Title: Dimensions of secant varieties

Abstract:  A variety  is the set of solutions of a polynomial system of equations. Considering the union of all linear subspaces spanned by k points on a variety V, one obtains the k-th secant variety of V. Determining the dimension of a secant variety is an interesting and challenging problem. We illustrate this in two instances. The first one concerns the Waring rank. Any homogeneous polynomial f of degree d can be written as a sum of d-th powers of linear forms. The minimum number of summands in such a decomposition is the Waring rank of f. It admits a geometric interpretation using secant varieties. In the second instance we use linear algebra to solve the problem in some cases. The general problem (of decomposing tensors as sums of pure tensors) is open.

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Among the many fascinating issues in Heraclitus’ discourse, this paper focuses on two: the predominance of the theme of wakefulness in contrast to “those asleep,” and the striking predominance of the hearing metaphor for knowing, largely replacing for Heraclitus the already more common sight and grasping metaphors.  By putting the two together, the paper will attempt to show what a different conception of philosophic thinking Heraclitus is proposing, and even what a different experience of the world he espouses.

"Merleau-Ponty and the Phenomenology of Natural Time"

Like Bergson and Heidegger before him, Merleau-Ponty argues for an intimate link between time and our distinctive character as finite, historical beings.  Merleau-Ponty likewise holds that our familiar conception of objective or clock time—a uniform, quantifiable time that purports to be indifferent to and independent of our lived experience—is in the end founded upon the temporality peculiar to the internal dynamics of experience itself.  However, Merleau-Ponty arguably adds a distinctive new layer to this phenomenological approach to thinking about time.  For in the Phenomenology of Perception there are the traces of a phenomenology of the cyclical time of nature, and this cyclical time is arguably different from, and irreducible to, both objective time and the uniquely historical time characteristic of human experience. 

My goal is to offer a basic reconstruction of Merleau-Ponty’s phenomenology of natural time, and to show that it shapes Merleau-Ponty’s account of human experience in substantial and interesting ways.  It turns out, for instance, that on Merleau-Ponty’s account we are always experiencing the present “now” in terms of different time-scales:  while, from certain narrower time scales, the present is historically unique and neatly individuated (as in typical empiricist conceptions of experience), there are also broader, more cyclical time scales operative within experience, and these abstract from individuated details and capture only the more generic, repeating structures at play in the phenomenal world.  For Merleau-Ponty, I suggest, the most fruitful way to understand the link between human subjectivity and the natural world is to explore the ways in which human experience itself negotiates these differing ways of engaging with time that are internal to it.  

At the beginning of the first millennium CE, Maya people in northern Yucatan, Mexico, constructed a causeway that linked several towns and villages with a large city: Ucí. This talk presents the results of six seasons of archaeological research that have explored the impact of this integration on communities in Ucí’s hinterland.