#### Event Details

As a joint activity developed by Instituto de Ciencias Matemáticas (ICMAT), the mixed center of CSIC, Universidad Autónoma de Madrid, Universidad Carlos III and Universidad Complutense de Madrid, and Instituto de Matemáticas de la Universidad de Granada (IMAG),we offer 20 full grants for attending a 3 weeks graduate intensive Doc-Course in Functional Analysis (12-30 June 2023).

# ICMAT-IMAG DOC-COURSE IN FUNCTIONAL ANALYSIS

The program will consist of three parts:

#### PART 1. SPECIFIC COURSES to be taught in ICMAT

**Course 1: Logic and descriptive set theory in functional analysis**(ICMAT, June 12-16)

Notions of descriptive set theory and combinatorics will be presented with applications to solving problems in functional analysis in recent research.

Taught by Antonio Avilés (Universidad de Murcia).**Course 2: Metric embeddings and invariants**(ICMAT, June 12-16)

In this mini-course we will explain how various curvature inequalities in metric spaces can place stringent restrictions on faithful embeddings of certain families of graphs. We will pay particular attention to the notion of uniform convexity in Banach spaces, and to the geometry of Hamming cubes, trees, and diamond or Laakso graphs, and related invariants (e.g.,Enflo type, Markov convexity, umbel convexity...). These topics are part of a far-reaching program, the Ribe Program, whose original aim was to find purely metric characterization of local properties of Banach spaces.

Taught by Florent Baudier (Texas A&M University, USA).**Course 3: Stable phase retrieval in function spaces**(ICMAT, June 12-16)

A subspace \(E\) of a function space or Banach lattice is said to do stable phase retrieval (SPR) if there is a constant \(C\) such that for any f,g in \(E\) whose moduli are close, then necessarily f is \(C\)-close to a multiple of g. Phase retrieval appears in several applied circumstances, ranging from crystallography to quantum mechanics. In this Doc-course, we will motivate the phase retrieval problem, present some elementary examples of subspaces of \(L_{p}(μ)\) which do SPR, and discuss the structure of this class of subspaces. In particular, we will discuss a recent joint work with M. Christ and B. Pineau, which uses completely elementary machinery to construct SPR subspaces of \(L_{2}[0, 1]\). After presenting these examples, we will cover a portion of a joint work with D. Freeman, B. Pineau and T. Oikhberg, where we identify the maximizers of instabilities in phase retrieval problems and uncover various interesting connections between SPR and more classical topics in functional and harmonic analysis.

Taught by Mitchell A. Taylor (University of California at Berkeley, USA).**Course 4: Recent advances in Banach lattices**(ICMAT, June 12-16)

Banach lattices were introduced as an abstract framework to deal with many of the function spaces arising in classical analysis. They are Banach spaces which are equipped with a compatible lattice structure so that geometrical and order properties are nicely related. In this course we will present some of the recent developments in the theory, focusing in particular on free Banach lattices associated to Banach spaces and how this construction can be useful to address some open questions in the area.

Taught by Pedro Tradacete (ICMAT).

#### PART 2. SPECIFIC COURSES to be taught in Facultad de Ciencias Granada

**Course 1: Nonlinear geometry and asymptotic properties of Banach spaces**(Facultad de Ciencias Granada, June 19-23)

The celebrated Ribe program aims at metrically characterizing the local properties of Banach spaces (i.e., isomorphic properties of their finite dimensional subspaces). Let us say, for simplification, that asymptotic properties of Banach spaces are those that can be read on their finite codimensional subspaces. For instance, the properties of the weakly null sequences or nets, or more generally of weakly null trees. Between 2000 and 2010, Nigel Kalton and his coauthors exhibited many asymptotic properties that are stable under nonlinear embeddings or equivalences, as well as the corresponding metric invariants. We will describe them, as well as the more recent developments and applications of this line of research, which is sometimes called the asymptotic Ribe program, but could as well be called the Kalton program.

Taught by Gilles Lancien (Université Franche-Comté, France).**Course 2: Daugavet property, big slice phenomena, and local versions**(Facultad de Ciencias Granada, June 19-23)

A Banach space \(X\) has the Daugavet property if every rank-one operator \(T\) on \(X\) satisfies the Daugavet equation \(||Id-T||=1+||T||\). In this case, all weakly compact operators also satisfy DE. This property takes its roots from the 1963 result by I. Daugavet showing that \(\mathbb{C}[0,1]\) enjoys it, and there has been an extensive study of this property in the last 25 years. We will provide an overview of the main examples of spaces with the Daugavet property and of the main geometric and topological implications of it. For instance, a Banach space with this property satisfies that all of the slices, all the relative weakly open subsets, and all the convex combinations of slices of its unit ball have diameter two (the maximum possible). These three properties do not imply the Daugavet property, and they are different each other’s. We will present the main counterexamples on this. We will also present the “local” or pointwise versions of all the four properties and present the relations between them. Some focus will be put on the study of this properties for Lipchitz free spaces, spaces of Lipschitz functions.

Taught by Miguel Martín and Abraham Rueda Zoca (IMAG).**Course 3: Lipschitz-free spaces and their strcucture**(Facultad de Ciencias Granada, June 19-23)

To every metric space \(M\), it is possible to assign a Banach space \(F(M)\) generated by \(M\) in such a way that Lipschitz maps between metric spaces are converted into bounded linear operators between the corresponding Banach spaces. We call Banach space \(F(M)\) the Lipschitz-free space over \(M\) (also known as the Arens-Eells space or Transportation Cost Space).

In this course, we will introduce the Lipschitz-free spaces, their basic properties, and their role in the study of the metric classification of Banach space geometry. We will then present some recent results on the structure of Lipschitz-free spaces, in particular their approximation properties and the relation of their elements to measures.

Taught by Eva Pernecka (Czech Technical University in Prague, Czech Republic).**Course 4: Transportation cost spaces and their embeddings into \(L_{1}\)**(Facultad de Ciencias Granada, June 19-23)

After introducing the Lipschitzfree space \(F(X)\), aka Transportation cost space, Wasserstein space, Arens-Eels space, over a metric space \((X,d)\), we consider the \(L_{1}\)-distortion \(dist_{L_{1}}(F(X))\) of \(F(X)\),which is defined by

\(dist_{L_{1}}(F(X))=\inf\left\lbrace dist(f) / f:F(X)→ L_{1} \text{expansive}\right\rbrace\)

where for an expansive map \(f:F(X)→ L1 (i.e. ||x-y||≤ ||f(x)-f(y)||\) for all \(x,y\) in \(X)\) \(dist(f)\) denotes its best Lipschitz constant. We are interested to find lower as well as upper estimates of \(dist_{L_{1}}(F(X))\).

To do that, we will first recall some relevant results from Computer Science about stochastic embeddings of finite metric spaces into trees. These results imply that \(dist_{L_{1}}(F(X))\) is at most of the order \(\log(n)\) if X is of cardinality n. According to a result of Koth and Naor, this upper estimate is optimal in the case of Hamming cubes. Following an argument of Kysliakov, and its discretization by Naor and Schechtman, we willalso provide lower estimates for \(dist_{L_{1}}(F(X))\) for certain metric spaces \(X\).

Taught by Thomas Schlumprecht (Texas A&M University, USA).

#### PART 3. SUPERVISED RESEARCH PROGRAM

PhD students attending the Doc-Course will also participate in short research projects supervised by experts, in which they will learn some of the most up-to-date techniques in Functional Analysis. This will take place both at ICMAT and IMAG, half of the students in each center) during June 26-30.

### APPLICATION/REGISTRATION AND GRANTS FOR STUDENTS

Students with special interest and background on Functional Analysis are encouraged to apply for a full grant to attend this Doc-Course. This Doc-Course is addressed to students enrolled in a doctoral program. Postdoc researcher with a doctoral thesis obtained in the last two years are also encouraged to apply.

Interested student in a full grant should send to imag@ugr.es with the subject **Doc-Course application**, the following documents:

- Complete academic record
- CV (free format)
- Recommendation letter
- Short motivation letter (explain why you are interested and your expectations)
- Choose your priority for the research projects between the list at the end of this document

The deadline for registration is April 20, 2023. The 20 selected students will be contacted before May 6th with instructions.

The 20 grants will cover board and lodging in Granada and Madrid. The 20 selected students will travel from Madrid to Granada on June 18, and only ten of them will travel back from Granada to Madrid on June 25. Only these travels will be covered by the grants.

**Supervised Research Projects**

- The approximation properties of Lipschitz-free spaces following Godefroy and Kalton (IMAG).
- Differences between some of the big slice phenomena properties (IMAG).
- Applications of set theory and logic in functional analysis problems (ICMAT)
- Open problems in stable phase retrieval (ICMAT)

#### ORGANIZERS

Ginés López (IMAG)

Miguel Martín (IMAG)

Pedro Tradacete (ICMAT)

Ignacio Villanueva (ICMAT)