In this paper, we show that the quantum disk, i.e. the quantum space corresponding to the Toeplitz C∗-algebra does not admit any compact quantum group structure. We prove that if such a structure existed the resulting compact quantum group would simultaneously be of Kac type and not of Kac type. The main tools used in the solution come from the theory of operators on Hilbert spaces.

In this paper, we study the moduli space of frozen planet orbits in the Helium atom for an interpolation between instantaneous and mean interactions and show that this moduli space is compact.

We prove uniqueness, up to diffeomorphism, of symplectically aspherical fillings of certain unit cotangent bundles, including those of higher-dimensional tori.

In this paper, we consider the Galois covers of algebraic surfaces of degree 6, with all associated planar degenerations. We compute the fundamental groups of those Galois covers, using their degeneration. We show that for 8 types of degenerations, the fundamental group of the Galois cover is non-trivial and for 20 types it is trivial. Moreover, we compute the Chern numbers of all the surfaces with

The goal of the present work is to introduce new metric space techniques to study the intersection properties of families of balls. These new techniques add, on the one hand, to results due to Lindenstrauss on the extension of uniformly continuous functions and compact linear operators, and answer, on the other hand, questions raised by Aronszajn and Panitchpakdi on hyperconvex metric spaces. The present

For any g≥3, we construct genus-g Lefschetz fibrations over the two-sphere whose slopes are arbitrarily close to 2. The total spaces of the Lefschetz fibrations can be chosen to be minimal and simply connected. It is also shown that the infimum and the supremum of slopes of all Lefschetz fibrations over the two-sphere are not realized as slopes.

Let MΣ be an n-dimensional Thom–Mather stratified space of depth 1. We denote by βM the singular locus and by L the associated link. In this paper, we study the problem of when such a space can be endowed with a wedge metric of positive scalar curvature. We relate this problem to recent work on index theory on stratified spaces, giving first an obstruction to the existence of such a metric in terms

We relate the quantum Steenrod square to Seidel’s equivariant pair-of-pants product for open convex symplectic manifolds that are either monotone or exact, using an equivariant version of the PSS isomorphism. We define continuation maps between different Hamiltonians in ℤ/2-equivariant Floer cohomology, and prove expected properties of them. We prove a symplectic Cartan relation for the equivariant

For an element in BS(1,n)=〈t,a|tat−1=an〉 written in the normal form t−uavtw with u,w≥0 and v∈ℤ, we exhibit a geodesic word representing the element and give a formula for its word length with respect to the generating set {t,a}. Using this word length formula, we prove that there are sets of elements of positive density of positive, negative and zero conjugation curvature, as defined by Bar Natan,

We study the nonlinear embeddability of Banach spaces and the equi-embeddability of the family of Kalton’s interlaced graphs ([ℕ]k,d𝕂)k into dual spaces. Notably, we define and study a modification of Kalton’s property 𝒬 that we call property 𝒬p (with p∈(1,+∞]). We show that if ([ℕ]k,d𝕂)k equi-coarse Lipschitzly embeds into X∗, then the Szlenk index of X is greater than ω, and that this is optimal

We show that the Torelli group of a closed surface of genus ≥3 acts nontrivially on the rational cohomology of the space of 3-element subsets of that surface. This implies that for a Riemann surface of genus ≥3, the mixed Hodge structure on the space of its positive, reduced divisors of degree 3 does in general not split over ℚ.

We generalize the notions of asymptotic dimension and coarse embeddings from metric spaces to quantum metric spaces in the sense of Kuperberg and Weaver [A von Neumann algebra approach to quantum metrics, Mem. Am. Math. Soc.215(1010) (2012) 1–80]. We show that quantum asymptotic dimension behaves well with respect to metric quotients and direct sums, and is preserved under quantum coarse embeddings

We show that the integral foliated simplicial volume of a compact oriented smooth manifold with a regular foliation by circles vanishes.

We study the homotopy type of the space of the unitary group U1(Cu∗(|ℤn|)) of the uniform Roe algebra Cu∗(|ℤn|) of ℤn. We show that the stabilizing map U1(Cu∗(|ℤn|))→U∞(Cu∗(|ℤn|)) is a homotopy equivalence. Moreover, when n=1,2, we determine the homotopy type of U1(Cu∗(|ℤn|)), which is the product of the unitary group U1(C∗(|ℤn|)) (having the homotopy type of U∞(ℂ) or ℤ×BU∞(ℂ) depending on the parity

For an orientable surface of finite type equipped with a flat metric with holonomy of finite order q, the set of maximal embedded cylinders can be empty, non-empty, finite, or infinite. The case when q≤2 is well-studied as such surfaces are (semi-)translation surfaces. Not only is the set always infinite, the core curves form an infinite diameter subset of the curve complex. In this paper we focus

In this paper, we introduce a fast and memory efficient approach to compute the Persistent Homology (PH) of a sequence of simplicial complexes. The basic idea is to simplify the complexes of the input sequence by using strong collapses, as introduced by Barmak and Miniam [DCG (2012)], and to compute the PH of an induced sequence of reduced simplicial complexes that has the same PH as the initial one

We show that closed arithmetic hyperbolic 3-dimensional orbifolds with larger and larger volumes give rise to triangulations of the underlying spaces whose 1-skeletons are harder and harder to embed nicely in Euclidean space. To show this we generalize an inequality of Gromov and Guth to hyperbolic n-orbifolds and find nearly optimal geodesic triangulations of arithmetic hyperbolic 3-orbifolds.

In this paper, we introduce and study a notion of asymptotic expansion in measure for measurable actions. This generalizes expansion in measure and provides a new perspective on the classical notion of strong ergodicity. Moreover, we obtain structure theorems for asymptotically expanding actions, showing that they admit exhaustions by domains of expansion. As an application, we recover a recent result

Suppose a finitely generated group G is hyperbolic relative to 𝒫 a set of proper finitely generated subgroups of G. Established results in the literature imply that a “visual” metric on ∂(G,𝒫) is “linearly connected” if and only if the boundary ∂(G,𝒫) has no cut point. Our goal is to produce linearly connected metrics on ∂(G,𝒫) that are “piecewise” visual when ∂(G,𝒫) contains cut points. Our main

Suppose a sequence Mj of Alexandrov spaces collapses to a space X with only weak singularities. Yamaguchi constructed a map fj:Mj→X called an almost Lipschitz submersion for large j. We prove that if Mj has a uniform positive lower bound for the volumes of spaces of directions, which is sufficiently large compared to the weakness of singularities of X, then fj is a locally trivial fibration. Moreover

In this paper, we study symplectic embedding questions for the ℓp-sum of two discs in ℝ4, when 1≤p≤∞. In particular, we compute the symplectic inner and outer radii in these cases, and show how different kinds of embedding rigidity and flexibility phenomena appear as a function of the parameter p.

This paper is part of the program of studying large-scale geometric properties of totally disconnected locally compact groups, TDLC-groups, by analogy with the theory for discrete groups. We provide a characterization of hyperbolic TDLC-groups, in terms of homological isoperimetric inequalities. This characterization is used to prove the main result of this paper: for hyperbolic TDLC-groups with rational

In this paper, we study the asymptotic Plateau problem in ℍ2×ℝ. We construct the first examples of non-fillable finite curves with no thin tail in S∞1×ℝ.

We discover a bifurcation of the perturbations of non-generic closed self-shrinkers. If the perturbation is outward, then the next mean curvature flow singularity is cylindrical and collapsing from outside; if the perturbation is inward, then the next mean curvature flow singularity is cylindrical and collapsing from inside.

An n-dimensional knot Sn⊂Sn+2 is called doubly slice if it occurs as the cross section of some unknotted (n+1)-dimensional knot. For every n it is unknown which knots are doubly slice, and this remains one of the biggest unsolved problems in high-dimensional knot theory. For ℓ>1, we use signatures coming from L(2)-cohomology to develop new obstructions for (4ℓ−3)-dimensional knots with metabelian knot

An estimate for the modulus of convexity and characteristic of convexity of a general direct sum of Banach spaces is established. Using direct sums, we construct a space with given characteristic of convexity and the value of the modulus of convexity at 2. A result on uniform convexity of spaces obtained with the general discrete interpolation method is proved.

This paper uses relative symplectic cohomology, recently studied by Varolgunes, to understand rigidity phenomena for compact subsets of symplectic manifolds. As an application, we consider a symplectic crossings divisor in a Calabi–Yau symplectic manifold M whose complement is a Liouville manifold. We show that, for a carefully chosen Liouville structure, the skeleton as a subset of M exhibits strong

We give explicit examples of pairs of one-ended, open 4-manifolds whose end-sums yield uncountably many manifolds with distinct proper homotopy types. This answers strongly in the affirmative a conjecture of Siebenmann regarding nonuniqueness of end-sums. In addition to the construction of these examples, we provide a detailed discussion of the tools used to distinguish them; most importantly, the

We initiate the study of perturbation of von Neumann algebras relatively to the Banach–Mazur distance. We first prove that the type decomposition is continuous, i.e. if two von Neumann algebras are close, then their respective summands of each type are close. We then prove that, under some vanishing conditions on its Hochschild cohomology groups, a von Neumann algebra is Banach–Mazur stable, i.e. any

We construct a geodesic net in the plane with four unbalanced (boundary) vertices that has sixteen balanced vertices and does not contain nontrivial subnets. This is the first example of an irreducible geodesic net in the Euclidean plane with four boundary vertices that contain cycles of balanced vertices.

We define a variant of Benjamini–Schramm convergence for finite simplicial complexes with the action of a fixed finite group G which leads to the notion of unimodular random rooted simplicial G-complexes. For every unimodular random rooted simplicial G-complex we define a corresponding ℓ2-homology and the ℓ2-multiplicity of an irreducible representation of G in the homology. The ℓ2-multiplicities generalize

The tail of a quantum spin network in the two-sphere is a q-series associated to the network. We study the existence of the head and tail functions of quantum spin networks colored by 2n. We compute the q-series for an infinite family of quantum spin networks and give the relation between the tail of these networks and the tail of the colored Jones polynomial. Finally, we show that the family of quantum

Lazarev and Lieb showed that finitely many integrable functions from the unit interval to ℂ can be simultaneously annihilated in the L2 inner product by a smooth function to the unit circle. Here, we answer a question of Lazarev and Lieb proving a generalization of their result by lower bounding the equivariant topology of the space of smooth circle-valued functions with a certain W1,1-norm bound.

We introduce a notion of Ricci curvature for Cayley graphs that can be thought of as “medium-scale” because it is neither infinitesimal nor asymptotic, but based on a chosen finite radius parameter. We argue that it gives the foundation for a definition of Ricci curvature well adapted to geometric group theory, beginning by observing that the sign can easily be characterized in terms of conjugation

Any knot K in genus-11-bridge position can be moved by isotopy to lie in a union of n parallel tori tubed by n−1 tubes so that K intersects each tube in two spanning arcs, which we call a leveling of the position. The minimal n for which this is possible is an invariant of the position, called the level number. In this work, we describe the leveling by the braid group on two points in the torus, which

Let SLn(ℤ) be the special linear group over integers and Mr=Sr1×Sr2,Tr1×Sr2, or Tr0×Sr1×Sr2, products of spheres and tori. We prove that any group action of SLn(ℤ) on Mr by diffeomorphims or piecewise linear homeomorphisms is trivial if r

We introduce the notion of controlled products on metric spaces as a generalization of Gromov products, and construct boundaries by using controlled products, which we call the Gromov boundaries. It is shown that the Gromov boundary with respect to a controlled product on a proper metric space is the ideal boundary of a coarse compactification of the space. It is also shown that there is a bijective

We study smooth isotopy classes of complex curves in complex surfaces from the perspective of the theory of bridge trisections, with a special focus on curves in ℂℙ2 and ℂℙ1×ℂℙ1. We are especially interested in bridge trisections and trisections that are as simple as possible, which we call efficient. We show that any curve in ℂℙ2 or ℂℙ1×ℂℙ1 admits an efficient bridge trisection. Because bridge trisections

We develop a local index theory for a class of operators associated with non-proper and non-isometric actions of Lie groupoids on smooth submersions. Such actions imply the existence of a short exact sequence of algebras, relating these operators to their non-commutative symbol. We then compute the connecting map induced by this extension on periodic cyclic cohomology. When cyclic cohomology is localized

We define a system of groups, a duality system of groups and a PD(n)-system of groups, generalizing the corresponding concepts of pairs of groups. We give several characterizations of duality systems of groups and apply these results to derive an important characterization of PD(n)-systems of groups.

We prove that for 4-manifolds M with residually finite fundamental group and non-spin universal covering M̃, the inequality dimmcM̃≤3 implies the inequality dimmcM̃≤2. This allows us to complete the proof of Gromov’s Conjecture for 4-manifolds with abelian fundamental group.

We consider the problem whether for a group G there exists a constant Λ(G)>1 such that for any (r,s)-matrix A over the integral group ring ℤG the Fuglede–Kadison determinant of the G-equivariant bounded operator L2(G)r→L2(G)s given by right multiplication with A is either one or greater or equal to Λ(G). If G is the infinite cyclic group and we consider only r=s=1, this is precisely Lehmer’s problem

For a nonorientable surface, the twist subgroup is an index 2 subgroup of the mapping class group generated by Dehn twists about two-sided simple closed curves. In this paper, we consider involution generators of the twist subgroup and give generating sets of involutions with smaller number of generators than the ones known in the literature using new techniques for finding involution generators.

We prove that for a free product G with free factor system 𝒢, any automorphism ϕ preserving 𝒢, atoroidal (in a sense relative to 𝒢) and none of whose power send two different conjugates of subgroups in 𝒢 on conjugates of themselves by the same element, gives rise to a semidirect product G⋊ϕℤ that is relatively hyperbolic with respect to suspensions of groups in 𝒢. We recover a theorem of Gautero–Lustig

In this paper, we give a comprehensive treatment of a “Clifford module flow” along paths in the skew-adjoint Fredholm operators on a real Hilbert space that takes values in KO∗(ℝ) via the Clifford index of Atiyah–Bott–Shapiro. We develop its properties for both bounded and unbounded skew-adjoint operators including an axiomatic characterization. Our constructions and approach are motivated by the principle

Let G be a group acting properly and by isometries on a metric space X; it follows that the quotient or orbit space X/G is also a metric space. We study the Vietoris–Rips and Čech complexes of X/G. Whereas (co)homology theories for metric spaces let the scale parameter of a Vietoris–Rips or Čech complex go to zero, and whereas geometric group theory requires the scale parameter to be sufficiently large

Let G be a semisimple Lie group with finite center, K⊂G a maximal compact subgroup, and P⊂G a parabolic subgroup. Following ideas of P. Y. Gaillard, one may use G-invariant differential forms on G/K×G/P to construct G-equivariant Poisson transforms mapping differential forms on G/P to differential forms on G/K. Such invariant forms can be constructed using finite-dimensional representation theory.

In this paper, we introduce the notion of almost flatness for (stably) relative bundles on a pair of topological spaces and investigate basic properties of it. First, we show that almost flatness of topological and smooth sense are equivalent. This provides a construction of an almost flat stably relative bundle on enlargeable manifolds. Second, we show the almost monodromy correspondence, that is

Let Γ be either the mapping class group of a closed surface of genus ≥2, or the automorphism group of a free group of rank ≥3. Given any homological representation ρ of Γ corresponding to a finite cover, and any term ℐk of the Johnson filtration, we show that ρ(ℐk) has finite index in ρ(ℐ), the Torelli subgroup of Γ. Since [ℐ:ℐk]=∞ for k>1, this implies for instance that no such representation is faithful

A subset S of a group G invariably generates G if G is generated by {sg(s)|s∈S} for any choice of g(s)∈G,s∈S. A topological group G is said to be ℐ𝒢 if it is invariably generated by some subset S⊆G, and 𝒯ℐ𝒢 if it is topologically invariably generated by some subset S⊆G. In this paper, we study the problem of (topological) invariable generation for linear groups and for automorphism groups of trees

Let G be a matrix group. Topological G-manifolds with Palais-proper action have the G-homotopy type of countable G-CW complexes (3.2). This generalizes Elfving’s dissertation theorem for locally linear G-manifolds (1996). Also, we improve the Bredon–Floyd theorem from compact Lie groups G to arbitrary Lie groups G.

Let X be a topological space admitting an amenable cover of multiplicity k∈ℕ. We show that, for every n≥k and every α∈Hn(X;ℝ), the image of α in the ℓ1-homology module Hnℓ1(X;ℝ) vanishes. This strengthens previous results by Gromov and Ivanov, who proved, under the same assumptions, that the ℓ1-seminorm of α vanishes.

We prove that on a surface of negative Euler characteristic, two real-analytic Finsler metrics have the same unparametrized oriented geodesics, if and only if they differ by a scaling constant and addition of a closed 1-form.

Following proposals of Ostrover and Polterovich, we introduce and study “coarse” and “fine” versions of a symplectic Banach–Mazur distance on certain open subsets of ℂn and other open Liouville domains. The coarse version declares two such domains to be close to each other if each domain admits a Liouville embedding into a slight dilate of the other; the fine version, which is similar to the distance

We represent the universal Menger curve as the topological realization |𝕄| of the projective Fraïssé limit 𝕄 of the class of all finite connected graphs. We show that 𝕄 satisfies combinatorial analogues of the Mayer–Oversteegen–Tymchatyn homogeneity theorem and the Anderson–Wilson projective universality theorem. Our arguments involve only 0-dimensional topology and constructions on finite graphs

We study the Kasparov product on (possibly non-compact and incomplete) Riemannian manifolds. Specifically, we show on a submersion of Riemannian manifolds that the tensor sum of a regular vertically elliptic operator on the total space and an elliptic operator on the base space represents the Kasparov product of the corresponding classes in KK-theory. This construction works in general for symmetric

In this paper, using ordered total K-theory, we give a K-theoretic classification for the real rank zero inductive limits of direct sums of generalized dimension drop interval algebras.

We investigate properties of some spherical functions defined on hyperbolic groups using boundary representations on the Gromov boundary endowed with the Patterson–Sullivan measure class. We prove sharp decay estimates for spherical functions as well as spectral inequalities associated with boundary representations. This point of view on the boundary allows us to view the so-called property RD (also

We show that a 1-parameter family of real analytic map germs ft:(ℝ2,0)→(ℝ3,0) with isolated instability is topologically trivial if it is excellent and the family of double point curves D(ft) in (ℝ2,0) is topologically trivial. In particular, we deduce that ft is topologically trivial when the Milnor number μ(D(ft)) is constant.

We provide the converses to two results of Roe [Warped cones and property A, Geom. Topol.9 (2005) 163–178, https://doi.org/10.2140/9t.2005.9.163]: first, the warped cone associated to a free action of an a-T-menable group admits a fibered coarse embedding into a Hilbert space, and second, a free action yielding a warped cone with property A must be amenable. We construct examples showing that in both