Frontiers of Quantitative Symplectic and Contact Geometry

June 27 - July 1, 2022

A fundamental interest driving symplectic geometry research is the dichotomy between rigidity and flexibitility: is the field more like differential geometry or topology? The goal of this workshop is to focus on quantifiable rigid aspects of this dichotomy.

$C^0$-symplectic topology is a prime example. If a sequence of symplectomorphisms $C^0$-converges to a diffeomorphism, then the diffeomorphism is symplectic. This type of rigidity is surprising because the symplectic condition is on the differential, for which the $C^0$-convergence seems to say nothing. Nonetheless, many related results have since been proven. In particular, new invariants for approximating Hamiltonian dynamics have been constructed, and long-standing simplicity conjectures in dimension 2 are now answered. These results broadly fall under the field of quantitative symplectic geometry, which also studies the displacement energy and Gromov width of Lagrangian submanifolds and symplectic domains.

For contact geometry, the odd-dimensional analogue of symplectic geometry, one can ask the same set of questions. Contact geometry has overtwisted contact structures and loose Legendrians on the flexible side, and tight contact structures and fillable Legendrians on the rigid side. $C^0$-convergence questions, displacement energy, Gromov width, contact Hamiltonian/Reeb mechanics, etc are natural ideas in this contact set-up. But there is less progress: while the symplectic form is canonical, the contact form is not; the even number of dimensions in symplectic geometry naturally connect it to the powerful and long-established tools of complex geometry. Still, there is much activity and success answering these questions, such as translated points, positive loops and orderability, some $C^0$-rigidity, Legendrian displacement, and Lagrangian cobordism length/widths.

The tools behind these results rest on pseudo-holomorphic curves. But unlike usual Gromov-Witten-Floer-type theories which package these curves into algebraic invariants, the curves also keep track of the symplectic action of underlying dynamical objects, namely, the Reeb and Hamiltonian orbits. So in a sense, these results rely on a more refined filtered version of the more-studied Floer theories. These filtrations can be captured by symplectic/contact capacities to obstruct symplectic embeddings or to define spectrums. And more recently, these curves produce the barcodes popularized by data science. A related current trend is to combine the triangulated structure encoded in Fukaya type categories with the filtered nature of the relevant Floer type structures and define new algebraic structures and measurements.

While the questions are diverse, the commonality of the tools that can answer them will bring together the different experts on the cutting-edge of developments and progress in this field.

 

Seminars Scroll to the next upcoming seminar

  • Topological entropy, barcodes and Floer theory June 27 09:45 - 10:45

    Topological entropy, barcodes and Floer theory

  • On the volume of Lagrangian submanifolds June 27 11:00 - 12:00

    On the volume of Lagrangian submanifolds

  • Genericity of Birkhoff sections for Reeb flows in dimension three June 27 13:30 - 14:30

    Genericity of Birkhoff sections for Reeb flows in dimension three

  • Conformal points of area preserving maps and related topics June 27 15:00 - 16:00

    Conformal points of area preserving maps and related topics

  • Nearby Lagrangians, fixed points, and flux June 27 16:15 - 17:15

    Nearby Lagrangians, fixed points, and flux

  • C^0 rigidity of the reduction for coisotropic manifolds June 28 09:45 - 10:45

    C^0 rigidity of the reduction for coisotropic manifolds

  • Groups of area-preserving homeomorphisms, spectral estimators, and Sikorav’s trick June 28 11:00 - 12:00

    Groups of area-preserving homeomorphisms, spectral estimators, and Sikorav’s trick

  • Capacities of Lagrangian complements June 28 13:30 - 14:30

    Capacities of Lagrangian complements

  • Almost toric fibrations and symplectic embeddings June 28 15:00 - 16:00

    Almost toric fibrations and symplectic embeddings

  • Infinite Staircases in Hirzebruch surfaces June 28 16:15 - 17:15

    Infinite Staircases in Hirzebruch surfaces

  • Quasimorphisms and area-preserving homeomorphisms of the sphere June 29 09:45 - 10:45

    Quasimorphisms and area-preserving homeomorphisms of the sphere

  • Groups of area preserving homeomorphisms and subleading asymptotics of link spectral invariants June 29 11:00 - 12:00

    Groups of area preserving homeomorphisms and subleading asymptotics of link spectral invariants

  • Legendrian persistence modules and dynamics June 30 09:45 - 10:45

    Legendrian persistence modules and dynamics

  • Symplectic topology and ideal-valued measures June 30 11:00 - 12:00

    Symplectic topology and ideal-valued measures

  • Lagrangian Floer theory on symplectic orbifolds June 30 13:30 - 14:30

    Lagrangian Floer theory on symplectic orbifolds

  • A closing lemma for ellipsoids in any dimension June 30 15:00 - 16:00

    A closing lemma for ellipsoids in any dimension

  • Interlevel persistence and Floer theory June 30 16:15 - 17:15

    Interlevel persistence and Floer theory

  • Problem Session June 30 20:00 - 21:00

    Problem Session

  • Quantitative K-theory in symplectic geometry July 01 09:00 - 10:00

    Quantitative K-theory in symplectic geometry

  • On the Rouquier dimension of wrapped Fukaya categories July 01 10:15 - 11:15

    On the Rouquier dimension of wrapped Fukaya categories

Participants

Name University Country Arrival and Departure
Peter
Albers
University: Heidelberg University City: Germany Arrival and Departure: -
Johan
Asplund
University: Columbia University City: United States Arrival and Departure: -
Lev
Buhovsky
University: Tel Aviv University City: Israel Arrival and Departure: -
Baptiste
Chantraine
University: University of Nantes City: France Arrival and Departure: -
Jean-Philippe
Chassé
University: Université de Montréal City: Canada Arrival and Departure: -
Erman
Çineli
University: Sorbonne University City: France Arrival and Departure: -
Vincent
Colin
University: Nantes Université City: France Arrival and Departure: -
Octav
Cornea
University: University of Montreal City: Canada Arrival and Departure: -
Laurent
Cote
University: Harvard University City: United States Arrival and Departure: -
Dan
Cristofaro-Gardiner
University: University of Maryland, College Park City: United States Arrival and Departure: -
Come
Dattin
University: Uppsala University City: Sweden Arrival and Departure: -
Yash Uday
Deshmukh
University: Columbia University City: United States Arrival and Departure: -
Georgios
Dimitroglou Rizell
University: Uppsala University City: Sweden Arrival and Departure: -
Milica
Đukić
University: Uppsala University City: Sweden Arrival and Departure: -
Michael
Entov
University: Technion - Israel Institute of Technology City: Israel Arrival and Departure: -
Maia
Fraser (Online)
University: University of Ottawa City: Canada Arrival and Departure: -
Urs
Fuchs
University: Monash University City: Australia Arrival and Departure: -
Viktor
Ginzburg
University: University of California, Santa Cruz City: United States Arrival and Departure: -
Basak
Gurel
University: University of Central Florida City: United States Arrival and Departure: -
Richard
Hind
University: University of Notre Dame City: United States Arrival and Departure: -
Vincent
Humiliere
University: Sorbonne Université City: France Arrival and Departure: -
Alexandre
Jannaud
University: University of Heidelberg City: Germany Arrival and Departure: -
Dušan
Joksimović (Online)
University: IMJ-PRG City: France Arrival and Departure: -
Yasin
Karacan
University: University of Massachusetts Amherst City: United States Arrival and Departure: -
Yusuke
Kawamoto
University: Université de Montréal City: Canada Arrival and Departure: -
Remi
Leclerq
University: Université Paris-Saclay City: France Arrival and Departure: -
Noémie
Legout
University: Uppsala University City: Sweden Arrival and Departure: -
Samuel
Lisi (Online)
University: University of Mississippi City: United States Arrival and Departure: -
Cheuk Yu
Mak
University: University of Edinburgh City: United Kingdom Arrival and Departure: -
Irena
Matkovic (Online)
University: Uppsala University City: Sweden Arrival and Departure: -
Dusa
McDuff (Online)
University: Barnard College City: United States Arrival and Departure: -
Francesco
Morabito
University: École Polytechnique City: France Arrival and Departure: -
Lukas
Nakamura
University: Uppsala University City: Sweden Arrival and Departure: -
Yong-Geun
Oh
University: IBS Center for Geometry and Physics / POSTECH City: South Korea Arrival and Departure: -
Kaoru
Ono
University: RIMS, Kyoto University City: Japan Arrival and Departure: -
Emmanuel
Opshtein
University: Strasbourg University City: France Arrival and Departure: -
Leonid
Polterovich
University: Tel Aviv University City: Israel Arrival and Departure: -
Semen
Rezchikov
University: Harvard University City: United States Arrival and Departure: -
Daniel
Rosen (Online)
University: Ruhr-Universität Bochum City: Germany Arrival and Departure: -
Johan
Rydholm
University: Uppsala University City: Sweden Arrival and Departure: -
Josh
Sabloff
University: Haverford College City: United States Arrival and Departure: -
Murat
Saglam (Online)
University: Universitaet zu Koeln City: Germany Arrival and Departure: -
Felix
Schlenk
University: Université de Neuchatel City: Switzerland Arrival and Departure: -
Joel
Schmitz
University: Université de Neuchâtel City: Switzerland Arrival and Departure: -
Sobhan
Seyfaddini
University: Sorbonne University Paris City: France Arrival and Departure: -
Egor
Shelukhin (Online)
University: University of Montreal City: Canada Arrival and Departure: -
Maksim
Stokic
University: Tel Aviv University City: Israel Arrival and Departure: -
Michael
Sullivan
University: University of Massachusetts Amherst City: United States Arrival and Departure: -
Yuhan
Sun (Online)
University: Rutgers university City: United States Arrival and Departure: -
Shira
Tanny
University: Institute for Advanced Study City: United States Arrival and Departure: -
Lisa
Traynor (Online)
University: Bryn Mawr College City: United States Arrival and Departure: -
Michael
Usher
University: University of Georgia City: United States Arrival and Departure: -