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We prove the Gross-Deligne conjecture on CM periods for motives
associated with $H^2$ of certain surfaces fibered over the projective
line. Then we prove for the same motives a formula which expresses
the $K_1$-regulators in terms of hypergeometric functions ${}_3F_2$,
and obtain a new example of non-trivial regulators.
We investigate the isomorphic structure of the Cesàro spaces
and their duals, the Tandori spaces.
The main result states that the Cesàro function space $Ces_{\infty}$
and its sequence counterpart
$ces_{\infty}$ are isomorphic, which answers the question posted
previously.
This is rather surprising
since $Ces_{\infty}$ (like the known Talagrand's example)
has no natural lattice predual.
We prove that $ces_{\infty}$ is not isomorphic to ${\ell}_{\infty}$
nor is $Ces_{\infty}$ isomorphic to the
Tandori space $\widetilde{L_1}$ with the norm $\|f\|_{\widetilde{L_1}}=
\|\widetilde{f}\|_{L_1},$ where
$\widetilde{f}(t):= \operatorname{esssup}_{s \geq t} |f(s)|.$ Our investigation
involves also an examination of the
Schur and Dunford-Pettis properties of Cesàro and Tandori
spaces.
In particular, using results of Bourgain we show that a wide
class of Cesàro-Marcinkiewicz and
Cesàro-Lorentz spaces have the latter property.
In this paper, we investigate Dirichlet spaces $\mathcal{D}_\mu$ with
superharmonic weights induced by positive Borel measures $\mu$
on
the open unit disk. We establish the Alexander-Taylor-Ullman
inequality for $\mathcal{D}_\mu$ spaces and we characterize the cases where
equality occurs.
We define a class of weighted Hardy spaces $H_{\mu}^{2}$ via
the balayage of the measure $\mu$.
We show that $\mathcal{D}_\mu$
is equal to $H_{\mu}^{2}$ if and only if $\mu$ is a
Carleson measure for $\mathcal{D}_\mu$. As an application, we obtain the
reproducing kernel of $\mathcal{D}_\mu$ when $\mu$ is an infinite
sum of point mass measures. We consider the boundary
behavior and inner-outer factorization of functions in $\mathcal{D}_\mu$.
We also characterize the boundedness and
compactness of composition operators on $\mathcal{D}_\mu$.
We study the geometry of the component of the origin
in the uniform spanning forest of $\mathbb{Z}^d$
and give bounds on the size of balls in the intrinsic metric.
We study w*-semicrossed products over actions of the free semigroup
and the free abelian semigroup on (possibly non-selfadjoint)
w*-closed algebras.
We show that they are reflexive when the dynamics are implemented
by uniformly bounded families of invertible row operators.
Combining with results of Helmer we derive that w*-semicrossed
products of factors (on a separable Hilbert space) are reflexive.
Furthermore we show that w*-semicrossed products of automorphic
actions on maximal abelian selfadjoint algebras are reflexive.
In all cases we prove that the w*-semicrossed products have the
bicommutant property if and only if the ambient algebra of the
dynamics does also.
If the Hasse invariant of a $p$-divisible group is small enough,
then one can construct a canonical subgroup inside its $p$-torsion.
We prove that, assuming the existence of a subgroup of adequate
height in the $p$-torsion with high degree, the expected properties
of the canonical subgroup can be easily proved, especially the
relation between its degree and the Hasse invariant. When one
considers a $p$-divisible group with an action of the ring of
integers of a (possibly ramified) finite extension of $\mathbb{Q}_p$,
then much more can be said. We define partial Hasse invariants
(they are natural in the unramified case, and generalize a construction
of Reduzzi and Xiao in the general case), as well as partial
degrees. After studying these functions, we compute the partial
degrees of the canonical subgroup.
Let $\Theta ^{[j]}$ be an analogue of the Ramanujan theta operator
for Siegel modular forms.
For a given prime $p$, we give the weights of elements of mod
$p$ kernel of $\Theta ^{[j]}$,
where the mod $p$ kernel of $\Theta ^{[j]}$ is the set of all
Siegel modular forms $F$ such that $\Theta ^{[j]}(F)$ is congruent
to zero modulo $p$.
In order to construct examples of the mod $p$ kernel of $\Theta
^{[j]}$ from any Siegel modular form,
we introduce new operators $A^{(j)}(M)$ and show the modularity
of $F|A^{(j)}(M)$ when $F$ is a Siegel modular form.
Finally, we give some examples of the mod $p$ kernel of $\Theta
^{[j]}$ and the filtrations of some of them.
The object of this paper is to prove a version of the Beurling-Helson-Lowdenslager
invariant subspace theorem for operators on certain Banach spaces
of functions on a multiply connected domain in $\mathbb{C}$. The norms
for these spaces are either the usual Lebesgue and Hardy space
norms or certain continuous gauge norms.
In the Hardy space case the expected corollaries include the
characterization of the cyclic vectors as the outer functions
in this context, a demonstration that the set of analytic multiplication
operators is maximal abelian and reflexive, and a determination
of the closed operators that commute with all analytic multiplication
operators.
The paper constitutes a comprehensive study of ten classes of
self-maps on metric spaces $\langle X,d\rangle$ with the local
and pointwise (a.k.a. local radial) contraction properties.
Each of those classes appeared previously in the literature in
the context of fixed point theorems.
We begin with presenting an overview
of these fixed point results, including concise self contained
sketches of their proofs.
Then, we proceed with a discussion of the relations among the
ten classes of self-maps with domains $\langle X,d\rangle$
having various topological properties which often appear in the
theory of fixed point theorems: completeness, compactness, (path)
connectedness, rectifiable path connectedness, and $d$-convexity.
The bulk of the results presented in this part consists of examples
of maps that show non-reversibility of the previously established
inclusions between theses classes.
Among these examples, the most striking is a differentiable auto-homeomorphism
$f$ of a compact perfect subset $X$ of $\mathbb R$
with $f'\equiv 0$, which constitutes also a minimal dynamical
system.
We finish with discussing a few remaining open problems on weather
the maps with specific pointwise contraction
properties must have the fixed points.
A space X is said to be Lipschitz 1-connected if every L-Lipschitz loop in X bounds a O(L)-Lipschitz disk. A Lipschitz 1-connected space admits a quadratic isoperimetric inequality, but it is unknown whether the converse is true. Cornulier and Tessera showed that certain solvable Lie groups have quadratic isoperimetric inequalities, and we extend their result to show that these groups are Lipschitz 1-connected.
We characterize the Local Langlands Correspondence (LLC) for
inner forms of $\operatorname{GL}_n$ via the Jacquet-Langlands Correspondence
(JLC) and compatibility with the Langlands Classification. We
show that LLC satisfies a natural compatibility with parabolic
induction and characterize LLC for inner forms as a unique family
of bijections $\Pi(\operatorname{GL}_r(D)) \to \Phi(\operatorname{GL}_r(D))$ for each $r$,
(for a fixed $D$) satisfying certain properties. We construct
a surjective map of Bernstein centers $\mathfrak{Z}(\operatorname{GL}_n(F))\to
\mathfrak{Z}(\operatorname{GL}_r(D))$
and show this produces pairs of matching distributions in the
sense of Haines. Finally, we construct explicit Iwahori-biinvariant
matching functions for unit elements in the parahoric Hecke
algebras
of $\operatorname{GL}_r(D)$, and thereby produce many explicit pairs of matching
functions.
Given a link $L$, the Blanchfield pairing $\operatorname{Bl}(L)$ is a pairing
which is defined on the torsion submodule of the Alexander module
of $L$. In some particular cases, namely if $L$ is a boundary
link or if the Alexander module of $L$ is torsion, $\operatorname{Bl}(L)$
can be computed explicitly; however no formula is known in general.
In this article, we compute the Blanchfield pairing of any link,
generalizing the aforementioned results. As a corollary, we obtain
a new proof that the Blanchfield pairing is hermitian. Finally,
we also obtain short proofs of several properties of $\operatorname{Bl}(L)$.
Let $W$ be a compact simply connected triangulated manifold with
boundary and $K\subset W$ be a subpolyhedron. We construct an
algebraic model of the rational homotopy type of $W\backslash K$ out of
a model of the map of pairs $(K,K \cap \partial W)\hookrightarrow
(W,\partial W)$ under some high codimension hypothesis.
We deduce the rational homotopy invariance of the configuration
space
of two points in a compact manifold with boundary under 2-connectedness
hypotheses. Also, we exhibit
nice explicit models of these configuration spaces for a large
class
of compact manifolds.
We characterize the class of RFD $C^*$-algebras as those containing
a dense subset of elements that attain their norm under a finite-dimensional
representation. We show further that this subset is the whole
space precisely when every irreducible representation of the
$C^*$-algebra is finite-dimensional, which is equivalent to the
$C^*$-algebra having no simple infinite-dimensional AF subquotient.
We apply techniques from this proof to show the existence of
elements in more general classes of $C^*$-algebras whose norms
in finite-dimensional representations fit certain prescribed
properties.
Is there some absolute $\varepsilon > 0$ such that for any claw-free
graph $G$, the chromatic number of the square of $G$ satisfies
$\chi(G^2) \le (2-\varepsilon) \omega(G)^2$, where $\omega(G)$ is the
clique number of $G$? Erdős and Nešetřil asked this
question for the specific case of $G$ the line graph of a simple graph
and this was answered in the affirmative by Molloy and Reed. We show
that the answer to the more general question is also yes, and moreover
that it essentially reduces to the original question of Erdős and
Nešetřil.
We will give a representation-theoretic proof for the multiplication
formula
in the Ringel-Hall algebra
$\mathfrak{H}_\Delta(n)$ of a cyclic quiver $\Delta(n)$. As a first
application, we see immediately the existence of Hall polynomials
for cyclic quivers, a fact established
by J. Y. Guo and C. M. Ringel,
and derive a recursive formula
to compute them.
We will further use the formula and the construction of a certain
monomial base for $\mathfrak{H}_\Delta(n)$ given
by Deng, Du, and Xiao
together with the double Ringel--Hall algebra realisation of
the quantum loop algebra $\mathbf{U}_v(\widehat{\mathfrak{g}\mathfrak{l}}_n)$
given by
Deng, Du, and Fu
to develop some algorithms and to compute the canonical basis
for $\mathbf{U}_v^+(\widehat{\mathfrak{g}\mathfrak{l}}_n)$. As examples,
we will show explicitly the part of the canonical basis
associated with modules of Lowey length at most $2$ for the quantum
group $\mathbf{U}_v(\widehat{\mathfrak{g}\mathfrak{l}}_2)$.
We study criteria for the uniform convergence of trigonometric
series with general monotone coefficients.
We also obtain necessary and sufficient conditions for a given
rate of convergence of partial Fourier sums of such series.
This paper provides some evidence for conjectural
relations between extensions of (right) weak order on Coxeter
groups, closure operators on root systems, and Bruhat order.
The conjecture
focused upon here refines an earlier question as to whether the
set of initial sections of reflection orders, ordered by inclusion,
forms a complete lattice.
Meet and join in weak order are described in terms of a suitable
closure operator. Galois connections are defined from the
power set of $W$ to itself, under which maximal subgroups of
certain groupoids correspond
to certain complete meet subsemilattices of weak order. An analogue
of weak order for standard parabolic subsets of any rank
of the root system
is defined, reducing to the usual weak order in rank zero, and
having some analogous properties in rank one (and conjecturally
in general).
We address the classification problem for graph $C^*$-algebras of
finite graphs (finitely many edges and vertices), containing
the class of Cuntz-Krieger algebras as a
prominent special case. Contrasting earlier work, we do not assume
that the graphs satisfy the standard condition (K), so that the
graph
$C^*$-algebras may come with uncountably many ideals.
We find that in this generality, stable isomorphism of graph
$C^*$-algebras does not coincide with the geometric notion of Cuntz
move equivalence. However, adding a modest condition on the
graphs, the two notions are proved to be mutually equivalent and
equivalent to the $C^*$-algebras having isomorphic $K$-theories. This
proves in turn that under this condition, the graph
$C^*$-algebras are in fact classifiable by $K$-theory, providing in
particular complete classification when the $C^*$-algebras in question
are either of real rank zero or type I/postliminal. The key ingredient
in obtaining these results is a characterization of Cuntz move
equivalence using the adjacency matrices of the graphs.
Our results are applied to discuss the classification problem
for the quantum lens spaces defined by Hong and Szymański,
and to complete the classification of graph $C^*$-algebras associated to
all simple graphs with four vertices or less.
We define Schwartz functions, tempered functions and tempered
distributions on (possibly singular) real algebraic varieties.
We prove that all classical properties of these spaces, defined
previously on affine spaces and on Nash manifolds, also hold
in the case of affine real algebraic varieties, and give partial
results for the non-affine case.
Order three elements in the exceptional groups of type $G_2$
are classified up to conjugation over arbitrary fields. Their
centralizers are computed, and the associated classification
of idempotents in symmetric composition algebras is obtained.
Idempotents have played a key role in the study and classification
of these algebras.
Over an algebraically closed field, there are two conjugacy classes
of order three elements in $G_2$ in characteristic not $3$ and
four of them in characteristic $3$. The centralizers in characteristic
$3$ fail to be smooth for one of these classes.
In this paper we discuss the range of a co-analytic Toeplitz
operator. These range spaces are closely related to de Branges-Rovnyak
spaces (in some cases they are equal as sets). In order to understand
its structure, we explore when
the range space decomposes into the range of an associated analytic
Toeplitz operator and an identifiable orthogonal complement.
For certain cases, we compute this orthogonal complement in terms
of the kernel of a certain Toeplitz operator on the Hardy space
where we focus on when this kernel is a model space (backward
shift invariant subspace).
In the spirit of Ahern-Clark, we also discuss the non-tangential
boundary behavior in these range spaces. These results give us
further insight into the description of the range of a co-analytic
Toeplitz operator as well as its orthogonal decomposition. Our
Ahern-Clark type results, which are stated in a general abstract
setting, will also have applications to related sub-Hardy Hilbert
spaces of analytic functions such as the de Branges-Rovnyak spaces
and the harmonically weighted Dirichlet spaces.
We consider ideals in a polynomial ring that are generated by regular sequences of homogeneous polynomials and are stable under the action of the symmetric group permuting the variables. In previous work, we determined the possible isomorphism types for these ideals. Following up on that work, we now analyze the possible degrees of the elements in such regular sequences. For each case of our classification, we provide some criteria guaranteeing the existence of regular sequences in certain degrees.
Fix a von Neumann algebra $\mathcal{N}$ equipped with a suitable trace
$\tau$. For a path of self-adjoint Breuer-Fredholm operators, the
spectral flow measures the net amount of spectrum which moves from
negative to non-negative. We consider specifically the case of paths
of bounded perturbations of a fixed unbounded self-adjoint
Breuer-Fredholm operator affiliated with $\mathcal{N}$. If the unbounded
operator is p-summable (that is, its resolvents are contained in the
ideal $L^p$), then it is possible to obtain an integral formula which
calculates spectral flow. This integral formula was first proven by
Carey and Phillips, building on earlier approaches of Phillips. Their
proof was based on first obtaining a formula for the larger class of
$\theta$-summable operators, and then using Laplace transforms to
obtain a p-summable formula. In this paper, we present a direct proof
of the p-summable formula, which is both shorter and simpler than
theirs.
We introduce a new invariant describing the structure of sets of lengths in atomic monoids and domains. For an atomic monoid $H$, let $\Delta_{\rho} (H)$ be the set of all positive integers $d$ which occur as differences of arbitrarily long arithmetical progressions contained in sets of lengths having maximal elasticity $\rho (H)$. We study $\Delta_{\rho} (H)$ for transfer Krull monoids of finite type (including commutative Krull domains with finite class group) with methods from additive combinatorics, and also for a class of weakly Krull domains (including orders in algebraic number fields) for which we use ideal theoretic methods.
We provide general inequalities that compare the surface area
$S(K)$ of a convex body $K$ in ${\mathbb R}^n$
to the minimal, average or maximal surface area of its hyperplane
or lower dimensional projections. We discuss the
same questions for all the quermassintegrals of $K$. We examine
separately the dependence of the constants
on the dimension in the case where $K$ is in some of the classical
positions or $K$ is a projection body.
Our results are in the spirit of the hyperplane problem, with
sections replaced by projections and volume by
surface area.
We prove completeness for the main examples
of infinite-dimensional Lie groups and some related topological
groups.
Consider a sequence
$G_1\subseteq G_2\subseteq\cdots$ of topological groups~$G_n$
such that~$G_n$ is a subgroup of $G_{n+1}$ and the latter induces
the given topology on~$G_n$,
for each $n\in\mathbb{N}$.
Let $G$ be the direct limit of the sequence in the category of
topological groups.
We show that $G$ induces the given topology on each~$G_n$ whenever
$\bigcup_{n\in \mathbb{N}}V_1V_2\cdots V_n$ is an identity neighbourhood
in~$G$
for all identity neighbourhoods $V_n\subseteq G_n$. If, moreover,
each $G_n$ is complete, then~$G$ is complete.
We also show that the weak direct product $\bigoplus_{j\in J}G_j$
is complete for
each family $(G_j)_{j\in J}$ of complete Lie groups~$G_j$.
As a consequence, every strict direct limit $G=\bigcup_{n\in
\mathbb{N}}G_n$ of finite-dimensional
Lie groups is complete, as well as the diffeomorphism group
$\operatorname{Diff}_c(M)$
of a paracompact finite-dimensional smooth manifold~$M$
and the test function group $C^k_c(M,H)$, for each $k\in\mathbb{N}_0\cup\{\infty\}$
and complete Lie group~$H$
modelled on a complete locally convex space.
Suppose that $\mathbb{N}$ is $2$-coloured. Then there are infinitely
many monochromatic solutions to $x + y = z^2$. On the other hand,
there is a $3$-colouring of $\mathbb{N}$ with only finitely many monochromatic solutions to this equation.
By previous work of Giordano and the author, ergodic
actions of $\mathbf Z$ (and other discrete groups) are completely classified
measure-theoretically by their dimension space, a construction
analogous to the dimension group used in C*-algebras and topological
dynamics. Here we investigate how far from AT (approximately
transitive) can actions be which derive from circulant (and related)
matrices. It turns out not very: although non-AT actions can
arise from this method of construction, under very modest additional
conditions, ATness arises; in addition, if we drop the positivity
requirement in the isomorphism of dimension spaces, then all
these ergodic actions satisfy an analogue of AT. Many examples
are provided.
Consider a finite sequence of linear contractions $S_{j}(x)=\varrho
x+d_{j}$ and
probabilities $p_{j}\gt 0$ with $\sum p_{j}=1$. We are interested
in the
self-similar measure $\mu =\sum p_{j}\mu \circ S_{j}^{-1}$, of
finite type.
In this paper we study the multi-fractal analysis of such measures,
extending the theory to measures arising from non-regular probabilities
and
whose support is not necessarily an interval.
Under some mild technical assumptions, we prove that there exists
a subset
of supp$\mu $ of full $\mu $ and Hausdorff measure, called the
truly
essential class, for which the set of (upper or lower) local
dimensions is a
closed interval. Within the truly essential class we show that
there exists
a point with local dimension exactly equal to the dimension of
the support.
We give an example where the set of local dimensions is a two
element set,
with all the elements of the truly essential class giving the
same local
dimension. We give general criteria for these measures to be
absolutely
continuous with respect to the associated Hausdorff measure of
their support
and we show that the dimension of the support can be computed
using only
information about the essential class.
To conclude, we present a detailed study of three examples. First,
we show
that the set of local dimensions of the biased Bernoulli convolution
with
contraction ratio the inverse of a simple Pisot number always
admits an
isolated point. We give a precise description of the essential
class of a
generalized Cantor set of finite type, and show that the $kth$
convolution
of the associated Cantor measure has local dimension at $x\in
(0,1)$ tending
to 1 as $k$ tends to infinity. Lastly, we show that within a
maximal loop
class that is not truly essential, the set of upper local dimensions
need
not be an interval. This is in contrast to the case for finite
type measures
with regular probabilities and full interval support.
Let $k$ be a number field. We describe the category of Laumon
$1$-isomotives over $k$ as the universal category in the sense
of Nori associated with a quiver representation built out of
smooth proper $k$-curves with two disjoint effective divisors
and a notion of $H^1_\mathrm{dR}$ for such "curves with modulus".
This result extends and relies on the theorem of J. Ayoub
and L. Barbieri-Viale that describes Deligne's category
of $1$-isomotives in terms of Nori's Abelian category of motives.
We derive a forcing axiom from the conjunction
of square and diamond, and present a few applications,
primary among them being the existence of super-Souslin trees.
It follows that for every uncountable cardinal $\lambda$, if
$\lambda^{++}$ is not a Mahlo cardinal in Gödel's constructible
universe,
then $2^\lambda = \lambda^+$ entails the existence of a $\lambda^+$-complete
$\lambda^{++}$-Souslin tree.
In this paper, we study the boundary quotient C*-algebras associated
to products of odometers. One of our main results
shows that the boundary quotient C*-algebra of the standard product
of $k$ odometers
over $n_i$-letter alphabets ($1\le i\le k$) is always nuclear,
and that
it is a UCT Kirchberg algebra
if and only if $\{\ln n_i: 1\le i\le k\}$ is rationally independent,
if and only if the associated single-vertex $k$-graph C*-algebra
is simple.
To achieve this, one of our main steps is to construct a topological
$k$-graph such that
its associated Cuntz-Pimsner C*-algebra is isomorphic to the
boundary quotient C*-algebra.
Some relations between the boundary quotient C*-algebra and the
C*-algebra $\mathrm{Q}_\mathbb{N}$ introduced by Cuntz are also
investigated.
In this paper, we prove the spherical fundamental lemma for
metaplectic group $Mp_{2n}$ based on the formalism of endoscopy
theory by J.Adams, D.Renard and Wen-Wei Li.
We investigate the smoothing problem of limit linear series of
rank one on an enrichment of the notions of nodal curves and
metrized complexes called saturated metrized complexes. We give
a finitely verifiable full criterion for smoothability of a limit
linear series of rank one on saturated metrized complexes, characterize
the space of all such smoothings, and extend the criterion to
metrized complexes. As applications, we prove that all limit
linear series of rank one are smoothable on saturated metrized
complexes corresponding to curves of compact-type, and prove
an analogue for saturated metrized complexes of a theorem of
Harris and Mumford on the characterization of nodal curves contained
in a given gonality stratum. In addition, we give a full combinatorial
criterion for smoothable limit linear series of rank one on saturated
metrized complexes corresponding to nodal curves whose dual graphs
are made of separate loops.
In this paper we complete the ADE-like
classification
of simple transitive $2$-representations
of Soergel bimodules
in finite dihedral type, under the assumption of gradeability.
In particular, we use bipartite
graphs and zigzag algebras of ADE type to give an explicit
construction of a graded (non-strict)
version of all these $2$-representations.
Moreover,
we give simple combinatorial
criteria for when two such $2$-representations are
equivalent and for when their Grothendieck groups
give rise to isomorphic representations.
Finally, our construction
also gives a large class of simple transitive $2$-representations
in infinite dihedral type for general bipartite graphs.
Culler and Vogtmann defined a simplicial space $O(g)$ called
outer space to study the outer automorphism group
of the free group $F_g$. Using representation theoretic methods,
we give an embedding of $O(g)$ into the analytification of $\mathcal{X}(F_g,
SL_2(\mathbb{C})),$ the $SL_2(\mathbb{C})$ character variety
of $F_g,$ reproving a result of Morgan and Shalen. Then we show
that every point $v$ contained in a maximal cell of $O(g)$ defines
a flat degeneration of $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ to
a toric variety $X(P_{\Gamma})$. We relate $\mathcal{X}(F_g,
SL_2(\mathbb{C}))$ and $X(v)$ topologically by showing that there
is a surjective, continuous, proper map $\Xi_v: \mathcal{X}(F_g,
SL_2(\mathbb{C})) \to X(v)$. We then show that this map is a
symplectomorphism on a dense, open subset of $\mathcal{X}(F_g,
SL_2(\mathbb{C}))$ with respect to natural symplectic structures
on $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ and $X(v)$. In this
way, we construct an integrable Hamiltonian system in $\mathcal{X}(F_g,
SL_2(\mathbb{C}))$ for each point in a maximal cell of $O(g)$,
and we show that each $v$ defines a topological decomposition
of $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ derived from the decomposition
of $X(P_{\Gamma})$ by its torus orbits. Finally, we show that
the valuations coming from the closure of a maximal cell in $O(g)$
all arise as divisorial valuations built from an associated projective
compactification of $\mathcal{X}(F_g, SL_2(\mathbb{C})).$
We prove many simultaneous congruences mod 2 for elliptic and
Hilbert modular forms
among forms with different Atkin--Lehner eigenvalues. The proofs
involve the notion of quaternionic $S$-ideal classes and the
distribution of Atkin--Lehner signs among
newforms.
We study a relation between distinction and special values of
local invariants for representations of the general linear group
over a quadratic extension of $p$-adic fields.
We show that the local Rankin-Selberg root number of any pair
of distinguished representation is trivial and as a corollary
we obtain an analogue for the global root number of any pair
of distinguished cuspidal representations. We further study the
extent to which the gamma factor at $1/2$ is trivial for distinguished
representations as well as the converse problem.
We prove that for every surface $\Sigma$ of Euler genus $g$,
every edge-maximal embedding of a graph in $\Sigma$ is at most
$O(g)$ edges short of a triangulation of $\Sigma$. This provides
the first answer to an open problem of Kainen (1974).
We show that the partial transposes of complex Wishart
random matrices are asymptotically free. We also investigate
regimes where the number of blocks is fixed but the size of
the blocks increases. This gives a example where the partial
transpose produces freeness at the operator level. Finally
we investigate the case of real Wishart matrices.
We prove that digital sequences modulo $m$ along squares are
normal,
which covers some prominent sequences like the sum of digits
in base $q$ modulo $m$, the Rudin-Shapiro sequence and some generalizations.
This gives, for any base, a class of explicit normal numbers
that can be efficiently generated.
We introduce the tracial Rokhlin property for a conditional expectation
for an inclusion
of unital C*-algebras $P \subset A$ with index finite, and show
that an action $\alpha$
from a finite group $G$ on a simple unital C*-algebra $A$ has
the tracial Rokhlin property
in the sense of N. C. Phillips
if and only if the canonical conditional expectation $E\colon
A \rightarrow A^G$ has the tracial
Rokhlin property.
Let $\mathcal{C}$ be a class of infinite dimensional stably
finite separable unital C*-algebras
which is closed under the following conditions:
(1)
If $A \in {\mathcal C}$ and $B \cong A$, then $B \in \mathcal{C}$.
(2)
If $A \in \mathcal{C}$ and $n \in \mathbb{N}$, then $M_n(A) \in \mathcal{C}$.
(3)
If $A \in \mathcal{C}$ and $p \in A$ is a nonzero projection,
then $pAp \in \mathcal{C}$.
Suppose that any C*-algebra in $\mathcal{C}$ is weakly semiprojective.
We prove that if $A$ is a local tracial $\mathcal{C}$-algebra
in the sense of Fan and Fang and a conditional expectation
$E\colon A \rightarrow P$ is of index-finite type with the tracial
Rokhlin property, then $P$ is a unital
local tracial $\mathcal{C}$-algebra.
The main result is that if $A$ is simple, separable, unital
nuclear, Jiang-Su absorbing
and $E\colon A \rightarrow P$ has the tracial Rokhlin property,
then $P$ is Jiang-Su absorbing.
As an application, when an action $\alpha$
from a finite group $G$ on a simple unital C*-algebra $A$ has
the tracial Rokhlin property,
then for any subgroup $H$ of $G$ the fixed point algebra $A^H$
and the crossed product algebra
$A \rtimes_{\alpha_{|H}} H$ is Jiang-Su absorbing.
We also show that the strict comparison property for a Cuntz
semigroup $W(A)$
is hereditary to $W(P)$ if $A$ is simple, separable, exact,
unital,
and $E\colon A \rightarrow P$ has the tracial Rokhlin property.
We consider three point positive masses moving on $S^2$ and $H^2$.
An Eulerian-relative equilibrium, is a relative equilibrium where
the three masses are on the same geodesic, in this paper we analyze
the spectral stability of these kind of orbits where the mass
at the middle is arbitrary and the masses at the ends are equal
and located at the same distance from the central mass. For the
case of $S^2$, we found a positive measure set in the set of
parameters where the relative equilibria are spectrally stable,
and we give a complete classification of the spectral stability
of these solutions, in the sense that, except on an algebraic
curve in the space of parameters, we can determine if the corresponding
relative equilibria is spectrally stable or unstable.
On $H^2$, in the elliptic case, we prove that generically all
Eulerian-relative equilibria are unstable; in the particular
degenerate case when the two equal masses are negligible we get
that the corresponding solutions are spectrally stable. For the
hyperbolic case we consider the system where the mass in the
middle is negligible, in this case the Eulerian-relative equilibria
are unstable.
This paper investigates regularity in Lorentz
spaces of weak solutions of a class of divergence form quasi-linear
parabolic equations with singular divergence-free drifts. In
this class of equations, the principal terms are vector field
functions which are measurable in $(x,t)$-variable, and nonlinearly
dependent on both unknown solutions and their gradients. Interior,
local boundary, and global regularity estimates in Lorentz spaces
for gradients of weak solutions are established assuming that
the solutions are in BMO space, the John Nirenberg space.
The results are even new when the drifts are identically zero
because they do not require solutions to be bounded as in the
available literature. In the linear setting, the results of
the paper also improve the standard Calderón-Zygmund regularity
theory to the critical borderline case. When the principal term
in the equation does not depend on the solution as its variable,
our results recover and sharpen known, available results. The
approach is based on the perturbation technique introduced by
Caffarelli and Peral together with a "double-scaling parameter"
technique, and the maximal function free approach introduced
by Acerbi and Mingione.
We define a homology theory of virtual links built out of the
direct sum of the standard Khovanov complex with itself, motivating
the name doubled Khovanov homology. We demonstrate that
it can be used to show that some virtual links are non-classical,
and that it yields a condition on a virtual knot being the connect
sum of two unknots. Further, we show that doubled Khovanov homology
possesses a perturbation analogous to that defined by Lee in
the classical case and define a doubled Rasmussen invariant.
This invariant is used to obtain various cobordism obstructions;
in particular it is an obstruction to sliceness. Finally, we
show that the doubled Rasmussen invariant contains the odd writhe
of a virtual knot, and use this to show that knots with non-zero
odd writhe are not slice.
Since Rob Pollack and Glenn Stevens used overconvergent
modular symbols to construct $p$-adic $L$-functions for non-critical
slope rational modular forms, the theory has been extended to
construct $p$-adic $L$-functions for non-critical slope automorphic
forms over totally real and imaginary quadratic fields by the
first and second authors respectively. In this paper, we give
an analogous construction over a general number field. In particular,
we start by proving a control theorem stating that the specialisation
map from overconvergent to classical modular symbols is an isomorphism
on the small slope subspace. We then show that if one takes the
modular symbol attached to a small slope cuspidal eigenform,
then one can construct a ray class distribution from the corresponding
overconvergent symbol, that moreover interpolates critical values
of the $L$-function of the eigenform. We prove that this distribution
is independent of the choices made in its construction. We define
the $p$-adic $L$-function of the eigenform to be this distribution.
We compute cup product pairings in the integral cohomology ring of the moduli space of rank two stable bundles with odd determinant over a Riemann surface using methods of Zagier. The resulting formula is related to a generating function for certain skew Schur polynomials. As an application, we compute the nilpotency degree of a distinguished degree two generator in the mod two cohomology ring. We then give descriptions of the mod two cohomology rings in low genus, and describe the subrings invariant under the mapping class group action.
We provide an explicit construction of representations in the
discrete spectrum of two $p$-adic symmetric spaces. We consider
$\mathbf{GL}_n(F) \times \mathbf{GL}_n(F) \backslash \mathbf{GL}_{2n}(F)$
and $\mathbf{GL}_n(F) \backslash \mathbf{GL}_n(E)$, where $E$
is a quadratic Galois extension of a nonarchimedean local field
$F$ of characteristic zero and odd residual characteristic. The
proof of the main result involves an application of a symmetric
space version of Casselman's Criterion for square integrability
due to Kato and Takano.
We give a new proof of the Hansen-Mullen irreducibility conjecture.
The proof relies on an application of a (seemingly new) sufficient
condition for the existence of
elements of degree $n$ in the support of functions on finite
fields.
This connection to irreducible polynomials is made via the least
period of the discrete Fourier transform (DFT) of functions with
values in finite fields.
We exploit this relation and prove, in an elementary fashion,
that a relevant function related to the DFT of characteristic
elementary symmetric functions (which produce the coefficients
of characteristic polynomials)
satisfies a simple requirement on the least period.
This bears a sharp contrast to previous techniques in literature
employed to tackle existence
of irreducible polynomials with prescribed coefficients.
Let $\mathcal{C}$ be a curve of genus at least $2$ embedded in $E_1
\times \cdots \times E_N$ where the $E_i$ are elliptic curves
for $i=1,\dots, N$. In this article we give an explicit sharp
bound for the Néron-Tate height of the points of $\mathcal{C}$ contained
in the union of all algebraic subgroups of dimension
$\lt \max(r_\mathcal{C}-t_\mathcal{C},t_\mathcal{C})$
where $t_\mathcal{C}$, respectively $r_\mathcal{C}$, is the minimal dimension
of a translate, respectively of a torsion variety, containing
$\mathcal{C}$.
As a corollary, we give an explicit bound for the height of
the rational points of special curves, proving new cases of
the explicit Mordell Conjecture and in particular making explicit
(and slightly more general in the CM case) the Manin-Dem'janenko
method in products of elliptic curves.
Let $(M^n,g)$ be a Riemannian manifold without
boundary. We study the amount of initial regularity is required
so that the solution to free Schrödinger equation converges
pointwise to its initial data. Assume the initial data is in
$H^\alpha(M)$. For Hyperbolic Space, standard Sphere and the
2 dimensional Torus, we prove that $\alpha\gt \frac{1}{2}$ is enough.
For general compact manifolds, due to lacking of local smoothing
effect, it is hard to beat the bound $\alpha\gt 1$ from interpolation.
We managed to go below 1 for dimension $\leq 3$. The more interesting
thing is that, for 1 dimensional compact manifold, $\alpha\gt \frac{1}{3}$
is sufficient.
For a line arrangement $\mathcal{A}$ in the complex projective
plane $\mathbb{P}^2$, we investigate the compactification $\overline{F}$
in $\mathbb{P}^3$ of the affine Milnor fiber $F$ and its minimal
resolution $\widetilde{F}$. We compute the Chern numbers of $\widetilde{F}$
in terms of the combinatorics of the line arrangement $\mathcal{A}$.
As applications of the computation of the Chern numbers, we show
that the minimal resolution is never a quotient of a ball; in
addition, we also prove that $\widetilde{F}$ is of general type
when the arrangement has only nodes or triple points as singularities;
finally, we compute all the Hodge numbers of some $\widetilde{F}$
by using some knowledge about the Milnor fiber monodromy of the
arrangement.
The $\operatorname{SL}(2,\mathbb C)$-representation varieties of punctured surfaces
form natural families parameterized by monodromies at the punctures.
In this paper, we compute the loci where these varieties are
singular for the cases of one-holed and two-holed tori and the
four-holed sphere. We then compute the de Rham cohomologies
of these varieties of the one-holed torus and the four-holed
sphere when the varieties are smooth via the Grothendieck theorem.
Furthermore, we produce the explicit Gauss-Manin connection
on the natural family of the smooth $\operatorname{SL}(2,\mathbb C)$-representation
varieties of the one-holed torus.
In this paper we prove that decomposable forms,
or homogeneous polynomials $F(x_1, \cdots, x_n)$ with integer
coefficients which split completely into linear factors over
$\mathbb{C}$, take on infinitely many square-free values subject
to simple necessary conditions and $\deg f \leq 2n + 2$ for all
irreducible factors $f$ of $F$. This work generalizes a theorem
of Greaves.
In this paper we study a class of second order fully nonlinear
elliptic equations
containing gradient terms on compact Hermitian manifolds and
obtain a priori estimates under
proper assumptions close to optimal.
The analysis developed here should
be useful to deal with other Hessian equations containing gradient
terms in other contexts.
For a Shimura variety of Hodge type with hyperspecial level
structure at a prime~$p$, Vasiu and Kisin constructed a smooth
integral model (namely the integral canonical model) uniquely
determined by a certain extension property. We define and study
the Ekedahl-Oort stratifications on the special fibers of those
integral canonical models when $p\gt 2$. This generalizes
Ekedahl-Oort stratifications defined and studied by Oort on moduli
spaces of principally polarized abelian varieties and those
defined and studied by Moonen, Wedhorn and Viehmann on good
reductions of Shimura varieties of PEL type. We show that the
Ekedahl-Oort strata are parameterized by certain elements $w$ in
the Weyl group of the reductive group in the Shimura datum. We
prove that the stratum corresponding to $w$ is smooth of dimension
$l(w)$ (i.e. the length of $w$) if it is non-empty. We also
determine the closure of each stratum.