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We derive a formula expressing the average number $E_n$ of real lines on a random hypersurface of degree $2n-3$ in $\mathbb{R}\textrm{P}^n$ in terms of the expected modulus of the determinant of a special random matrix. In the case $n=3$ we prove that the average number of real lines on a random cubic surface in $\mathbb{R}\textrm{P}^3$ equals: $$E_3=6\sqrt{2}-3.$$ Our technique can also be used to express the number $C_n$ of complex lines on a generic hypersurface of degree $2n-3$ in $\mathbb{C}\textrm{P}^n$ in terms of the determinant of a random Hermitian matrix. As a special case we obtain a new proof of the classical statement $C_3=27.$ We determine, at the logarithmic scale, the asymptotic of the quantity $E_n$, by relating it to $C_n$ (whose asymptotic has been recently computed D. Zagier). Specifically we prove that: $$\lim_{n\to \infty}\frac{\log E_n}{\log C_n}=\frac{1}{2}.$$ Finally we show that this approach can be used to compute the number $R_n=(2n-3)!!$ of real lines, counted with their intrinsic signs, on a generic real hypersurface of degree $2n-3$ in $\mathbb{R}\textrm{P}^n$.

In this work we consider infinite dimensional extensions of some finite dimensional Gaussian geometric functionals called the Gaussian Minkowski functionals. These functionals appear as coefficients in the probability content of a tube around a convex set $D\subset\mathbb{R}^k$ under the standard Gaussian law $N(0,I_{k\times k})$. Using these infinite dimensional extensions, we consider geometric properties of some smooth random fields in the spirit of [Random Fields and Geometry (2007) Springer] that can be expressed in terms of reasonably smooth Wiener functionals.

Random fields are useful mathematical objects in the characterization of non-deterministic complex systems. A fundamental issue in the evolution of dynamical systems is how intrinsic properties of such structures change in time. In this paper, we propose to quantify how changes in the spatial dependence structure affect the Riemannian metric tensor that equips the model's parametric space. Defining Fisher curves, we measure the variations in each component of the metric tensor when visiting different entropic states of the system. Simulations show that the geometric deformations induced by the metric tensor in case of a decrease in the inverse temperature are not reversible for an increase of the same amount, provided there is significant variation in the system entropy: the process of taking a system from a lower entropy state A to a higher entropy state B and then bringing it back to A, induces a natural intrinsic one-way direction of evolution. In this context, Fisher curves resemble mathematical models of hysteresis in which the natural orientation is pointed by an arrow of time.

The short- and long-scale behaviour of tangled wave vortices (nodal lines) in random three-dimensional wave fields is studied via computer experiment. The zero lines are tracked in numerical simulations of periodic superpositions of three-dimensional complex plane waves. The probability distribution of local geometric quantities such as curvature and torsion are compared to previous analytical and new Monte Carlo results from the isotropic Gaussian random wave model. We further examine the scaling and self-similarity of tangled wave vortex lines individually and in the bulk, drawing comparisons with other physical systems of tangled filaments.

1Told in the Coffee House

By Cyrus Adler and Allan Ramsay

In the course of a number of visits to Constantinople, I became much interested in the tales that are told in the coffee houses. These are usually little more than rooms, with walls made of small panes of glass. The furniture consists of a tripod with a contrivance for holding the kettle, and a fire to keep the coffee boiling. A carpeted bench traverses the entire length of the room. This is occupied by turbaned Turks, their legs folded under them, smoking nargilehs or chibooks or cigarettes, and sipping coffee. A few will be engaged in a game of backgammon, but the majority enter into conversation, at first only in syllables, which gradually gives rise to a general discussion. Finally, some sage of the neighborhood comes in, and the company appeals to him to settle the point at issue. This he usually does by telling a story to illustrate his opinion. Some of the stories told on these occasions are adaptations of those already known in Arabic and Persian literature, but the Turkish mind gives them a new setting and a peculiar philosophy. They are characteristic of the habits, customs, and methods of thought of the people, and for this reason seem worthy of preservation. - Summary by Preface

2Essentials of Spirituality

By Felix Adler

This recording contains four addresses of Felix Adler (1851-1933) concerning spirituality from the perspective of Ethical Culture, an educational and religious movement founded and first led by Adler, based on the belief that living in accordance with ethical principles is central to a meaningful and fulfilling life. – Lee Smalley “Spirituality is morality carried out to the finish….The ultimate end (of spirituality) itself is to elicit worth in others, and, by so doing, in one's self. The indispensable condition of this attitude is to ascribe worth to every human being before even we observe it, to cast as it were a mantle of glory over him, to take toward every fellow human being the expectant attitude, to seek the worth in him until we find it.” – Felix Adler, Dec. 11, 1904