Advances in Mathematics最新文献

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Sliced skein algebras and geometric Kauffman bracket

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-01-30 DOI: 10.1016/j.aim.2025.110118

Charles D. Frohman , Joanna Kania-Bartoszynska , Thang T.Q. Lê

The sliced skein algebra of a closed surface of genus g with m punctures,

S = Σ_{g, m}

, is the quotient of the Kauffman bracket skein algebra

S_{ξ} (S)

corresponding to fixing the scalar values of its peripheral curves. We show that the sliced skein algebra of a finite type surface is a domain if the ground ring is a domain. When the quantum parameter ξ is a root of unity we calculate the center of the sliced skein algebra and its PI-degree. Among applications we show that any smooth point of a sliced character variety is an Azumaya point of the skein algebra

S_{ξ} (S)

For any

S L_{2} (C)

-representation ρ of the fundamental group of an oriented connected 3-manifold M and a root of unity ξ with the order of

ξ^{2}

odd, we introduce the ρ-reduced skein module

S_{ξ, ρ} (M)

. We show that

S_{ξ, ρ} (M)

has dimension 1 when M is closed and ρ is irreducible. We also show that if ρ is irreducible the ρ-reduced skein module of a handlebody, as a module over the skein algebra of its boundary, is simple and has the dimension equal to the PI-degree of the skein algebra of its boundary.

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引用次数: 0

The stabilizer bitorsors of the module and algebra harmonic coproducts are equal

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-01-28 DOI: 10.1016/j.aim.2025.110128

Benjamin Enriquez , Hidekazu Furusho

In earlier work, we constructed a pair of “Betti” and “de Rham” Hopf algebras and a pair of module-coalgebras over this pair, as well as the bitorsors related to both structures (which will be called the “module” and “algebra” stabilizer bitorsors). We showed that Racinet's torsor constructed out of the double shuffle and regularization relations between multiple zeta values is essentially equal to the “module” stabilizer bitorsor, and that the latter is contained in the “algebra” stabilizer bitorsor. In this paper, we show the equality of the “algebra” and “module” stabilizer bitorsors. We reduce the proof to showing the equality of the associated “algebra” and “module” graded Lie algebras. The argument for showing this equality involves the relation of the “algebra” Lie algebra with the kernel of a linear map, the expression of this linear map as a composition of three linear maps, the relation of one of them with the “module” Lie algebra and the computation of the kernel of the other one by discrete topology arguments.

引用次数: 0

Revisiting the moduli space of 8 points on P1

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-01-28 DOI: 10.1016/j.aim.2025.110126

Klaus Hulek , Yota Maeda

The moduli space of 8 points on

P^{1}

, a so-called ancestral Deligne-Mostow space, is, by work of Kondō, also a moduli space of K3 surfaces. We prove that the Deligne-Mostow isomorphism does not lift to a morphism between the Kirwan blow-up of the GIT quotient and the unique toroidal compactification of the corresponding ball quotient. Moreover, we show that these spaces are not K-equivalent, even though they are natural blow-ups at the unique cusps and have the same cohomology. This is analogous to the work of Casalaina-Martin-Grushevsky-Hulek-Laza on the moduli space of cubic surfaces. The moduli spaces of ordinary stable maps, that is the Fulton-MacPherson compactification of the configuration space of points on

P^{1}

, play an important role in the proof. We further relate our computations to new developments in the minimal model program and the recent work of Odaka. We briefly discuss other cases of moduli space of points on

P^{1}

where a similar behaviour can be observed, hinting at a more general, but not yet fully understood phenomenon.

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引用次数: 0

Generalised Whittaker models as instances of relative Langlands duality

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-01-28 DOI: 10.1016/j.aim.2025.110129

Wee Teck Gan, Bryan Peng Jun Wang

The recent proposal by Ben-Zvi, Sakellaridis and Venkatesh of a duality in the relative Langlands program, leads, via the process of quantization of Hamiltonian varieties, to a duality theory of branching problems. This often unexpectedly relates two a priori unrelated branching problems. We examine how the generalised Whittaker (or Gelfand-Graev) models serve as the prototypical example for such branching problems. We give a characterization, for the orthogonal and symplectic groups, of the generalised Whittaker models possibly contained in this duality theory. We then exhibit an infinite family of examples of this duality, which, provably at the local level via the theta correspondence, satisfy the conjectural expectations of duality.

引用次数: 0

Pseudolocality theorems of Ricci flows on incomplete manifolds

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-01-24 DOI: 10.1016/j.aim.2025.110127

Liang Cheng

In this paper we study the pseudolocality theorems of Ricci flows on incomplete manifolds. We prove that if a relatively compact ball in an incomplete manifold has the small scalar curvature lower bound and almost Euclidean isoperimetric constant, or almost Euclidean local ν constant, then we can construct a solution of Ricci flow in a smaller ball for which the pseudolocality theorems hold on a uniform time interval. We also give two applications. First, we prove the short-time existence of Ricci flows on complete manifolds with scalar curvature bounded below uniformly and almost Euclidean isoperimetric inequality holds locally. Second, we obtain a rigidity theorem that any complete manifold with nonnegative scalar curvature and Euclidean isoperimetric inequality must be isometric to the Euclidean space.

引用次数: 0

Nielsen realization in dimension four and projective twists

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-01-22 DOI: 10.1016/j.aim.2025.110112

Mihail Arabadji, R. İnanç Baykur

We demonstrate the existence of numerous non-spin 4–manifolds for which the smooth Nielsen realization problem fails; namely, there exist finite subgroups of their mapping class groups that cannot be realized by any group of diffeomorphisms. This extends and complements recent results for spin 4–manifolds. Our examples span virtually all possible intersection forms, both even and odd, indefinite and definite, and include many irreducible 4–manifolds. To derive these examples, we study multi-twists, projective twists, and multi-reflections, which are all mapping classes supported around collections of embedded spheres and projective planes. Our obstructions to Nielsen realization are based on the work of Konno. We investigate projective twists in further detail, and notably, employ them to show that, for many closed symplectic 4–manifolds, the symplectic Torelli group is not generated by squared Dehn twists.

引用次数: 0

Mean field coupled dynamical systems: Bifurcations and phase transitions

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-01-21 DOI: 10.1016/j.aim.2025.110115

Wael Bahsoun , Carlangelo Liverani

We develop a bifurcation theory for infinite dimensional systems satisfying abstract hypotheses tailored for applications to mean field coupled chaotic maps. Our abstract theory can be applied to many cases, from globally coupled expanding maps to globally coupled Axiom A diffeomorphisms. We analyze an explicit example consisting of globally coupled Anosov diffeomorphisms. For such an example, we classify all the invariant measures as the coupling strength varies; we show which invariant measures are physical, and we prove that the existence of multiple invariant physical measures is an infinite dimensional phenomenon, i.e., the model exhibits phase transitions in the sense of statistical mechanics.

引用次数: 0

Strong maximum principle for generalized solutions to equations of the Monge-Ampère type

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-01-21 DOI: 10.1016/j.aim.2025.110116

Huaiyu Jian , Xushan Tu

In this paper, we investigate the strong maximum principle for generalized solutions of Monge-Ampère type equations. We prove that the strong maximum principle holds at points where the function is strictly convex but not necessarily

C^{1, 1}

smooth, and show that it fails at non-strictly convex points. The results we obtain can be applied to various Minkowski type problems in convex geometry by the virtue of the Gauss image map.

引用次数: 0

BPS Lie algebras and the less perverse filtration on the preprojective CoHA

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-01-20 DOI: 10.1016/j.aim.2025.110114

Ben Davison

The affinization morphism for the stack

M (Π_{Q})

of representations of a preprojective algebra

Π_{Q}

is a local model for the morphism from the stack of objects in a general 2-Calabi–Yau category to the good moduli space. We show that the derived direct image of the dualizing complex along this morphism is pure, and admits a decomposition in the sense of the Beilinson–Bernstein–Deligne–Gabber decomposition theorem.

We introduce a new perverse filtration on the Borel–Moore homology of

M (Π_{Q})

, using this decomposition. We show that the zeroth piece of the resulting filtration on the cohomological Hall algebra built out of the Borel–Moore homology of

M (Π_{Q})

is isomorphic to the universal enveloping algebra of an associated BPS Lie algebra

g_{Π_{Q}}

. This Lie algebra is defined via the Kontsevich–Soibelman theory of critical cohomological Hall algebras for 3-Calabi–Yau categories. We then lift this Lie algebra to a Lie algebra object in the category of perverse sheaves on the coarse moduli space of

Π_{Q}

-modules, and use this algebra structure to prove results about the summands appearing in the above decomposition theorem. In particular, we prove that the intersection cohomology of singular spaces of semistable

Π_{Q}

-modules provide “cuspidal cohomology” – a conjecturally complete canonical subspace of generators for

g_{Π_{Q}}

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引用次数: 0

Bumpless pipe dreams meet puzzles

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-01-16 DOI: 10.1016/j.aim.2025.110113

Neil J.Y. Fan , Peter L. Guo , Rui Xiong

Knutson and Zinn-Justin recently found a puzzle rule for the expansion of the product

G_{u} (x, t) \cdot G_{v} (x, t)

of two double Grothendieck polynomials indexed by permutations with separated descents. We establish its triple Schubert calculus version in the sense of Knutson and Tao, namely, a formula for expanding

G_{u} (x, y) \cdot G_{v} (x, t)

in different secondary variables. Our rule is formulated in terms of pipe puzzles, incorporating the structures of both bumpless pipe dreams and classical puzzles. As direct applications, we recover the separated-descent puzzle formula by Knutson and Zinn-Justin (by setting

y = t

) and the bumpless pipe dream model of double Grothendieck polynomials by Weigandt (by setting

v = id

and

x = t

). Moreover, we utilize the formula to partially confirm a positivity conjecture of Kirillov about applying a skew operator to a Schubert polynomial.

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引用次数: 0

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