# Manchester Geometry Seminar 2014/2015

**Thursday 5 February 2015. ** *The Frank Adams Room (Room 1.212), the Alan Turing Building. 4.15pm*
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Adjoints for Nonlinear Operators and $L_{\infty}$-Bialgebroids

Theodore Voronov (University of Manchester)

`theodore.voronov@manchester.ac.uk`

We have introduced a generalization of smooth maps of manifolds (or supermanifolds) called "thick morphisms". Such morphisms are defined via formal canonical relations between cotangent bundles and make a formal category, a "thickening" of the usual category of smooth manifolds with the same class of objects.

In the talk, we shall explain how this new construction makes it possible to obtain an analog of the adjoint operator for the case when the initial operator is nonlinear. (We consider maps of vector spaces or fiberwise maps of vector bundles $\Phi:\, E_1\to E_2$.) This gives a "nonlinear pushforward map" of the spaces of functions on the dual bundles $\Phi_*:\, \mathbf{C}^{\infty}(E_1^*)\to \mathbf{C}^{\infty}(E_2^*)$. (Since the mapping of functions is itself nonlinear, the functions should be even or "bosonic"; there is a parallel "fermionic" construction.)

Time permitting, we shall give an application to homotopy Poisson algebras and homotopy Lie (bi)algebroids.

(See preprint arXiv:1411.6720 [math.DG].)