The quaternionic KP hierarchy and conformally immersed 2-tori in the 4-sphere

The quaternionic KP hierarchy is the integrable hierarchy of p.d.e obtained by replacing the complex numbers with the quaternions, mutatis mutandis, in the standard construction of the KP hierarchy equations and solutions; it is equivalent to what is often called the Davey-Stewartson II hierarchy. This article studies its relationship with the theory of quaternionic holomorphic 2-tori in HP^1 (which are equivalent to conformally immersed 2-tori in S^4). After describing how the Sato-Segal-Wilson construction of KP solutions (particularly solutions of finite type) carries over to this quaternionic setting, we compare three different notions of "spectral curve": the QKP spectral curve, which arises from an algebra of commuting differential operators; the (unnormalised) Floquet multiplier spectral curve for the related Dirac operator; and the curve parameterising Darboux transforms of a conformal 2-torus in S^4 (in the sense of Bohle, Leschke, Pedit and Pinkall). The latter two are shown to be images of the QKP spectral curve, which need not be smooth. Moreover, it is a singularisation of this QKP spectral curve, rather than the normalised Floquet multiplier curve, which determines the classification of conformally immersed 2-tori of finite spectral genus.Comment: 35 pages. Edited heavily from the original version, mainly to shorten, but also to make minor corrections. To appear in: Tokyo Mathematical Journa

Cubic Differentials in the Differential Geometry of Surfaces

We discuss the local differential geometry of convex affine spheres in \re^3 and of minimal Lagrangian surfaces in Hermitian symmetric spaces. In each case, there is a natural metric and cubic differential holomorphic with respect to the induced conformal structure: these data come from the Blaschke metric and Pick form for the affine spheres and from the induced metric and second fundamental form for the minimal Lagrangian surfaces. The local geometry, at least for main cases of interest, induces a natural frame whose structure equations arise from the affine Toda system for $\mathfrak a^{(2)}_2$. We also discuss the global theory and applications to representations of surface groups and to mirror symmetry.Comment: corrected published editio

Minimal Lagrangian surfaces in CH^2 and representations of surface groups into SU(2,1)

We use an elliptic differential equation of Tzitzeica type to construct a minimal Lagrangian surface in the complex hyperbolic plane CH^2 from the data of a compact hyperbolic Riemann surface and a small holomorphic cubic differential. The minimal Lagrangian surface is invariant under an SU(2,1) action of the fundamental group. We further parameterise a neighborhood of the R-Fuchsian representations in the representation space by pairs consisting of a point in Teichmuller space and a small cubic differential. By constructing a fundamental domain, we show these representations are complex-hyperbolic quasi-Fuchsian, thus recovering a result of Guichard and Parker-Platis. Our proof involves using the Toda lattice framework to construct an SU(2,1) frame corresponding to a minimal Lagrangian surface. Then the equation of Tzitzeica type is an integrability condition. A very similar equation to ours governs minimal surfaces in hyperbolic 3-space, and our paper can be interpreted as an analog of the theory of minimal surfaces in quasi-Fuchsian manifolds, as first studied by Uhlenbeck

Political Pilgrimage during China’s Cultural Revolution: The Case of Dazhai

During China’s Cultural Revolution (1966-76) political pilgrimage was a tool for promoting the communist ideal of collectivism. This paper explores the creation in the 1960s by Chairman Mao Zedong of one of the foremost sites of political pilgrimage, the small agricultural commune of Dazhai in north-central China. I argue that in the days of militant atheism, all the factors usually associated with the creation of a religious pilgrimage site were present and utilized to great effect by the communists to create a ‘super symbol’ of China’s desired future. These factors included a miracle, charismatic leadership, altruism, poignant sites of historical significance, rituals of remembrance, and a sacred mission that would inspire the impoverished Chinese masses to strive unceasingly towards the revolution’s goals. Through careful marketing (and propaganda), national leaders sold the Dazhai story to the Chinese people through political speeches and media reporting, and persuasive posters, banners and songs. At its peak, millions of peasants visited the commune, either voluntarily or at the invitation of the state, to learn about Dazhai’s agricultural techniques and the peoples’ spirit of self-sacrifice. In this paper, I describe my own visit to Dazhai from Australia in 1977 and speculate on the long-term legacy of the Dazhai agricultural experiment, and of this political pilgrimage. In contemporary Dazhai, tourists have the opportunity to reflect on the commune’s heroic past, but also to consider the environmental damage caused by a model of development that was rich in revolutionary zeal but poor in science

Xuanzang and Bodhidharma: Pilgrimage and peace-building in Buddhist China and India

In the study of pilgrimage and peace-building, case studies deemed to be influential in initiating positive change in the world are in the spotlight. A novel example in which people with different ethnicities, cultures and walks of life have come together with a common purpose is associated with the spread of Buddhism from India to China beginning in the early centuries of the Common Era. Unlike other instances of the expansion of a world religion beyond its homeland, the spread of Buddhism along the maritime and overland ‘silk roads’ was not linked to domination or conquest. Rather, it was linked to the uplifting of the human spirit, the search for enlightenment, and the betterment of society. Facilitated by ‘dharma-seeking’ and ‘preaching’ monks, commercial caravans, and diplomats, bridges of understanding and cooperation were forged between India and China that, despite the decline in Buddhism in later years, would last well into the 20th century. In this paper, attention is focused on two astonishing figures from this period of economic, cultural and religious exchange - Chinese scholar-monk Xuanzang (602-664 CE) and Indian Chan/Zen master Bodhidharma (~6th century CE). I examine both their lives and contributions as peacebuilders, and also the work of bricoleurs (or myth-makers) who have transformed their earthly pilgrimages into fantastic odysseys. Why the need for mythopoesis? Anthropologist Claude Levi-Strauss coined the term ‘intellectual bricolage’ to describe this process of myth-making; just as a handyman (or bricoleur) will draw upon ‘odds and ends’ to construct ‘things’, so too does the myth-maker. My goal is to analyse a sample of narratives and related pilgrimage sites associated with Xuanzang and Bodhidharma in order to show how the bricoleur (by incorporating in myth the ruminations of countless pilgrims) inspired generations to travel to places associated with our ‘hero monks’ and to pursue the ideals they embraced. Myth-enhancement has empowered and given meaning to the lives of pilgrims and, in so doing, helped build a transnational relationship that has endured for more than a thousand years

Working Paper Part 1 : The Very First Pilgrimage - An Inspired Trajectory Out of Africa

One hundred thousand years ago, give or take, the forebears of some of Australia’s First Nations—by some accounts as few as 150 people—left Africa on an immense journey. After some 2,000 generations, the passage through new and unfamiliar territories of these first modern human beings terminated in a supercontinent that included Australia, Papua, and Tasmania. By some estimates, no more than 150 people—the same number that had originally left Africa—made the final sea crossing that separates Indonesia and Australia. Research on such ancient migrations emphasises population growth, the ‘selfish gene,’ and the territorial imperative, as key drivers of mobility. This working paper speculates that the numinous was the equal to any other factor in migration, which is why this vast trek is called an inspired journey or the very first pilgrimage