Jack polynomials and orientability generating series of maps

We study Jack characters, which are the coefficients of the power-sum expansion of Jack symmetric functions with a suitable normalization. These quantities have been introduced by Lassalle who formulated some challenging conjectures about them. We conjecture existence of a weight on non-oriented maps (i.e., graphs drawn on non-oriented surfaces) which allows to express any given Jack character as a weighted sum of some simple functions indexed by maps. We provide a candidate for this weight which gives a positive answer to our conjecture in some, but unfortunately not all, cases. In particular, it gives a positive answer for Jack characters specialized on Young diagrams of rectangular shape. This candidate weight attempts to measure, in a sense, the non-orientability of a given map.Comment: v2: change of title, substantial changes of the content v3: substantial changes in the presentatio

The Multi-Orientable Random Tensor Model, a Review

After its introduction (initially within a group field theory framework) in [Tanasa A., J. Phys. A: Math. Theor. 45 (2012), 165401, 19 pages, arXiv:1109.0694], the multi-orientable (MO) tensor model grew over the last years into a solid alternative of the celebrated colored (and colored-like) random tensor model. In this paper we review the most important results of the study of this MO model: the implementation of the $1/N$ expansion and of the large $N$ limit ($N$ being the size of the tensor), the combinatorial analysis of the various terms of this expansion and finally, the recent implementation of a double scaling limit

Embedding Riemannian Manifolds by the Heat Kernel of the Connection Laplacian

Given a class of closed Riemannian manifolds with prescribed geometric conditions, we introduce an embedding of the manifolds into $\ell^2$ based on the heat kernel of the Connection Laplacian associated with the Levi-Civita connection on the tangent bundle. As a result, we can construct a distance in this class which leads to a pre-compactness theorem on the class under consideration

3. The Non-Orientability of the Mechanical in Thomas Carlyle’s Early Essays

Thomas Carlyle’s early writings epitomise the critical stance towards the utilitarian culture of the age, which Carlyle condemns for glorifying the ‘outward’, i.e. the physical world of machinery, governed by automaticity and mechanical principles, at the expense of the ‘inward’, i.e. the spiritual realm of the human individual, whose hallmarks are instead free volition and agency, and to which automaticity and mechanical principles are foreign. By the 19th century, however, the distinction between human and machine was becoming increasingly problematic. Drawing from Thomas Carlyle’s essays “Signs of the Times” (1829) and “Characteristics” (1831), and from earlier physiological texts with which they engage—chiefly David Hartley’s Observations on Man (1749)—, I explore the tension between the understanding of the human in terms of free will and agency, and the physiological evidence that human thought and behaviour are partly automatic. I argue that the understanding of human nature as partly automatic destabilises Carlyle’s categories of ‘outward’ and ‘inward’ by disrupting their underlying assumption of a clear boundary between man and machine based on their functioning (the latter) or not (the former) according to mechanical principles, entailing instead a fluid connotation whereby the ‘mechanical’ is defined as much by identification with the ‘human’ as in opposition to it. I conclude by offering a geometric illustration of the destabilised inward-outward boundary through the metaphor of non-orientable surfaces (e.g. Möbius strip and Klein bottle), in which it is impossible to distinguish an inside and an outside.