### Admission

Requirements : first year master's, with a major in mathematics, computer science or logic – or equivalent

Application file : Registration will open on May 1st, 2020.

Requirements : first year master's, with a major in mathematics, computer science or logic – or equivalent

Application file : Registration will open on May 1st, 2020.

The LMFI naturally leads to pursuing a PhD, either in mathematical logic or in (fundamental) computer science.http://pubmaster.math.univ-paris-diderot.fr/admin/root:annee:m2-lmfi/preview

One term of core classes, one term of advanced classes and a research internship.

- Director of studies : Antonio Bucciarelli , Boban Velickovic
- Administrative contact : Mme Chatoux

LMFI consists of:

- During the first term:
- an
**intensive introductory class**in logic (30h), optional; - four
**core classes**(three 48h classes, one 84h class, 20 ECTS); - core classes
**exercices session**(36h each, 8 ECTS each).

- an
- During the second term:
- a choice of eight
**advanced classes**(48h each); - a choice of three
**ouverture classes**(24h chacun); - an
**research internship (Master's thesis)**, supervised by an academic.

- a choice of eight

*Classes may be taught in English, if so requested by the students.*

The Master's internship can take place, subject to approval by the Master's directors, either:

- in one of the two host departments: the Institut de mathématiques de Jussieu - Paris Rive Gauche (and more particulary its Logic team) or the Institut de recherche en informatique fondamentale;
- in one the of the many Parisian universities where research in logic is conducted: the École Normale Supérieure, the Université Paris-Saclay...
- in other French universities: the Université Claude Bernard, in Lyon, the Université Aix-Marseille...
- in other European universities : Torino, Pisa, Münster, Freiburg, Düsseldorf, Florence...

contact: Tamara Servi

LMFI is in partnership with Logic Groups at several European universities (Turin, Münster, Pisa, Freiburg, Florence...). With the Erasmus+ exchange programme, students and teachers from partner universities can participate in some of the LMFI activities, and students and teachers from LMFI can participate in some of the activities of partner universities. The list of all universities which have a partnership with Université Paris Diderot is available here.

*Erasmus+ incoming*:
If you are a Master's student at one of the partner universities, you can apply for a study abroad period at LMFI (1st term, 2nd term or both). The application procedure and deadlines depend on your university (ask a logic teacher or the person responsible for international exchanges at your home university).

*Erasmus+ outgoing*:
If you are an M1 student at Paris Diderot or an LMFI student, you can apply for a study abroad period at one of the partner universities (4 to 10 months). For more details (application procedure and deadlines, the best partner destinations to study logic, etc.), ask Tamara Servi.

*Incoming*:
If you are a Master's student at any foreign university and you wish to do a research internship (Master's thesis) in mathematical logic or theoretical computer science at Université Paris Diderot, then you should contact Boban Velickovic and Christine Tasson.

*Outgoing*:
If you are an LMFI student and you wish to write your Master's thesis (stage) under the supervision of a researcher at some foreign university (or under the joint supervision of a researcher in Paris and a foreign researcher), then you should contact Tamara Servi.

**August 30 to September 10th, 2021**: intensive introductory class;**September 13th to December 3rd, 2021**: core classes;**December 13h to 17th, 2021**: first term exams;**January 3rd to March 25th, 2022**: advanced and ouverture classes;**April 4th to 8th, 2022**: second term exams;**from April 11th, up to September 30th, 2022**: introduction to research intership/dissertation

The first term schedule can be found here.

To obtain the LMFI degree, a 2nd year Master's degree, students must obtain 60 ECTS distributed as follows:

- the four core classes (20 ECTS);
- two advanced classes (8 ECTS each);
- 8 ouverture ECTS obtained either:
- by taking two ouverture classes (4 ECTS each);
- by taking a third advanced class (8 ECTS);

- the internship/dissertation (16 ECTS).

Ouverture classes can be chosen form the LMFI ouverture classes, or, subject to approval by the Master's directors, among the classes of others 2nd years Master's, for example in the Fundamental Mathematics master's or the MPRI (Master Parisien de recherche en informatique).

0 ECTS, semestre 1

Requirements | |

Program requirements | sans |

Teacher | Patrick Simonetta et Pierre Letouzey |

Weekly hours | 18 h CM |

- Propositional calculus: truth tables, tautologies, normal forms, compactness.
- Predicate calculus: first-order languages, terms, formulas, models; satisfaction of a formula in a model; substructures; isomorphisms; elementary equivalence.
- Set theory: axioms of Zermelo-Frænkel; cardinals; Cantor and Cantor-Bernstein theorems; finite sets, countable sets.
- Introduction to programming: introduction to Ocaml functional programming; connection to lambda-calculus, recursivity, ML typing; common data structures (Boolean, integers, lists, options, trees, etc.).

4 ECTS, semestre 1

Requirements | |

Program requirements | examen |

Teacher | Tamara Servi |

Weekly hours | 2 h CM , 2 h TD |

- 1st-order languages, structures, theories
- Ultraproducts, compactness.
- Elementary extensions, Lowenheim-Skolem theorems, elementary chains.
- Preservation theorems.
- Back-and-forth arguments.
- Quantifier elimination, model completeness
- The space of types.
- (If time allows it) Realized and omitted types, atomic models.

4 ECTS, semestre 1

Requirements | |

Program requirements | examen |

Teacher | Alessandro Vignati |

Weekly hours | 2 h CM , 2 h TD |

- Axioms of ZF
- Ordinals, cardinals, transfinite recursion
- Ordinal and cardinal arithmetic
- The Axiom of Choice and equivalents, filters and ultrafilters
- Cofinality, regular/singular cardinals, König's theorem
- Stationary and club sets, Fodor's lemma
- Absoluteness and reflection theorems
- The constructible universe

4 ECTS, semestre 1

Requirements | |

Program requirements | examen |

Teacher | Alexis Saurin |

Weekly hours | 2 h CM , 2 h TD |

- Completeness theorem of the LK sequent calculus with equality by Henkin's witnesses.
- Sequent calculus: cut elimination of and median sequent theorem in LK. Herbrand's theorem. LJ sub-calculation: intuitionist logic and its BHK interpretation. Properties of the sub-formula and existential witnesses in LJ.
- Natural deduction: NK and NJ systems. Cut elimination in NJ. Properties of the sub-formula and existential witnesses in NJ, then in HA (intuitionist arithmetic).
- Lambda-calculus: Confluence and standardization properties. Representation of recursive functions. T system. Curry-Howard correspondence. Realizability, strong standardization and program correctness.

8 ECTS, semestre 1

Requirements | |

Program requirements | examen |

Teacher | Paul Rozière et Hervé Fournier |

Weekly hours | 4 h CM , 2 h TD |

- Computability: recursive functions and functions computable by machines; logical characterization of computable functions; s-m-n theorem and fixed point theorems; the concept of reduction and undecidable problems.
- Introduction to complexity: time and space complexity classes, hierarchy theorems, reductions, completeness, Boolean circuits, introduction to algebraic complexity.
- Formal arithmetic: Peano axioms and weak subsystems; arithmetization of logic; undecidability theorems; Gödel's incompleteness theorems.

4 ECTS, semestre 1

Requirements | |

Program requirements | Examen |

Teacher | Francois Metayer |

Weekly hours | 2 h CM |

The course presents the fundamental concepts of category theory, accompanied by numerous examples. The main goal is to pave the way towards the modern applications of category theory in logic, theoretical computer science and homotopy theory.

8 ECTS, semestre 1

Requirements | |

Program requirements | projet |

Teacher | Pierre Letouzey |

Weekly hours | 2 h CM , 2 h TP |

One half of this module will consist of course work, the other half will consist of practical work on a machine. The course will finish with a project to be carried out in Coq. The first part of this course is a prerequisite for the Homotopy Type Theory course.

8 ECTS, semestre 2

Requirements | Besides the notions and results of the first semester course, a general mathematical background (at Bachelor's level) will be useful to understand some examples and applications. |

Program requirements | examen |

Teacher | Tomás Ibarlucía |

Weekly hours | 4 h CM |

This course is a natural continuation of the first semester Model Theory course. It will seek to understand and classify the models of a given 1st order theory through the types that can be realized or omitted.

8 ECTS, semestre 2

Requirements | Galois theory. Rudiments of algebraic geometry and model theory would be welcome. Reminders will be done in class if necessary. |

Program requirements | examen |

Teacher | Silvain Rideau |

Weekly hours | 4 h CM |

From its early development, valued fields have always played an important role in model theory. One reason for this interest is that their strong connection to arithmetic and geometry has allowed the introduction of model theoretic techniques in other fields of mathematics, resulting most often in the resolution of open questions in that field.

One of the first example of such an interaction can be found in the work of Ax-Kochen and independently Ershov who gave a solution to Artin’s conjecture on the existence of solution to homogeneous equations over p-adic fields. One of the core concept of their proof is a reduction of a question on certain characteristic zero Henselian fields, in that specific case the description of its theory, to this very same question on their residue field and value group. The idea of this reduction can be found time and time again in later work on the model theory of valued fields.

The goal of this class will be, starting with questions of quantifier elimination and then moving on to Shelah’s classification theory and more « geometric » model theory, to show the ubiquity of this Ax-Kochen-Ershov principle in the model theory of valued fields. We will also try, as much as it is possible, to give an insight into some of the recent applications of the model theory of valued fields.

8 ECTS, semestre 2

Requirements | |

Program requirements | examen |

Teacher | Mirna Dzamonja |

Weekly hours | 4 h CM |

Set theory means many things to many people, but most will agree that it is a subject of mathematics that has two main roles: a foundational role in the way it gives axioms to most (although not all) aspects of modern mathematics, and a mathematical role in the way that it provides a firm theory of infinity. In its most common axiomatisation, Zermelo-Fraenkel with Choice (ZFC), set theory has been able to join these two roles to the point that each describes the strength and the limits of the other. These are best understood through the study of inner and outer models of set theory, notably the constructible universe L and the method of forcing.

This course will open the first pages of the advanced set theory, assuming that the student already knows basic axiomatic set theory. It will describeforcing, go through the classical proof that the Continuum Hypothesis is not a consequence of the axioms of ZFC and then continue to give some more classical forcing notions, for example the Lévy Collapse. This will naturally lead to the study of iterated forcing and Martin’s Axiom, some applications and limitations.

8 ECTS, semestre 2

Requirements | |

Program requirements | Examen |

Teacher | Boban Velickovic |

Weekly hours | 4 h CM |

Large cardinal axioms postulate the existence of cardinals with a given degree of transcendence over smaller cardinals and provide a superstructure for the analysis of strong mathematical statements. The investigation of these axioms is indeed a mainstream of modern set theory. For instance, they play a crucial role in the study of definable sets of reals and their regularity properties such as Lebesgue measurability. Although formulated at various stages in the development of set theory and with different motivation, these hypotheses were found to form a linear hierarchy reaching up to the inconsistency. All known set-theoretic propositions can be gauged in this hierarchy in terms of their consistency strength, and the emerging structure of implications provides a remarkably rich, detailed and coherent picture of the strongest propositions of mathematics as embedded in set theory.

8 ECTS, semestre 2

Requirements | |

Program requirements | examen |

Teacher | Antonio Bucciarelli and Claudia Faggian |

Weekly hours | 4 h CM |

Proof theory has undergone at least two major developments over the past century as a result of Gödel's incompleteness theorems. The first took place in the 1930s, immediately after the results on incompleteness, with the introduction and study of natural deduction and sequent calculus s by Gentzen and lambda-calculus by Church. Church then showed the undecidability of predicate calculus via lambda-calculus while introducing a universal computation model while Gentzen deduced the consistency of various logical systems as a corollary of cut elimination of breaks in sequent calculus.

The second stage took place in the 1960s with the gradual highlighting, through the Curry-Howard correspondence, of the profound links between proofs and programs, from the correspondence between simply typed lambda-calculus and minimal propositional natural deduction to the various extensions of this correspondence to the second order, to classical logic and to the emergence of the notion of linearity in proof theory. Linear logic has profoundly renewed the links between the formal semantics of programming languages on one hand and proof theory on the other. Linear algebra is the third pole of this correspondence, focusing on the notion of computational resource.

The basic course covered the first step. This course will be devoted to some more recent developments.

4 ECTS, semestre 2

Requirements | Basic knowledge in category theory |

Program requirements | CC+examen |

Teacher | Pierre-Louis Curien |

Weekly hours | 2 h CM |

Homotopy theory, which is devoted to the study of spaces up to deformation, has given rise to a branch of algebra called homotopical algebra, in which tools are developed for dealing with structures in which laws like associativity do not hold exactly like in classical algebra, but up to homotopy, these homotopies being themselves subject to coherences, etc.

Homotopy theory has also a logical side, in which types, proofs of equality, and proofs of equality of proofs of equality are interpreted as spaces, paths, and homotopies between paths, respectively. The notion of fibration, that plays an essential rôle in homotopy theory, is tightly related with substitution in dependent type theory. This interplay has led to a new version of type theory called homotopy type theory.

The course is a follow-up of that on category theory taught in the first semester, but can be followed by students who have already some basic background in category theory. We shall introduce the important notions underlying the subject: enriched categories, model categories, as well as different approaches to the definition of higher catégories, notably via simplicial sets. We shall also hint at connected subjects such as operads and ∞-operads, that also have arisen from topology. The lectures will partly follow the flow of exposition found in recent books (Categorical homotopy theory by Emily Riehl, The homotopy theory of (∞, 1)-categories by Julia Bergner, From categories to homotopy theory by Birgit Richter, Higher categories and homotopical algebra by Denis-Charles Cisinski, Simplicial methods for higher categories by Simona Paoli, which all offer opportunities to the interested students for learning more), with an eye on the links with homotopy type theory.

4 ECTS, semestre 2

Requirements | |

Program requirements | examen |

Teacher | Vincent Danos |

Weekly hours | 2 h CM |

This course aims to present the logical and computational foundations of blockchains (communication protocols, game theory), as well as examples of protocols implemented in particular in crypto-currencies and smart-contracts. We will devote a substantial part of the course to the examination of "decentralized finance", i.e. all the existing chain contracts which postpone and in some cases extend financial practices.

4 ECTS, semestre 2

Requirements | |

Program requirements | CC+examen |

Teacher | Brice Halimi |

Weekly hours | 2 h CM |

The course will be devoted to this question. In particular, we will examine three main reasons for declaring logic "formal": because it uses discursive resources that can be said to be formal (schematic); because it concerns forms (whose status is to be specified: "logical constants" for Russell, "forms derived from something in general" for Husserl, to mention two important examples); and because it aims at an independent validity of any particular content (logic as universal science).

These three main reasons are not necessarily compatible. Moreover, the examination of the question raised will of course imply taking into account the history of logic, and a reflection on the situation of logic between philosophy and mathematics. This will be an opportunity to examine the issue of "absolute generality", that is, the possibility of a theory covering absolutely everything in general.

**Aim of the class**: knowledge of the philosophical issues of the history of logic in the 20th century.

Candidates must have a 1st year master's degree (M1), or an equivalent degree, with a major in mathematics, computer science or logic.

In order to make the application process easier for international students, the University of Paris Diderot follows the Campus France procedure. Foreign students should find all relevant information on the Campus France website. Foreign students from countries involved in the "Étude en France" procedure should register on that platform before March 2019.

Students must apply on the university website from Mai the 1st to July the 10th.

There are possibilities of scholarships for prospective M1 or M2 students, and particularly for foreigners:

- List of scholarship programs on Campus France;
- PGSM program of the Fondation des Sciences Mathématiques de Paris.

**March 2021**: deadline for foreign students who apply via the "Étude en France" procedure, see the Campus France website for details. This does not apply to students already enrolled in a university establishment in France or European Union citizens.**May 3rd to July 9th, 2021**: application on the E-candidate website.**August 23th to September 15th, 2021**: application on the E-candidate website, for a review in the September session.**early September**: optional introductory class.**mid September**: start of the core classes.

The LMFI naturally leads to pursuing a PhD, either in mathematical logic or in (fundamental) computer science. Phd's in computer science can also be pursued in a compagny or a public research institute (INRIA, CEA, ONERA, etc.). In recent years, more than half of the students that obtained the LMFI Master's degree have continued with a PhD thesis.

The main career prospects after a PhD thesis are in research in a broad sense:

- in academia (French or foreign) or public research institutes (CNRS, INRIA, CEA, ONERA, etc.);
- in private sector research and development departments (EDF, France Telecom, Siemens, EADS, etc.). Research and development departments are particularly interested in recruiting people with strong mathematical, logical and computer skills, allowing them to supervise engineers in software certification, program and protocol verification and more generally in cyber security. In some cases, recruitment may take place directly after the Master's degree.