The LIGAT is supported through the following sources:

**MICINN Grant PID2020-116481GB-I00. Teoría de homotopía de estructuras combinatorias y algebraicas.**

**Project leaders: Wolfgang Pitsch ang Joachim Kock
Start: 2021-09-01
**

While traditionally algebraic topology uses discrete, algebraic methods to tackle topological problems, we are as much concerned with the applications of homotopy methods to understand combinatorial and algebraic structures. We explore applications to posets, decomposition spaces, incidence algebras, finite groups, representations, and other structures.

We develop in particular the field of homotopy combinatorics, where the homotopy theory of groupoids and infinity-groupoids is applied to problems in algebraic combinatorics. Specifically we develop the theory of decomposition spaces, certain simplicial infinity groupoids, which can be regarded as a far-reaching homotopical generalisation of posets, as a setting for incidence algebras and Möbius inversion. In this project we study decomposition spaces with a view towards combinatorial Hopf algebras, with applications to the theory of symmetric functions and to noncommutative probability.

We continue driving forward the theory of fusion systems, both finite and compact. We focus our attention more specifically on the associated linking systems. In this context we attack several important open problems, some of which are of interest in group theory. We pursue the program towards the classification of homotopy classes of maps between classifying spaces, also with the aim of developing complex homotopy representation theory. In a more algebraic setting, we aim to understand systematically exotic examples of fusion systems and construct new ones.

We also pursue a general study of homotopy augmented functors and localisations, with applications to topology and representation theory. We aim to investigate to which extent homotopy augmented functors applied to different objects either detect or preserve their properties. The combinatorics relating different localisation functors on a well-behaved category (e.g. tensor triangulated), is captured by the Balmer spectrum. We will pursue the study of this object using two techniques developed by our group. On the one hand a point-free point of view on this space and on the other hand descent techniques. In particular descent techniques should open the path to build a good theory of local (co)homology functors on the Balmer spectrum.

Lastly, we shall continue our work using techniques from equivariant stable homotopy to study the so-called conjugations spaces. Through our study of the equivariant Steenrod algebra, we aim at finding equivariant couterparts of some of the fundemental tools of classical algebraic topology, as for instance a good unstability condition for equivariant cohomology modules.

**Collaborators:** Roger Bergadà, Carles Broto Blanco, Guillermo Carrión Santiago, Natàlia Castellana Vila, Wilson Forero Baquero, Joachim Kock, Thomas Mikhail, Wolfgang Pitsch and Albert Ruiz Cirera.

**MICINN Grant PID2020-113047GB-I00. Rings, modules, C*-algebras, and dynamics: Classification, Fine structure, and Regularity.**

**Project leaders: Pere Ara and Francesc Perera
Start: 2021-09-01
**

In this project several questions and problems related to the structure of rings, algebras, and C*-algebras will be studied using invariants associated to these objects and their module categories.

Classification of C*-algebras in both the simple and non-simple settings is a very active area of research. One of the invariants that plays a prominent role, particularly in the non-simple case, is the Cuntz semigroup. Many significant examples come from topological dynamics, and have been transported to crossed products and groupoid C*-algebras, which have since become central objects in the development of the theory. We will introduce and study a dynamical Cuntz semigroup and elaborate a version of the Toms-Winter Conjecture for C*-dynamical systems. We will also analyze versions of the Cuntz semigroup for general rings and study its impact in classificacion problems. The recently developed unitary Cuntz semigroup will be used to classify a class of non-simple C*-algebras with K-Theory obstructions. We will develop tools to tackle Matui’s conjecture on the relationship between K-Theory and homology for groupoid C*-algebras. In particular, we will consider this conjecture in the settings of Exel-Pardo algebras, Deaconu-Renault groupoids, and groupoids associated to adaptable separated graphs. The study of non-Hausdorff ample groupoids will be considered, including conditions on the groupoid under which the corresponding C*-algebra is simple or purely infinite.

Approximation theory roughly consists of approximating classes of modules using covers and envelopes. Classical results from the 1950s-1960s show that, in module categories, injective envelopes always exist, while projective covers just exist in few cases. The brilliant resolution of the flat cover conjecture by Bican, El Bashir, and Enochs (2001), showed that any module over any ring has a flat cover, and thus it becomes clear that the theory of cotorsion pairs is a suitable framework to develop the subject. The theory of approximations will be pushed forward by analysing right-hand classes in cotorsion theories.

Another main objective of our project seeks to develop the theory of fair-sized projective modules in the case of integral group

representations and, in general, of orders over Dedekind domains or, even more generally, over Gorenstein rings. We will also investigate the correspondence between TTF triples, recollements, and idempotent ideals.

The Yang-Baxter equation is a fundamental question in Mathematical Physics. Drinfeld raised the question of computing all the set-theoretical solutions of this equation. This has led to a burst of activity in the last few years. We will construct indecomposable and irretractable involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation whose cardinalities are the square of a prime, and we will study whether said solutions are simple. The structure monoid and structure group of not necessarily bijective solutions will also be scrutinized. In the course of our investigations, the structure of semi-braces will play an important role.

**Collaborators:** Román Álvarez, Ramon Antoine, Pere Ara, Joan Bosa, Laurent Cantier, Ferran Cedó, Joan Claramunt, Dolors Herbera, Enrique Pardo, Francesc Perera, Álvaro Sánchez, Eduard Vilalta and Simone Virili.

**MCINN Grant PGC2018-095998-B-I00. Local and global invariants in geometry
Project leaders: Gil Solanes and Florent Balacheff
Start: 2019-01-01
**

This project is devoted to the study of invariant objects and functionals in several areas of differential geometry. It is structured along the following lines

- Integral geometry and valuations. We will investigate some local and global extensions of the classical Lipschitz-Killing invariants in the context of Kähler, quaternionic-Kähler and pseudo-riemannian manifolds. We will also deal with geometric inequalities for convex sets.
- Universal inequalities between invariants in Riemannian geometry. We will study universal inequalities on manifolds that describe how length spectrum interacts with Riemannian invariants.
- Group actions and moduli spaces for holomorphic geometric structures. We will study symmetry properties of holomorphic foliations and k-webs on algebraic surfaces. Deformations and moduli spaces of compact homogeneous complex manifolds will also be investigated.
- Invariants of geometric structures on three-manifolds. We will study the algebraic and geometric properties of the variety of characters of a 3-manifold, as well as asymptotic properties of Reidemeister torsion.

**Collaborators:** Florent Balacheff, Eduardo Gallego Gómez, Davin Marín Pérez, Marcel Nicolau Reig, Joan Porti Piqué, Agustí Reventós Tarrida and Gil Solanes Farrés.

**MCINN Grant MTM2017-83487-P. Structure and classification of rings, modules and C*-algebras: interactions with dynamics, combinatorics and topology.
Project leader: Pere Ara
Period: 2018-01-01 to 2020-12-31
**

We will study several questions and problems concerning the structure and the classification of rings and algebras, through the use of suitable invariants associated with these objects and their module categories. We will initiate a study of Homological Algebra for the category Cu of Cuntz semigroups, studying the structure of the injective objects, and duality theory with respect to them. Several other aspects of the category Cu will be analyzed, including the computation of the Cuntz semigroup of products and ultraproducts of C*-algebras. We will analyze the type semigroup of dynamical systems on the Cantor set, and the relationships with the non-stable K-theory of the crossed product algebras. Another dynamical invariant will be introduced in order to capture Cuntz semigroup information from the crossed product. Several combinatorially defined algebras will be studied, notably algebras associated to self-similar graphs and to separated graphs. Motivated by important existing examples, such as the algebras associated to the Grigorchuk group, we will undertake a careful study of non-Hausdorff étale groupoids, characterizing the simplicity of their algebras. We will study the structure group of a finite multipermutation solution of the Yang-Baxter equation, and advance in the problem of classifying the finite simple left braces. We will explore the structure of the pure projective modules over commutative noetherian rings, with special emphasis in the class of direct sums of Maximal Cohen-Macaulay modules. Various problems on the embeddings of rings into division rings will also be treated.

**Collaborators:** Ramon Antoine, Pere Ara, David Bachiller, Joan Bosa, Laurent Cantier, Ferran Cedó, Joan Claramunt, Warren Dicks, Dolors Herbera, Enrique Pardo, Francesc Perera, Álvaro Sánchez, Javier Sánchez and Eduard Vilalta

**MICINN Grant MTM2016-80439-P. Teoría de homotopía de estructuras algebraicas.**

**Project leaders: Natàlia Castellana Vila and Joachim Kock
Period: 2017-01-01 to 2020-06-30
**

This project belongs to the field of Algebraic Topology, in a broad sense, and with applications going both ways between algebra and topology. While originally, algebraic topology meant the study of homotopy types of topological spaces by the means of algebraic invariants assigned to them functorially, the past decade has seen a proliferation of applications of homotopical methods in various areas of mathematics, such as notably algebraic geometry, representation theory and aspects of mathematical physics. The project is concerned more specifically with applications of methods of homotopy theory to the understanding of fusion systems both finite and compact modeling conjugacies in finite groups and Lie groups. In this context we attack several important open problems, some of which are of interest in group theory.

We pursue the program for the classification of homotopy classes of maps between classifying spaces, also with the aim of developing complex homotopy representation theory. In a more algebraic setting, we aim to understand systematically exotic examples of fusion systems and construct new ones. As a new line of research we will start the study of stable homotopy theory of p-local compact groups, which is the starting point for extending statements on generalized cohomology theories for Lie groups.

The techniques of localisation and cellularisation belong properly both to algebraic topology and to algebra. We aim to investigate to which extent homotopy idempotent functors applied to classifying spaces detect algebraic structure. A more geometric aspect of the project goes in the direction of properties of the mapping class group, the group of isotopies of an orientable surface, and its relationship with the invariants of homology spheres.

We develop the theory of homotopy combinatorics, where the homotopy theory of groupoids and infinity-groupoids is applied to problems in enumerative and algebraic combinatorics, with special emphasis on applications to quantum field theory. In particular we develop a unifying framework for incidence algebras, combinatorial Hopf algebras, inductive data types, and related structures, uncovering common patterns in a wide range of application areas.

**Collaborators:** Jaume Aguadé, Carles Broto, José María Cantarero López, Louis Carlier, Carlos Andrés Giraldo, Álex González, Wolfgang Pitsch, Ricard Riba, Albert Ruiz and Jérôme Scherer.

**MICINN Grant MTM2015-66165-P. Geometric Structures**

** Project leaders: Joan Porti and David Marin**

The project studies geometric structures on manifolds and it is organized in three research lines:

1. Representations in Lie groups and their moduli spaces. We shall study representation varieties of the fundamental group of a two or three-dimensional manifold in a Lie group. We aim to compute new representation varieties. We also aim to study geometric

structures associated to those representations and to determine their moduli spaces.

2. Moduli spaces of foliations and other geometric structures. We are interested in holomorphic foliations in dimension two, both locally (we seek to identify the moduli space of their singularities) and globally, on the projective plane. We will also deal with the relationship of foliations with complex structures on compact manifolds and we shall study the structure of their group of automorphisms.

3. Integral geometry of isotropic spaces and Kähler manifolds. We intend to determine the kinematic formulas in affine, projective and hyperbolic quaternionic spaces. We will also study the integral geomerty of isotropic non-riemannian spaces. We will investigate the existence of a universal algebra canonically asociated to Kähler manifolds.

**Collaborators:** Judit Abardia, Andreas Bernig, Juan Luis Durán, Eduardo Gallego, Teresa Gálvez, Michael Heusener, Jean-François Mattei, Laurent Meersseman, Marcel Nicolau, Agustí Reventós and Gil Solanes.

**MICINN Grant MTM2014-53644-P. Structure and classification of rings, modules and C*-algebras. **

**Project leader: Pere Ara **

We will study several questions and problems concerning the structure and the classification of rings and algebras, through the use of suitable invariants associated with these objects and their module categories. We will undertake a study of non-commutative dimension theory of Z-stable C*-algebras, and we will apply it to get classification results for non-simple C*-algebras. The Cuntz semigroup of a crossed product C*-algebra will be analysed, and a bivariant version of the Cuntz semigroup, along the lines of KK-theory, will be defined and studied. We will also try to solve Hazrat’s Conjecture on the classification, up to graded isomorphism, of Leavitt path algebras of finite graphs by means of their graded, ordered K-theory. We will use tools coming from the theory of dynamical systems in the study of the structure of several algebras. In particular, the theory of partial actions of groups on topological spaces and the theory of topological groupoids will be used to analyse several problems concerning the structure of C*-algebras of separated graphs and of self-similar graph C*-algebras, a class of C*-algebras generalizing both Katsura C*-algebras and Nekrashevych’s self-similar C*-algebras. In addtition we will undertake a deep analysis of a general notion of entropy, which subsumes most of notions of entropy in Mathematics. We will investigate various questions concerning group and semigroup algebras, as well as the nondegenerated involutive set-theoretic solutions of the Yang-Baxter equation, using a new algebraic structure called left brace, introduced by Rump. We will use torsion theories and localization techniques to analyse categories obtained inverting certain classes of quasi-isomorphisms in the category of unbounded complexes of left modules over a ring. We also plan to undertake a deep study of the direct summands of infinite direct sums of Maximal Cohen-Macaulay modules.

**Collaborators:** Ramon Antoine, David Bachiller, Joan Bosa, Ferran Cedó, Warren Dicks, Dolors Herbera, Enrique Pardo, Francesc Perera, Javier Sánchez and Simone Virili.