Let M be a closed 3-manifold which is a finite cover of degree n along the fibers over the unit tangent bundle of an oriented connected closed surface of genus g>1. We prove that if n is odd, there is only one Anosov flow on M up to orbital equivalence, and if n is even, there are exactly two orbital equivalence classes of Anosov flows on M.
We construct a sheaf theoretic and derived geometric machinery to study nonlinear partial differential equations and their singular supports. We establish a notion of derived microlocalization for solution spaces of non-linear equations and develop a formalism to pose and solve singular non-linear Cauchy problems globally. Using this approach we estimate the domains of propagation for the solutions of non-linear systems. It is achieved by exploiting the fact that one may greatly enrich and simplify the study of derived non-linear PDEs over a space X by studying its derived linearization which is a module over the sheaf of functions on the S1-equivariant derived loop stack LX.
One handle stabilization adds a handle to an embedded surface. It is known that any pair of smoothly knotted surfaces in a 4-manifold become smoothly isotopic after sufficiently many one handle stabilizations. An internal stabilization is a homotopically trivial one handle stabilization. In this paper, we show that there is no upper bound on the number of internal stabilizations required. In fact, this behavior is fairly generic. The definition of a subtly smoothly knotted pair of surfaces is given, and it is shown that many surfaces may be modified to obtain subtly knotted surfaces with large internal stabilization distance. Similarly, a pair of smoothly knotted surfaces in a 4-manifold become equivalent after sufficiently many finger-Whitney moves. The surfaces with large internal stabilization distance will also have large finger-Whitney distance. Furthermore, it is shown that after stabilizing, any simply connected 4-manifold will contain topologically isotopic, smoothly related, non-isotopic copies of any 3-manifold having positive first betti number.
In a given complete metric measure space of homogeneous type (X,ρ,μ) supporting a Poincaré inequality, by introducing two families of lifted Sobolev sharp maximal operators {Mθ♯}θ∈(0,1] and {fθ⁎}θ∈(0,1], we prove that, for p∈[1,∞), γ∈R in an optimal range, and f∈Lip(X),supθ∈(0,1]‖Mθ♯(f)(x,t)tγ‖Lp,∞(X×(0,∞),tγp−1dtdμ(x))∼‖lipf‖Lp(X) andsupθ∈(0,1]‖fθ⁎(x,y)[U(x,y)]γ‖Lp,∞(X×X,[U(x,y)]γp−1dμ(y)dμ(x))∼‖lipf‖Lp(X), where lip f denotes the pointwise lower Lipschitz-constant function of f,U(x,y):=min{μ(B(x,ρ(x,y))),μ(B(y,ρ(x,y)))}, and the positive equivalence constants are independent of f. The key ingredient in their proofs is to establish two optimal estimates involving dyadic cubes and BV functions by using η-good dyadic cubes, the coarea formula, and the isoperimetric inequality. As applications, we establish new weak-type representations for the Sobolev seminorms, including endpoint estimates. In particular, utilizing the geometry of Grushin-type vector fields, we identify the aforementioned optimal range of γ. This gives an affirmative answer to the question in Remark 2.13 of Dai et al. (2022) [31]. As further applications, we obtain several Gagliardo–Nirenberg type inequalities and an estimate for thresholding operators related to Haar wavelets, both of which are optimal when X:=Rn.
In this paper we study the Birkhoff billiards inside cones in Rn. We prove that every trajectory inside a cone over a C3 strictly convex closed hypersurface embedded in Rn-1 with nondegenerate second fundamental form has a finite number of reflections. Using this result we prove that the billiard admits first integrals whose values uniquely determine all billiard trajectories. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
We study a closed differential form on the symmetric space of positive definite matrices, which is defined using the Pfaffian and is GL2n(Z) invariant up to a sign. It gives rise to an infinite family of unstable classes in the compactly-supported cohomology of the locally symmetric space for GL2n(Z) with coefficients in the orientation bundle. Furthermore, by applying the Pfaffian forms to the dual Laplacian of graphs, and integrating them over the space of edge lengths, we construct an infinite family of cocycles for the odd commutative graph complex. By explicit computation, we show that the first such cocycle gives a non-trivial class in H−6(GC3).
We confirm the conjecture posed by Guedon, Hinrichs, Litvak, and Prochno in 2017 that E(a(ij)g(ij))(i) <= m,j( )<= n: & ell;(n ->)(p) & ell;(q)(q) is comparable, up to constants depending only on p and q, to max (i) ||(a(ij))(j)||(p* )+ max (j )||(a(ij))(i)||(q )+ E max( i,j )|a(ij)g(ij)| provided that 1 <= p <= 2 <= q <= infinity. This was known before only in the case p = 1 or q = infinity, and in the spectral case p = 2 = q. We also reprove the conjecture in the case p = 2 = q without using spectral theory (which was employed in the previously known proof). (c) 2026 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC license (http:// creativecommons.org/licenses/by-nc/4.0/).
Let D1 subset of D2 be (phi, f)-modules of rank 2 over the Robba ring, and 7r(D1), 7r(D2) be the associated locally analytic representations of GL2(Qp) via the p-adic local Langlands correspondence. We describe the relation between 7r(D1) and 7r(D2). (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
We explore the relationship between multigraded Castelnuovo-Mumford regularity, truncations, Betti numbers, and virtual resolutions on a product of projective spaces X. After proving a uniqueness theorem for certain virtual resolutions, we show that the multigraded regularity region of a module M is determined by the minimal graded free resolutions of the truncations M >= d for d is an element of Pic X. Further, by relating the minimal graded free resolutions of M and M >= d we provide a new bound on multigraded regularity of Min terms of its Betti numbers. Using this characterization of regularity and this bound we also compute the multigraded Castelnuovo-Mumford regularity for a wide class of complete intersections in products of projective spaces. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
We explore some connections between vectors of integers and integer partitions seen as bi-infinite words. On the one hand, this methodology enables us to obtain enumerations connecting products of hook lengths and vectors of integers. On the other hand, this yields a combinatorial interpretation of the Macdonald identities for affine root systems of the 7 infinite families in terms of Schur functions, symplectic and special orthogonal Schur functions with respect to the type of the considered root system. From these results, we are able to derive q-Nekrasov-Okounkov formulas associated to each type. The latter for limit cases of q yield Nekrasov-Okounkov type formulas corresponding to all the specializations given by Macdonald. When q goes to 1, one can derive combinatorial developments of Euler product, answering an open problem from Han. (c) 2026 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).
By applying a small perturbation via continuous Steiner symmetrization, we calculate the first variation of the energy functional for the given bounded open set in Euclidean space Rdin two distinct ways, which enables us to conclude that hypersurfaces with prescribed nonlocal mean curvature are spheres. The energy functional is linked to the fractional perimeter as first identified by Caffarelli, Roquejoffre, and Savin [5]. In fact, studying sets with prescribed nonlocal mean curvature is also associated with the symmetry properties of stationary patch solutions of the generalized surface quasigeostrophic equations, where the velocity fields are influenced by a potential that is more singular than the Riesz potential. (c) 2026 Published by Elsevier Inc.
In this paper, we construct the multicomponent super boson-fermion correspondence of type B to derive the multicomponent super BKP (SBKP) hierarchy. Algebraic foundations are established through neutral free superfermions within a Clifford superalgebra and super bosonic fields satisfying Lie superalgebra relations. Using vertex operator, we formulate the hierarchy in super fermionic Fock space and derive its bosonic realization via the correspondence. For a positive integer s denoting the number of components, the main results include derivation of bilinear identities in both super fermionic and bosonic Fock spaces, formulation of super Hirota bilinear equations with explicit examples for s = 1 (recovering BKP and super KdV) and s = 2 (2-component BKP) equations, and the construction of super Baker-Akhiezer functions via super pseudo-differential operators. We further extend these results to the multicomponent super modified BKP (SmBKP) hierarchy with tau-functions and present dressing operator formulations for both hierarchies.
We prove that for each d > 3 and k > 2, the set of limit points of the first k eigenvalues of sequences of d-regular graphs is {(mu(1), . . . , mu(k)) : d = mu(1) >= & centerdot;& centerdot;& centerdot; >= mu(k) >= 2 root d-1}. The result for k = 2 was obtained by Alon and Wei, and our result confirms a conjecture of theirs. Our proof uses an infinite random graph sampled from a distribution that generalizes the random regular graph distribution. To control the spectral behavior of this infinite object, we show that Huang and Yau's proof of Friedman's theorem bounding the second eigenvalue of a random regular graph generalizes to this model. We also bound the trace of the non-backtracking operator, as was done in Bordenave's separate proof of Friedman's theorem. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
We show that if M is a closed, connected, oriented surface, and two Anosov magnetic systems on Mare conjugate by a volume-preserving conjugacy isotopic to the identity, with their magnetic forms in the same cohomology class, then the metrics are isometric. This extends the recent result by Guillarmou, Lefeuvre, and Paternain to the magnetic setting. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Recent works have initiated the study of open r--spin and open Fan-Jarvis-Ruan-Witten (FJRW) intersection theories, and related them to integrable hierarchies and mirror symmetry. This paper uses a new technique, the point insertion technique, to define new open r--spin and open FJRW intersection theories. These new constructions provide candidates for theories whose existence was conjectured before: 1. Hori predicted the existence of open r--spin theory with [ r r2 j types of boundary states. The previously constructed open r--spin theory (Buryak-Clader-Tessler 2018) has only boundary states of one type. In this work we describe [ [r [r2 j open r--spin theories, labelled by Cj E {0, ... , [ r r2 j-1}, where the Cjth one has Cj + 1 types of boundary states. We prove that the Cj = 0 theory is equivalent to the previous construction, and calculate all intersection numbers for all these theories. 2. Aleshkin and Liu conjectured the existence of a quintic Fermat FJRW theory. We construct such an FJRW theory, and provide evidence that this is the conjectured theory. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
In this paper we study higher level Deligne-Lusztig representations of reductive groups over discrete valuation rings with finite residue field Fq. In previous work we proved that at even levels, these geometrically constructed representations are isomorphic to certain algebraically constructed representations (referred to as the algebraisation theorem at even levels). In this paper we work with an arbitrary level > 1. Our main result is (1) the algebraisation theorem at all levels > 1 (with the sign being explicitly determined for q >= 7). As consequences, we obtain (2) the regular semisimplicity of orbits of generic higher level Deligne-Lusztig representations, and the dimension formula; in the course of the proof, we give (3) an induction formula of higher level Deligne-Lusztig representations, and a new proof of the character formula at regular semisimple elements. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
We settle the problem of finding the sharp constant in the log Sobolev inequality on the n-cycle for all n≥ 4, by showing that it is equal to half of the spectral gap. We deduce this result from an optimal cubic Sobolev inequality.
An important research topic in fluid dynamics equations is to study the stability of nonlinear waves. With small perturbations of the initial data around nonlinear waves, there are many stability results for scalar equations and systems. However, for large initial perturbations around nonlinear waves, there are few global stability results on systems except for scalar equations. In this paper, we study the global stability of rarefaction waves to the Cauchy problem/initialboundary value problem for a hyperbolic-elliptic coupled system describing radiating gas flows. We prove that the solutions converge to the corresponding rarefaction waves as time tends to infinity, where both the initial perturbation around rarefaction waves and the wave strength |u-- u+| can be arbitrarily large. To the best of our knowledge, it seems to be the first result on the global stability of nonlinear waves for hyperbolic-structured fluid dynamics equations with arbitrarily large initial perturbations and arbitrarily large wave strength in the classical sense. The proof is based on the fact that the hyperbolic-elliptic coupled equations can be rewritten into a scalar equation form with convolution terms. It is the scalar equation form that enables us to show the monotonicity of the solution by using the maximum principle, under suitable conditions upon the initial data. This allows us to handle the growth of nonlinear terms resulting from large perturbation in H2-energy estimates. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
