pinecone/core-2020-05-10-deduplication huggingface dataset:
Positive loops and $L^{\infty}$-contact systolic inequalities

We prove an inequality between the $L^{\infty}$-norm of the contact
Hamiltonian of a positive loop of contactomorphims and the minimal Reeb period.
This implies that there are no small positive loops on hypertight or Liouville
fillable contact manifolds. Non-existence of small positive loops for
overtwisted 3-manifolds was proved by Casals-Presas-Sandon in [CPS16].
  As corollaries of the inequality we deduce various results. E.g. we prove
that certain periodic Reeb flows are the unique minimizers of the
$L^\infty$-norm. Moreover, we establish $L^\infty$-type contact systolic
inequalities in the presence of a positive loop.Comment: 26 pages, 6 figures; v2: corrected an error, changed statements of
  main theorems; v3: accepted version, to appear in Selecta Mathematic
Free factorization algebras and homology of configuration spaces in
  algebraic geometry

We provide a construction of free factorization algebras in algebraic
geometry and link factorization homology of a scheme with coefficients in a
free factorization algebra to the homology of its (unordered) configuration
spaces. As an application, this construction allows for a purely
algebro-geometric proof of homological stability of configuration spaces.Comment: The final publication is available at Springer via
  http://dx.doi.org/doi:10.1007/s00029-017-0339-1, Selecta Mathematica (N.S.)
  201
Abelian duality and propagation of resonance

We explore the relationship between a certain "abelian duality" property of
spaces and the propagation properties of their cohomology jump loci. To that
end, we develop the analogy between abelian duality spaces and those spaces
which possess what we call the "EPY property." The same underlying homological
algebra allows us to deduce the propagation of jump loci: in the former case,
characteristic varieties propagate, and in the latter, the resonance varieties.
We apply the general theory to arrangements of linear and elliptic hyperplanes,
as well as toric complexes, right-angled Artin groups, and Bestvina-Brady
groups. Our approach brings to the fore the relevance of the Cohen-Macaulay
condition in this combinatorial context.Comment: 30 page
Grothendieck ring of varieties, D- and L-equivalence, and families of
  quadrics

We discuss a conjecture saying that derived equivalence of simply connected
smooth projective varieties implies that the difference of their classes in the
Grothendieck ring of varieties is annihilated by a power of the affine line
class.
  We support the conjecture with a number of known examples, and one new
example. We consider a smooth complete intersection $X$ of three quadrics in
${\mathbf P}^5$ and the corresponding double cover $Y \to {\mathbf P}^2$
branched over a sextic curve. We show that as soon as the natural Brauer class
on $Y$ vanishes, so that $X$ and $Y$ are derived equivalent, the difference
$[X] - [Y]$ is annihilated by the affine line class.Comment: Exposition improved, main conjecture slightly update
Higher level vertex operators for $U_q (\hat{\mathfrak{sl}}_2)$

We study graded nonlocal $\underline{\mathsf{q}}$-vertex algebras and we
prove that they can be generated by certain sets of vertex operators. As an
application, we consider the family of graded nonlocal
$\underline{\mathsf{q}}$-vertex algebras $V_{c,1}$, $c\geq 1$, associated with
the principal subspaces $W(c\Lambda_0)$ of the integrable highest weight $U_q
(\hat{\mathfrak{sl}}_2)$-modules $L(c\Lambda_0)$. Using quantum integrability,
we derive combinatorial bases for $V_{c,1}$ and compute the corresponding
character formulae.Comment: 28 pages, 1 figur
The algebraisation of higher Deligne–Lusztig representations.

In this paper we study higher Deligne–Lusztig representations of reductive\ud
groups over finite quotients of discrete valuation rings. At even levels, we show that\ud
these geometrically constructed representations, defined by Lusztig, coincide with\ud
certain explicit induced representations defined by Gérardin, thus giving a solution\ud
to a problem raised by Lusztig. In particular, we determine the dimensions of these\ud
representations. As an immediate application we verify a conjecture of Letellier for\ud
GL2 and GL3
A 2-categorical extension of Etingof-Kazhdan quantisation

Let k be a field of characteristic zero. Etingof and Kazhdan constructed a
quantisation U_h(b) of any Lie bialgebra b over k, which depends on the choice
of an associator Phi. They prove moreover that this quantisation is functorial
in b. Remarkably, the quantum group U_h(b) is endowed with a Tannakian
equivalence F_b from the braided tensor category of Drinfeld-Yetter modules
over b, with deformed associativity constraints given by Phi, to that of
Drinfeld-Yetter modules over U_h(b). In this paper, we prove that the
equivalence F_b is functorial in b.Comment: Small revisions in Sections 2 and 6. An appendix added on the
  equivalence between admissible Drinfeld-Yetter modules over a QUE and modules
  over its quantum double. To appear in Selecta Math. 71 page
On a theory of the $b$-function in positive characteristic

We present a theory of the $b$-function (or Bernstein-Sato polynomial) in
positive characteristic. Let $f$ be a non-constant polynomial with coefficients
in a perfect field $k$ of characteristic $p>0.$ Its $b$-function $b_f$ is
defined to be an ideal of the algebra of continuous $k$-valued functions on
$\mathbb{Z}_p.$ The zero-locus of the $b$-function is thus naturally
interpreted as a subset of $\mathbb{Z}_p,$ which we call the set of roots of
$b_f.$ We prove that $b_f$ has finitely many roots and that they are negative
rational numbers. Our construction builds on an earlier work of Musta\c{t}\u{a}
and is in terms of $D$-modules, where $D$ is the ring of Grothendieck
differential operators. We use the Frobenius to obtain finiteness properties of
$b_f$ and relate it to the test ideals of $f.$Comment: Final versio
Donaldson-Thomas theory of $[\mathbb{C}^2/\mathbb{Z}_{n+1}]\times
  \mathbb{P}^1$

We study the relative orbifold Donaldson-Thomas theory of
$[\mathbb{C}^2/\mathbb{Z}_{n+1}]\times \mathbb{P}^1$. We establish a
correspondence between the DT theory relative to 3 fibers to quantum
multiplication by divisors in the Hilbert scheme of points on
$[\mathbb{C}^2/\mathbb{Z}_{n+1}]$. This determines the whole theory if a
further nondegeneracy condition is assumed. The result can also be viewed as a
crepant resolution correspondence to the DT theory of $\mathcal{A}_n\times
\mathbb{P}^1$.Comment: 44 pages, 1 figure. Minor changes, Selecta Mathematica New Series
  (2018
Semiclassical asymptotics of $\operatorname{GL}_N(\mathbb{C})$ tensor
  products and quantum random matrices

The Littlewood--Richardson process is a discrete random point process arising
from the isotypic decomposition of tensor products of irreducible
representations of $\operatorname{GL}_N(\mathbb{C})$. Biane--Perelomov--Popov
matrices are quantum random matrices obtained as the geometric quantization of
random Hermitian matrices with deterministic eigenvalues and uniformly random
eigenvectors. As first observed by Biane, correlation functions of certain
global observables of the LR process coincide with correlation functions of
linear statistics of sums of classically independent BPP matrices, thereby
enabling a random matrix approach to the statistical study of
$\operatorname{GL}_N(\mathbb{C})$ tensor products. In this paper, we prove an
optimal result: classically independent BPP matrices become freely independent
in any semiclassical/large-dimension limit. This proves and generalizes a
conjecture of Bufetov and Gorin, and leads to a Law of Large Numbers for the
BPP observables of the LR process which holds in any and all semiclassical
scalings.Comment: 52 page
Plane partitions with a "pit": generating functions and representation
  theory

We study plane partitions satisfying condition $a_{n+1,m+1}=0$ (this
condition is called "pit") and asymptotic conditions along three coordinate
axes. We find the formulas for generating function of such plane partitions.
  Such plane partitions label the basis vectors in certain representations of
quantum toroidal $\mathfrak{gl}_1$ algebra, therefore our formulas can be
interpreted as the characters of these representations. The resulting formulas
resemble formulas for characters of tensor representations of Lie superalgebra
$\mathfrak{gl}_{m|n}$. We discuss representation theoretic interpretation of
our formulas using $q$-deformed $W$-algebra $\mathfrak{gl}_{m|n}$.Comment: 30 pages, v2. 32 pages, new subsection 4.4 included, v3 36 pages,
  many corrections, references added, v4 38 pages many corrections, references
  added, version to appear in Sel. Math. New Ser. (2018
Weak Separation, Pure Domains and Cluster Distance

Following the proof of the purity conjecture for weakly separated
collections, recent years have revealed a variety of wider examples of purity
in different settings. In this paper we consider the collection $\mathcal
A_{I,J}$ of sets that are weakly separated from two fixed sets $I$ and $J$. We
show that all maximal by inclusion weakly separated collections $\mathcal
W\subset\mathcal A_{I,J}$ are also maximal by size, provided that $I$ and $J$
are sufficiently "generic". We also give a simple formula for the cardinality
of $\mathcal W$ in terms of $I$ and $J$. We apply our result to calculate the
cluster distance and to give lower bounds on the mutation distance between
cluster variables in the cluster algebra structure on the coordinate ring of
the Grassmannian. Using a linear projection that relates weak separation to the
octahedron recurrence, we also find the exact mutation distances and cluster
distances for a family of cluster variables.Comment: 44 pages, 11 figure
Geometry of Hessenberg varieties with applications to Newton-Okounkov
  bodies

In this paper, we study the geometry of various Hessenberg varieties in type
A, as well as families thereof, with the additional goal of laying the
groundwork for future computations of Newton-Okounkov bodies of Hessenberg
varieties. Our main results are as follows. We find explicit and
computationally convenient generators for the local defining ideals of
indecomposable regular nilpotent Hessenberg varieties, and then show that all
regular nilpotent Hessenberg varieties are local complete intersections. We
also show that certain families of Hessenberg varieties, whose generic fibers
are regular semisimple Hessenberg varieties and the special fiber is a regular
nilpotent Hessenberg variety, are flat and have reduced fibres. This result
further allows us to give a computationally effective formula for the degree of
a regular nilpotent Hessenberg variety with respect to a Pl\"ucker embedding.
Furthermore, we construct certain flags of subvarieties of a regular nilpotent
Hessenberg variety, obtained by intersecting with Schubert varieties, which are
suitable for computing Newton-Okounkov bodies. As an application of our
results, we explicitly compute many Newton-Okounkov bodies of the
two-dimensional Peterson variety with respect to Pl\"ucker embeddings.Comment: 25 pages. Arguments substantially streamlined thanks to comments from
  anonymous referee. Readers who want more leisurely explanations may wish to
  consult our first version on the ArXi
Derived coisotropic structures I: affine case

We define and study coisotropic structures on morphisms of commutative dg
algebras in the context of shifted Poisson geometry, i.e. $P_n$-algebras.
Roughly speaking, a coisotropic morphism is given by a $P_{n+1}$-algebra acting
on a $P_n$-algebra. One of our main results is an identification of the space
of such coisotropic structures with the space of Maurer--Cartan elements in a
certain dg Lie algebra of relative polyvector fields. To achieve this goal, we
construct a cofibrant replacement of the operad controlling coisotropic
morphisms by analogy with the Swiss-cheese operad which can be of independent
interest. Finally, we show that morphisms of shifted Poisson algebras are
identified with coisotropic structures on their graph.Comment: 49 pages. v2: many proofs rewritten and the paper is split into two
  part
Derived coisotropic structures II: stacks and quantization

We extend results about $n$-shifted coisotropic structures from part I of
this work to the setting of derived Artin stacks. We show that an intersection
of coisotropic morphisms carries a Poisson structure of shift one less. We also
compare non-degenerate shifted coisotropic structures and shifted Lagrangian
structures and show that there is a natural equivalence between the two spaces
in agreement with the classical result. Finally, we define quantizations of
$n$-shifted coisotropic structures and show that they exist for $n>1$.Comment: 45 pages. Contains the second half of arXiv:1608.01482v1 with new
  material adde
Quivers with relations for symmetrizable Cartan matrices IV: Crystal
  graphs and semicanonical functions

We generalize Lusztig's nilpotent varieties, and Kashiwara and Saito's
geometric construction of crystal graphs from the symmetric to the
symmetrizable case. We also construct semicanonical functions in the
convolution algebras of generalized preprojective algebras. Conjecturally these
functions yield semicanonical bases of the enveloping algebras of the positive
part of symmetrizable Kac-Moody algebras.Comment: 50 pages. Version 2: A few typos fixed. Final version published in
  Selecta Mathematic
Transition formulas for involution Schubert polynomials

The orbits of the orthogonal and symplectic groups on the flag variety are in
bijection, respectively, with the involutions and fixed-point-free involutions
in the symmetric group $S_n$. Wyser and Yong have described polynomial
representatives for the cohomology classes of the closures of these orbits,
which we denote as $\hat{\mathfrak{S}}_y$ (to be called involution Schubert
polynomials) and $\hat{\mathfrak{S}}^{\tt FPF}_y$ (to be called
fixed-point-free involution Schubert polynomials). Our main results are
explicit formulas decomposing the product of $\hat{\mathfrak{S}}_y$
(respectively, $\hat{\mathfrak{S}}^{\tt FPF}_y$) with any $y$-invariant linear
polynomial as a linear combination of other involution Schubert polynomials.
These identities serve as analogues of Lascoux and Sch\"utzenberger's
transition formula for Schubert polynomials, and lead to a self-contained
algebraic proof of the nontrivial equivalence of several definitions of
$\hat{\mathfrak{S}}_y$ and $\hat{\mathfrak{S}}^{\tt FPF}_y$ appearing in the
literature. Our formulas also imply combinatorial identities about involution
words, certain variations of reduced words for involutions in $S_n$. We
construct operators on involution words based on the Little map to prove these
identities bijectively. The proofs of our main theorems depend on some new
technical results, extending work of Incitti, about covering relations in the
Bruhat order of $S_n$ restricted to involutions.Comment: 31 pages; v2: updated references and acknowledgments; v3: added
  references, minor corrections; v4: a few more references, examples, and
  corrections, final versio
The Elliptic Hall algebra and the deformed Khovanov Heisenberg category

We give an explicit description of the trace, or Hochschild homology, of the
quantum Heisenberg category defined by Licata and Savage. We also show that as
an algebra, it is isomorphic to "half" of a central extension of the elliptic
Hall algebra of Burban and Schiffmann, specialized at $\sigma = \bar\sigma^{-1}
= q$. A key step in the proof may be of independent interest: we show that the
sum (over $n$) of the Hochschild homologies of the positive affine Hecke
algebras $\mathrm{AH}_n^+$ is again an algebra, and that this algebra injects
into both the elliptic Hall algebra and the trace of the $q$-Heisenberg
category. Finally, we show that a natural action of the trace algebra on the
space of symmetric functions agrees with the specialization of an action
constructed by Schiffmann and Vasserot using Hilbert schemes.Comment: 49 pages, numerous figure
A cohomological approach to immersed submanifolds via integrable systems

A geometric approach to immersion formulas for soliton surfaces is provided
through new cohomologies on spaces of special types of $\mathfrak{g}$-valued
differential forms. This leads us to introduce Poincar\'e-type lemmas for these
cohomologies, which appropriately describe the integrability conditions of Lax
pairs associated with systems of PDEs. Our methods clarify the structure and
properties of the deformations and soliton surfaces for the aforesaid Lax
pairs. Our findings also allow for the generalization of the theory of soliton
surfaces in Lie algebras to general soliton submanifolds. Techniques from the
theory of infinite-dimensional jet manifolds and diffieties enable us to
justify certain common assumptions of the theory of soliton surfaces.
Theoretical results are illustrated through $\mathbb{C}P^{N-1}$ sigma models.Comment: 30 pages. Several typos corrected and some comments adde
On classical upper bounds for slice genera

We introduce a new link invariant called the algebraic genus, which gives an
upper bound for the topological slice genus of links. In fact, the algebraic
genus is an upper bound for another version of the slice genus proposed here:
the minimal genus of a surface in the four-ball whose complement has infinite
cyclic fundamental group. We characterize the algebraic genus in terms of
cobordisms in three-space, and explore the connections to other knot invariants
related to the Seifert form, the Blanchfield form, knot genera and unknotting.
Employing Casson-Gordon invariants, we discuss the algebraic genus as a
candidate for the optimal upper bound for the topological slice genus that is
determined by the S-equivalence class of Seifert matrices.Comment: 29 pages, 5 figures, comments welcome! V2: Improved exposition and
  figures, added an example, implemented referee's recommendations. Accepted
  for publication in Selecta Mathematic
Bounded-From-Below Solutions of the Hamilton-Jacobi Equation for Optimal
  Control Problems with Exit Times: Vanishing Lagrangians, Eikonal Equations,
  and Shape-From-Shading

We study the Hamilton-Jacobi equation for undiscounted exit time control
problems with general nonnegative Lagrangians using the dynamic programming
approach. We prove theorems characterizing the value function as the unique
bounded-from-below viscosity solution of the Hamilton-Jacobi equation which is
null on the target. The result applies to problems with the property that all
trajectories satisfying a certain integral condition must stay in a bounded
set. We allow problems for which the Lagrangian is not uniformly bounded below
by positive constants, in which the hypotheses of the known uniqueness results
for Hamilton-Jacobi equations are not satisfied. We apply our theorems to
eikonal equations from geometric optics, shape-from-shading equations from
image processing, and variants of the Fuller Problem.Comment: 29 pages, 0 figures, accepted for publication in NoDEA Nonlinear
  Differential Equations and Applications on July 29, 200
Weak KAM theorem on non compact manifolds

In this paper, we consider a time independent $C^2$ Hamiltonian, sa\-tisfying
the usual hypothesis of the classical Calculus of Variations, on a non-compact
connected manifold. Using the Lax-Oleinik semigroup, we give a proof of the
existence of weak KAM solutions, or viscosity solutions, for the associated
Hamilton-Jacobi Equation. This proof works also in presence of symmetries. We
also study the role of the amenability of the group of symmetries to understand
when the several critical values that can be associated with the Hamiltonian
coincide.Comment: arXiv admin note: text overlap with arXiv:1004.0086 by other author
Large deviations for the stochastic shell model of turbulence

The article of record as published may be located at http://dx.doi.org/10.1007/s00030-009-0023-zIn this work, we first prove the existence and uniqueness of a strong solution to stochastic GOY model of turbulence with a small multiplicative noise.  Then using the weak convergence approach, Laplace principle for solutions of the stochastic GOY model is established in certain Polish space.  Thus a Wentzell-Freidlin type large deviation principle is established utilizing certain results by Varadhan and Bryc
Local bifurcation-branching analysis of global and "blow-up" patterns for a fourth-order thin film equation

Countable families of global-in-time and blow-up similarity sign-changing patterns of the Cauchy problem for the fourth-order thin film equation (TFE-4) ut= -del . (|u|(n)del Delta u) in R(N) x R(+), where n > 0, are studied. The similarity solutions are of standard "forward" and "backward" forms u(+/-)(x, t) = (+/- t)(-alpha)f(y), y = x/(+/- t)(beta), beta = 1-alpha n/4, +/- t>0, where f solve Bn+(alpha, f)equivalent to-del.(|f|n del Delta f)+/-beta y.del f +/-alpha f = 0 in RN, (0.1) and alpha is an element of R is a parameter (a "nonlinear eigenvalue"). The sign "+", i.e.,t > 0, corresponds to global asymptotics as t -> +infinity, while "-" (t < 0) yields blow-up limits t -> 0(-) describing possible "micro-scale" (multiple zero) structures of solutions of the PDE. To get a countable set of nonlinear pairs {f(gamma), alpha(gamma)}, a bifurcation-branching analysis is performed by using a homotopy path n -> 0(+) in (0.1), where B(0)(+/-) (alpha, f) become associated with a pair {B, B*} of linear non-self-adjoint operators B=-Delta(2) + 1/4 y.del+N/4 I and B*=-Delta(2) - 1/4y.( so (B)*(L2) = B*), which are known to possess a discrete real spectrum, sigma(B) = sigma(B*) = {lambda(gamma) = - |gamma/4|}(|gamma|>0) (gamma is a multiindex in R(N)). These operators occur after corresponding global and blow-up scaling of the classic bi-harmonic equation u(t) = -Delta(2)u. This allows us to trace out the origin of a countable family of n-branches of nonlinear eigenfunctions by using simple or semi-simple eigenvalues of the linear operators {B, B*} leading to important properties of oscillatory sign-changing nonlinear patterns of the TFE, at least, for small n > 0
Finite reduction and Morse index estimates for mechanical systems

A simple version of exact finite dimensional reduction for the variational
setting of mechanical systems is presented. It is worked out by means of a
thorough global version of the implicit function theorem for monotone
operators. Moreover, the Hessian of the reduced function preserves all the
relevant information of the original one, by Schur's complement, which
spontaneously appears in this context. Finally, the results are
straightforwardly extended to the case of a Dirichlet problem on a bounded
domain.Comment: 13 pages; v2: minor changes, to appear in Nonlinear Differential
  Equations and Application
Convergence of approximate solutions to an elliptic-parabolic equation without the structure condition

International audienceWe study the Cauchy-Dirichlet problem for the elliptic-parabolic equation $$b(u)_t +\div F(u) - \Delta u=f$$ in a bounded domain. We do not assume the structure condition ''$b(z)=b(\hat z) \Rightarrow F(z)=F(\hat z)$''. Our main goal is to investigate the problem of continuous dependence of the solutions on the data of the problem and the question of convergence of discretization methods. As in the work of Ammar and Wittbold \cite{AmmarWittbold} where existence was established, monotonicity and penalization are the main tools of our study. In the case of a Lipschitz continuous flux $F$, we justify the uniqueness of $u$ (the uniqueness of $b(u)$ is well-known) and prove the continuous dependence in $L^1$ for the case of strongly convergent finite energy data. We also prove convergence of the $\varepsilon$-discretized solutions used in the semigroup approach to the problem; and we prove convergence of a monotone time-implicit finite volume scheme. In the case of a merely continuous flux $F$, we show that the problem admits a maximal and a minimal solution