim Akademisi tarafından sağlanmaktadır./Zoom link is provided by The Science Academy.