27,698 research outputs found
Toric geometry of the 3-Kimura model for any tree
In this paper we present geometric features of group based models. We focus
on the 3-Kimura model. We present a precise geometric description of the
variety associated to any tree on a Zariski open set. In particular this set
contains all biologically meaningful points. Our motivation is a conjecture of
Sturmfels and Sullivant on the degree in which the ideal associated to 3-Kimura
model is generated
Whitney tower concordance of classical links
This paper computes Whitney tower filtrations of classical links. Whitney
towers consist of iterated stages of Whitney disks and allow a tree-valued
intersection theory, showing that the associated graded quotients of the
filtration are finitely generated abelian groups. Twisted Whitney towers are
studied and a new quadratic refinement of the intersection theory is
introduced, measuring Whitney disk framing obstructions. It is shown that the
filtrations are completely classified by Milnor invariants together with new
higher-order Sato-Levine and higher-order Arf invariants, which are
obstructions to framing a twisted Whitney tower in the 4-ball bounded by a link
in the 3-sphere. Applications include computation of the grope filtration, and
new geometric characterizations of Milnor's link invariants.Comment: Only change is the addition of this comment: This paper subsumes the
entire preprint "Geometric Filtrations of Classical Link Concordance"
(arXiv:1101.3477v2 [math.GT]) and the first six sections of the preprint
"Universal Quadratic Forms and Untwisting Whitney Towers" (arXiv:1101.3480v2
[math.GT]
Local description of phylogenetic group-based models
Motivated by phylogenetics, our aim is to obtain a system of equations that
define a phylogenetic variety on an open set containing the biologically
meaningful points. In this paper we consider phylogenetic varieties defined via
group-based models. For any finite abelian group , we provide an explicit
construction of phylogenetic invariants (polynomial equations) of
degree at most that define the variety on a Zariski open set . The
set contains all biologically meaningful points when is the group of
the Kimura 3-parameter model. In particular, our main result confirms a
conjecture by the third author and, on the set , a couple of conjectures by
Bernd Sturmfels and Seth Sullivant.Comment: 22 pages, 7 figure
Higher Order Intersections in Low-Dimensional Topology
We show how to measure the failure of the Whitney trick in dimension 4 by
constructing higher- order intersection invariants of Whitney towers built from
iterated Whitney disks on immersed surfaces in 4-manifolds. For Whitney towers
on immersed disks in the 4-ball, we identify some of these new invariants with
previously known link invariants like Milnor, Sato-Levine and Arf invariants.
We also define higher- order Sato-Levine and Arf invariants and show that these
invariants detect the obstructions to framing a twisted Whitney tower. Together
with Milnor invariants, these higher-order invariants are shown to classify the
existence of (twisted) Whitney towers of increasing order in the 4-ball. A
conjecture regarding the non- triviality of the higher-order Arf invariants is
formulated, and related implications for filtrations of string links and
3-dimensional homology cylinders are described. This article is an announcement
and summary of results to be published in several forthcoming papers
The Jacobian Conjecture as a Problem of Perturbative Quantum Field Theory
The Jacobian conjecture is an old unsolved problem in mathematics, which has
been unsuccessfully attacked from many different angles. We add here another
point of view pertaining to the so called formal inverse approach, that of
perturbative quantum field theory.Comment: 22 pages, 13 diagram
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