The LIGAT is supported through the following sources:

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

**Project leaders: Natàlia Castellana Vila & Joachim Kock**

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.

**(2) MICINN Grant MTM2015-66165-P. Geometric Structures**

** Project leaders: Joan Porti i 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.

**(3) 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.