Winter School

Geometric and Topological Methods in Dynamics of PDEs

February 13-15, 2023

venue: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University in Toruń, ul. Chopina 12/18, Toruń, Poland

 

Organizers

Juliusz P. Schauder Center for Nonlinear Studies, Nicolaus Copernicus University in Toruń

Faculty of Mathematics and Computer Science, Nicolaus Copernicus University in Toruń

University Centre of Excellence “Dynamics, Mathematical Analysis and Artificial Intelligence”, Nicolaus Copernicus University in Toruń

Organizing Committee

Grzegorz Gabor - Chairman

Bartosz Bieganowski

Aleksander Ćwiszewski

Piotr Kokocki

Lecturers

Tomasz Cieślak - Institute of Mathematics, Polish Academy of Science

Title: Spiral vortex sheets and 2D Euler

Abstract: 

After an introduction to the 2D Euler equation and 2D steady Euler flows, we shall focus on vortex sheets, particularly important irregular flows. Both Prandtl's  and Kaden's spirals will be introduced. We shall show that Prandtl's spirals solve 2D Euler in a weak sense (a recent result with P. Kokocki and W.S. Ożański) as well as study the inhomogeneous Euler equations satisfied by Kaden's spirals (work in progress with B. Bieganowski and J. Siemianowski).

Aleksander Ćwiszewski - Faculty of Mathematics and Computer Science, Nicolaus Copernicus University in Toruń

Title: Standing waves for Schrodinger equation with Kato-Rellich type potential

Abstract: 

We shall deal with standing waves for nonlinear Schrodinger equations. By use of topological degree and Conley index in the infinite dimensional version due to Rybakowski we shall provide criteria for the existence (recent results with P. Kokocki) and bifurcation from infinity (recent results with W. Kryszewski). The topological methods require analysis of the asymptotic compactness properties of the related nonlinear parabolic semiflow as well as its geometry determined by the nonlinear part. 

Piotr Kokocki - Faculty of Mathematics and Computer Science, Nicolaus Copernicus University in Toruń

Title: Painlevé II transcendents and applications to PDEs

Abstract:

We provide formulas expressing total integrals of Painlevé II transcendents. It appears that the formulas have interesting applications to partial differential equations. In particular, we prove that arbitrary logarithmic spiral is a finite time singularity developed by a geometric flow related with vortex patch dynamics. We also apply the formulas to show the existence of smooth solutions to the mKdV equation with some particular type of critical initial data.

Maciej Starostka - Faculty of Applied Physics and Mathematics, Gdańsk University of Technology, Martin Luther University Halle-Wittenberg

Title: Applications of the Conley index theory

Abstract: We start by recalling the definition and properties of the classical Conley index and compare it with other invariants like topological degree or Morse cohomology.  We discuss how it is generalized so it can be applied to strongly indefinite functionals. In particular how it was applied to prove the Arnold conjecture on T^2n (Conley and Zehnder, 1983) or triangulation conjecture (Manolescu 2013). We finish by stating open problems.

 

Schedule

The schedule will be available here soon.

Accomodation

Accomodation on your own.

Nevertheless several rooms in the dormitories are available to participants. Price: 70 PLN per night (single room without breakfast). To reserve the room please contact iszalaty@umk.pl Payable in cash upon check-in. Attention: number of places is limited.

We also recommend Hotel Ibis, which is very close to the faculty building. When booking, please provide a phrase "Szkoła zimowa 2023" to get the promotional price of 130 PLN per night (single room with breakfast).

Participation fee

Participation fee is 250 PLN. The first 30 students and PhD students from Polish universities as well as mathematicians from NCU in Toruń are exempt from payment.

PhD students from outside Poland who intend to present a poster may apply for financial support (travel and accommodation costs) under the PROM programme. Details are available at prom-umk.pl. Registration to the programme is possible by filling the form at http://prom-umk.pl/?page_id=76. The application deadline is December 16, 2022.

The participation fee should be paid by bank transfer to:

Uniwersytet Mikołaja Kopernika w Toruniu

Bank Millenium S.A. Warszawa

45 1160 2202 0000 0000 3174 8579

with transfer title: WS2023, name and family name of the participant

For payments from abroad one should add:

SWIFT: BIGBPLPWXXX

IBAN: PL45 1160 2202 0000 0000 3174 8579

 

Registration

To register for the Winter School please fill the REGISTRATION FORM.

Important dates

  • Deadline for registration: 31 January 2023
  • Dates for paying the participation fee: 1 December 2022 - 31 January 2023