Lorentz Center - Integrability and Isomonodromy in Mathematical Physics from 7 Jul 2014 through 11 Jul 2014
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    Integrability and Isomonodromy in Mathematical Physics
    from 7 Jul 2014 through 11 Jul 2014

 

Integrability and Isomonodromy in Mathematical Physics

7-11 July 2014


Topic

The interest in integrability started to grow in the middle of the twentieth century. In that

time Fermi, Pasta and Ulam performed numerical experiments with an anharmonic one-

dimensional lattice and observed to their surprise not a steady continuous flow of energy

from the first mode to the higher ones, but more an exchange of energy, essentially, among

only a certain few.

Their work inspired, on one hand, M.Toda to start the search for nonlinear lattices that

are susceptible for further mathematical analysis. He considered systems of one-dimensional

particles where the equations of motion of each particle are determined by a potential de-

pending only of the distance to the particle on the left and the distance to the particle on the

right. On the other hand, Kruskal and Zabrusky made a continuous model to understand

the phenomenon observed by Fermi, Pasta and Ulam, where the equations of motions in

the approximation up to the second order in the distance between the springs, reduced to

a known nonlinear partial differential equation: the Korteweg-de Vries equation. Starting

with these two lines of research, the impact of integrability has been growing ever, since

with a great impact on mathematical physics.

The theory of isomonodromy deformations of rational connections over P1(C) even has

a longer history. Pioneering work was done in the beginning of the twentieth century by R.

Fuchs and L. Schlesinger, then it was dormant for around fifty years, till the work of the

Kyoto school of Sato made it fully alive in the late seventies, culminating in their series of

papers on holonomic quantum fields.

Both Integrability and Isomonodromy are topics that still play an important role in

various areas of Mathematical Physics. We have chosen to focus on five topics, to which

members of the Universities of Amsterdam and Utrecht, the International Laboratory of

Representation Theory and Mathematical Physics (HSE) in Moscow and the Bogoliubov

Institute in Kiev have contributed significantly.

 

Program

The various days of the conference were each centered around a common theme. To make

the workshop also accessible for young researchers, we asked experts in each of these topics

to give an instructive presentations that prepared them for the more specialized lectures.

Besides that we also organized working sessions for young researchers to support these

addresses. All in all, the ratio between instructional and specialized lectures was one to one.

The themes were:

1) Associativity equations

2) Quivers, gauge theories and integrable systems

4) KZ-equations and their quantum version

5) Classical and quantum spin chains

 

Participants

The program was attended by 59 participants from England, France, Germany, Italy, Japan,

the Netherlands, Russia, Ukraine and the US, showing the broad international interest in and

scope of the workshop. Among them were 25 young researchers and 10 female participants.

 

Outcome

At the end of the conference and afterwards at later encounters, we obtained very positive

reactions on our workshop: many participants told us that this workshop formed a source

of inspiration for them, leading to new ideas and yielding various new cooperations that

will result in joint papers and activities. More concretely, this fall it led to an application

in Brussels for a joint program in which the theme of the conference formed a key issue.

Furthermore, we were offered the possibility to bring out a special issue of Theoretical and

Mathematical Physics on the topic of the conference.

 

Acknowledgements

The conference was financially made possible by the support of the following institutions or

organizations:

         The International Laboratory of Representation Theory and Mathematical Physics at

the National Research University, HSE, in Moscow

         Foundation Compositio

         Cluster Geometry and Quantum Theory

         Research School Wonder

 

Last, but not least we like to express our gratitude to the staff of the Lorentz Center,

in particular Henriette Jensenius, Mieke Schutte and Yousra Jaddour for their guidance,

help and support at the whole process of organizing this workshop. All participants were

impressed by the pleasant ambiance at the Center and the excellent support from its sta.

Gleb Arutyunov (Utrecht, The Netherlands)
Gerard Helminck (Amsterdam, The Netherlands)
Pavel Pyatov (Dubna, Russia)
Vitalij Shadura (Kiev, Ukraine)



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