Beautiful. The idea of complex analysis as the special case of a much weirder general analysis is delightful.
So if I understand (please correct me if I'm wrong) Berkovich found that there was this beautiful space that you could embed the p-adics in that make analysis possible, and you're trying to show that this space is a completion of... what? The p-adics?
Also, you said that all fields with metrics are either subfields of the complex plane or have an ultrametric. Well... are there a lot of fields out there with ultrametrics, besides the p-adics?
On Wed, Nov 14, 2012 at 11:45 PM, Andrew Dudzik adudzik@gmail.com wrote:
I thought I'd give a quick overview of my research area because a) it's math, and b) somebody might find it interesting. It also might give others some idea of what I'm interested in and what I know about, which seems like useful information for a math discussion list. I'll try to organize it well enough that you can ignore the boring or incomprehensible parts. I'm happy to elaborate on anything that I rushed over.
*What It's Called* My research area goes by a few different names, of which Nonarchimedean Geometry is my favorite. Rigid Analysis is another common one, or sometimes Nonarchimedean Analysis. One might also hear of Berkovich Theory, Tropical Geometry, or Nonarchimedean Dynamics, which are closely related areas.
*Ultrametric Spaces* A metric space formalizes the notion of distance. An ultrametric space adds the axiom that all triangles are isosceles. To see why this can be a meaningful axiom, consider the metric space X of all living organisms, where d(A,B) is defined to be the number of years since the most recent common ancestor of A and B was born (or 0, if A=B). Then X is an ultrametric space, because, for any three organisms, the closest two must be equally far from the third.
We can also regard the universe as an ultrametric space by taking "ballpark" distances. For example, if I regard all points in London to be equally far from myself (which, in practical terms, is almost true) but closer than points on Mars, then I see, again, that for any three points, the closest two are equally far from the third.
These both illustrate a deep fact about ultrametric spaces: that they are very tree-like—even though, in both examples, the vertices of the tree are not necessarily points in the space. One also detects a relationship between ultrametric spaces and the notion of an order of magnitude.
*p-Adic Analyis* To cut wildly away from the vaguely practical: The p-adics have been important in number theory for quite some time. But those used to the idyllic world of complex analysis have long been frustrated by the poor analytic properties of the p-adics—as a metric space, they are totally disconnected, which means that the only connected subspaces are single points. The situation gets worse if we pass to the algebraic closure; then they fail to even be locally compact (even after completion). Both of these things are a disaster for traditional approaches to analysis, but that hasn't stopped the gods of math from posing problems that require analytical tools.
A number of technical tools were developed to get around these problems, but the key intuition came in the 80s from Vladimir Berkovich, who saw that fields like the p-adics could be embedded in larger spaces which *are *connected and locally compact. These spaces serve the same role in p-adic analysis as the complex plane has served in traditional analysis—which is to say that they're a really, really big deal.
The p-adics, as a metric space, are in fact ultrametric, the deeper fact responsible for the problems mentioned above. Passing to the Berkovich space involves adding points which correspond to the vertices of the trees that we imagined in the previous section; for example, for the space of life (which isn't a field, but the geometric issues are similar), we would add a point corresponding to Chordata.
Clearly, these spaces bear little resemblance to the complex plane. Yet an astonishing number of theorems from complex analysis hold true with almost no modification in this new Berkovich setting. When there are modifications, they usually take the form of adding juicy combinatorial and graph-theoretic elements. My advisor is written a whole bookhttp://arxiv.org/pdf/math/0407433v1.pdfon this.
I'll finish here with a brief theorem. If F is a field with a metric, then either F is a subfield of the complex numbers, or F is ultrametric. So if we think of analysis on ultrametric fields, then the complex numbers are the special case, a pathology within a wider, deeper, and weirder theory.
*Metric Graphs* There's a lot I'll omit here for brevity (and because I'm procrastinating far more important things), but there is a theory of chip-firing on metric graphs that has played the role of the theory of divisors on algebraic curves. This is a joyful mingling of analysis and algebra with combinatorics, and most of the basic definitions have not yet been fully worked out. Other things, like the graph-theoretic Laplacian, begin to lose their distinction from the classical Laplacian. Metaphors from the theory of electrical networks abound.
*Tropical Geometry* Tropical geometry emerged out of study of the min-plus algebra, which is, roughly, what happens if you take the logarithm of the usual real number system. Tropical gadgets are these piecewise-linear doohickeys (e.g.http://www.crcg.de/wiki/images/c/c9/Ninecubics.jpg) which have been entertaining mathematicians for years. But it has become more and more clear over the past two decades that they are all cross-sections of something much larger: the Berkovich space.
Connected to this fact is the important observation that Berkovich spaces are polyhedral in nature. Though they are generally infinitely-branching monstrosities, they have a well-behaved polyhedral skeleton.
*Dynamical Systems* This is something I know next to nothing about, but Berkovich spaces have recently become a hot topic in the dynamical systems crowds. Somehow, these are the right spaces in which to explore certain iterative phenomena.
*Algebraic Geometry* The problem of understanding Berkovich spaces in two dimensions is deeply connected to an old concern in algebraic geometry of the resolution of singularities on surfaces. There is also a nonarchimedean version of the old correspondence between complex algebraic curves and Riemann surfaces.
*Connections to Logic* Some high-powered model theorists have become interested in nonarchimedean geometry lately. Model theorists are to mathematicians what mathematicians are to everyone else. The strange language of quantifier elimination and stably dominated types brings the almost mystical ability to shuffle proofs around as if math were a three-shell game. I have been studying this for the past 10 weeks and I have no idea what it's about.
*Number Theory* One shouldn't believe the hype that this stuff will prove the Riemann Hypothesis, but there are many promising developments. For one, many of the classical constructions in number theory can be seem in simple geometric terms on an appropriate Berkovich-style space. Rigid analysis is quickly becoming ubiquitous in algebraic number theory.
*What I'm Doing* My own project has been leaning heavily on the theory of distributive lattices, which has beautiful connections to topology. If we imbue the Berkovich space with extra information about tangent directions (which are more subtle in non-archimedi-land than in classical analysis) then we get a strange space—sometimes called the 'adic' space—which is related to a natural lattice and hides a few untapped secrets. My contribution, using this point of view, has been showing that the Berkovich space is, in some sense, a "completion", akin to producing the reals from the rationals (I use filters on lattices which generalize Cauchy sequences).
The nature of this work is mainly to clarify the current state of discourse in nonarchimedean geometry. Many theorems have been proved in a technical fashion, but need to be redone in a way that exposes the underlying intuitions.
OK, it's time for me to stop procrastinating. Hope that this spiced things up a bit, and wasn't too bewildering.
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