David Carchedi Higher & Derived Differential Geometry MPI-Oberseminar 2012 Symmetry Lie Groups Lie Group Actions Lie Groupoids (Global Symmetries) (Local Symmetries) Higher Lie Groupoids There are arrows between the arrows and so on... Differentiable

Stacks Higher

Differentiable

Stacks (an infinity topos) Action Groupoid "Stacky quotients" of manifolds by local symmetries. Examples: Orbifolds Leaf spaces of foliated manifolds "Spaces" whose points possess intrinsic symmetries, and these symmetries can have symmetries themselves and so on... "Spaces" whose points possess intrinsic

symmetry groups. Intersection Theory Pontryagin-Thom construction Unoriented cobordism ring Thom spectrum Let X be a compact n-manifold. Main Idea: Embed the of smooth manifolds of . Non-transversal intersections of smooth manifolds exist as derived manifolds and have into an good cohomological properties. universal bundle If is transversal to the zero-section is a smooth manifold cobordant to X. The cobordism ring of derived manifolds is equivalent to that of ordinary manifolds. is derived cobordant to X. Models Spivak Carchedi

and Roytenberg Comes from a model category structure on differential Calculations are easy! Uses weakly Is "good for doing intersection theory" Hard to work with. Is equivalent to Spivak's Uses simplicial Spivak defined axioms for an to satisfy to be "good for doing intersection theory." Any such is called an of . spaces. Borisov

and

Noel . (So is affine.) . (So is affine.) Still need to show it's "good for intersection theory" and compare to the other models. graded super- Objects consist entirely of dg-manifolds, which have been widely studied. Could provide a geometric framekwork for

BV-BRST quantization. I'd like to learn about this. Finally, one should combine these two subjects and study derived differentiable stacks. THANK YOU! Moreover, for any manifold T, and any two submanifolds in the derived cobordism ring the following cup-product formula holds: A central theme to all existing models for derived manifolds is the concept of a . Informally, a is a commutative ring , which additionally carries operations for each smooth function subject to natural compatibility conditions. , Joyce (of finite type) Uses derived . Is a strict 2-category rather than an . ("almost" a 2-truncation of Spivak's model.) Cup-product formula still holds. . More formally, a is a

product-preserving functor , where is the category of manifolds of the form . Étale Stacks "spaces" whose intrinsic symmetry groups are discrete. Analogous to Deligne-Mumford stacks from algebraic geometry.

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