Ramón J. Aliaga
Title: Topics in isometric Lipschitz-free space theory
Abstract: Lipschitz-free spaces are canonical linearizations of metric spaces. Their complicated Banach space structure is the subject of much recent research. This lecture will focus on their isometric structure. We will introduce some of the main tools used for the analysis of elements of Lipschitz-free spaces: supports and De Leeuw representations. We will show how they can be exploited to obtain information about the extremal structure of Lipschitz-free spaces. In particular, we shall present some elements of the very recent solution to the problem of characterizing extreme points of the unit ball.

The notes for this course can be read and downloaded at https://shorturl.at/50JvR

Michal Doucha
Title: Complexity of separable Banach spaces and variants of the Mazur rotation problem
Abstract: This series of three talks aims to connect two different research directions in Banach space theory.

The first concerns computations of complexities of isometry classes of separable infinite-dimensional Banach spaces in an appropriate Polish space. For instance, I will show that the Hilbert space is characterized as the unique space whose isometry class is closed and whose isomorphism class is F_sigma, while other L_p's have G_delta isometry classes as well as the Gurarii space.

The second concerns Fraisse Banach spaces, a notion introduced very recently by Ferenczi--Lopez-Abad-Mbombo-Todorcevic to provide examples satisfying a relaxed version of the Mazur rotation problem. These examples are, besides the Hilbert space, again the Gurarii space and most, but not all, of the L_p's.

The first two talks will introduce these two directions and the third talk will connect them. I will show that having a G_delta isometry class is equivalent with an even further relaxation of the Mazur rotation problem and I will exactly characterize for which values of p and q, the space L_p(L_q) has this property. Based on joint works with Cuth, de Rancourt, Dolezal, Kurka.

Miguel Martín
Title:  How minimal can be the linear structure of the set of norm attaining functionals of a Banach  space?
Abstract: 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 series of talks is to present a detailed account of the very recent construction of a Banach space whose set of norm attaining functionals contains no nontrivial cones (see http://arxiv.org/abs/2406.07273). 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.

There is no needed background apart of basic knowledge of functional analysis and of the geometry of Banach spaces.

You can download the final version of the paper of the series of talks: https://authors.elsevier.com/a/1kPv~51yEl-se

Antonín Procházka
Title: Gliding hump methods in Lipschitz free spaces
Abstract: The Lipschitz-free construction provides a functor from the category of metric spaces and Lipschitz maps into the category of Banach spaces and bounded linear maps. The aim of this lecture is to study to which extent isomorphic properties of Lipschitz free spaces are influenced by metric properties of the corresponding metric spaces. After a brief overview of properties studied so far by various authors we will focus on properties for which the application of gliding hump methods is appropriate. In particular we will examine the technique of compact reduction, the characterization of metric spaces $M$ whose free space has the Schur property (equivalently the Radon-Nikodým property), and the extension of these techniques which provides a characterization of those Lipschitz maps whose free linearization is Dunford-Pettis (equivalently Radon-Nikodým).

Juan B. Seoane Sepúlveda
Title: Convex Analysis in Polynomial Spaces and Applications. A brief story.
Abstract: We shall present a brief (although global) perspective on the geometry of a variety of spaces of polynomials, focusing on providing tools to explicitly describe the set of extreme points in their corresponding unit balls. As we will see, this study has quite a number of applications, some of them aimed to obtain sharp classical polynomial inequalities, which we will present.

This work is partly based on the recent research monographs [1] and [2] which are the first ever complete account on the geometry of the unit ball of polynomial spaces.

Nowadays there are many research papers on this topic and this set of talks intend to gather the state of the art of the main and/or more relevant results up to now, some of which were originally started by authors such as the Markov brothers, S. Bernstein, R. M. Aron, Y. S. Choi, D. García, M. Klimek, M. Maestre, S. Révész, or Y. Sarantopoulos (among others). We shall also comment on some new results from several joint works with D. García, Manwook Han, Sun Kwang Kim, Mingu Jung, J. Llorente, M. Maestre, G. Muñoz, D.L Rodríguez, and Hyung-Joon Tag (see, e.g., the very recent works [3, 4]).

[1] Ferrer, J.; García, D.; Maestre M.; Muñoz-Fernández, G.A.; Rodríguez-Vidanes, D. L.; and Seoane-Sepu ́lveda, J.B., Geometry of the Unit Sphere in Polynomial Spaces, Springer Briefs in Mathematics. Springer (2023), ISBN 978-3-031-23675-4.
[2] García, D.; Jung, M.; Maestre M.; Muñoz-Fernández, G.A.; and Seoane-Sepúlveda, J.B., Convex Analysis in Polynomial Spaces with Applications, CRC Press - Taylor & Francis (2025), in press - ISBN 9781032967653.
[3] Kim, S.J.; Han, M.; Muñoz-Fernández, G.A.; and Seoane-Sepúlveda, J.B., Geometry of homogeneous polynomials on nonsymmetric convex bodies and applications, (work in progress).
[4] Llorente, J; Rodríguez-Vidanes, D.L.; Muñoz-Fernández, G.A.; Seoane-Sepúlveda, J.B.; and Tag, H.J, Two homogeneous polynomials on translations of unbalanced convex bodies: When classical inequalities fail to hold, (work in progress).

Geunsu Choi
Title: Approximations of maximal derivative attaining Lipschitz maps
Abstract: This talk is divided into two parts. First, we study maximal derivative attaining Lipschitz maps defined on Banach spaces, which combine the concepts of differentiability and norm-attainment of Lipschitz maps. In particular, we prove that if every Lipschitz map can be approximated by maps that either strongly attain their norm or attain their maximal derivative for every renorming of the range space, then the range space must have the Radon-Nikodým property. Second, we extend the approximation by affine property in terms of maximal derivative, and it is shown that every Lipschitz functional defined on the real line can be locally approximated by maximal affine functions, while such an approximation cannot be guaranteed in the context of uniform approximation.

Sheldon Dantas
Title: A collection of strong subdifferentiable norms
Abstract: We focus on a concept of differentiability of the norm of a Banach space, which is called the strong subdifferentiable (SSD, for short) norms. To put it simply, studying when the norm of a Banach space is strongly subdifferentiable is essentially the same as investigating what is lacking for a Gâteaux differentiable norm to become a Fréchet differentiable norm. In fact, SSD is a strictly weaker concept than Fréchet differentiability (indeed, every point of a finite-dimensional Banach space is SSD). During the talk, we will try to motivate the problem as well as present classical and recent results on the topic.

Helena Del Río Fernández
Title: On a new notion of norm-attaining operators on Banach spaces
Abstract: We present a new notion of norm-attainment in which the maximum is required to be strongly attained with respect to the range space. This is a less restrictive concept than the absolute strong exposition introduced by Bourgain in 1977, although it is also closely related to the Radon-Nikodým Property. Using this new concept, we improve some classical results by Uhl (1976) and Schchermayer (1983) and get some analogous results to those obtained by Bourgain (1977). Joint work with Geunsu Choi, Audrey Fovelle, Mingu Jung and Miguel Martín.

Audrey Fovelle
Title: Norm attaining operators into asymptotically uniformly convex Banach spaces
Abstract: In this talk, we will show that if $Y$ is an $\ell_p$ sum of finite dimensional spaces ($1 <p < \infty$), we can always find a Banach space $X$ and an operator $T :X \to Y$ that cannot be approximated by norm-attaining ones. In other words, an $\ell_p$ sum of finite dimensional spaces does not have Lindenstrauss’ Property B. This will be a consequence of a more general result: any locally asymptotically uniformy convex Banach space which contains a sequence "looking like the canonical basis of some $\ell_p$" fails Lindenstrauss’ Property B.

Manwook Han
Title: M-ideals of compact operators with numerical radius norm
Abstract: For a Banach space $X$, if its numerical index is not zero, the numerical radius on $\mathcal{L}(X)$ defines a norm equivalent to the operator norm. In this research, we focus on conditions under which $\mathcal{K}_v(X)$ is an M-ideal in $\mathcal{L}_v(X)$, where $\mathcal{K}_v(X)$ and $\mathcal{L}_v(X)$ denote the space of compact operators and the space of bounded linear operators, respectively, equipped with the numerical radius as the norm.

Tatsuhiro Honda
Title: Bohr's radii for holomorphic mappings
Abstract: In this talk, we will discuss to generalize some results about Bohr's radii for a holomorphic mapping to higher dimensional complex Banach spaces. This is a joint work with Hidetaka Hamada [cf. Hamada-Honda, Results in Mathematics. 79  (2024), , no. 7, Paper No. 239, 20 pp]

Vladimir Kadets
Title: Strong Law of Large Numbers for Random Sets in Banach spaces
Abstract: The Strong Law of Large Numbers (SLLN) for random variables or random vectors in Banach spaces with different mathematical expectations easily reduces by means of shifts to SLLN for random variables or random vectors whose mathematical expectations are equal to zero. The situation changes for random sets, where shifts cannot reduce sets of more than one point to the set consistion of the point 0. In this talk I am going to present our recent joint results with Olesia Zavarzina (arXiv:2410.04832) about effects that appear because of this difference.
Also, I am going to speak about  joint results with Zvi Artstein (to appear in J. of Convex Analysis) on connection between SLLN for convex and non-convex random sets in Banach spaces.

Sun Kwang Kim
Title: The Bishop-Phelps-Bollobás property for numerical radius
Abstract: Based on the recent results by Han Ju Lee, Miguel Martín, Óscar Roldán and myself, I would like to introduce the new results on the Bishop-Phelps-Bollobás property for numerical radius. Among other, the main observations are the followings.
- $\ell_1\oplus_1 c_0$ and $\ell_1 \oplus_\infty c_0$ do not have the Bishop-Phelps-Bollobás property for numerical radius.
- A real Banach space $\ell_\infty$ has the Bishop-Phelps-Bollobás property for numerical radius.
It is worth to note that for the complex space $X=\ell_1\oplus_1 c_0$ the set of numerical radius attaining operators is dense in the whole space $\mathcal{L}(X)$ of linear operators. As far as we know, item (1) is the first example which fails Bishop-Phelps-Bollobás property for the numerical radius such that the set of numerical radius attaining operators is dense in the whole space of linear operators without using renormings. 

Jaagup Kirme
Title: A note on the Bishop–Phelps–Bollobás theorem
Abstract: The Bishop–Phelps theorem states that for any Banach space $X$ the set of norm-attaining functionals is dense in the dual $X^\ast = \mathcal L(X, \mathbb K)$, and the Bishop–Phelps–Bollobás theorem sharpens this result. If we instead view (norm-attaining) operators between two Banach spaces $X$ and $Y$, such a general theorem will not hold. In the presentation we will define a generalisation of the BP and BPB theorems (having now become properties to check for) and bring both positive and negative examples in different pairs of Banach spaces.

Seung-Hyeok Kye
Title: Bilinear pairings between linear mapping spaces and tensor products of matrices arising in quantum information theory
Abstract: Bilinear pairings between mapping spaces and tensor products play the central roles in current quantum information theory. It is essential to detect entanglement and determine Schmidt numbers of bi-partite states which measure the degree of entanglement. We examine two bilinear pairings for this purposes in the literature and find all the possible bilinear pairings to fit our purposes. Through the discussion, variants of Choi matrices also play important roles.

Rubén Medina
Title: On a problem of Dilworth, Kutzarova and Ostrovskii
Abstract: We are going to give a partial solution to a problem of Dilworth, Kutzarova and Ostrovskii, namely, find a condition on a finite metric space equivalent to its Lispchitz free space being close (for the Banach-Mazur distance) to $\ell_1^N$ of the corresponding dimension. We will use this characterization to give new examples of families of finite metric spaces whose Lipschitz free space is uniformly close to their corresponding $\ell_1^N$'s.

Óscar Roldán
Title: Points that localize diameter 2 properties in vector-valued function spaces
Abstract: A well-known geometrical characterization of Daugavet property in a Banach space $X$ is that given any point $x\in S_X$ in its sphere and any slice $S\subset B_X$ of the ball, $\sup_{y\in S} d(x,y)=2$. Actually, this is also true if slices are replaced with weakly open sets and with convex combinations of slices. In recent years, those points of $S_X$ that satisfy such properties, or versions of them, have been studied in detail. In spaces like $C(K)$, $L_1(\mu)$, and uniform algebras $A(K)$, some of those types of points have been recently characterized. In this talk, we will extend the known results further and characterize several of these point notions in vector-valued function spaces such as $C_0(L,X)$, $L_p(\mu, X)$, and $A(K,X)$ (which is a vector-valued function space over a base uniform algebra $A(K)$). Our results partially answer an open question posed by M. Martín, Y. Perreau, and A. Rueda Zoca, and they also have some consequences in the scalar-valued cases.

This talk is based in a recent joint work with Han Ju Lee and Hyung Joon Tag (see [1]). The author has been supported by MICIU / AEI / 10.13039 / 501100011033 and ERDF/EU through the grants PID2021-122126NB-C31 and PID2021-122126NB-C33, and also by the Basic Science Research Program, National Research Foundation of Korea through the grant NRF-2020R1A2C1A01010377.

[1] H. J. Lee, Ó. Roldán, and H. J. Tag, On various diametral notions of points in the unit ball of some vector-valued function spaces. Submitted preprint, available on https://arxiv.org/abs/2410.04706.

Daniel L. Rodríguez-Vidanes
Title: Isometric structures of $L^p$-spaces over infinite-dimensional measures
Abstract: This talk presents a detailed study of the $L^p$-spaces associated with an infinite-dimensional analogue of Lebesgue measure, as introduced by R. Baker. We explore the structural differences between these spaces and their finite-dimensional counterparts, focusing on their unique properties and behavior. The analysis highlights key functional insights into infinite-dimensional measure theory, offering a new perspective on the isometric classification of $L^p$-spaces.

More talks will be added. Last updated on Jan 8, 2025.