2025 CIMPA School

Virtual Preliminary Courses.

Week -1:  July 14-18, 2025  

1. Title: An introduction to C*-algebras.

Lecturer: Damián Ferraro (Universidad de la República)

Description: This course covers the basic theory of C*-algebras. We will present the abstract definition of such objects; the functional calculus for normal elements and use states to construct (faithful) representations on Hilbert spaces., *-homomorphisms, algebras with generators and relations, tensor products.

2. Title: Inverse semigroups, groupoids and Steinberg algebras.
Lecturer: Guillermo Cortiñas (Universidad de Buenos Aires)
Description: We will give a concise introduction to the subject of the title. Topics include:

-Inverse semigroups, examples: groups, graph semigroups, partially defined injections.
-Actions of semigroups on spaces, examples: partial actions of groups, graphs and shift spaces, universal semigroup action.
-Groupoids, examples: groupoid of germs associated to a semigroup action; graph groupoids. Slice semigroup of a groupoid.
–Properties of groupoids: ampleness, effectivity, minimality, etc. Groupoid gradings.
–The Steinberg algebra of a groupoid; gradings. Examples: Leavitt algebras, semigroup algebras. Ideal structure and simplicity theorems.

3. Title: Representations of compact groups

Lecturer: Gastón García (Universidad Nacional de La Plata)

Description: Roughly speaking, a topological group is a group which is also a topological space such that the group structure maps are continuous.
A topological group G is a compact group if its underlying space is compact and Hausdorff.
Representations of compact groups share several properties with that of finite groups, for the compactness works as a finiteness condition.
In this short course we discuss the main properties and important notions of representations of compact groups, as complete reducibility of finite-dimensional representations, the Haar measure, Schur's lemma and the Peter-Weyl Theorem, among others, which we illustrate through the examples of U(1), SU(2) and SO(3).

4.Title: Category Theory and Homological Algebra

Lecturers: Guido Arnone (Universidad de Buenos Aires) and Devarshi Mukherjee (Universität Münster)

Description: In this course, we will review basic concepts from category theory and homological algebra. The first part of this will entail definitions and examples of categories, functors, naturality, (co)limits, and adjunctions. In the second part, we will specialise to the category of modules over a ring and define the homotopy category of chain complexes and its derived category. Finally, we will introduce the notion of projective modules, leading to the category of perfect complexes and its relation to the derived category of a ring.

Course 1 will provide preliminaries to Prof. Gardella's course, Course 2 to Prof. Li's, Courses 1, 3 and 4 to Prof. Voigt's and Course 4 to Prof. Mukherjee's.

In-person Courses

Week 1: July 28-August 1, 2025, Venue: Facultad de Cs. Exactas, UNLP.

C1: The classification of C*-algebras associated to dynamical systems                      

Lecturer:  Eusebio Gardella 

Description: Given an action of a discrete group on a compact space, there is a canonical way to construct a C*-algebra called the \emph{crossed product}. For example, the noncommutative torus, which is arguably the C*-algebra which historically received the most attention, can be described as the crossed product of the irrational rotation on the circle. Given the recent advances in the classification of simple, nuclear C*-algebras using K-theory and traces, a very interesting question is to determine, in dynamical terms, when the crossed product of a topological dynamical  system is classifiable with said invariants. Minimality, (essential) freeness and amenability of the action are necessary conditions, but not  sufficient in general. Although we are far from having a complete picture, all  evidence indicates that the assumptions that need to be added to get classifiability depend on the amenability of the group. In the amenable case, the relevant condition  is mean dimension zero, which is a notion introduced by  Lindenstrauss and Weiss in a completely different setting.  Perhaps paradoxically, the nonamenable setting seems to  be tamer, and we expect minimality, essential freeness  and amenability of the action to in fact be sufficient for classifiability.  This course will give an introduction to crossed products, as well as an overview of the results described above,  putting them into context and stating some of the most active conjectures in this setting. No previous knowledge on dynamical systems will be assumed, and only basic C*-algebra theory will be needed to follow the course.

C2: Steinberg algebras of ample groupoids

Lecturer:  Huanhuan Li

Description: Steinberg algebras were introduced in the context of discrete inverse semigroup algebras and independently as a model for Leavitt path algebras. In the following talks, we talk on graded representations of Steinberg algebras.The Steinberg algebra of the skew-product groupoid is isomorphic to the corresponding smash product. This allows us to relate the category of graded modules of the algebra to the category of modules of its smash product. We also consider the minimal (that is, irreducible) representations in the category of graded modules of a Steinberg algebra, and establish a connection between the annihilator ideals of these minimal representations, and effectiveness of the groupoid. 

C3: Quantum symmetries and applications                      

Lecturer: Christian Voigt (University of Glasgow)

Description: This course is an introduction to the notion of quantum symmetry, a topic with applications ranging from operator algebras, graph theory, to quantum information theory. We will cover key concepts in the area, including the definition of a quantum group, and illustrate everything with concrete examples as we go along.   

C4: Homotopical algebra in functional analytic settings                      

Lecturer: Devarshi Mukherjee (Universität Münster)

Description: Topological vector spaces and rings have classically not interacted well with homological algebra, stemming from the foundational issue that modules over topological algebras are never abelian. This has prevented the use of categorical (or more precisely, homotopy-theoretic) methods in the study of topological algebras - which constitute the fundamental objects of analytic and noncommutative geometry. In this lecture series, we introduce two alternative approaches to functional analysis that replace topological vector spaces with categories that have better properties, namely, bornological and condensed modules. We demonstrate that both approaches yield well-behaved derived categories of suitably complete modules. Finally, we construct the subcategory of nuclear modules, which plays the role of perfect complexes over a ring in the analytic setting. In particular, the category of nuclear modules could be used as input for localising invariants such as K-theory and cyclic homology, defined in the generality of dualisable stable \infty-categories, The resulting invariants of topological algebras are presently being investigated in the context of analytic geometry and geometric topology. 

Week 2: August 4-8, 2025. Venue: Facultad de Cs. Exactas y Naturales, UBA

C1: Stability for C*-algebras and groups                      

Lecturer: Tatiana Shulman (Chalmers University of Technology and Universiteit Gothenborg)

Description: Group-theoretical stability requires that approximate representations of a group are close to actual representations, where closeness is measured using a chosen metric (e.g., the Hilbert-Schmidt distance or the operator norm). In recent years, there has been a surge of interest in group stability. This is motivated in part by the fact that group stability provides a promising new angle to attack appoximation conjectures (e.g. Connes embedding problem for groups). Operator algebraic methods come up naturally in the study of stability for groups. Moreover aspects of stability appeared in C*-algebra theory long ago and play an important role in structure theory of C*-algebras.                      

C2: Bergman Algebras                      

Lecturer: Roozbeh Hazrat (University of Western Sydney)Description: A half a century ago Bergman introduced a stunning machinery which realized any conical monoid as a nonstable K-theory of a ring. The ring constructed was “universal”. Many subsequent algebras constructed were a special case of Bergman algebras. We start by describing Bergman’s machinery and extend it to the case of Gamma-monoids. This allows us to realize graded Grothendick groups, which are thought to be a complete invariants for certain algebras.                      

C3: Quantum groups, tensor categories and conformal field theory                      

Lecturer: Claudia Pinzari (Università di Roma La Sapienza)

Description: The course aims to give a general theory weak quasi-Hopf algebras and Tannakian reconstruction. Weak quasi-Hopf algebras form a class of quantum groups extending Woronowicz compact quantum groups and incorporating aspects of Drinfeld-Jimbo quantum groups at roots of unity in regards to unitary structures. We shall describe applications to construction of tensor equivalences between tensor categories arising from quantum groups and to constructions of unitary structures. The course is based on pioneering and important results by Hans Wenzl, and builds on the work of many people working in the areas of quantum groups, tensor categories, vertex operator algebras and CFT, that we shall attempt to explain.                    

C4: Efimov K-theory and assembly maps                       

Lecturer: Thomas Nikolaus (Universität Münster)

Description: We want to use Efimov K-Theory to give a model for `assembly maps' and explain some certain applications in geometric topology. In order to do this we will define certain dualisable stable infinity categories of (completed) sheaves on a topological space X with values in Mod_R (R a ring spectrum). The main fact about those categories is that their Efimov K-theory is equivalent to to K(R)-homology of the space X. Using this, our model for the assembly map is simply a functor to local systems on X. We will explain how certain canonical objects of this categories give rise to classes in K(R)-homology and what this has to do with simple homotopy types, surgery and assembly conjectures.                        

School Coordinators

Guillermo Cortiñas (Universidad de Buenos Aires)

Enrique Pardo (Universidad de Cádiz)

Gisela Tartaglia (Universidad Nacional de La Plata)

Organizing Committee

Guillermo Cortiñas (Universidad de Buenos Aires)

Eugenia Ellis (Universidad de la República)

Gastón García (Universidad Nacional de La Plata)

María Eugenia Rodríguez (Universidad de Buenos Aires)

Emanuel Rodríguez Cirone (Universidad de Buenos Aires)

Gisela Tartaglia  (Universidad Nacional de La Plata)