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This article discusses the concept of rational equivalence
in tropical
geometry
(and replaces an older and imperfect version).
We give the basic definitions in the context of tropical varieties
without boundary points and prove some basic properties.
We then compute the ``bounded'' Chow groups of $\mathbb{R}^n$ by showing
that they are isomorphic
to the group of fan cycles. The main step in the proof is of
independent interest:
We show that every tropical cycle in $\mathbb{R}^n$ is a sum of (translated)
fan cycles. This also
proves that the intersection ring of tropical cycles is generated
in codimension 1 (by hypersurfaces).
We prove that the number of right ideals of codimension $n$ in
the algebra of noncommutative Laurent polynomials in two variables over the finite field $\mathbb F_q$ is equal to
$(q-1)^{n+1} q^{\frac{(n+1)(n-2)}{2}}\sum_\theta q^{inv(\theta)}$,
where the
sum is over all indecomposable permutations in $S_{n+1}$ and
where $inv(\theta)$
stands for the number of inversions of $\theta$.
In \cite{RatTOFL}, the second author showed that a tunnel of a tunnel
number one, fibered link in $S^3$ can be isotoped to lie as a properly
embedded arc in the fiber surface of the link. In this paper, we
observe that this is true for fibered links in any 3-manifold, we
analyze how the arc behaves under the monodromy action, and we show
that the tunnel arc is nearly clean, with the possible exception of
twisting around the boundary of the fiber.
We prove a Torelli theorem for the moduli space of semistable
parabolic Higgs bundles over a smooth complex projective algebraic
curve under the assumption that the parabolic weight system is
generic. When the genus is at least two, using this result
we also
prove a Torelli theorem for the moduli space of semistable
parabolic
bundles of rank at least two with generic parabolic weights.
The key
input in the proofs is a method of
J.C. Hurtubise.
We obtain bounds for the number of variables required to establish
Hasse principles, both for existence of solutions and for asymptotic
formulæ, for systems of additive equations containing forms
of differing degree but also multiple forms of like degree.
Apart from the very general estimates of Schmidt and Browning--Heath-Brown,
which give weak results when specialized to the diagonal situation,
this is the first result on such "hybrid" systems. We also obtain
specialised results for systems of quadratic and cubic forms,
where we are able to take advantage of some of the stronger methods
available in that setting. In particular, we achieve essentially
square root cancellation for systems consisting of one cubic
and $r$ quadratic equations.
Let $p \gt 2$ be a prime and let $X$ be a compactified PEL Shimura
variety of type (A) or (C) such that $p$ is an unramified prime
for the PEL datum and such that the ordinary locus is dense in
the reduction of $X$. Using the geometric approach of Andreatta,
Iovita, Pilloni, and Stevens we define the notion of families
of overconvergent locally analytic $p$-adic modular forms of
Iwahoric level for $X$. We show that the system of eigenvalues
of any finite slope cuspidal eigenform of Iwahoric level can
be deformed to a family of systems of eigenvalues living over
an open subset of the weight space. To prove these results, we
actually construct eigenvarieties of the expected dimension that
parameterize finite slope systems of eigenvalues appearing in
the space of families of cuspidal forms.
An equivariant map queer Lie superalgebra is the Lie superalgebra
of regular maps from an algebraic variety (or scheme) $X$ to
a queer Lie superalgebra $\mathfrak{q}$ that are equivariant with respect
to the action of a finite group $\Gamma$ acting on $X$ and $\mathfrak{q}$.
In this paper, we classify all irreducible finite-dimensional
representations of the equivariant map queer Lie superalgebras
under the assumption that $\Gamma$ is abelian and acts freely
on $X$. We show that such representations are parameterized
by a certain set of $\Gamma$-equivariant finitely supported maps
from $X$ to the set of isomorphism classes of irreducible finite-dimensional
representations of $\mathfrak{q}$. In the special case where $X$ is the
torus, we obtain a classification of the irreducible finite-dimensional
representations of the twisted loop queer superalgebra.
In this paper we give sharp norm estimates for the Bergman operator
acting from weighted
mixed-norm spaces to weighted Hardy spaces in the ball,
endowed with natural norms.
Letting $p$ vary over all primes and $E$ vary over all elliptic
curves over the finite field $\mathbb{F}_p$, we study the frequency to
which a given group $G$ arises as a group of points $E(\mathbb{F}_p)$.
It is well-known that the only permissible groups are of the
form $G_{m,k}:=\mathbb{Z}/m\mathbb{Z}\times \mathbb{Z}/mk\mathbb{Z}$.
Given such a candidate group, we let $M(G_{m,k})$ be the frequency
to which the group $G_{m,k}$ arises in this way.
Previously, the second and fourth named authors determined an
asymptotic formula for $M(G_{m,k})$ assuming a conjecture about primes
in short arithmetic progressions. In this paper, we prove several
unconditional bounds for $M(G_{m,k})$, pointwise and on average. In
particular, we show that $M(G_{m,k})$ is bounded above by a constant
multiple of the expected quantity when $m\le k^A$ and that the
conjectured asymptotic for $M(G_{m,k})$ holds for almost all groups
$G_{m,k}$ when $m\le k^{1/4-\epsilon}$.
We also apply our methods to study the frequency to which a given
integer $N$ arises as the group order $\#E(\mathbb{F}_p)$.
If $f,g:\mathbb{R}^n\longrightarrow\mathbb{R}_{\geq0}$ are non-negative measurable
functions, then the Prékopa-Leindler inequality asserts that
the integral of the Asplund sum (provided that it is measurable)
is greater or equal than the $0$-mean of the integrals of $f$
and $g$.
In this paper we prove that under the sole assumption that $f$
and $g$ have
a common projection onto a hyperplane, the Prékopa-Leindler
inequality admits a linear refinement. Moreover, the same inequality
can be obtained when assuming that both projections (not necessarily
equal as functions) have the same integral. An analogous approach
may be also carried out for the so-called Borell-Brascamp-Lieb
inequality.
We review some simple techniques based on monotone mass transport
that allow to obtain transport-type inequalities for any log-concave
probability measure. We discuss quantitative forms of these inequalities,
with application to the variance Brascamp-Lieb inequality.
We collect evidence in support of a conjecture of Griffiths,
Green
and Kerr
on the arithmetic of extension classes of
limiting
mixed Hodge structures arising from semistable degenerations
over
a number field. After briefly summarizing how a result of Iritani
implies this conjecture for a collection of hypergeometric
Calabi-Yau threefold examples studied by Doran and Morgan,
the authors investigate a sequence of (non-hypergeometric) examples
in dimensions $1\leq d\leq6$ arising from Katz's theory of the
middle
convolution.
A crucial role is played by the Mumford-Tate
group (which is $G_{2}$) of the family of 6-folds, and the theory
of boundary components of Mumford-Tate domains.
We show that the assignment of the (left) completely bounded
multiplier algebra
$M_{cb}^l(L^1(\mathbb G))$ to a locally compact quantum group
$\mathbb G$, and
the assignment of the intrinsic group, form functors between
appropriate
categories. Morphisms of locally compact quantum
groups can be described by Hopf $*$-homomorphisms between universal
$C^*$-algebras, by bicharacters, or by special sorts of coactions.
We show that the whole
theory of completely bounded multipliers can be lifted to the
universal
$C^*$-algebra level, and that then the different pictures of
both multipliers
(reduced, universal, and as centralisers)
and morphisms interact in extremely natural ways. The intrinsic
group of a
quantum group can be realised as a class of multipliers, and
so our techniques
immediately apply. We also show how to think of the intrinsic
group using
the universal $C^*$-algebra picture, and then, again, show how
the differing
views on the intrinsic group interact naturally with morphisms.
We show that
the intrinsic group is the ``maximal classical'' quantum subgroup
of a locally
compact quantum group, show that it is even closed in the strong
Vaes sense,
and that the intrinsic group functor is an adjoint to the inclusion
functor
from locally compact groups to quantum groups.
By a tiling of a topological linear space $X$ we mean a
covering of $X$ by at least two closed convex sets,
called tiles, whose nonempty interiors are
pairwise disjoint.
Study of tilings of infinite-dimensional spaces initiated in
the
1980's with pioneer papers by V. Klee.
We prove some general properties of tilings of locally convex
spaces,
and then apply these results to study existence of tilings of
normed and Banach spaces by tiles possessing
certain smoothness or rotundity properties. For a Banach space
$X$,
our main results are the following.
1. $X$ admits no tiling by Fréchet smooth bounded tiles.
2. If $X$ is locally uniformly rotund (LUR), it does not admit
any tiling by balls.
3. On the other hand, some $\ell_1(\Gamma)$ spaces, $\Gamma$
uncountable, do admit
a tiling by pairwise disjoint LUR bounded tiles.
Fontaine described the category of formal groups over the ring
of Witt vectors over a finite field
of characteristic $p$ with the aid of triples consisting of the
module of logarithms,
the Dieudonné module and the morphism from the former to the
latter. We propose
an explicit construction for the kernels in this category in
term of Fontaine's triples.
The construction is applied to the formal norm homomorphism in
the case of an unramified extension
of $\mathbb{Q}_p$ and of a totally ramified extension of degree less
or equal than $p$. A similar
consideration applied to a global extension allows us to establish
the existence of a strict
isomorphism between the formal norm torus and a formal group
law coming from $L$-series.
For an appropriate class of Fano complete intersections in toric
varieties, we prove that there is a concrete relationship between
degenerations to specific toric subvarieties and expressions
for Givental's Landau-Ginzburg models as Laurent polynomials.
As a result, we show that Fano varieties presented as complete
intersections in partial flag manifolds admit degenerations to
Gorenstein toric weak Fano varieties, and their Givental Landau-Ginzburg
models can be expressed as corresponding Laurent polynomials.
We also use this to show that all of the Laurent polynomials
obtained by Coates, Kasprzyk and Prince by the so called Przyjalkowski
method correspond to toric degenerations of the corresponding
Fano variety. We discuss applications to geometric transitions
of Calabi-Yau varieties.
We study in detail two row Springer fibres of even orthogonal
type from an algebraic as well as topological point of view.
We show that the irreducible components and their pairwise intersections
are iterated $\mathbb{P}^1$-bundles. Using results of Kumar and Procesi
we compute the cohomology ring with its action of the Weyl group.
The main tool is a type $\operatorname D$ diagram calculus labelling the
irreducible components in a convenient way which relates to a
diagrammatical algebra describing the category of perverse sheaves
on isotropic Grassmannians based on work of Braden. The diagram
calculus generalizes Khovanov's arc algebra to the type
$\operatorname
D$ setting and should be seen as setting the framework for generalizing
well-known connections of these algebras in type $\operatorname A$ to other
types.
In this paper we introduce two rearrangement optimization
problems, one being a maximization and the other a minimization
problem, related to a nonlocal boundary value problem of Kirchhoff
type. Using the theory of rearrangements as developed by
G. R. Burton we are able to show that both problems are solvable,
and derive the corresponding optimality conditions. These conditions
in turn provide information concerning the locations of the
optimal
solutions. The strict convexity of the energy functional plays
a
crucial role in both problems. The popular case in which the
rearrangement class (i.e., the admissible set) is generated
by a
characteristic function is also considered. We show that in
this
case, the maximization problem gives rise to a free boundary
problem
of obstacle type, which turns out to be unstable. On the other
hand,
the minimization problem leads to another free boundary problem
of
obstacle type, which is stable. Some numerical results are
included
to confirm the theory.
Let $\mathcal{C}$ denote the Fermat curve over $\mathbb{Q}$ of prime
exponent $\ell$. The Jacobian $\operatorname{Jac}(\mathcal{C})$
of~$\mathcal{C}$ splits over $\mathbb{Q}$ as the product of Jacobians
$\operatorname{Jac}(\mathcal{C}_k)$, $1\leq k\leq \ell-2$, where
$\mathcal{C}_k$ are curves obtained as quotients of $\mathcal{C}$ by
certain subgroups of automorphisms of $\mathcal{C}$. It is well known
that $\operatorname{Jac}(\mathcal{C}_k)$ is the power of an absolutely
simple abelian variety $B_k$ with complex multiplication. We call
degenerate those pairs $(\ell,k)$ for which $B_k$ has degenerate CM
type. For a non-degenerate pair $(\ell,k)$, we compute the Sato-Tate
group of $\operatorname{Jac}(\mathcal{C}_k)$, prove the generalized
Sato-Tate Conjecture for it, and give an explicit method to compute
the moments and measures of the involved distributions. Regardless of
$(\ell,k)$ being degenerate or not, we also obtain Frobenius
equidistribution results for primes of certain residue degrees in the
$\ell$-th cyclotomic field. Key to our results is a detailed study of
the rank of certain generalized Demjanenko matrices.
The aim of this paper is to prove that a K3 surface is the minimal
model of the quotient of an Abelian surface by a group $G$ (respectively
of a K3 surface by an Abelian group $G$) if and only if a certain
lattice is primitively embedded in its Néron-Severi group.
This allows one to describe the coarse moduli space of the K3
surfaces which are (rationally) $G$-covered by Abelian or K3
surfaces (in the latter case $G$ is an Abelian group).
If either $G$ has order 2 or $G$ is cyclic and acts on an Abelian
surface, this result was already known, so we extend it to the
other cases.
Moreover, we prove that a K3 surface $X_G$ is the minimal model
of the quotient of an Abelian surface by a group $G$ if and only
if a certain configuration of rational curves is present on $X_G$.
Again this result was known only in some special cases, in particular
if $G$ has order 2 or 3.
Let $G$ be a finite group. A faithful $G$-variety $X$ is called
strongly incompressible if every dominant $G$-equivariant rational
map of $X$ onto another faithful $G$-variety $Y$ is birational.
We settle the problem of existence of strongly incompressible
$G$-curves for any finite group $G$ and any base field $k$ of
characteristic zero.
The representation theory of semisimple algebraic groups over
the complex numbers (equivalently, semisimple complex Lie algebras
or Lie groups, or real compact Lie groups) and the question of
whether a
given complex representation is symplectic or orthogonal has
been solved since at least the 1950s. Similar results for Weyl
modules of split reductive groups over fields of characteristic
different from 2 hold by
using similar proofs. This paper considers analogues of these
results for simple, induced and tilting modules of split reductive
groups over fields of prime characteristic as well as a complete
answer for Weyl modules over fields of characteristic 2.
Let $K/\mathbb{Q}$ be Galois and let $\eta\in K^\times$ be such that
$\operatorname{Reg}_\infty (\eta) \ne 0$.
We define the local $\theta$-regulators $\Delta_p^\theta(\eta)
\in \mathbb{F}_p$
for the $\mathbb{Q}_p\,$-irreducible characters $\theta$ of
$G=\operatorname{Gal}(K/\mathbb{Q})$. A linear representation ${\mathcal L}^\theta\simeq \delta \,
V_\theta$ is associated with
$\Delta_p^\theta (\eta)$ whose nullity is equivalent to $\delta
\geq 1$.
Each $\Delta_p^\theta (\eta)$ yields $\operatorname{Reg}_p^\theta (\eta)$
modulo $p$ in the factorization
$\prod_{\theta}(\operatorname{Reg}_p^\theta (\eta))^{\varphi(1)}$ of
$\operatorname{Reg}_p^G (\eta) := \frac{ \operatorname{Reg}_p(\eta)}{p^{[K : \mathbb{Q}\,]}
}$ (normalized $p$-adic regulator).
From $\operatorname{Prob}\big (\Delta_p^\theta(\eta) = 0 \ \& \ {\mathcal
L}^\theta \simeq \delta \, V_\theta\big )
\leq p^{- f \delta^2}$ ($f \geq 1$ is a residue degree) and the
Borel-Cantelli heuristic,
we conjecture that, for $p$ large enough, $\operatorname{Reg}_p^G (\eta)$
is a $p$-adic unit or that
$p^{\varphi(1)} \parallel \operatorname{Reg}_p^G (\eta)$ (a single $\theta$
with $f=\delta=1$); this obstruction may be lifted assuming the
existence of a binomial probability law
confirmed through numerical studies
(groups $C_3$, $C_5$, $D_6$).
This conjecture would imply that, for all $p$ large enough,
Fermat quotients, normalized $p$-adic
regulators are $p$-adic units and that
number fields are $p$-rational.
We recall some deep cohomological results that
may strengthen such conjectures.
We use the method of Ash and Stevens to prove the existence of
small slope $p$-adic families of cohomological modular forms
for an indefinite quaternion algebra $B$. We prove that the
Jacquet-Langlands correspondence relating modular forms on
$\textbf{GL}_2/\mathbb{Q}$
and cohomomological modular forms for $B$ is compatible with
the formation of $p$-adic families. This result is an analogue
of a theorem of Chenevier concerning definite quaternion algebras.
Let $T_{\Omega}$ be the singular integral operator with kernel
$\frac{\Omega(x)}{|x|^n}$, where $\Omega$ is homogeneous of degree
zero, has mean value zero and belongs to $L^q(S^{n-1})$ for
some
$q\in (1,\,\infty]$. In this paper, the authors establish the
compactness on weighted $L^p$ spaces, and the Morrey spaces,
for the commutator generated by $\operatorname{CMO}(\mathbb{R}^n)$ function
and $T_{\Omega}$. The associated maximal operator and the discrete
maximal operator are also considered.
Let $\mathfrak{g}$ be a compact, simple Lie algebra of dimension
$d$. It is
a classical result that the convolution of any $d$ non-trivial,
$G$-invariant,
orbital measures is absolutely continuous with respect to
Lebesgue measure on $\mathfrak{g}$ and the sum of any $d$ non-trivial
orbits
has non-empty interior. The number $d$ was later reduced to the
rank of the
Lie algebra (or rank $+1$ in the case of type $A_{n}$). More
recently, the
minimal integer $k=k(X)$ such that the $k$-fold convolution of
the orbital
measure supported on the orbit generated by $X$ is an absolutely
continuous
measure was calculated for each $X\in \mathfrak{g}$.
In this paper $\mathfrak{g}$ is any of the classical, compact,
simple Lie
algebras. We characterize the tuples $(X_{1},\dots,X_{L})$, with
$X_{i}\in
\mathfrak{g},$ which have the property that the convolution of
the $L$-orbital
measures supported on the orbits generated by the $X_{i}$ is
absolutely continuous and, equivalently, the sum of their orbits
has
non-empty interior. The characterization depends on the Lie type
of
$\mathfrak{g}$ and the structure of the annihilating roots of
the $X_{i}$.
Such a characterization was previously known only for type $A_{n}$.
We continue the investigation of the isomorphism problem for
multiplier algebras of reproducing kernel
Hilbert spaces with the complete Nevanlinna-Pick property.
In contrast to previous work in this area,
we do not study these spaces by identifying them with restrictions
of a universal space, namely the Drury-Arveson space.
Instead, we work directly with the Hilbert spaces and their
reproducing kernels. In particular,
we show that two multiplier algebras of Nevanlinna-Pick spaces
on the same set are equal if and only if the Hilbert
spaces are equal. Most of the article is devoted to the study
of a special class of
complete Nevanlinna-Pick spaces on homogeneous varieties. We
provide a complete
answer to the question of when two multiplier algebras of spaces
of this type
are algebraically or isometrically isomorphic. This generalizes
results of Davidson, Ramsey, Shalit,
and the author.
The morphism $f:\mathbb{P}^N\to\mathbb{P}^N$ is called post-critically finite
(PCF) if the forward image of the critical locus, under iteration
of $f$, has algebraic support. In the case $N=1$, a result of
Thurston implies that there are no algebraic families of PCF
morphisms, other than a well-understood exceptional class known
as the flexible Lattès maps. A related arithmetic result
states that the set of PCF morphisms corresponds to a set of
bounded height in the moduli space of univariate rational functions.
We prove corresponding results for a certain subclass of the
regular polynomial endomorphisms of $\mathbb{P}^N$, for any $N$.
We study unitary quotients of the free product unitary pivotal
category $A_2*T_2$.
We show that such quotients are parametrized by an integer $n\geq
1$ and an $2n$-th root of unity $\omega$.
We show that for $n=1,2,3$, there is exactly one quotient and
$\omega=1$.
For $4\leq n\leq 10$, we show that there are no such quotients.
Our methods also apply to quotients of $T_2*T_2$, where we have
a similar result.
The essence of our method is a consistency check on jellyfish
relations.
While we only treat the specific cases of $A_2 * T_2$ and $T_2
* T_2$, we anticipate that our technique can be extended to a
general method for proving nonexistence of planar algebras with
a specified principal graph.
During the preparation of this manuscript, we learnt of Liu's
independent result on composites of $A_3$ and $A_4$ subfactor
planar algebras
(arxiv:1308.5691).
In 1994, Bisch-Haagerup showed that the principal graph of a
composite of $A_3$ and $A_4$ must fit into a certain family,
and Liu has classified all such subfactor planar algebras.
We explain the connection between the quotient categories and
the corresponding composite subfactor planar algebras.
As a corollary of Liu's result, there are no such quotient categories
for $n\geq 4$.
In this paper, we study the global regularity for
regular
Monge-Ampère type equations associated with semilinear Neumann
boundary conditions.
By establishing a priori estimates for second order derivatives,
the
classical solvability of the Neumann boundary value problem is
proved under natural conditions.
The techniques build upon the delicate and intricate treatment
of the standard Monge-Ampère case
by Lions, Trudinger and Urbas in 1986 and the recent barrier
constructions and second derivative bounds
by Jiang, Trudinger and Yang for the Dirichlet problem. We also
consider more general oblique boundary
value problems in the strictly regular case.
We study a natural discrete Bochner-type inequality on graphs,
and explore its merit as a notion of ``curvature'' in discrete
spaces.
An appealing feature of this discrete version of the so-called
$\Gamma_2$-calculus (of Bakry-Émery) seems to be that it is
fairly
straightforward to compute this notion of curvature parameter
for
several specific graphs of interest -- particularly, abelian
groups, slices of the hypercube, and the symmetric group under
various sets of generators.
We further develop this notion by deriving Buser-type inequalities
(à la Ledoux), relating functional and isoperimetric constants
associated with a graph.
Our derivations provide a tight bound on the Cheeger constant
(i.e., the edge-isoperimetric constant) in terms of
the spectral gap, for graphs with nonnegative curvature, particularly,
the class of abelian Cayley graphs -- a result of independent
interest.
This paper concerns free function theory. Free maps are free
analogs of analytic functions in several complex variables,
and are defined in terms of freely noncommuting variables.
A function of $g$ noncommuting variables is a function on $g$-tuples
of square matrices of all sizes that respects direct sums and
simultaneous conjugation.
Examples of such maps include noncommutative polynomials, noncommutative
rational functions and convergent noncommutative power series.
In sharp contrast to the existing literature in free analysis, this article
investigates free maps with involution --
free analogs of real analytic functions.
To
get a grip on these,
techniques and tools from invariant theory are developed and
applied to free analysis. Here is a sample of the results obtained.
A characterization of polynomial free maps via properties of
their finite-dimensional slices is presented and then used to
establish power series expansions for analytic free maps about
scalar and non-scalar points; the latter are series of generalized
polynomials for which an invariant-theoretic characterization
is given.
Furthermore,
an inverse and implicit function theorem for free maps with
involution is obtained.
Finally, with a selection of carefully chosen examples
it is shown that
free maps with involution
do not exhibit strong rigidity properties
enjoyed by their involution-free
counterparts.
Let $E/\mathbb{Q}$ be an elliptic curve of conductor
$N$, and
let $K$ be an imaginary quadratic field such that the root
number of
$E/K$ is $-1$. Let $\mathscr{O}$ be an order in $K$ and assume that
there
exists an odd prime $p$, such that $p^2 \mid\mid N$, and $p$
is inert in
$\mathscr{O}$. Although there are no Heegner points on $X_0(N)$
attached to $\mathscr{O}$, in this article we construct such points on
Cartan non-split curves. In order to do that we
give a method to compute Fourier expansions for forms on Cartan
non-split curves, and prove that the constructed points form a
Heegner system as in the classical case.
Let $\mathbf{k}$ be an algebraically closed field complete with respect
to a non-Archimedean absolute value of arbitrary characteristic.
Let $D_1,\dots, D_n$ be effective nef divisors intersecting
transversally in an $n$-dimensional nonsingular projective variety
$X$.
We study the degeneracy of non-Archimedean analytic maps from
$\mathbf{k}$ into $X\setminus \cup_{i=1}^nD_i$ under various geometric
conditions. When $X$ is a rational ruled surface and $D_1$ and
$D_2$ are ample, we obtain a necessary and sufficient condition
such that
there is no non-Archimedean analytic map from $\mathbf{k}$ into $X\setminus
D_1 \cup D_2$.
Using the dictionary between non-Archimedean Nevanlinna theory
and Diophantine approximation that originated in
earlier work with T. T. H. An, %
we also study arithmetic analogues of these problems, establishing
results on integral points on these varieties over $\mathbb{Z}$
or the ring of integers of an imaginary quadratic field.
We consider Schubert problems with respect to flags osculating
the rational normal curve. These problems are of special interest
when the osculation points are all real -- in this case, for
zero-dimensional Schubert problems, the solutions are "as real
as possible". Recent work by Speyer has extended the theory
to the moduli space
$
\overline{\mathcal{M}_{0,r}}
$,
allowing the points to collide.
These give rise to smooth covers of
$
\overline{\mathcal{M}_{0,r}}
(\mathbb{R})
$, with structure
and monodromy described by Young tableaux and jeu de taquin.
In this paper, we give analogous results on one-dimensional Schubert
problems over
$
\overline{\mathcal{M}_{0,r}}
$.
Their (real) geometry turns out to
be described by orbits of Schützenberger promotion and a
related operation involving tableau evacuation. Over
$\mathcal{M}_{0,r}$,
our results show that the real points of the solution curves
are smooth.
We also find a new identity involving "first-order" K-theoretic
Littlewood-Richardson coefficients, for which there does not
appear to be a known combinatorial proof.
The closest infinite dimensional relatives of compact Lie algebras are Hilbert-Lie algebras, i.e. real Hilbert spaces with a Lie
algebra
structure for which the scalar product is invariant.
Locally affine Lie algebras (LALAs)
correspond to double extensions of (twisted) loop algebras
over simple Hilbert-Lie algebras $\mathfrak{k}$, also called
affinisations of $\mathfrak{k}$.
They possess a root space decomposition
whose corresponding root system is a locally affine root system
of one of the $7$ families $A_J^{(1)}$, $B_J^{(1)}$, $C_J^{(1)}$,
$D_J^{(1)}$, $B_J^{(2)}$, $C_J^{(2)}$ and $BC_J^{(2)}$ for some
infinite set $J$. To each of these types corresponds a ``minimal"
affinisation of some simple Hilbert-Lie algebra $\mathfrak{k}$,
which we call standard.
In this paper, we give for each affinisation $\mathfrak{g}$ of
a simple Hilbert-Lie algebra $\mathfrak{k}$ an explicit isomorphism
from $\mathfrak{g}$ to one of the standard affinisations of $\mathfrak{k}$. The existence of such an isomorphism could also be derived from
the classification
of locally affine root systems, but
for representation theoretic purposes it is crucial to obtain
it explicitly
as a deformation between two twists which is compatible
with the root decompositions.
We illustrate this by applying our isomorphism theorem to the
study of positive energy highest weight representations of $\mathfrak{g}$.
In subsequent work, the present paper will be used to obtain
a complete classification
of the positive energy highest weight representations of affinisations
of $\mathfrak{k}$.
Continua $X$ and $Y$ are monotone equivalent
if there exist monotone onto maps $f:X\longrightarrow Y$ and
$g:Y\longrightarrow X$. A continuum $X$ is isolated with respect
to monotone maps if every continuum that is monotone equivalent
to $X$ must also be homeomorphic to
$X$. In this paper we show that a dendrite $X$ is isolated with
respect to
monotone maps if and only if the set of ramification points of
$X$ is
finite. In this way we fully characterize the classes of dendrites
that are
monotone isolated.
We give a normal form of the cuspidal edge
which uses only diffeomorphisms on the source
and isometries on the target.
Using this normal form, we study differential
geometric invariants of
cuspidal edges which determine them up to order three.
We also
clarify relations between these invariants.
Let $\mathcal A$, $\mathcal B$ be large subsets of $\{1,\ldots,N\}$.
We study the number of pairs $(a,b)\in\mathcal A\times\mathcal B$ such that
the sum of binary digits of $a+b$ is fixed.
We complete Mori's program with symmetric divisors for the moduli
space of stable seven-pointed rational curves. We describe all
birational models in terms of explicit blow-ups and blow-downs.
We also give a moduli theoretic description of the first flip,
which has not appeared in the literature.
For a fixed $K\gg 1$ and
$n\in\mathbb{N}$, $n\gg 1$, we study metric
spaces which admit embeddings with distortion $\le K$ into each
$n$-dimensional Banach space. Classical examples include spaces
embeddable
into $\log n$-dimensional Euclidean spaces, and equilateral spaces.
We prove that good embeddability properties are preserved under
the operation of metric composition of metric spaces. In
particular, we prove that $n$-point ultrametrics can be
embedded with uniformly bounded distortions into arbitrary Banach
spaces of dimension $\log n$.
The main result of the paper is a new example of a family of
finite metric spaces which are not metric compositions of
classical examples and which do embed with uniformly bounded
distortion into any Banach space of dimension $n$. This partially
answers a question of G. Schechtman.
We formulate and prove a ``winding number'' index
theorem for certain ``Toeplitz'' operators in the same spirit
as Gohberg-Krein, Lesch and others. The ``number'' is replaced
by a self-adjoint operator in a subalgebra $Z\subseteq Z(A)$
of a unital $C^*$-algebra, $A$. We assume a faithful $Z$-valued
trace $\tau$ on $A$ left invariant under an action $\alpha:{\mathbf
R}\to Aut(A)$ leaving $Z$ pointwise fixed.If $\delta$ is the
infinitesimal generator of $\alpha$ and $u$ is invertible in
$\operatorname{dom}(\delta)$ then the
``winding operator'' of $u$ is $\frac{1}{2\pi i}\tau(\delta(u)u^{-1})\in
Z_{sa}.$ By a careful choice of representations we extend $(A,Z,\tau,\alpha)$
to a von Neumann setting
$(\mathfrak{A},\mathfrak{Z},\bar\tau,\bar\alpha)$ where $\mathfrak{A}=A^{\prime\prime}$
and $\mathfrak{Z}=Z^{\prime\prime}.$
Then $A\subset\mathfrak{A}\subset \mathfrak{A}\rtimes{\bf R}$, the von
Neumann crossed product, and there is a faithful, dual $\mathfrak{Z}$-trace
on $\mathfrak{A}\rtimes{\bf R}$. If $P$ is the projection in $\mathfrak{A}\rtimes{\bf
R}$
corresponding to the non-negative spectrum of the generator of
$\mathbf R$ inside $\mathfrak{A}\rtimes{\mathbf R}$ and
$\tilde\pi:A\to\mathfrak{A}\rtimes{\mathbf R}$
is the embedding then we define for $u\in A^{-1}$, $T_u=P\tilde\pi(u)
P$
and show it is Fredholm in an appropriate sense and the $\mathfrak{Z}$-valued
index of $T_u$ is the negative of the winding operator.
In outline the proof follows the proof of the scalar case done
previously by the authors. The main difficulty is making sense
of the constructions with the scalars replaced by $\mathfrak{Z}$ in
the von Neumann setting. The construction of the dual $\mathfrak{Z}$-trace
on $\mathfrak{A}\rtimes{\mathbf R}$ required the nontrivial development
of a $\mathfrak{Z}$-Hilbert Algebra theory. We show that certain of
these Fredholm operators fiber as a ``section'' of Fredholm operators
with scalar-valued index and the centre-valued index fibers as
a section of the scalar-valued indices.
The notion of positive-definite functions over locally compact
quantum
groups was recently introduced and studied by Daws and Salmi.
Based
on this work, we generalize various well-known results about
positive-definite
functions over groups to the quantum framework. Among these are
theorems
on "square roots" of positive-definite functions, comparison
of
various topologies, positive-definite measures and characterizations
of amenability, and the separation property with respect to compact
quantum subgroups.
In this note we introduce and study a new class of maps called
oriented colored broken submersions. This is the simplest class
of maps that satisfies a version of the b-principle and in dimension
$2$ approximates the class of oriented submersions well in the
sense that
every oriented colored broken submersion of dimension $2$ to
a closed simply connected manifold is bordant to a submersion.
We show that the Madsen-Weiss theorem (the standard Mumford Conjecture)
fits a general setting of the b-principle. Namely, a version
of the b-principle for
oriented colored broken submersions together with the Harer
stability theorem and Miller-Morita theorem implies the Madsen-Weiss
theorem.
The notion of families of quantum invertible maps (C$^*$-algebra
homomorphisms satisfying Podleś' condition) is employed to strengthen
and reinterpret several results concerning universal quantum
groups acting on finite quantum spaces. In particular Wang's
quantum automorphism groups are shown to be universal with respect
to quantum families of invertible maps. Further the construction
of the Hopf image of Banica and Bichon is phrased in the purely
analytic language and employed to define the quantum subgroup
generated by a family of quantum subgroups or more generally
a family of quantum invertible maps.
Assuming Lang's conjectured lower bound on the heights of non-torsion
points on an elliptic curve, we show that there exists an absolute
constant $C$ such that for any elliptic curve $E/\mathbb{Q}$ and non-torsion
point $P \in E(\mathbb{Q})$, there is at most one integral multiple
$[n]P$ such that $n \gt C$. The proof is a modification of a proof
of Ingram giving an unconditional but not uniform bound. The
new ingredient is a collection of explicit formulae for the
sequence $v(\Psi_n)$ of valuations of the division polynomials.
For $P$ of non-singular reduction, such sequences are already
well described in most cases, but for $P$ of singular reduction,
we are led to define a new class of sequences called elliptic
troublemaker sequences, which measure the failure of the Néron
local height to be quadratic. As a corollary in the spirit of
a conjecture of Lang and Hall, we obtain a uniform upper bound
on $\widehat{h}(P)/h(E)$ for integer points having two large
integral multiples.
We develop a derivative version of the relative trace formula
on $\operatorname{PGL}(2)$ studied in our previous work,
and derive an asymptotic formula of an average of central values
(derivatives)
of automorphic $L$-functions for Hilbert cusp forms.
As an application, we prove the existence of Hilbert cusp forms
with non-vanishing central values (derivatives)
such that the absolute degrees of their Hecke fields are arbitrarily
large.
In this paper, we introduce the anisotropic
Sobolev capacity with fractional order and develop some basic
properties for this new object. Applications to the theory of
anisotropic fractional Sobolev spaces are provided. In particular,
we give geometric characterizations for a nonnegative Radon
measure $\mu$ that naturally induces an embedding of the anisotropic
fractional Sobolev class $\dot{\Lambda}_{\alpha,K}^{1,1}$ into
the $\mu$-based-Lebesgue-space $L^{n/\beta}_\mu$ with $0\lt \beta\le
n$. Also, we investigate the anisotropic fractional $\alpha$-perimeter.
Such a geometric quantity can be used to approximate the anisotropic
Sobolev capacity with fractional order. Estimation on the constant
in the related Minkowski inequality, which is asymptotically
optimal as $\alpha\rightarrow 0^+$, will be provided.
In this work type II Hermite-Padé approximants for a vector
of Cauchy transforms of smooth Jacobi-type densities are considered.
It is assumed that densities are supported on mutually disjoint
intervals (an Angelesco system with complex weights). The formulae
of strong asymptotics are derived for any ray sequence of multi-indices.
We study the Chern-Ricci flow, an evolution equation of Hermitian
metrics, on a family of Oeljeklaus-Toma (OT-) manifolds which
are non-Kähler compact complex manifolds with negative Kodaira
dimension. We prove that, after an initial conformal change,
the flow converges, in the
Gromov-Hausdorff sense, to a torus with a flat Riemannian metric
determined by the OT-manifolds themselves.
We establish an infinitesimal version of the
Jacquet-Rallis trace formula for unitary groups.
Our formula is obtained by integrating a
truncated kernel à la Arthur.
It has a geometric side which is a
sum of distributions $J_{\mathfrak{o}}$ indexed by classes of
elements
of the Lie algebra of $U(n+1)$ stable by $U(n)$-conjugation
as well as the "spectral side"
consisting of the Fourier transforms
of the aforementioned distributions.
We prove that the distributions $J_{\mathfrak{o}}$
are invariant and depend only on the choice of
the Haar measure on $U(n)(\mathbb{A})$.
For regular semi-simple classes $\mathfrak{o}$, $J_{\mathfrak{o}}$
is
a relative orbital integral of Jacquet-Rallis.
For classes $\mathfrak{o}$ called relatively regular semi-simple,
we express $J_{\mathfrak{o}}$
in terms of relative orbital integrals regularised by means of
zêta functions.