Spring 2024
The course starts at 10 am in Seminar Room IA 1/75 .
Seminar will be broadcasted live on zoom and recorded. Recorded versions of the
talks will be uploaded on youtube. If you want to attend remotely please click here:
Zoom link
Description of the seminar
Goal of the seminar will be to discuss general facts about generating functions and some of the classical (and more recent) applications.
18.07.2024 Luca Asselle "The contact non-squeezing theorem."
Abstract: We prove a contact version of the celebrated non-squeezing theorem of Gromov. References: [4],[8].
11.07.2024 Pierre-Alexandre Arlove "Uniqueness of gfqi."
Abstract: We prove uniqueness (up to equivalence) of gfqi for Lagrangian submanifolds of T∗M resp. Legendrian submanifolds of J1(M, R). References: [6],[7].
04.07.2024 Jacobus de Pooter "Existence of gfqi, Part 3."
Abstract: In this talk we prove the existence of gfqi for Legendrian submanifolds in spherizations of cotangent bundles. Reference: [5].
27.06.2024 Lucas Dahinden "Existence of gfqi, Part 2."
Abstract: In this talk we prove the existence of gfqi for Legendrian submanifolds of J1(M,R). Reference: [3].
13.06.2024 Lars Kelling "Existence of gfqi, Part 1."
Abstract: In this talk we prove the existence of gfqi for Lagrangian submanifolds of T∗M. Reference: [1].
16.05.2024 Alberto Abbondandolo "Geodesics in the Hofer norm and gene- rating functions."
Abstract: Reference: [2].
02.05.2024 Michael Vogel "Generating functions quadratic at infinity (gfqi), Part 3."
Abstract: This is the third of three talks in which we discuss general facts about gfqi: Spectral invariants; applications to the geometry of the Hamiltonian group; Viterbo capacity and Gromov’s non-squeezing theorem. Reference: [9], Sections 11,12,14.
Link to video here
25.04.2024 Jonas Fritsch "Generating functions quadratic at infinity (gfqi), Part 2."
Abstract: This is the second of three talks in which we discuss general facts about gfqi: Gfqi for Hamiltonian symplectomorphisms of R2n; composition formulas; equivalence of gfqi; the symplectic action. Reference: [9], Sections 7-10.
Link to video here
18.04.2024 Paul Anton Wilke "Generating functions quadratic at infinity (gfqi), Part 1."
Abstract: This is the first of three talks in which we discuss general facts about gfqi: Symplectic and Hamiltonian diffeomorphisms; Lagrangian submanifolds; coisotropic submanifolds, characteristic foliations and symplectic reduction; generating functions for Lagrangian submanifolds. Reference: [9], Sections 2-5.
Link to video here
11.04.2024 Stefan Nemirovski "Introduction to generating functions."
Abstract: This is an overview talk on the dfferent flavours of generating functions and where to find them.
Link to video here
Notes
Literature
The main references for the seminar are:
[1] M. Brunella - On a theorem of Sikorav, L’Ens. Math ́ematique 37 (1991), 83–87.
[2] M. Bialy, L. Polterovich - Geodesics of Hofer’s metric on the group of Hamiltonian
diffeomorphisms, Duke Math. J. 76 (1994), 273–292.
[3] E. Ferrand - On a theorem of Chekanov, Symplectic singularities and geometry of
gauge fields, Banach center Publications 39 (1997).
[4] M. Fraser, S. Sandon, B. Zhang - Contact non-squeezing at large scale via generating
functions (2023), https://arxiv.org/pdf/2310.11993.pdf
[5] P. Pushkar - Chekanov-type theorem for spherized cotangent bundles (2021),
arXiv:1602.08743v1
[6] D. Th ́eret - Utilisation des fonctions g ́en ́eratrices en g ́eom ́etrie symplectique globale, Ph.D. Thesis, Universit ́e Denis Diderot (1995).
[7] D. Th ́eret - A complete proof of Viterbo’s uniqueness theorem on generating functi- ons, Topology Appl. 96 (1999), 249–266.
[8] S. Sandon - Contact homology, capacity and non-squeezing in R2n ×S1 via generating functions, Ann. Inst. Fourier (Grenoble) 61 (2011), 145–185.
[9] S. Sandon - Generating functions in symplectic topology (2014),
http://members.unine.ch/felix.schlenk/Santiago/cours.margherita.pdf
Other references are:
• F. Laudenbach, and J.-C. Sikorav, Persistance d’intersection avec la section nulle au cours d’une isotopie hamiltonienne dans un fibr ́e cotangent, Invent. Math. 82 (1985), 349–357.
• J.C. Sikorav - Sur les immersions lagrangiennes dans un fibr ́e cotangent admettant une phase g ́en ́eratrice globale, C.R. Acad. Sci. Paris, S ́er. I Math. 302 (1986), 119–122.
• J.C. Sikorav, Probl ́emes d’intersections et de points fixes en g ́eom ́etrie hamiltonienne,
Comment. Math. Helv. 62 (1987), 62–73.
• C. Viterbo - Symplectic topology as the geometry of generating functions, Math.
Ann. 292 (1992), 685–710.