Teichm\"uller spaces of Riemann surfaces with orbifold points of arbitrary order and cluster variables

We generalize a new class of cluster type mutations for which exchange transformations are given by reciprocal polynomials. In the case of second-order polynomials of the form $x+2\cos{\pi/n_o}+x^{-1}$ these transformations are related to triangulations of Riemann surfaces of arbitrary genus with at least one hole/puncture and with an arbitrary number of orbifold points of arbitrary integer orders $n_o$. We propose the dual graph description of the corresponding Teichm\"uller spaces, construct the Poisson algebra of the Teichm\"uller space coordinates, propose the combinatorial description of the corresponding geodesic functions and find the mapping class group transformations.Comment: 20 pages, notations and many essential typos corrected, most significantly, formulae 2.3, 2.5, proof of Lemmata 2.6 and 4.5. Journal reference is added (published version contains typos

Matrix Models and Geometry of Moduli Spaces

We give the description of discretized moduli spaces (d.m.s.) \Mcdisc introduced in \cite{Ch1} in terms of discrete de Rham cohomologies for moduli spaces \Mgn. The generating function for intersection indices (cohomological classes) of d.m.s. is found. Classes of highest degree coincide with the ones for the continuum moduli space \Mc. To show it we use a matrix model technique. The Kontsevich matrix model is the generating function in the continuum case, and the matrix model with the potential N\alpha \tr {\bigl(- \fr 14 \L X\L X -\fr12\log (1-X)-\fr12X\bigr)} is the one for d.m.s. In the latest case the effects of Deligne--Mumford reductions become relevant, and we use the stratification procedure in order to express integrals over open spaces \Mdisc in terms of intersection indices, which are to be calculated on compactified spaces \Mcdisc. We find and solve constraint equations on partition function $\cal Z$ of our matrix model expressed in times for d.m.s.: t^\pm_m=\tr \fr{\d^m}{\d\l^m}\fr1{\e^\l-1}. It appears that $\cal Z$ depends only on even times and {\cal Z}[t^\pm_\cdot]=C(\aa N) \e^{\cal A}\e^{F(\{t^{-}_{2n}\}) +F(\{-t^{+}_{2n}\})}, where $F(\{t^\pm_{2n}\})$ is a logarithm of the partition function of the Kontsevich model, $\cal A$ being a quadratic differential operator in \dd{t^\pm_{2n}}.Comment: 40pp., LaTeX, no macros needed, 8 figures in tex

Quantum ordering for quantum geodesic functions of orbifold Riemann surfaces

We determine the explicit quantum ordering for a special class of quantum geodesic functions corresponding to geodesics joining exactly two orbifold points or holes on a non-compact Riemann surface. We discuss some special cases in which these quantum geodesic functions form sub--algebras of some abstract algebras defined by the reflection equation and we extend our results to the quantisation of matrix elements of the Fuchsian group associated to the Riemann surface in Poincar\'e uniformization. In particular we explore an interesting relation between the deformed $U_q(\mathfrak{sl}_2)$ and the Zhedanov algebra AW(3).Comment: 22 pages; 6 figures in LaTeX; contribution to AMS volume dedicated to the 75th birthday of S.P.Noviko

‘A new kind of conversation’: Michael Chekhov's ‘turn to the crafts’

Dartington Hall, which was the home of the Chekhov Theatre Studio between 1936 and 1938, also accommodated other performing artists including the Ballets Jooss and Hans Oppenheim's music school as well as artist-craftsmen such as the painter Mark Tobey, the potter Bernard Leach and the sculptor Willi Soukop. This essay examines the training undertaken in Chekhov's studio in dialogue with the practice of these artists (who also worked with his students) and theories of practice articulated by the wider constructive movement in the arts in the 1930s. It goes on to propose that Chekhov's technique be considered as a means of achieving theatre-artistry through craftsmanship, and as an artistic technique whose reach extends far beyond the confines of actor training