[math] Nonarchimedean Geometry

Andrew Dudzik adudzik at gmail.com
Wed Nov 14 22:45:18 CST 2012


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
book<http://arxiv.org/pdf/math/0407433v1.pdf>on 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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