Olivier   Mathieu

Research Topic: Representation Theory and Applications

RECENT PREPRINTS

RECENT CONFERENCES

Josè Adem Memorial Lecture Series, two lectures, November 2019,
Josè Adem Memorial

Conference on Lie and Jordan Algebras, their Representations and Applications IX, Sichuan University, January 2020
LieJor-IX

RECENT RESEARCH COURSES

Cours de Master 2: "Polynomes Harmoniques" (in French) UdL, Mars 2020 Supported by Labex MILYON/ANR-10-LABX-0070.

Sauf indication contraire, les videos sont au format webm.

Ch. 1. "Fonctions harmoniques"
Video1a: Fonctions Harmoniques 24min,
Video1b: Digression 30min.

Ressource: FonctionsHarmoniques.pdf.

Ch. 2. "Compléments d'algèbre commutative"
Video2a:Lemme de Nakayama.mp4 36min,
Video2b: Théorème de Kaplansky 23min,
Video2c: Critère de liberté 26min.

Rq: La video "Lemme de Nakayama", en mp4, commence par un écran noir de 9s.

Ch. 3. "Compléments d'algèbre commutative "
Video3a: Théorème de finitude d'Hilert 35min,
Video3b: Nilcones 20min.

Rq: Wave front= front d'onde

Ressource: Elementary Commutative Algebra.pdf

Ch. 4. "Fonctions G-harmoniques"
Video4a:Fonctions G-Harmoniques 45min,
Video4b: Structure de l'espace des polynomes harmoniques 52min.

Ch. 5. "Théorème de Chevalley"
Video5a: Enoncé du Théorème de Chevalley 16min,
Video5b:(1)=>(3) 25min,
Video5c:(3)=>(2) 37min,
Video5d:(2)=>(1) 32min,
Video5e:Conclusion 7min,
Video5f:Deux Exos 2min.

Ressource: Lie.pdf

Ch. 6: "L'algèbre A des coinvariants"
Video6a: Suites regulières 30min,
Video6b: Preuve du critère de régularité 42min,
Video6c: Algèbre de Frobenius 40min,
Video6d: Preuve que A est Frobenius 47min,
Video6e: Composante maximale de A 23min,
Video6f: Complément: Le lemme de Bezout 10min.

Rq: Mauvaise terminologie. Le lemme de Bezout est en fait le théorème de Bezout, calculant le nombre de points d'intersections de n hypersurfaces dans l'espace projectif de dimension n. Le lemme de Bezout concerne la forme normale des formes symplectiques.

Ressource: Advanced Commutative Algebra1.pdf Advanced Commutative Algebra2.pdf

Ch. 7: "La cohomologie de K/T"
Video7a: Décomposition de Bruhat 20min,
Video7b: L'action de Springer 29 min,
Video7c: La cohomologie de K/T.mp4 39 min,
Video7d: La transgression c 24 min,
Video7e: Le théorème de Soergel 25 min.

Rq: La video "La cohomologie de K/T", en mp4, commence par un écran noir de 10s.

Ch. 8: "Base des polynomes harmoniques"
Video8a:Description élémentaire 18 min,
Video8b: Base topologique des polynomes harmoniques 20 min,
Video8c: Conclusion 10 min.

Rq: En raison de la longueur du cours, la description combinatoire des bases n'a pu être abordée. Par exemple, les opérateurs θ_s introduits dans la Video 5b permettent de construire la base la plus simple, dite de Bernstein-Gelfand-Gelfand.

Ressource: K-Theory of G/B.pdf

Forum:
Isaac's Question.pdf
Luca's Question1.pdf
Luca's Question2.pdf

Bibliographie:
Fonctions harmoniques: Axler, Bourdou & Ramey "Harmonic Function Theory" ch. 1, voire ch.1-2
Algèbre commutative: Atiyah & Macdonald "Introduction to Commutative Algebra"
Théorème de Chevalley: Bourbaki "Lie 4-5-6", voir Ch.5 (paragraphe 5)
Suites regulières: Matsumara "Commutative Algebra" (ch.6) ou Bourbaki "Algèbre commutative 10".
K-Théorie de G/B: Mathieu "Positivity of some intersections in K(G/B)"

Research course: "Weyl Algebras" Kiev, July 2020 Kyev Conference

Ch. 1. "Generality on Growth"
Video1: Numeric Growth 23min,
Video2: Growth of Algebras 42min.

Ch. 2. "GK dimension and exponent"
Video3: Growth of Modules 5min,
Video4: Hilbert Functions 28min,
Video5: GK Dimension and exponent 35min.

Ch. 3. "Characteristic Varieties"
Video6: Almost Commutative Algebras 40min,
Video7: Characteristic varieties 22min.

Ch. 4. "Poisson Structures"
Video8: Poisson Structures 35min,
Video9: Gabber's Theorem 50min.

Ch. 5. "Holonomic Modules"
Video10: Holonomic Modules 31min.

Ch. 6. "Free Resolutions"
Video11: Nakayama Lemma 15min,
Video12: Hilbert's Syzygies 18min,
Video13: Serre-Stafford's Syzygies 28min.

Ch. 7. "Projective non-Free Ideals"
Video14: Generalities 20min,
Video15: An Example for the Weyl Algebras 22min.
Video16: An Example for the Envelopping Algebras 15min.

Ch. 8. "A Quick Review of the Berest and Wilson Work"
Video17: A Technical Lemma 20min,
Video18: An Abstract of the Berest-Wilson Works 10min.

The first part of the course is about the notion of holonomicity. Ch.6 and later are about projective modules for the weyl Algebra. Unfortunately, the beautiful works of Berest and Wilson about the ideal group of the Weyl algebra A1 is only quickly explained.

ATTENTION : il existe plusieurs Olivier Mathieu.
Ce site en construction est celui d'un paisible mathématicien. Je désaprouve les idèes racistes de mon homonyme nazillon. Celui-ci a d'ailleurs été condamné par la justice belge.

WARNING : there are some homonymous Olivier Mathieu.
The present unfinished website belongs to a peaceful French mathematician. I disapprove the racist claim of the homonymous Olivier Mathieu. Besides he has been condemned by a court in Belgium.





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