The recovery of objects obscured by scattering is an important goal in imaging and has been approached by exploiting, for example, coherence properties, ballistic photons or penetrating wavelengths. Common methods use scattered light transmitted through an occluding material, although these fail if the occluder is opaque. Light is scattered not only by transmission through objects, but also by multiple reflection from diffuse surfaces in a scene. This reflected light contains information about the scene that becomes mixed by the diffuse reflections before reaching the image sensor. This mixing is difficult to decode using traditional cameras. Here we report the combination of a time-of-flight technique and computational reconstruction algorithms to untangle image information mixed by diffuse reflection. We demonstrate a three-dimensional range camera able to look around a corner using diffusely reflected light that achieves sub-millimetre depth precision and centimetre lateral precision over 40 cm× 40 cm×40 cm of hidden space.

We show that the zeroth cohomology of M. Kontsevich's graph complex is isomorphic to the Grothendieck-Teichmüller Lie algebra grt 1 . The map is explicitly described. This result has applications to deformation quantization and Duflo theory. We also compute the homotopy derivations of the Gerstenhaber operad. They are parameterized by grt 1 , up to one class (or two, depending on the definitions). More generally, the homotopy derivations of the (non-unital) En operads may be expressed through the cohomology of a suitable graph complex. Our methods also give a second proof of a result of H. Furusho, stating that the pentagon equation for grt 1 -elements implies the hexagon equation.Theorem 1.2. The automorphism of the Gerstenhaber operad up to homotopy are given by grt 1 and one class, i.e., there is an isomorphisms of Lie algebras H 0 (Der(hoe 2 )) ∼ = grt 1 ⋊ K =: grt where K acts on grt 1 by multiplication with the degree with respect to the grading on grt 1 .

We analyze multi-bounce propagation of light in an unknown hidden volume and demonstrate that the reflected light contains sufficient information to recover the 3D structure of the hidden scene. We formulate the forward and inverse theory of secondary and tertiary scattering reflection using ideas from energy front propagation and tomography. We show that using careful choice of approximations, such as Fresnel approximation, greatly simplifies this problem and the inversion can be achieved via a backpropagation process. We provide a theoretical analysis of the invertibility, uniqueness and choices of space-time-angle dimensions using synthetic examples. We show that a 2D streak camera can be used to discover and reconstruct hidden geometry. Using a 1D high speed time of ight camera, we show that our method can be used recover 3D shapes of objects "around the corner".

Abstract. We study categorial properties of the operadic twisting functor Tw . In particular, we show that Tw is a comonad. Coalgebras of this comonad are operads for which a natural notion of twisting by Maurer-Cartan elements exists. We give a large class of examples, including the classical cases of the Lie, associative and Gerstenhaber operads, and their infinity-counterparts Lie∞, As∞, Ger∞. We also show that Tw is well behaved with respect to the homotopy theory of operads. As an application we show that every solution of Deligne's conjecture is homotopic to a solution that is compatible with twisting.

Abstract. We show that Kontsevich's formality of the little disk operad, obtained using graphs, is homotopic to Tamarkin's formality, for a special choice of a Drinfeld associator. The associator is given by parallel transport of the Alekseev-Torossian connection.

The configuration space of points on a D-dimensional smooth framed manifold may be compactified so as to admit a right action over the framed little D-disks operad. We construct a real combinatorial model for these modules, for compact smooth manifolds without boundary. Contents 1. Introduction 1 2. Compactified configuration spaces 5 3. The Cattaneo-Felder-Mnev graph complex and operad 7 4. Twisting Gra M and the co-module * Graphs M 10 5. Cohomology of the CFM (co)operad 14 6. The non-parallelizable case 21 7. A simplification of * Graphs M and relations to the literature 23 8. The real homotopy type of M and FM M 29 9. The framed case in dimension D = 2 33 Appendix A. Remark: Comparison to the Lambrechts-Stanley model through cyclic C ∞ algebras 36 Appendix B. Example computation: The partition function of the 2-sphere 38 Appendix C. Pushforward of PA forms 41 References 43 * Graphs M ⊂ * Graphs M that still comes with a map of dg Hopf collections * Graphs M → Ω PA (FM M ).Our first main result is the following.1 A (dg) Hopf collection C for us is a sequence C(r) of dg commutative algebras, with actions of the symmetric groups S r . A (dg) Hopf cooperad is a cooperad in dg commutative algebras.

We compute the homotopy derivations of the properads governing even and odd Lie bialgebras as well as involutive Lie bialgebras. The answer may be expressed in terms of the Kontsevich graph complexes. In particular, this shows that the Grothendieck-Teichmüller group acts faithfully (and essentially transitively) on the completions of the properads governing even Lie bialgebras and involutive Lie bialgebras, up to homotopy. This shows also that by contrast to the even case the properad governing odd Lie bialgebras admits precisely one non-trivial automorphism -the standard rescaling automorphism, and that it has precisely one non-trivial deformation which we describe explicitly.

Abstract. It is noted that the higher version of M. Kontsevich's Formality Theorem is much easier than the original one. Namely, we prove that the higher Hochschild-Kostant-Rosenberg map is already a hoe n+1 -formality quasi-isomorphism whenever n ≥ 2.