Topic
The Grothendieck-Katz p-curvature conjecture asserts that an ordinary linear differential equation with polynomial
coefficients over Z has a basis of algebraic solutions if and only if its p-curvature
is 0 for almost all primes p.
This latter condition is equivalent to the existence of a basis of algebraic solutions of the induced equation
obtained by reduction modulo p.
The general conjecture is still wide open, but many intriguing phenomena and insights have been observed since
its formulation in the 1960’s by Grothendieck and its proof by Katz for Picard-Fuchs equations.
This setting will be the inspiring background of the workshop at the Schrödinger Institute.
More specifically, we wish to address, among others, the following questions:
• How does the Fuchs-Frobenius theory about the solutions of ordinary differential equations carry over to the case of equations in positive characteristic?
• For order one equations, the Grothendieck-Katz conjecture is equivalent to a special case
(proved by Kronecker) of Chebotarev’s density theorem. What about equations of order 2 and their relation to number theory?
• There exist several variants of the conjecture, e.g. by André, Christol, Matzat, Bézivin, concerning also the integrality of the coefficients and diagonals of rational functions. How do these relate to each other?
• Is it possible to decide algorithmically if a given differential equation has algebraic solutions or if a given series solution is algebraic? How can one compute p-curvatures systematically?
We will approach the topic in a very accessible and explicit manner, focussing on the main ideas, arguments and constructions, while avoiding technicalities and heavy machinery.
Interested people are kindly asked to contact the organizers.
Program
Working Week 1 (October 7-11, 2024)
Four Mini-Courses (4 sessions each à 60-90 minutes, plus 2 discussion sessions each):
Duco van Straten: Phenomenology of Fuchsian differential equations.
Alin Bostan, Xavier Caruso, Julien Roques: p-Curvature Conjecture.
Michael Singer + NN: Differential Galois Theory and the Algebraicity of Solutions.
Masha Vlasenko + Daniel Vargas-Montoya : A p-adic Approach to Differential Equations.
Working Week 2 (October 15-18, 2024)
15-20 Selected Lectures (60-90 minutes) on topics of general interest, accessible for broad audience.
Cultural and Social Program
Viennese museums: Kunsthistorisches Museum, Naturhistorisches Museum, Albertina, Belvedere, Leopold Museum, Mumok, Wien Museum, ...
Sight-Seeing: Schatzkammer, Hofburg, Hofreitschule, Schloss Schönbrunn, Schloss Belvedere, Ringstrasse, Museumsquartier, ...
Music and Theatre: Staatsoper, Theater an der Wien, Volksoper, Burgtheater, Akademietheater, Theater in der Josefstadt, Musikverein, Konzerthaus, ...
Classical coffee shops: Palmenhaus, Heiner, Eiles, Prückel, Sacher, Sluka, Hawelka, Sperl, Aida, Schwarzes Kameel, Café Museum, Café Landtmann, Demel, Café de l’Europe, Kleines Kaffee, ...
Participants
Organizers & Scientific Committee:
Herwig Hauser, Vienna (local host)
Alin Bostan, Inria Saclay
Francis Brown, Oxford
Hiraku Kawanoue, Chubu
Shihoko Ishii, Tokyo
Michael Singer, North Carolina
Confirmed participants (as of April 25, 2024)
- Mariemi Alonso (Madrid)
- Hiroki Aoki (Tokyo)
- Matthias Aschenbrenner (Vienna)
- Frits Beukers (Utrecht)
- Gregor Böhm (Vienna)
- Jean-Benoît Bost (Paris)
- Alin Bostan (Paris)
- Francis Brown (Oxford)
- Francisco Castro (Sevilla)
- Antoine Chambert-Loir (Paris)
- Gilles Christol (Paris)
- Eric Delaygue (Lyon)
- Eleonore Faber (Graz)
- Javier Fresán (Paris)
- Florian Fürnsinn (Vienna)
- Luisa Gietl (Vienna)
- Vasily Golyshev (Trieste)
- Yoshishige Haraoka (Tokyo)
- Charlotte Hardouin (Toulouse)
- Herwig Hauser (Vienna)
- Shinobu Hikami (Okinawa)
- Shihoko Ishii (Tokyo)
- Manuel Kauers (Linz)
- Hiraku Kawanoue (Chubu/Kyoto)
- Maxim Kontsevich (Paris)
- Christian Krattenthaler (Vienna)
- Anton Mellit (Vienna)
- Luis Narváez (Sevilla)
- Hadrien Notarantonio (Paris-Saclay)
- Peter Paule (Linz)
- Eric Pichon (Paris-Saclay)
- Andrea Pulita (Grenoble)
- Armin Rainer (Vienna)
- Julien Roques (Lyon)
- Michael Singer (North Carolina)
- Yunqing Tang (Princeton/Berkeley)
- Marius van der Put (Groningen)
- Duco van Straten (Mainz)
- Daniel Vargas-Montoya (Toulouse)
- Masha Vlasenko (Warsaw)
- Shoji Yokura (Kagoshima)
- Masahiko Yoshinaga (Osaka)
- Sergey Yurkevich (Vienna)
- Don Zagier (Bonn)
- Wadim Zudillin (Nijmegen)