## Jacques Duparc

### Coordonnées

Professeur ordinaire
Département des opérations Contact Jacques.Duparc@unil.ch Anthropole, bureau 3082 Tél 021.692.35.88 Adresse postale Université de Lausanne Quartier UNIL-Chamberonne Bâtiment Anthropole 1015 Lausanne Page personnelle : people.epfl.ch/jacques.duparc |

### Recherches

### Axes de recherche

**Borel mappings via games and representation theorems**

We characterize some classes of Borel functions over the Baire space as strategies in suitable games. This is a way towards both obtaining representation theorems and elaborating a fine classifications of Borel functions. Representation theorems come as a representation of some classes of functions very different from their definitions. For instance, a cornerstone for this type of result is the Baire Grand Theorem which states that the following are equivalent for any function f on a Polish space:

- f is the pointwise limit of countably many continuous functions;

- on every non-empty closed subset f admits a point of continuity.

**Continuous Reductions on Quasi-Polish spaces**

Quasi-Polish spaces is a novel unifying theory due to Matthew de Brecht. It brings together topological structures that were previously unrelated. It connects closely topology in mathematical analysis -- which is usually Hausdorff (T_2) -- to topology in computer science -- which is rather Kolmogorov} (T_0) -- by offering the Polish spaces as well as the omega-algebraic or omega-continuous domains a common roof. Quasi-Polish spaces are derived from Polish spaces -- which are separable completely metrizable topological spaces -- by simply relaxing the symmetry condition in the definition of a metric.

We propose to design and make use of game theoretical tools to study the reductions between these sets and explore the underlying ordering as well as

the natural hierarchies that would arise. We intend to do this in a similar manner as the way we studied the Wadge hierarchy of Borel subsets of the Cantor space.

**Quotients of Projective Fraïssé Limits**

The idea of studying infinite structures via approximation by finite structures is a well rooted concept in mathematics. In particular, the Fraïssé limit is an extensively studied tool in many areas of mathematics.

In 2006 T. Irwin and S. Solecki introduced the projective Fraïssé limit of topological structures. Many applications have since been found in continua theory and descriptive dynamics.

We propose to isolate and study the class of all compact metric spaces that are obtainable as a quotient of a projective Fraïssé limit by the interpretation of a binary relation symbol from the language. Our hope is to describe a natural way of obtaining such spaces.

**The Wadge Hierarchy**

Over a century ago, the modern theory of integration, based on measure theory induced a fundamental interest in the study of well-behaved subsets of the real line or the real plane. Topology, which developed about the same time yielded the mathematical framework for such a study. For instance, the σ-algebra generated by the open subsets proved to be central in measure theory, for the sets it defines bear all desired nice properties.

The most refined classification of these sets is the so-called Wadge hierarchy whose study involves methods from (set theoretical) game theory

**Topological Complexity, Games, Logic and Automata**

We try to unravel the fine topological structure of omega-regular tree languages which are the infinitary languages of trees recognized by automata. In other words, we exhibit the Wadge hierarchy of non deterministic omega-tree automata.

### Assistants

Carla Perez Cortes
carla.perezcortes@unil.ch |
Beatrice Pitton
beatrice.pitton@unil.ch Tél: (021 692) 61.10 Bureau: ANT 3061 page personnelle |
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Thanusya Thambirajah
thanusya.thambirajah@unil.ch |

### Publications

please refer to the author in unisciences

### Curriculum

### Compétences

**Cours donnés à l'EPFL**

**Logique mathématique**

Fondement des mathématiques (théorie des ensembles) et fondement de l'informatique (informatique théorique).

### Formations

**Doctorat de mathématiques**

Université Paris VII-Denis Diderot, (1995)

### Mots-clés

- informatique théorique
- logique mathématique
- théorie descriptive des ensembles