60,254 research outputs found
Finite reflection groups and graph norms
Given a graph on vertex set and a function , define \begin{align*} \|f\|_{H}:=\left\vert\int
\prod_{ij\in E(H)}f(x_i,x_j)d\mu^{|V(H)|}\right\vert^{1/|E(H)|}, \end{align*}
where is the Lebesgue measure on . We say that is norming if
is a semi-norm. A similar notion is defined by
and is said to be weakly norming if
is a norm. Classical results show that weakly norming graphs
are necessarily bipartite. In the other direction, Hatami showed that even
cycles, complete bipartite graphs, and hypercubes are all weakly norming. We
demonstrate that any graph whose edges percolate in an appropriate way under
the action of a certain natural family of automorphisms is weakly norming. This
result includes all previously known examples of weakly norming graphs, but
also allows us to identify a much broader class arising from finite reflection
groups. We include several applications of our results. In particular, we
define and compare a number of generalisations of Gowers' octahedral norms and
we prove some new instances of Sidorenko's conjecture.Comment: 29 page
Tropical Fukaya Algebras
We introduce a tropical version of the Fukaya algebra of a Lagrangian
submanifold and use it to show that tropical Lagrangian tori are weakly
unobstructed. Tropical graphs arise as large-scale behavior of
pseudoholomorphic disks under a multiple cut operation on a symplectic manifold
that produces a collection of cut spaces each containing relative normal
crossing divisors, following works of Ionel and Brett Parker. Given a
Lagrangian submanifold in the complement of the relative divisors in one of the
cut spaces, the structure maps of the broken Fukaya algebra count broken disks
associated to rigid tropical graphs. We introduce a further degeneration of the
matching conditions (similar in spirit to Bourgeois' version of symplectic
field theory) which results in a tropical Fukaya algebra whose structure maps
are sums of products over vertices of tropical graphs. We show the tropical
Fukaya algebra is homotopy equivalent to the original Fukaya algebra. In the
case of toric Lagrangians contained in a toric component of the degeneration,
an invariance argument implies the existence of projective Maurer-Cartan
solutions.Comment: 167 pages, 17 figures. We fixed some issues with framings of broken
maps pointed out to us by Mohammad F. Tehrani, whom we than
NIP omega-categorical structures: the rank 1 case
We classify primitive, rank 1, omega-categorical structures having
polynomially many types over finite sets. For a fixed number of 4-types, we
show that there are only finitely many such structures and that all are built
out of finitely many linear orders interacting in a restricted number of ways.
As an example of application, we deduce the classification of primitive
structures homogeneous in a language consisting of n linear orders as well as
all reducts of such structures.Comment: Substantial changes made to the presentation, especially in sections
3 and
- β¦