Abstract
The title—Map and Territory—that Shyam Iyengar chose for this volume is, of course, rich in possible interpretations. The word map suggests any contrivance—perhaps of ephemeral utility—meant to model the geography of any territory. I’ll take the title to be an invitation to write about the manner in which we fashion structures of thought—the ‘maps,’—in order to understand, and negotiate our way through, whatever realm it is that encompasses the objects of our thought—the ‘territories.’
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Notes
- 1.
For example, in Feynman’s book QED see the classical law: angle of incidence equals angle of reflection proved by dealing with all possible reflecting paths.
- 2.
To allude to an example of this switch of focus, I might mention Shimura varieties, these being moduli spaces that classify a certain species of mathematical object interesting enough in its own right. But the Shimura varieties themselves play a key role in a significantly different project: establishing a bridge between two disparate mathematical fields: representation theory of reductive groups and algebraic number theory.
- 3.
On the other hand, this moduli space has some interesting dynamics, tending toward that bottom edge. I’m thankful to Curt McMullen, Susan Holmes and Persi Diaconis for telling me about this, and about the relevant literature regarding the statistics of the operation of performing a barycentric subdivision of a triangle. That is, if \(\Delta =ABC\) is our triangle, let D denote the barycenter of \(\Delta \) and decompose \(\Delta \) into a union of three triangles
$$\Delta _1 =ABD,\ \ \ \Delta _2=BCD,\ \ \ \Delta _3 = CAD.$$We can think of
$$\Delta \mapsto \{\Delta _1, \Delta _2, \Delta _3 \}$$as a many-valued transformation on similarity classes of triangles, and therefore on points in \({\mathcal M}\). Statistically, this transformation produces thinner triangles, i.e., represented by lower points in \({\mathcal M}\). So, iteration of this operation can be thought of as producing gentle cascades, sending points on our moduli space—statistically speaking—towards the bottom horizontal interval. This is a consequence of I. Barany, A. Beardon, and T. Carne: Barycentric subdivision of triangles and semigroups of Möbius maps, Mathematika, 43 165–171 (1996) and see Bob Hough’s Tesselation of a triangle by repeated barycentric subdivision’ in Elect. Comm. in Probab. 14 270–277 (2009) for a refinement of the result and a discussion of the literature about it.
- 4.
Quite a few important results about the general structure of moduli spaces come from a close examination of the boundary—for example: a close analysis of degenerate—i.e. singular—curves and the manner in which a smooth curve might degenerate to the boundary of \(M_g\), the moduli space of curves of genus g, leads to a proof of the irreducibility of that moduli space. To quote P. Deligne and D. Mumford in their paper The irreducibility of the space of curves of given genus: “The basis \({\ldots }\) is to construct families of curves X, some singular, \({\ldots }\) over non-singular parameter spaces, which in some sense contain enough singular curves to link together any two components that \(M_g\) might have.”.
- 5.
I’m grateful to Sarah Koch and Xavier Buff for this elegant picture.
- 6.
Computers nowadays (as we all know) can accumulate and manipulate massive data sets. But they also play the role of microscope for pure mathematics, allowing for a type of extreme visual acuity that is, itself, a powerful kind of evidence.
- 7.
“as the small pool by the elm ices over,” which is a line of Kevin Holden’s poem Julia Set that appeared in his book Solar (Fence Books, 2016).
- 8.
This is from Benoit Mandelbrot’s book Fractals and Chaos.
- 9.
I’m grateful to Sarah Koch and Xavier Buff for the diagrams and comments. Xavier Buff mentioned that if you think of the Mandelbrot set as an island in an ocean; and each filled Julia set J(c) corresponding to a point c of the Mandelbrot set, as an ‘inhabitant’ of this island, the main open question in the subject is whether there is a hidden landmass (component of the interior of M) where inhabitants are thin (have empty interior\(\ldots \)but have positive measure). As far as we know, says Xavier, all inhabitants that live inland have some interior. Thanks, as well, to Curt McMullen for helpful comments.
- 10.
One often deals rather with elliptic curves endowed with a bit of extra structure.
- 11.
One should stipulate that the part of the boundary of the shaded region consisting of
-
the right-hand vertical line—i.e., contained in the line with x-coordinate 0.5—and
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the part contained in the arc of the unit circle with positive x-coordinate
be not included.
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- 12.
This involved work of John Conway, Simon P. Norton, Igor Frenkel, James Lepowsky, Arne Meurman and Richard Borcherds.
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Mazur, B. (2018). To the Edge of the Map. In: Wuppuluri, S., Doria, F. (eds) The Map and the Territory. The Frontiers Collection. Springer, Cham. https://doi.org/10.1007/978-3-319-72478-2_21
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