Schedule for: 25w5347 - Functional and Metric Analysis and their Interactions

We define a notion of asymptotically spherical topological groups, which says that spheres of large radius with respect to any maximal length function are still spherical with respect to any other maximal length function. This is a strengthening of a related condition introduced by Sebastian Hurtado, which we call bounded eccentricity. Our main result is a partial characterization of which groups are asymptotically spherical, and we also give an example of a discrete, bounded eccentric group who fails to be asymptotically spherical. This is joint work with J. Zomback.

I will report on a work in progress concerning the existence of strictly convex renormings of Banach spaces that are invariant with respect to an action of a group by linear isometries. This problem appeared during a problem session at the CIRM meeting "Non-linear functional analysis" in 2018 in a question by Mikael de la Salle and had been already considered in works of Lancien, Ferenczi and Rosendal. I will present some new results showing that the solution depends not only on the Banach space in question, but also on the acting group and the type of action.

In this talk, we introduce the theory of metric spaces with small rough angles (SRA). A metric space $(X,d)$ is said to satisfy the $\text{SRA}(\alpha)$ condition, for $0\leq\alpha<1$, when it fulfills a strengthened form of the triangle inequality. This condition ensures, in particular, that all metric angles determined by triples of points in $X$ remain quantitatively bounded away from $\pi$, with the bound depending on $\alpha$. We explore the connection between this condition and the snowflaking operation, which transforms $(X,d)$ into $(X,d^{\varepsilon})$, for $0<\varepsilon<1$. Additionally, we consider metric spaces that are either $\text{SRA}(\alpha)$ free (meaning there exists a uniform upper bound on the size of any $\text{SRA}(\alpha)$ subset) or $\text{SRA}(\alpha)$ full (meaning the space contains an infinite $\text{SRA}(\alpha)$ subset). Examples of SRA free spaces include Euclidean spaces, finite-dimensional Alexandrov spaces of non-negative curvature, and Cayley graphs of virtually abelian groups. On the other hand, SRA full spaces include the sub-Riemannian Heisenberg group, Laakso graphs or Hilbert spaces. Finally, we apply this theory to investigate the rectifiability of a class of curves, so-called roughly self-contracting curves, which naturally arise in the theory of convex gradient flows.

The conformal dimension of a metric space measures its optimal shape from the perspective of quasisymmetric geometry. Dimension interpolation is an emerging program of research in fractal geometry which identifies geometrically natural one-parameter dimension functions interpolating between existing concepts. Two exemplars are the Assouad spectrum, which interpolates between box-counting and Assouad dimension, and the intermediate dimensions, which interpolate between Hausdorff and box-counting dimension. In this talk, I’ll discuss mapping-theoretic properties of intermediate dimensions and the Assouad spectrum, with applications to the quasiconformal classification of sets and to the range of conformal Assouad spectrum. Based on work with Efstathios Chrontsios Garitsis (Univ of Tennessee) and Jonathan Fraser (Univ of St. Andrews).

The nonlinear geometry of Banach spaces has been a subject of intense interest for decades, but the development of a corresponding nonlinear geometry for their noncommutative counterpart (that is, operator spaces) only started a few years ago. Initially, this theory had been restricted to constructing a large scale geometry for operator spaces. In this work, we begin the treatment of their small scale geometry. In previous work of the authors, it was shown that the natural large scale notion of a Lipschitz function on an operator space is trivial: it only yields linear functions. In sharp contrast, we now show that the small scale variant is much richer and covers, for example, polynomials on operator spaces that were considered by Dineen-Radu and Defant-Wiesner. Knowing that there are plenty of interesting such nonlinear maps, we then turn to study what the existence of such maps can tell us about the operator spaces involved. For example, for the class of Hilbertian operator spaces we introduce a simple parameter which we use to give lower bounds for an operator space version of the compression exponent of Guentner-Kaminker. This is joint work with Bruno M. Braga (IMPA, Brazil).

The Lipschitz-free space $\mathcal{F}(M)$ is a canonical linearization of a complete metric space $M$ whose dual is the space of Lipschitz functions on $M$. In this talk, we will review the properties of $\mathcal{F}(M)$ when the underlying space $M$ is purely 1-unrectifiable (that is, it contains no bi-Lipschitz copy of a subset of $\mathbb{R}$ with positive measure) and relate them to the properties of locally flat functions on $M$. On the isomorphic side, pure 1-unrectifiability of $M$ is equivalent to $\mathcal{F}(M)$ having the Radon-Nikodým and Schur properties and, in the compact case, also to admitting a predual. On the isometric side, every element of such a Lipschitz-free space can be expressed as a convex integral of elementary molecules, but it is currently unknown whether the converse holds. Based on joint works with Gartland, Pernecká, Petitjean, Procházka, and Smith.

There is considerable interest in the problem whether a metric space of bounded geometry coarsely embeds into a uniformly convex Banach space due to the work of Kasparov and Yu that established a connection between such embeddings and the Novikov conjecture. Brown and Guentner was able to prove that a metric space with bounded geometry coarsely embeds into a reflexive Banach space. Kalton significantly extended this result to stable metric spaces and asked whether these classes are coarsely equivalent, i.e. whether every reflexive Banach space coarsely embeds into a stable metric space. Kalton also introduced a new invariant called Property $Q$, which he used to show that $\mathrm{c}_0$ does not coarsely embed into a reflexive space. More recently, Baudier introduced the notion of upper stability, a relaxation of stability, for metric spaces as a new invariant in the hope to distinguish reflexive spaces from stable metric spaces. In this talk we show that in fact every reflexive space is upper stable, and moreover upper stability and Property $Q$ are equivalent. We also establish a connection of upper stability to the asymptotic structure of Banach spaces. This is joint work with F. Baudier and Th. Schlumprecht.

TBA

Differentiability of Lipschitz mappings and functions, and the questions to what extent one can claim that such mappings have 'many' points of differentiability have been subjects of sustained interest in analysis for a long time, starting from the classical Rademacher theorem. In the recent joint work with Dymond we show that for any subset S of an arbitrary normed space X, and any Banach space Y, a typical 1-Lipschitz mapping f from S to Y is extremely non-differentiable at residual set of points in S. "Extremely" means that the derivative ratios approach every operator with norm at most 1. If X is finite-dimensional and S can be covered by countably many closed, purely unrectifiable sets, this extreme non-differentiability holds simultaneously at every point of S, for any choice of Y. In the 'positive' direction, our earlier results show that if S in a finite-dimensional space cannot be covered in this way, then a typical 1-Lipschitz real-valued function does admit at least one point of differentiability in S. I will discuss what can be said in this case when $dim Y > 1$.

Ten years ago, the late Charles Read constructed a Banach space which contains no proximinal subspaces of finite codimension greater than one, solving a long standing open problem by Singer of the 1970's. Shortly after that, Martin Rmoutil showed that this space also satisfies that its set of norm attaining functionals contains no two dimensional subspaces, solving thus a problem of Godefroy of 2000. There are extensions (and somehow simplifications) of Read's space by Kadets, Lopez, Martín, and Werner. None of these spaces is smooth, so their sets of norm attaining functionals contain nontrivial cones. Even though there are smooth "a posteriori" versions of Read spaces, their sets of norm attaining functionals also contain nontrivial cones. The aim of this talk is to present an account of the very recent construction of a Banach space whose set of norm attaining functionals contains no nontrivial cones [1]. Actually, the space satisfies that the intersection of any two dimensional subspace of its dual with the set of norm attaining functionals contains, at most, two straight lines (which is, of course, the minimal possibility). We will also show the relation between this example and the long standing open problem of whether finite-rank operators can be always approximated by norm attaining operators. [1] M. Martin, A Banach space whose set of norm-attaining functionals is algebraically , J. trivial, J. Funct. Anal. 288 (2025), 110815. https://doi.org/10.1016/j.jfa.2024.110815

The Ribe program is a far-reaching program whose goal is to reformulate in purely metric terms local properties of Banach spaces. The Kalton program has a similar goal, but for asymptotic properties instead. In this talk, we will revisit the profound connections between local and asymptotic properties of Banach spaces from a new probabilistic angle. This approach has many advantages and suggests a conceptual bridge between the Ribe and Kalton programs. It is also instrumental in attacking a wide range of fundamental problems in metric geometry, and we hope to discuss at least one of them.

Given a uniformly locally finite metric space $X$, we study one parameter automorphism groups, a.k.a. flows, on the associated uniform Roe algebra $C^*_u(X)$. If $\sigma $ is such a flow, we say that $\sigma $ is ``coarse'' if all of the finte propagation operators are differentiable with respect to $\sigma $. Observing that every flow on $C^*_u(X)$ is given by a (possibly unbounded) ``Hamiltonian'' operator $h$, we characterize precisely which operators occur, provided we assume that $X$ satisfies Yu's property $A$. Another aspect of the talk will be the discussion of cocycle equivalence and cocycle perturbation of flows, culminating with the result stating that, again under property $A$, every coarse flow is a cocycle perturbation of a coarse diagonal flow (i.e. a flow whose Hamiltonian is a diagonal operator whose diagonal entries are given by a coarse funcion on $X$). This is joint work with Bruno Braga and Alcides Buss.

Given a metric space, one can consider operators on it that have 'bounded displacement', that is, that only move point of the space up to a bounded distance away. These operators form an algebra first considered by Roe in the 1990's, and that inherits much of the large-scale geometric structure of the space. In this talk, based on joint work with Federico Vigolo, we will discuss why these algebras are always a coarse invariant, that is, why their isomorphism class does not depend on the local geometric aspects of the starting space.

We will give an overview of a solution to a problem of Dilworth, Kutzarova and Ovstrovskii, namely, to find (up to universal constant factor) the growth rate of the Banach-Mazur distance between the Lipschitz free space over a Laakso graph and $\ell_1^N$ of the corresponding dimension. This is closely related to obtaining nice decompositions of optimal transports in the aforementioned metric spaces, connection that we would hopefully have time to explain.

A norm one element $x$ of a Banach space is a Daugavet point (respectively, a $\Delta$-point) if every slice of the unit ball (respectively, every slice of the unit ball containing $x$) contains an element that is almost at distance 2 from $x$. We show that these notions coincide in Lipschitz-free spaces over subsets of $\mathbb{R}$-trees, and furthermore, they coincide with their "super" versions. We also take a look at some characterizations for these points in the aforementioned setting. The talk is based on joint work with T. A. Abrahamsen, R. Aliaga, V. Lima, A. Martiny, Y. Perreau, and A. Prochazka.

The cut of a finite graph X is the minimal size of a set E of edges such that after removing E, the size of remaining connected components of X is at most $|X|/2$. Now, given an infinite graph G (for instance a Cayley graph), one may look for every n at the maximal cut of a subgraph of size n. This yields the separation profile, introduced by Benjamini-Schramm and Timar. This provides a powerful obstruction to the existence of Lipshitz embeddings between Cayley graphs. An important application being the following theorem: an amenable group Lipschitz embeds in a hyperbolic group if and only if it is virtually nilpotent. Alternatively, one may focus on the cut of balls in G, which imposes restriction on subsets which coarsely separate G. An application is the following theorem, recently obtained with Oussama Bensaid and Anthony Genevois: a hyperbolic group is coarsely separated by a subgraph of subexponential growth if and only it virtually splits over a 2-ended subgroup.

The Maximum Distance Problem asks to find the shortest curve whose $r$-neighborhood contains a given set. Such curves are called $r$-maximum distance minimizers. We explore the limiting behavior of $r$-maximum distance minimizers as well as the asymptotics of their $1$-dimensional Hausdorff measures as $r$ tends to zero. Of note, we obtain results involving objects of fractal nature. This talk is based on is joint work with Enrique Alvarado, Louisa Catalano, and Tomás Merchán.

A measure $\mu$ on a metric space is called doubling if there is a constant $C>0$ such that $\mu(B(x,2r))\leq C \mu(B(x,r))$, where $B(x,r)$ denotes the open ball of center $x$ and radius $r>0$. The doubling constant of a measure is the best possible $C$ satisfying this inequality. The purpose of this talk is to study the set of doubling minimizers, that is those measures which minimize the doubling constant on a given metric space. This is based on joint work with J. Conde-Alonso and F. Benito.

For bounded linear operators acting on (the $\ell_2$ space of) a uniformly locally finite metric space, there are two notions of localness, finite-propagation and quasi-locality. The distinction of these two is similar to that of compactly supported functions and functions vanishing at infinity on that metric space. John Roe has asked whether quasi-local operators are approximately finite-propagation. It has been proved over the time that this is the case provided that the underlying space is sufficiently nice. On the other hand, I recently found the first example of a quasi-local operator that is not approximately finite-propagation. This is done by looking at embeddability of matrix algebras into the uniform C*-algebra consisting of approximately finite-propagation operators and the quasi-local version of it.

Curve-flat Lipschitz functions in a metric space $M$ are Lipschitz functions whose derivative vanishes almost everywhere along any curve fragment of $M$. In this presentation, we talk about some recent advances regarding these functions. Specifically, we will look at an important approximation theorem of curve-flat Lipschitz functions by locally almost flat functions, due to Bate; and we will see how to apply this result to the theory of linearizations of Lipschitz functions in the context of Lipschitz-free spaces.

Using an operator introduced by K. de Leeuw in the 1960s, it is possible to represent elements of Lipschitz-free spaces by Radon measures on a certain compact space. These representations have applications in the isometric theory of Lipschitz-free spaces. However, they are not unique and ``some representations are better than others''. In this talk I will present elements of a ``Choquet theory of Lipschitz-free spaces'', which seeks to make rigorous this informal notion and identify the ``best'' representations. This draws on the classical Choquet theory, but while in the classical case the Choquet-maximal measures are of interest, here the focus is on measures that are minimal with respect to a Choquet-like quasi-order. Using this new theory, the set of extreme points of the closed unit ball of every Lipschitz-free space can be characterised, solving a problem that can be traced to the mid-1990s. The talk is based partially on joint work with Ramón J. Aliaga (Universitat Politècnica de València) and Eva Pernecká (Czech Technical University, Prague).

An important classical way to study groups is through their possible affine isometric actions on Hilbert spaces, leading to the development of influential concepts such as Kazhdan's Property (T) and the Haagerup property. Other Banach spaces, and other types of action, are also natural and useful to study. For example, actions on $L^p$ spaces, or actions by uniformly Lipschitz affine maps, with breakthroughs in this area by Oppenheim and de Laat-de la Salle. We'll discuss some of the recent developments in this area, including joint work with Drutu where we find actions on $L^1$ spaces for groups with hyperbolic features.

Several important conjectures in topology and geometry can be solved if one knows enough about K-theory of appropriate C*-algebras, the so-called 'Roe algebras'. On the other hand, expanders (sequences of sparse, highly connected graphs) are known to lead to pathological behavior in these K-theory groups, with different properties of expanders leading to different types of pathology. I'll survey the different properties of expanders involved, discuss what is known, and ask some open questions. I will not assume any prior knowledge of K-theory or Roe C*-algebras.

For a Banach space $V$, a metric measure space $(X,d,\mu)$ is a $V$-Lipschitz differentiability space ($V$-LDS) if all Lipschitz maps from $X$ to $V$ are $\mu$-a.e. differentiable, in a generalized sense. Work by Cheeger, and later with Kleiner, established that any sufficiently connected metric measure space is a $V$-LDS, for every Banach space $V$ with the Radon-Nikodym property (RNP). In other words, sufficiently connected spaces are RNP-LDS. Remarkably, the converse is also true: Bate-Li and Eriksson-Bique prove that RNP-LDS satisfy the connectivity assumptions of the differentiability theorem of Cheeger and Kleiner. Hence, RNP-LDS are well-understood. However, this is not the case for general $V$-LDS. Indeed, solving an open problem, Schioppa constructed an LDS which is not RNP-LDS, but is an $l_2$-LDS. The existence of such a space is non-trivial and required a new idea to prove differentiability, as connectivity cannot be used. This example shows that general $V$-LDS can be very different from RNP-LDS. Beyond this, little is currently known. In this talk, I will provide an overview of the theory of LDS and present new examples which are not RNP-LDS. They answer a question of Schioppa and also raise further questions about the relationship between the geometry of the Banach space $V$ and the structure of $V$-LDS.

Given two metric spaces $M$ and $N$, a natural question is to determine under which conditions their corresponding Lipschitz-free spaces $\mathcal{F}(M)$ and $ \mathcal{F}(N)$ are isomorphic. One approach is to study the isomorphic properties of these spaces to disprove the existence of such an isomorphism. In this talk, attention will be given to two classical properties in Banach space theory: the Radon-Nikodým Property (RNP) and the Point of Continuity Property (PCP). More precisely, we investigate the behavior of two ordinal indices that quantify these properties - the dentability index $D$ and the weak fragmentability index - when applied to free spaces over various classes of metric spaces. For instance, we show that if $M$ is an infinite countable uniformly discrete metric space, $D(\mathcal{F}(M))$ is $\omega^2$ or $\omega^3$. In contrast, uncountably many mutually non-isomorphic free spaces over countable complete discrete metric spaces can be constructed, distinguished by the values of their dentability indices, which we show can take arbitrarily large countable values. In particular, we exhibit a free space over a countable complete discrete metric space that does not embed into the free space over any uniformly discrete or compact purely-1-unrectifiable metric space. This talk is based on joint work with Gilles Lancien and Antonín Procházka.

We shall report on a series of ongoing works with Manor Mendel and Assaf Naor. These include the development of new bi-Lipschitz invariants for metric spaces which characterize classical local properties of norms, the refutation of conjectured metric analogues of classical results from the linear theory, and the emergence of new, purely metric phenomena within the Ribe program dictionary. Time permitting, some algorithmic applications shall also be discussed.

Alberti, Csörnyei and Preiss introduced the notion of a tangent field for a subset of Euclidean space. They showed that in the plane any measure zero subset admits a tangent field that is at most one dimensional. In higher dimensions, the situation is much less clear. We show that any doubling subset of Hilbert space has a weak tangent field whose dimension is controlled by the Assouad dimension of the set. Moreover, we study a quantification of this result, introduce the notion of a coarse tangent field, and show existence of coarse tangent fields for doubling subsets of Hilbert space. This is joint work with Guy C. David and Ranaan Schul.

In 2008, in order to show that $L_p$ is not uniformly homeomorphic to $\ell_p \oplus \ell_2$ for $p \in (1,\infty)$ and $p \neq 2$, Kalton and Randrianarivony introduced a new technique based on a certain class of graphs and asymptotic smoothness ideas. More specifically, they proved that reflexive asymptotically uniformly smooth Banach spaces satisfy some concentration property for Lipschitz maps defined on the Hamming graphs. After introducing all the objects at stake and explaining their interest, we will see how one can construct the first example of a Banach space that has such concentration property without being asymptotically smooth.

In the first part of the talk we will discuss constructions and properties of discrete subgroups of Banach spaces. Dilworth, Odell, Schlumprecht, and Zs\'ak proved that every separable Banach space contains a subgroup that is $1$-separated and $(1+\varepsilon)$-dense. We explain how a modification of an argument due to Klee permits to obtain a shorter self-contained proof of said result and also to extend it to non-separable Banach spaces. In the second part of the talk, we focus on the non-separable Hilbert space $\ell_2(\Gamma)$, for $\Gamma^\omega= \Gamma$, where our construction yields a subgroup which is $(\sqrt{2}+)$-separated and $1$-dense. The main output of this construction is that there is a symmetric, bounded, convex body whose translates via the subgroup form a tiling of $\ell_2(\Gamma)$. This answers a question due to Fonf and Lindenstrauss. The same construction performed in $\ell_1(\Gamma)$ yields a tiling by balls of radius $1$ for $\ell_1(\Gamma)$, which compares to a celebrated construction due to Klee. Joint work with Carlo Alberto De Bernardi and Jacopo Somaglia.

A point x on the unit sphere of a Banach space X is called a Delta (respectively super Delta) point if every slice (respectively every relatively weakly open subset) of the unit ball of X containing x also contains points that are at distance arbitrarily close to 2 to the point x. In this talk we investigate Delta and super Delta points in Lipschitz-free spaces. In particular, we show that these two notions coincide for molecules, and characterize this property in a purely metric way by a specific connectability condition between the two defining points in the underlying metric space. As a consequence, we obtain a renorming of the space $l_1$ with a super Delta point, and get that every infinite dimensional Banach space can be renormed with a super Delta point.

Lipschitz-free spaces, denoted by $\mathcal{F}(M)$ for a complete metric space $M$, are the canonical linearization of metric spaces, providing a natural bridge between functional analysis and metric geometry. This talk focuses on two significant properties in this setting: weak sequential completeness and Pełczyński’s property (V$^*$). While these properties are not equivalent in general, they do coincide in some classes of Banach spaces. However, whether they are equivalent in the context of Lipschitz-free spaces remains an open problem. In this talk, we will present recent results concerning property (V$^*$) in Lipschitz-free spaces and discuss sufficient conditions under which $\mathcal{F}(M)$ enjoys this property -- for instance, when $M$ is locally compact and purely 1-unrectifiable, a Hilbert space, or a Carnot-Carathéodory space satisfying a bi-Hölder condition, such as a Carnot group. This presentation is based on joint work with Ramón J. Aliaga and Eva Pernecká.

M. Daher (1995) gave conditions so that the spheres of the spaces in the interior of a complex interpolation scale are uniformly homeomorphic. We shall discuss extrapolation theorems as well as notions of uniformity of scales, to find sufficient conditions for the extension of Daher's result to spaces on the extremes of the scale. Possible applications to uniform homeomorphism between spheres of Banach spaces and the sphere of the Hilbert space will be mentioned. Joint work with W. Corrêa (ICMC- U. São Paulo), R. Gesing (U. Münster), and P. Tradacete (ICM-Madrid).