Workshop Overview

We are pleased to announce Flows on the Saskatchewan: A Workshop on Integrability and Inverse Problems. This 3-day workshop will be held in celebration of the 65th birthday of Jacek Szmigielski, Professor of Mathematics at the University of Saskatchewan.

 

Jacek Szmigielski

 

The purpose of this 3-day workshop is to highlight new developments in the analysis and geometry of a broad range of nonlinear PDEs and their inverse problems. Speakers will be encouraged to provide historical context and "big picture" ideas in order to help younger attendees access the field. There will also be a mini-course on inverse problems, with a view to Camassa-Holm integrability, by Jacek himself on the Friday. Indeed, we intend the workshop to be a reflection of Prof. Szmigielski's many contributions to the field on the occasion of his 65th birthday in 2019.


While the workshop offers an opportunity to reflect upon Jacek's contributions, it is also meant to provide a glimpse into the future of these topics from the points of view of a number of distinguished experts in the field. As such, we plan to strongly encourage participation of junior attendees from across the network of the Pacific Institute for the Mathematical Sciences (PIMS) and beyond.



All of the workshop activities will take place at the University of Saskatchewan in the city of Saskatoon, Canada. The workshop will begin in the afternoon on Friday, April 5 with a two-lecture mini-course for students by Jacek Szmigielski and will conclude around lunchtime on Sunday, April 7. There will be a workshop dinner on the Saturday evening at Willows Golf Club.  Lunch on each of Saturday and Sunday will be held on campus in Marquis Hall.

 

Invited participants will stay in the Holiday Inn Express & Staybridge at 1838 College Drive, a 10-minute walk from most points on the University campus.

  • Walking directions from the hotel to the Friday talk venue (McLEAN building, Room 201)

  • Walking directions from the hotel to the Saturday / Sunday talk venue (PHYSICS building, Room 130)

Speakers

  • Richard Beals (Yale)

  • Josef Dorfmeister (TU-München)

  • Michael Gekhtman (Notre Dame)

  • Alex Himonas (Notre Dame)

  • Niky Kamran (McGill)

  • David Sattinger (Arizona)

  • Feride Tiglay (Ohio State)

Schedule and Abstracts

 

All talks in McLEAN 201





6:15 pm: Jacek Szmigielski (Saskatchewan)
  • Title: A first look at inverse problems


7:00 pm: Break



7:15 pm: Jacek Szmigielski (Saskatchewan)
  • Title: A second look at inverse problems


 

All talks in PHYSICS 130





9:00 am: Arrival at PHYSICS 130, refreshments



9:30 am: Alex Himonas (Notre Dame)
  • Title: On the analysis of integrable evolution equations

  • Abstract: In this talk we shall consider questions of existence, uniqueness, dependence on initial data, and regularity of solutions to the Cauchy problem of Camassa-Holm and related integrable equations in a variety of function spaces. Some of these equations can be thought as "toy" models for the Euler equations governing the motion of an incompressible fluid, and the analytic techniques developed for these equations have been in some cases transferable to the Euler equations. In particular, we shall discuss the phenomena of norm-inflation and non-uniquness that arise when attempting to prove well-posedness for these equations with low regularity data. Peakon traveling wave solutions are the basic ingredients. The talk is based on work in collaboration with C. Kenig, C. Holliman, D. Mantzavinos, G. Misiolek and G. Petronilho.


10:30 am: Break



11:00 am: Feride Tiglay (Ohio)
  • Title: Integrating the μHS equation

  • Abstract: The μHS equation is a Euler-Arnold equation corresponding to a natural right-invariant Sobolev metric. It is bihamiltonian and admits a Lax pair. We describe how to use Fredholm determinants to construct special functions and integrate this equation.


12:00 pm: Lunch (Marquis Hall)



2:00 pm: Richard Beals (Yale)
  • Title: Jacek, C. S. Meijer, and me

  • Abstract: This talk will discuss randomness, serendipity, an enjoyable collaboration, and our most popular paper.


3:00 pm: Break



3:30 pm: Niky Kamran (McGill)
  • Title: A survey of non-uniqueness results for the anisotropic Calderon problem with disjoint data

  • Abstract: The anisotropic Calderon problem is an inverse problem of geometric analysis which consists in recovering, up to some natural gauge equivalences, the metric of a compact Riemannian manifold with boundary from the knowledge of the Dirichlet-to-Neumann map at fixed energy. We shall give a motivated review of some recent non-uniqueness results obtained for the anisotropic Calderon problem in the case in which the Dirichlet and Neumann data are measured on disjoint subsets of the boundary. These examples consist in Riemannian manifolds diffeomorphic to toric cylinders endowed with suitably chosen conformal rescalings of warped product metrics, where the conformal factors are governed by non-linear second-order elliptic pdes of Yamabe type. These are joint results with Thierry Daude (Cergy-Pontoise) and Francois Nicoleau (Nantes).


6:15 pm: Leave from Holiday Inn for Dinner at Willows



 

All talks in PHYSICS 130





9:00 am: Josef Dorfmeister (TU-München)
  • Title: Simple occurrences of the non-trivial sine-Gordon equation

  • Abstract: In this talk we will start by explaining the occurrence of the "sine-Gordon equation" in geometry, physics, and textile manufacturing. In the second part of this talk, we will make the applications to geometry more explicit. Looking at surfaces of "Gauss curvature K = -1" in R3, i.e. surfaces looking locally like a saddle at each point, we will associate to each point a natural "tripod", two tangent vectors and a vector perpendicular to them. It will become clear (quite easily) how such a tripod induces the sine-Gordon equation, and, vice-versa, a solution to the sine-Gordon equation will induce a surface of Gauss curvature K = -1. We will at least indicate how all the solutions to the sine-Gordon equation discussed above can be constructed (using a generalization of the Gauss algorithm and the Gram-Schmidt algorithm). The discussion will be illustrated by several pictures of surfaces. In the (short) third part, time permitting, we will point out some open (to me) problems. Most of this talk should be accessible to any undergraduate student.


10:00 am: David Sattinger (Arizona)
  • Title: Einstein meets Huygens: an optical view of general relativity

  • Abstract: Huygens saw light as a wave-particle phenomenon — light was a wave propagated by "ethereal" particles. The mathematical basis for this viewpoint is the symplectic geometry developed by Hamilton and Jacobi in the 19th century. The relativistic version of Hamilton-Jacobi's theory is used to show that the Einstein field equations are equivalent to a Hamiltonian system with 4 degrees of freedom.


11:00 am: Break



11:30 am: Michael Gekhtman (Notre Dame)
  • Title: Cluster integrable systems

  • Abstract: Combinatorial structures embedded into a definition of cluster algebras proved instrumental in reimagining many important integrable models and helped to discover new instances of complete integrability. The talk will provide an overview of an interaction between theories of cluster algebras and integrable systems with example ranging from dilogarithm identities to pentagram maps and their generalizations to discrete Toda-like systems that are close relatives of the peakon system to which a significant part of Jacek’s celebrated work is devoted.



12:30 pm: Lunch (Marquis Hall)

 

 

For those participants who are still in Saskatoon on Monday morning, Hans Lundmark (Linköping University) will be giving a related talk titled Peakons in Pictures at 11:00 am in ARTS 108. Directions to the venue will be provided on Sunday.

Participants

  • Richard Beals (Yale)

  • Xiangke Chang (Chinese Academy of Sciences)

  • Robert Cornea (Saskatchewan)

  • Josef Dorfmeister (TU-München)

  • Michael Gekhtman (Notre Dame)

  • Brandon Gill (Saskatchewan)

  • Alex Himonas (Notre Dame)

  • Niky Kamran (McGill)

  • Matthew Koban (Saskatchewan)

  • Jennifer Lam (Saskatchewan)

  • Aidan Lindberg (Saskatchewan)

  • Hans Lundmark (Linköping)

  • Christopher Mahadeo (Saskatchewan)

  • Alessandro Malusà (Saskatchewan-PIMS)

  • Devin Patterson (Saskatchewan)

  • Sergio Peralta (Talca)

  • Steven Rayan (Saskatchewan)

  • Yi Ren (Saskatchewan)

  • David Sattinger (Arizona)

  • Artur Sowa (Saskatchewan)

  • Thomas Stanley (Saskatchewan)

  • Evan Sundbo (Saskatchewan)

  • Jacek Szmigielski (Saskatchewan)

  • Kaori Tanaka (Saskatchewan)

  • Jacob Taylor (Saskatchewan)

  • Feride Tiglay (Ohio State)

  • Nicole Zolkavich (Saskatchewan)

Organizers

For any questions regarding the workshop, please contact one of the organizers below:

Sponsors

We gratefully acknowledge the support of PIMS, NSERC, and the University of Saskatchewan.

Poster

You may download the workshop poster here.