We prove an equivariant version of the Cosmetic Surgery Conjecture for strongly invertible knots. Our proof combines a recent result of Hanselman with the Khovanov multicurve invariants and . We apply the same techniques to reprove a result of Wang about the Cosmetic Crossing Conjecture and split links. Along the way, we show that and detect if a Conway tangle is split.

In the present paper, we show that for an optimal class of elliptic operators with non-smooth coefficients on a 1-sided Chord-Arc domain, the boundary of the domain is uniformly rectifiable if and only if the Green function behaves like a distance function to the boundary, in the sense that is the density of a Carleson measure, where is a regularized distance adapted to the boundary of the domain. The main ingredient in our proof is a corona decomposition that is compatible with Tolsa's -number of uniformly rectifiable sets. We believe that the method can be applied to many other problems at the intersection of PDE and geometric measure theory, and in particular, we are able to derive a generalization of the classical F. and M. Riesz theorem to the same class of elliptic operators as above.

For , let be a random matrix, a real deterministic matrix, and the corresponding structured random matrix. We study the expected operator norm of considered as a random operator between and for . We prove optimal bounds up to logarithmic terms when the underlying random matrix has i.i.d. Gaussian entries, independent mean-zero bounded entries, or independent mean-zero ( ) entries. In certain cases, we determine the precise order of the expected norm up to constants. Our results are expressed through a sum of operator norms of Hadamard products and .

Given a closed smooth manifold of even dimension with finite fundamental group, we show that the classifying space of the diffeomorphism group of is of finite type and has finitely generated homotopy groups in every degree. We also prove a variant of this result for manifolds with boundary and deduce that the space of smooth embeddings of a compact submanifold of arbitrary codimension into has finitely generated higher homotopy groups based at the inclusion, provided the fundamental group of the complement is finite. As an intermediate result, we show that the group of homotopy classes of simple homotopy self-equivalences of a finite CW complex with finite fundamental group is up to finite kernel commensurable to an arithmetic group.

Being driven by the goal of finding edge modes and of explaining the occurrence of edge modes in the case of time-modulated metamaterials in the high-contrast and subwavelength regime, we analyse the topological properties of Floquet normal forms of periodically parameterized time-periodic linear ordinary differential equations . In fact, our main goal being the question whether an analogous principle as the bulk-boundary correspondence of solid-state physics is possible in the case of Floquet metamaterials, i.e., subwavelength high-contrast time-modulated metamaterials. This paper is a first step in that direction. Since the bulk-boundary correspondence states that topological properties of the bulk materials characterize the occurrence of edge modes, we dedicate this paper to the topological analysis of subwavelength solutions in Floquet metamaterials. This work should thus be considered as a basis for further investigation on whether topological properties of the bulk materials are linked to the occurrence of edge modes. The subwavelength solutions being described by a periodically parameterized time-periodic linear ordinary differential equation , we put ourselves in the general setting of periodically parameterized time-periodic linear ordinary differential equations and introduce a way to (topologically) classify a Floquet normal form ,   of the associated fundamental solution . This is achieved by analysing the topological properties of the eigenvalues and eigenvectors of the monodromy matrix and the Lyapunov transformation . The corresponding topological invariants can then be applied to the setting of Floquet metamaterials. In this paper these general results are considered in the case of a hexagonal structure. We provide two interesting examples of topologically non-trivial time-modulated hexagonal structures.

Recently, Debruyne and Tenenbaum proved asymptotic formulas for the number of partitions with parts in ( ) and good analytic properties of the corresponding zeta function, generalizing work of Meinardus. In this paper, we extend their work to prove asymptotic formulas if is a multiset of integers and the zeta function has multiple poles. In particular, our results imply an asymptotic formula for the number of irreducible representations of degree of . We also study the Witten zeta function , which is of independent interest.

The paper is devoted to the analysis of the global well-posedness and the interior regularity of the 2D Navier-Stokes equations with inhomogeneous stochastic boundary conditions. The noise, white in time and coloured in space, can be interpreted as the physical law describing the driving mechanism on the atmosphere-ocean interface, i.e. as a balance of the shear stress of the ocean and the horizontal wind force.

We consider a two-dimensional determinantal point process arising in the random normal matrix model and which is a two-parameter generalization of the complex Ginibre point process. In this paper, we prove that the probability that no points lie on any number of annuli centered at 0 satisfies large asymptotics of the form where is the number of points of the process. We determine the constants explicitly, as well as the oscillatory term which is of order 1. We also allow one annulus to be a disk, and one annulus to be unbounded. For the complex Ginibre point process, we improve on the best known results: (i) when the hole region is a disk, only were previously known, (ii) when the hole region is an unbounded annulus, only were previously known, and (iii) when the hole region is a regular annulus in the bulk, only was previously known. For general values of our parameters, even is new. A main discovery of this work is that is given in terms of the Jacobi theta function. As far as we know this is the first time this function appears in a large gap problem of a two-dimensional point process.

We prove the Manin-Peyre conjecture for the number of rational points of bounded height outside of a thin subset on a family of Fano threefolds of bidegree (1, 2).

We derive the variational formula of the Loewner driving function of a simple chord under infinitesimal quasiconformal deformations with Beltrami coefficients supported away from the chord. As an application, we obtain the first variation of the Loewner energy of a Jordan curve, defined as the Dirichlet energy of its driving function. This result gives another explanation of the identity between the Loewner energy and the universal Liouville action introduced by Takhtajan and Teo, which has the same variational formula. We also deduce the variation of the total mass of loops touching the Jordan curve under quasiconformal deformations.

We prove that twisted -Betti numbers of locally indicable groups are equal to the usual -Betti numbers rescaled by the dimension of the twisting representation; this answers a question of Lück for this class of groups. It also leads to two formulae: given a fibration with base space having locally indicable fundamental group, and with a simply-connected fiber , the first formula bounds -Betti numbers of in terms of -Betti numbers of and usual Betti numbers of ; the second formula computes exactly in terms of the same data, provided that is a high-dimensional sphere. We also present an inequality between twisted Alexander and Thurston norms for free-by-cyclic groups and 3-manifolds. The technical tools we use come from the theory of generalised agrarian invariants, whose study we initiate in this paper.

We prove that, in the flat torus and in any dimension, the volume-preserving mean curvature flow and the surface diffusion flow, starting -close to a strictly stable critical set of the perimeter ,  exist for all times and converge to a translate of exponentially fast as time goes to infinity.

We consider the varifold associated to the Allen-Cahn phase transition problem in (or -dimensional Riemannian manifolds with bounded curvature) with integral bounds on the Allen-Cahn mean curvature (first variation of the Allen-Cahn energy) in this paper. It is shown here that there is an equidistribution of energy between the Dirichlet and Potential energy in the phase field limit and that the associated varifold to the total energy converges to an integer rectifiable varifold with mean curvature in . The latter is a diffused version of Allard's convergence theorem for integer rectifiable varifolds.

Bohnenblust-Hille inequalities for Boolean cubes have been proven with dimension-free constants that grow subexponentially in the degree (Defant et al. in Math Ann 374(1):653-680, 2019). Such inequalities have found great applications in learning low-degree Boolean functions (Eskenazis and Ivanisvili in Proceedings of the 54th annual ACM SIGACT symposium on theory of computing, pp 203-207, 2022). Motivated by learning quantum observables, a qubit analogue of Bohnenblust-Hille inequality for Boolean cubes was recently conjectured in Rouzé et al. (Quantum Talagrand, KKL and Friedgut's theorems and the learnability of quantum Boolean functions, 2022. arXiv preprint arXiv:2209.07279). The conjecture was resolved in Huang et al. (Learning to predict arbitrary quantum processes, 2022. arXiv preprint arXiv:2210.14894). In this paper, we give a new proof of these Bohnenblust-Hille inequalities for qubit system with constants that are dimension-free and of exponential growth in the degree. As a consequence, we obtain a junta theorem for low-degree polynomials. Using similar ideas, we also study learning problems of low degree quantum observables and Bohr's radius phenomenon on quantum Boolean cubes.

Let be non-empty, open and proper. This paper is concerned with , the space of -integrable Borel measures on equipped with the transportation metric introduced by Figalli and Gigli that allows the creation and destruction of mass on . Alternatively, we show that is isometric to a subset of Borel measures with the ordinary Wasserstein distance, on the one point completion of equipped with the shortcut metric In this article we construct bi-Lipschitz embeddings of the set of unordered -tuples in into Hilbert space. This generalises Almgren's bi-Lipschitz embedding theorem to the setting of optimal partial transport.

This paper provides a characterization of expansive matrices generating the same anisotropic homogeneous Triebel-Lizorkin space for and . It is shown that if and only if the homogeneous quasi-norms associated to the matrices ,  are equivalent, except for the case with . The obtained results complement and extend the classification of anisotropic Hardy spaces , , in Bownik (Mem Am Math Soc 164(781):vi+122, 2003).

Let be either a free group or the fundamental group of a closed hyperbolic surface. We show that if is a finitely generated residually- group with the same pro- completion as , then two-generated subgroups of are free. This generalises (and gives a new proof of) the analogous result of Baumslag for parafree groups. Our argument relies on the following new ingredient: if is a residually-(torsion-free nilpotent) group and is a virtually polycyclic subgroup, then is nilpotent and the pro- topology of induces on its full pro- topology. Then we study applications to profinite rigidity. Remeslennikov conjectured that a finitely generated residually finite with profinite completion is necessarily . We confirm this when belongs to a class of groups that has a finite abelian hierarchy starting with finitely generated residually free groups. This strengthens a previous result of Wilton that relies on the hyperbolicity assumption. Lastly, we prove that the group is profinitely rigid within finitely generated residually free groups.

We show that a recently-derived model for the propagation of nonlinear waves in the atmosphere admits undular bores as travelling-wave solutions. These solutions represent waves consisting of a damped oscillation behind a front that is preceded by a uniform breeze-type flow. The generation of such wave profiles requires a jump in the heat source across the leading front of the wave, a feature that is consistent with observations.

This paper establishes power-saving bounds for Kloosterman sums associated with the long Weyl element for for arbitrary , as well as for another type of Weyl element of order 2. These bounds are obtained by establishing an explicit representation as exponential sums. As an application we go beyond Sarnak's density conjecture for the principal congruence subgroup of prime level. We also obtain power-saving bounds for all Kloosterman sums on .

We investigate norms of spectral projectors on thin spherical shells for the Laplacian on generic tori, including generic rectangular tori. We state a conjecture and partially prove it, improving on previous results concerning arbitrary tori.

The stable reduction theorem says that a family of curves of genus over a punctured curve can be uniquely completed (after possible base change) by inserting certain stable curves at the punctures. We give a new this result for curves defined over , using the Kähler-Einstein metrics on the fibers to obtain the limiting stable curves at the punctures.