8,916 research outputs found
Momentum polytopes of projective spherical varieties and related K\"ahler geometry
We apply the combinatorial theory of spherical varieties to characterize the
momentum polytopes of polarized projective spherical varieties. This enables us
to derive a classification of these varieties, without specifying the open
orbit, as well as a classification of all Fano spherical varieties. In the
setting of multiplicity free compact and connected Hamiltonian manifolds, we
obtain a necessary and sufficient condition involving momentum polytopes for
such manifolds to be K\"ahler and classify the invariant compatible complex
structures of a given K\"ahler multiplicity free compact and connected
Hamiltonian manifold.Comment: v1: 32 pages. v2: 47 pages, fixed errors, improved exposition,
expanded Section 7. v3: 47 pages, implemented changes and corrections
requested by refere
Finitary and Infinitary Mathematics, the Possibility of Possibilities and the Definition of Probabilities
Some relations between physics and finitary and infinitary mathematics are
explored in the context of a many-minds interpretation of quantum theory. The
analogy between mathematical ``existence'' and physical ``existence'' is
considered from the point of view of philosophical idealism. Some of the ways
in which infinitary mathematics arises in modern mathematical physics are
discussed. Empirical science has led to the mathematics of quantum theory. This
in turn can be taken to suggest a picture of reality involving possible minds
and the physical laws which determine their probabilities. In this picture,
finitary and infinitary mathematics play separate roles. It is argued that
mind, language, and finitary mathematics have similar prerequisites, in that
each depends on the possibility of possibilities. The infinite, on the other
hand, can be described but never experienced, and yet it seems that sets of
possibilities and the physical laws which define their probabilities can be
described most simply in terms of infinitary mathematics.Comment: 21 pages, plain TeX, related papers from
http://www.poco.phy.cam.ac.uk/~mjd101
Infinity
This essay surveys the different types of infinity that occur in pure and applied mathematics, with emphasis on: 1. the contrast between potential infinity and actual infinity; 2. Cantor's distinction between transfinite sets and absolute infinity; 3. the constructivist view of infinite quantifiers and the meaning of constructive proof; 4. the concept of feasibility and the philosophical problems surrounding feasible arithmetic; 5. Zeno's paradoxes and modern paradoxes of physical infinity involving supertasks
Correspondences between projective planes
We characterize integral homology classes of the product of two projective
planes which are representable by a subvariety.Comment: Improved readability, 14 page
Thurston obstructions and Ahlfors regular conformal dimension
Let be an expanding branched covering map of the sphere to
itself with finite postcritical set . Associated to is a canonical
quasisymmetry class \GGG(f) of Ahlfors regular metrics on the sphere in which
the dynamics is (non-classically) conformal. We show \inf_{X \in \GGG(f)}
\hdim(X) \geq Q(f)=\inf_\Gamma \{Q \geq 2: \lambda(f_{\Gamma,Q}) \geq 1\}.
The infimum is over all multicurves . The map
is defined by where the second sum is over all preimages
of freely homotopic to in , and is its Perron-Frobenius leading eigenvalue. This
generalizes Thurston's observation that if , then there is no
-invariant classical conformal structure.Comment: Minor revisions are mad
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