Timur Oikhberg
Title: Free Banach lattices: background and new developments
Abstract: Free Banach lattices have attracted much interest in the last dozen years. Specifically, suppose C is a class of Banach lattices, and E is a Banach space. We say that Z is the free Banach lattice of class C over E if (i) there exists an isometric embedding of E into Z, generating Z as a Banach lattice, and (ii) for any Banach lattice X from he class C, any bounded linear map T from E to X extends to a lattice homomorphism from Z to X, of the same norm.
In this series of lectures, we begin by surveying the known results, such as the existence of the free lattice for different classes C (such as the class of all Banach lattices; of p-convex lattices; or of lattices with an upper p-estimate), and its functional representation. We then move on to other topics, such as:
* The geometry of the unit ball of a free lattice, and its extreme points;
* Connections between the properties of a Banach space E and its free lattice;
* Lattice homomorphisms between free lattices.
Along the way, open problems will be mentioned.

Thomas Schlumprecht
Title: Asymptotic Properties of Banach Spaces
Abstract: Asymptotic properties of Banach spaces can be described by infinite asymptotic games, the properties of skipped block bases of a Markushevich basis, or by countably branching trees of infinite height.
Using asymptotic properties, we consider the following type of problems:
Let X be a separable Banach space with a certain asymptotic property P, Find a Banach space Z  with a basis or an FDD, also satsifying P, which contains a copy of X. 
Let  C be  a class of spaces with a basis (or an FDD)  having a certain property P:
(being unconditional, having upper or lower l_p-estimates, being shrinking, boundedly complete). Find an intrinsic characterization of the property of a Banach space X (which might not have a basis or FDD) to embed into a space in  Z in C.

Geunsu Choi
Title: Affine approximation of Lipschitz functions and its extension
Abstract: The local approximation of Lipschitz functions by affine functions was first investigated by Bates et al. (1999), where they demonstrated in particular that local uniform approximation is obtained if and only if either the domain or the range space is super-reflexive and the other is finite-dimensional. This talk presents recent research on this local approximation problem from the perspective of the maximal slope. Specifically, we prove that every Lipschitz functional on the real line can be approximated by maximal affine functions, whereas the uniform approximation fails.

Sheldon Dantas
Title: Revisiting Shvidkoy's characterization of the Daugavet property via the polynomial weak topology
Abstract: Taking advantage of a series of talks on the Daugavet property (DPr, for short) given by Miguel Martín, we will present the proof of the fact that the DPr implies the polynomial DPr. In order to do this, we show that a geometric characterization of the DPr due to Shvidkoy remains valid for the weak polynomial topology. Using similar techniques, we further establish that every linear Daugavet center is also a polynomial Daugavet center. This is recent joint work with Yoël Perreau and with Miguel Martín himself.

Juan Guerrero-Viu
Title: Optimal representations in Projective Tensor Products and $L_1$ spaces
Abstract: Given two Banach spaces $X$ and $Y$, an element of the projective tensor product $X\widehat{\otimes}_\pi Y$ is said to attain its norm if it admits an optimal representation as a series of elementary tensors. In this talk, we will analyze geometric conditions on the spaces $X$ and $Y$ that guarantee that every tensor attains its norm, or that the set of such tensors is dense in $X\widehat{\otimes}_\pi Y$. Furthermore, we will characterize the set of norm-attaining tensors in the case of $L_1\widehat{\otimes}_\pi Y$ with $Y$ strictly convex, as precisely those representable by Bochner integrable functions with pairwise disjoint supports. This is part of joint work with L. C. García-Lirola, T. Procházka, and A. Rueda Zoca.

Vladimir Kadets
Title:  Norms of partial sums operators for a basis with respect to a filter.
Abstract: Basis of a Banach space with respect to a filter F on N (F-basis for short) is a generalization of basis, where the ordinary convergence of series is substituted by convergence of partial sums with respect to the filter F. We study the behavior of the norms of partial sums operators for an F-basis, depending on the filter and on the space. One of the central results is:
The following properties of a sequence $(a_n) \subset (1, \infty)$ are equivalent:
(i) $\sum_{n \in N} a_n^{-1} = \infty$.
(ii) There are a free filter F on N, an infinite-dimensional Banach space X and an F-basis
$(u_k)$ of X such that the norms of the partial sums operators with respect to $(u_k)$ are equal to the corresponding $a_n$.
Joint results with Maryna Manskova

Ruben Medina
Title: On the separation modulus and the Nagata dimension of metric spaces
Abstract: A metric space has finite separation modulus if, for every D>0, it is possible to find a probability distribution over the D-bounded partitions of the space such that points that are close have low (linear with the distance) probability of being cut by a partition. This property has been intensively used in metric geometry and computer science since Y. bartal introduced it in 1998. Naor and Silberman proved in 2011 that a metric space with finite Nagata dimension has finite separation modulus, but the reverse implication remained unknown. In this talk we will find an example of a metric space with infinite Nagata dimension but finite separation modulus, concluding that the reverse implication of the latter result does not hold.

Óscar Roldán
Title: A minimum-norm version of the Bishop-Phelps-Bollobás theory
Abstract: Given a bounded linear operator T between two Banach spaces X and Y, we define its minimum norm as $m(T):=\inf\{\|T(x)\|:\, x\in X,\, \|x\|=1\}$. We say that T attains its minimum norm if that infimum is actually a minimum. In this talk we will discuss recent results regarding the density of the set MA(X, Y) of minimum-attaining operators from X to Y. Several Bishop-Phelps-Bollobás-type properties for m will also be discussed. The results include a characterization of the Radon-Nikodym Property and of finite-dimensional spaces. The talk is based on a recent joint work with Domingo García, Manuel Maestre, and Miguel Martín.

Jakub Rondoš
Title: C(K) spaces over countable compact spaces - distances and positive isomorphisms 
Abstract: There has been a substantial progress in the study of the Banach-Mazur distance between spaces of continuous functions over countable compact spaces in the last years. We survey the recent results and present a new contribution to the field. Also, extending the classical results of Bessaga and Pelczynski, we provide a classification of C(K) spaces over countable K by positive isomorphisms. Finally, we consider the related positive version of the Banach-Mazur distance between spaces of continuous functions, and we prove several novel results in this direction. The talk is based on a joint work in progress with M. Cuth, J. Havelka and B. Sari.

Juan Seoane Sepúlveda
Title: Lineability and spaceability. A general overview and open questions.
Abstract: Vladimir Gurariy showed (1966) that the set of Weierstrass' monsters (classical continuous nowhere differentiable functions) contains (up to the zero function) infinite dimensional linear spaces. On top of that, in 1999, he (jointly with V. Fonf and M. Kadets) showed that, when working within C[0,1], the above infinite dimensional linear space can be chosen to be closed in C[0,1]. These results led Gurariy (2005) to coin the terms “lineability” and “spaceability”. The idea behind it is, in a nutshell, to answer the following questions:  How common are “bad” properties? And… what do we mean by “common” in the previous question? And by “bad”?
Lately, quite a few works have been focusing on the search for large algebraic structures (linear spaces, closed subspaces, or infinitely generated algebras) composed of mathematical objects enjoying certain special properties. This trend has caught the eye of several researchers within  Real and Complex Analysis, Operator Theory, Polynomials in Banach spaces, Probability Theory, or general Functional Analysis.
We shall present a general overview of this topic, present some of the early results by Richard Aron, Vladimir Gurariy, or Per Enflo (among others), and provide some new developments, open questions,  and directions of research within several different areas of mathematics.

Hyung-Joon Tag
Title: Existence of vector space structures in nonlinear subsets of Orlicz-Lorentz spaces
Abstract:  Lineability phenomena, the presence of large linear structures in nonlinear subsets, have been investigated in various Banach function spaces such as $L_p$ and Orlicz spaces. In this talk, we study lineability within various nonlinear subsets of Orlicz-Lorentz spaces. Our results rely on a careful analysis of inclusion between Orlicz-Lorentz spaces, which serves as a main tool to show the existence of infinite-dimensional vector subspaces in those subsets. Moreover, using the necessary and sufficient conditions for the inclusion operators being disjointly strictly singular, we also identify other nonlinear subsets where the lineability problem is invalid. This talk is based on joint work with Luis Bernal-Gonz\'alez, Daniel L. Rodríguez-Vidanes, and Juan B. Seoane-Sepúlveda.   

Xu Zhendong
Title: Noncommutative Poincaré-type inequalities
Abstract: We establish $L_q$-Poincaré-type inequalities for ergodic quantum Markov semigroups on finite von Neumann algebras, derived directly from the $ L_2$-Poincaré inequality (PI) or from the modified logarithmic Sobolev inequality (MLSI). Crucially, we obtain the asymptotic behaviors of best constants on $L_q$-Poincaré-type inequalities: $O(\sqrt{q})$ under the MLSI, while $O(q)$ under the PI, as $q\rightarrow\infty$. Applications encompass broad classes of ergodic quantum Markov semigroups, including but not limited to those on quantum tori, mixed $q$-Gaussian algebras, group von Neumann algebras, and compact quantum groups of Kac-type.