Schedule for: 25w5355 - Geometric Nonlinear Functional Analysis

The classification of certain families of groups up to quasi-isometry has been the object of an intensive study in the last decades, culminating with the classification of lattices in Lie groups, and mapping class groups. The case of amenable, and more specifically solvable groups appears to be much harder, and has known very few developments in comparison. Until recently, only lamplighters over Z, Baumslag-Solitar groups, and very specific polycyclic groups had been treated. With Anthony Genevois, we introduce new tools to study the large scale geometry of certain families of groups obtained as a semi-direct product of a locally finite group with an arbitrary group H: for instance Lamplighter groups (wreath product of H with a finite group), or Lampshuffler groups (semi-direct product of H with permutations finitely supported on H), and other similar constructions. Under the assumption that H is one-ended and finitely presented, we obtain various rigidity/classification results. As a by-product of these techniques, I will construct the first example of an amenable bounded degree graph which is quasi-isometry-transitive, but not quasi-isometric to a homogeneous metric space (answering a question of Itai Benjamini).

The separation modulus of a metric space $X$ is defined as the minimal constant $K > 0$ for which, given any scale $R > 0$, one can obtain a random partition of $X$ into clusters with diameter no greater than $R$ such that for all points $x, y \in X$, the probability that they fall into different clusters does not exceed $(K/R)d(x,y)$. In this talk, by analysing an alternative formulation in terms of an isomorphic reverse isoperimetry phenomenon, we will see how one can obtain precise asymptotics of the separation modulus of unitarily invariant norms on high-dimensional matrix spaces. Further applications of these results in relation to the Lipschitz extension problem on these spaces will be given. Based on joint work with Naor.

The polynomial cluster value problem for a Banach space $X$ studies the limit behavior at a point in the closed unit ball of the bidual $X^{\ast\ast}$ of bounded holomorphic functions defined on the open unit ball of $X$, where convergence is understood in the polynomial-star topology. In this talk, I will discuss the definition and key properties of this topology, as well as outline steps to reduce the polynomial cluster value problem for all separable Banach spaces to the case of countable $\ell_1$-sums of finite-dimensional spaces.

A Banach space $X$ is said to have the Lipschitz-lifting property when the barycenter map $\beta\in L(F(X),X)$ admits a bounded linear right inverse, called a linear lifting for $X$. Such property plays a central role in the investigation of the structure of Lipschitz-free spaces over Banach spaces, notably in Godefroy and Kalton's seminal paper Lipschitz-free Banach spaces, Studia Math., 2003. In the present work, we consider a group $G$ acting on $X$ by linear isometries, and study the possible existence of a linear lifting $T\in L(X,F(X))$ that is moreover $G$-equivariant, in the sense that for each $g\in G$, $Tg=\tilde g T$, where $\tilde g$ is the isometry of $F(X)$ induced by $g$. In particular, we prove that such lifting exists when $G$ is compact in the strong operator topology, or an increasing union of such groups and $F(X)$ is complemented in its bidual by a $G$-equivariant projection.    Joint work with Valentin Ferenczi and Eva Pernecká.

In this talk, we will discuss the extent to which the nonexistence of uncountable biorthogonal systems can be generalized to the nonseparable context. We will show that the bounding cardinal captures the essence of separable Banach spaces from this perspective. This is based on joint work with Stevo Todorcevic.

In 2023, Tim de Laat and Mikael de la Salle posted a complete proof of a conjecture from 2007 by Bader, Furman, Gelander and Monod: higher rank groups and their lattices have the fixed point property for affine isometric actions on super-reflexive Banach spaces. We shall explain their proof in the case of the simple Lie group $SL_3(\mathbb{R})$: it involves ideas from optimal transportation, Gaussian measures on the real line, and actions of the Heisenberg group.

Differentiability of Lipschitz mappings is the focus of many strands of Analysis. As a starting point, we may always consider Rademacher theorem, which guarantees that the set of non-differentiability points of a Lipschitz mapping between two finite-dimensional Euclidean spaces is of Lebesgue measure zero. This has versions beyond finite-dimensional case, under reasonable assumptions on Banach spaces $X$ and $Y$. In recent joint work with Dymond, however, we show that for any subset S of a normed space X, there exists a residual set of 1-Lipschitz mappings to Y, each of which is extremely non-differentiable at residual set of points. "Extremely" means that the derivative ratios approach every operator of norm at most 1. Moreover, if X is finite-dimensional and S can be covered by countably many closed purely unrectifiable sets, extreme non-differentiability holds simultaneously at every point of S.

In the theory of normed spaces, it is well known that every finite-dimensional subspace is complemented in any superspace containing it. Building on this, a natural question arises: how can the norm of a projection onto a given subspace be quantified? A key concept to address this problem is the so-called projection constant of the space $X$, defined as the smallest constant $c > 0$ such that, for any superspace $Y$ containing $X$, there exists a projection from $Y$ onto $X$ with norm not exceeding $c$. In this talk, we provide explicit formulas for the projection constants of several important function spaces and discuss methods for computing them or estimating their asymptotic growth as the dimension of the ambient space tends to infinity. This is based on work done in collaboration with A. Defant, M. Mansilla, M. Mastyło, and S. Muro.

Given a finite metric space $M$ one can define the corresponding transportation cost space $F(M)$ as the normed linear space of transportation problems on $M$. Roughly speaking, a transportation problem can be understood as a supply/demand configuration on $M$ where the norm of the transportation problem is the lowest cost of transporting goods from locations with a surplus to those with shortages. In this setting, an important line of research is studying the relation between transportation cost spaces and $\ell_1$. A core problem posed by S. Dilworth, D. Kutzarova, and M. Ostrovskii is finding a condition on a metric space $M$ equivalent to $F(M)$ being Banach-Mazur close to $\ell_1^N$ in the corresponding dimension.  In this talk, we discuss our recent work where a partial solution to this problem is obtained by examining tree-like structure within the underlying metric space. Tangential to this result, we have also developed a new technique that, potentially, could serve as a step toward a complete solution to the problem of Dilworth, Kutzarova, and Ostrovskii. We conclude by discussing two applications of this technique: finding an asymptotically tight upper bound of the $\ell_1^N$-distortion of the Laakso graphs, and proving that finite hyperbolic approximations of doubling metric spaces have uniformly bounded $\ell_1^N$-distortion. This is joint work with Ruben Medina.

A major open problem due to Pisier and Mendel/Naor asks whether every regular expander graph $G$ satisfies a discrete Poincaré inequality for functions taking values in a Banach space $(X,\|\cdot\|_X)$ with finite cotype $q\geqslant 2$, that is, there exists a constant $\gamma>0$ such that for every $f\colon V(G)\to X$ we have, \[ \underset{x,y\in V(G)}{\mathbb{E}} \, \|f(x)-f(y)\|_X \leqslant \gamma \underset{\{x,y\}\in E(G)}{\mathbb{E}} \, \|f(x)-f(y)\|_X.\] Works of Odell--Schlumprecht (1994), Ozawa (2004) and Naor (2014) yield a positive answer for spaces having an unconditional basis, in addition to finite cotype. However, little is known with respect to quantitative estimates: the aforementioned results provide an estimate for the Poincaré constant that depends super-exponentially on $q$. In this talk, we shall present a novel combinatorial method for obtaining quantitative nonlinear spectral gaps that relies on a property of regular graphs that we call long-range expansion. In particular we shall discuss the following results. (1) Every regular graph with the long-range expansion property satisfies a discrete Poincaré inequality for functions taking values in a Banach space with an unconditional basis and cotype $q$, with a Poincaré constant proportional to $q^{10}$. This estimate is nearly optimal. (2) For any integer $d\geqslant 10$, a uniformly random $d$-regular graph satisfies the long-range expansion property with high probability. This is joint work with Dylan Altschuler, Pandelis Dodos and Konstantin Tikhomirov.

Let $G$ be a topological group. A $G$-Banach space is an ordered triple $(G, X, u)$, where $X$ is a Banach space, and $u$ is a bounded left action of $G$ on $X$ (i.e., $u$ is a map from $G$ into ${B}(X)$ such that: i) $u(e_G) = {id}_X$; ii) for each $(g, h) \in G \times G$, $u(g \cdot h) = u(g) \circ u(h)$; and iii) $u(G)$ is a bounded subset of ${B}(X)$). A $G$-Banach space $(G, X, u)$ is said to be a $G_{{T} op}$-Banach space if the action $u$ is $(\tau_G, SOT)$-continuous. Moreover, if $(G, X, u)$ and $(G, Y, v)$ are $G$-Banach spaces, we say that a linear and continuous map $T \colon X \to Y$ such that $T \circ u(g) = v(g) \circ T$ for any $g \in G$ is a $G$-operator. The category of $G$-Banach spaces is the category that has $G$-Banach spaces as objects and $G$-operators as morphisms. In this talk, we will show that, if \[ 0 \longrightarrow (G, X, u) \overset{\iota}{\longrightarrow} (G, Z, \lambda) \overset{q}{\longrightarrow} (G, Y, v) \longrightarrow 0 \] is an exact sequence in the category of $G$-Banach spaces such that $(G, X, u)$ and $(G, Y, v)$ are $G_{{T} op}$-Banach spaces, and such that $Z$ is either super-reflexive or reflexive and separable, then $(G, Z, \lambda)$ is also a $G_{{T} op}$-Banach space (such a $Z$ is called a twisted sum of $X$ and $Y$ --- which, in turn, explains the title of the talk). Afterwards, we will present a criterion for the equivalence of certain types of exact sequences in the category of $G_{{T} op}$-Banach spaces (which is simply the full subcategory of the category of $G$-Banach spaces that has the $G_{{T} op}$-Banach spaces as objects).

Let $X$ be an infinite dimensional Banach space with a symmetric basis.  We show  the existence of bounded bijective  polynomials $P:X\rightarrow X $  with a continuous inverse    which are not polynomial automorphisms, i.e., such that $P^{-1}$ is not a polynomial.   This result  contrasts with the finite-dimensional case where,  remarkably,   injective  polynomial maps   $P:\mathbb{C}^n \rightarrow \mathbb{C}^n$    are polynomial automorphisms.

Let $M,N$ be pointed metric spaces and $F(M)$, $F(N)$ their corresponding Lipschitz-free spaces. It is well known that every Lipschitz map $f:M \to N$ such that $f(0)=0$ admits the free linearization $\hat{f}:F(M) \to F(N)$, also called the Lipschitz operator associated to $f$. For example, it can be realized as the pre-adjoint of the composition operator $C_f: Lip_0(N) \to Lip_0(M)$. In this talk, based on a joint work with Gonzalo Flores, Mingu Jung, Gilles Lancien, Colin Petitjean and Andres Quilis, we will characterize those $f$ for which $\hat{f}$ is Dunford-Pettis as well as those $f$ for which $\hat{f}$ is Radon-Nikodým. It turns out that the corresponding metric property for $f$ is the same for both linear properties. This generalizes the recent result that in Lipschitz free spaces the Radon-Nikodým property and the Schur property coincide.

The talk is based on joint papers with S. J. Dilworth and M. Ostrovskii. We study transportation cost spaces associated to some unweighted graphs with the shortest path distance. Our main goal is to estimate the Banach-Mazur distance to an $\ell_1^n$ space of the same dimension. We use the method of invariant projections of Grunbaum, Rudin and Andrew to analyze projections that are invariant with respect to a certain group of isometries of the edge space. That allows us to find lower bounds for the Banach-Mazur distance. We use ad hoc arguments to get matching upper bounds in some particular cases. Recently, R. Medina and G. Tresch developed a general method for obtaining upper bounds.

This is a report on an ongoing project with D. Kutzarova and M. Ostrovskii. We investigate the transportation cost space associated to a finite metric space (an unweighted graph with the shortest path metric in the examples we consider). The transportation cost space is a finite-dimensional normed space whose dual is the space of Lipschitz functions on the vertex set. We are particularly interested in estimating the Banach-Mazur distance to an $\ell_1^n$ space. Analysis of the `invariant' projections from the edge space onto the space of Lipschitz functions (which embeds naturally into the edge space) that commute with a group of isometries can yield useful lower bounds, e.g. for the families of diamond and Laakso graphs [1,2]. Ad hoc arguments sometimes yield matching upper bounds, but whether there are good upper bounds in general is an open problem. More recently, we considered the case of Hamming graphs and discrete tori [3]. For example, we show that the projection constant of the space of Lipschitz functions on the $n$-dimensional Hamming cube is $(n+1)/2$. [1] S. J. Dilworth, Mikhail Ostrovskii, and Denka Kutzarova, Lipschitz free spaces on finite metric spaces, Canad. J. Math. 72 (2020), no. 3, 774–804. [2] S. J. Dilworth, Mikhail Ostrovskii, and Denka Kutzarova, Analysis on Laakso graphs with application to the structure of transportation cost spaces, Positivity 25 (2021), no. 4, 1403--1435. [3] S. J. Dilworth, Mikhail Ostrovskii, and Denka Kutzarova, Cycle Spaces: Invariant Projections and Applications to Transportation Cost, Geometry of Banach Spaces and Related Fields, Proceedings of Symposia in Pure Mathematics, vol. 106, American Mathematical Society, 97--131, 2024.

In this talk, we discuss the asymptotic analogue of Pisier’s classical renorming theorem from a new probabilistic framework. Pisier’s theorem states the equivalence between martingale type and martingale cotype to uniform smoothness and uniform convexity respectively, up to norm equivalence. In the classical theory, the geometry is encoded in binary trees, while in the asymptotic setting, the countably branching trees become the central geometric object. However, as we will see during the talk, this new probabilistic approach leads us to consider so-called ultra-martingales, which allow us to draw new parallels to Pisier’s renorming. Furthermore, we will explore connections of these notions to the geometry of countably branching diamond graphs and to diamond convexity, a metric invariant introduced by Eskenazis, Mendel, and Naor. This is joint work with Florent Baudier (Texas A&M University).

The talk is about Gleason parts and their interaction with the fibers for the spectrum of the Banach algebra $A_u(B_{\ell_p})$ (uniformly continuous holomorphic functions defined on the unit ball of $\ell_p$). Based on joint works with Richard Aron, Daniel Carando, Silvia Lassalle, Manuel Maestre and Tomás Rodríguez.

Optional excursion to Monte Albán

We study the problem of how well the transportation cost space over a finite metric space can embed into $L_1$. In this talk, we will present new Sobolev-type inequalities on certain self similar graphs, such as Laakso graphs and the 2-dimensional Euclidean grids $[0,1,\dots 2^n]^2$, and describe how the inequalities may be used to derive new lower bounds on the $L_1$-distortion of the transport cost spaces over these graphs. Based on joint work with Mikhail Ostrovskii.

In this talk, we consider the classical Anderson’s theorem: Let $(\mathbb{R}^n,\|\cdot\|)$ be a norm and $G$ an isotropic Gaussian vector. Then, for any $x\in\mathbb{R}^n$, it holds that $$\mathbb{E}\|G+x\|^2 \geq \mathbb{E}\|G\|^2.$$ Remarkably, when $\|\cdot\|$ is the $\ell_p$-norm for $p\in[1,2]$, an improved bound can be obtained: $$\mathbb{E}\|G+x\|^2 \geq \mathbb{E}\|G\|^2+c\|x\|^2.$$ where $c\in(0,1)$ is an absolute constant. We conjecture that this improved bound holds more generally for any cotype-2 norm in the appropriate position. We will discuss this conjectured quantitative version of Anderson’s theorem and explore its numerous applications in statistics and learning theory.

Given a set $S$ of $n$ points on the unit square $[0,1]^2$, the travelling salesman problem asks for the shortest length of the curve that passes through all the points in $S$. In the 1980s Bartholdi and Platzman introduced the universal travelling salesman heuristic which gives an approximate solution as follows. Take the Sierpinski space-filling curve and define a linear order on $[0,1]^2$ by setting $p$<$q$ when the curve visits first $p$ and then $q$. Given a set of $n$ points $S$ we will order them according to the space filling curve and visit them in that order. Bartholdi and Platzman proved that this approximate solution is at most an $O(\log(n))$-factor away from the optimal solution. In this talk, we give lower bounds: for any linear order on the unit square there exist sets $S$ of arbitrarily large size $n$ such that the approximate solution is a $\Omega((\log(n)/\log\log(n))^{1/2})$-factor away from the optimal solution. This improves the earlier lower bound of $\Omega((\log(n)/\log\log(n))^{1/6})$ by Hajiaghayi, Kleinberg and Leighton (2006). The proof establishes a dichotomy about any long walk on a cycle: the walk either zig-zags between two far away points, or else for a large amount of time it stays inside a set of small diameter.

Existing concentration inequalities for functions that take values in a general Banach space, such as the classical results of Pisier, are known only in very special settings, such as the Gaussian measure on $\mathbb{R}^n$ and the uniform measure on the discrete hypercube $\{-1,1\}^n$. We present a novel vector-valued concentration inequality for the uniform measure on the symmetric group which goes beyond the product setting of the prior known results. Furthermore, we discuss the implications of this result for the nonembeddability of the symmetric group into Banach spaces of nontrivial Rademacher type. The proof uses a variety of tools related to concentration of Markov semigroups, including discrete analogs of Ricci curvature. This talk is based on joint work with Ramon van Handel.

The talk is about the open question whether separable dual Banach spaces can be coarsely universal, that is, if $c_0$ could coarsely embed into such a space.   What seems to be the key case is to resolve the problem for the generalized James tree spaces. We show that a class of generalized James tree spaces, which includes the original James tree space, are not coarsely universal, in fact, the Kalton’s interlacing graphs do not equi-coarsely embed into them.   Joint work with Steve Jackson and Cory Krause.

By finite determination I mean results of the following type: Theorem (M.O. (2012)) Let $A$ be a locally finite metric space whose finite subsets admit bilipschitz embeddings into a Banach space $X$ with uniformly bounded distortions. Then $A$ admits a bilipschitz embedding into $X$. More recently, the quantitative aspect of finite determination was investigated. This investigation could be regarded as completed if two natural conjectures would be proved. The goal of the talk is to state the conjectures and possible approaches to their proofs.

In this talk, I will explore how the geometry and analysis of nonpositively curved, infinite-dimensional spaces can be applied to investigate the Novikov conjecture for diffeomorphism groups. My aim is to present the material in a way that is accessible to graduate students.

In their pioneering work, J. Farmer and W. B. Johnson (2009) introduced the Lipschitz $p$-summing operators. Since then, there has been an increasing interest in studying different classes of Lipschitz maps between Banach spaces that extend several Banach linear operator ideals. In this talk, we will present general methods for constructing diverse Banach Lipschitz operator ideals from a specific Banach operator ideal. This constructions allows us to describe Banach linear operators ideals in terms of Lipschitz mappings. As an application of these results, we first provide examples where a Lipschitz (hence nonlinear) factorization of linear operators leads to a linear factorization. We then present cases where a linear map belonging to a given operator ideal can be extended to a Lipschitz function in a corresponding ideal if and only if it admits a linear extension Joint work with N. Albarracín.

I will present some results supporting the conjecture that given a metric space of density at most continuum, the space of Lipschitz functions is weak* separable. The talk will be based mostly on a joint paper with L. Candido and B. Vejnar.