### 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)

Five Mini-Courses (3 sessions of 60-75 minutes, plus 2 discussion sessions each):

Julien Roques, Lyon: The p-Curvature Conjecture.

Michael Singer, Raleigh: Differential Galois Theory and the Algebraicity of Solutions.

Duco van Straten, Mainz: Differential Equations, Geometry and Arithmetic.

Daniel Vargas-Montoya, Toulouse, Masha Vlasenko, Warsaw: A p-adic Approach to Differential Equations.

Don Zagier, Bonn: Periods, Differential Equations, and Modular Forms.

#### Working Week 2 (October 14-18, 2024)

15-20 Selected Lectures (60-75 minutes) on topics of general interest, accessible for broad audience.

To be announced in due time.

#### 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, ...

### Week 1: Mini-Courses

#### Julien Roques, Lyon: *"The p-Curvature Conjecture"*

The primary objective of this mini-course is to explain the meaning of the following statement taken from the conference web page:

"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."

To understand this in detail, we start with an ordinary linear differential equation (*) Ly = 0 with coefficients rational functions in Q(x) (or even polynomials in Z[x]). The following topics will be covered (possibly in a different order):

We define, for primes p, the p-curvature of (*) and establish its basic properties. We will explain the link between the vanishing of a given p-curvature and the existence of rational solutions of the reduction modulo p of the differential equation we started with. This will allow us to formulate the Grothendieck-Katz p-curvature conjecture. To get a feeling, we will outline the proof of this conjecture in the (already non-trivial) case of first-order equations.

A second issue is to explain why the vanishing of almost all p-curvatures of (*) implies strong restrictions on the differential equation itself: for instance, all its singularities have to be regular (as defined by Fuchs). Regularity will be characterized in various ways in the course.

An interesting concrete situation is the study of the Grothendieck-Katz p-curvature conjecture for (generalized) hypergeometric equations. Time permitting, we will also discuss further topics, such as the analog of this conjecture for linear q-difference equations.

Reading suggestion: "Algebraic solutions of linear differential equations: an arithmetic approach". A. Bostan, X. Caruso, J. Roques. Bulletin of the AMS, to appear. arXiv:2304.05061.

#### Michael Singer, Raleigh: *"Differential Galois Theory and the Algebraicity of Solutions"*

Description: I will give a quick introduction to the basic Galois theory of linear differential equations. I will then discuss some of the basic algorithms concerning finding rational and exponential solutions as well as factoring these equations. I will then turn to the problem of deciding if a linear differential equation has algebraic solutions and how to decide if a given power series solution of an linear differential equation is algebraic. I will discuss Klein's theorem (and its generalizations) that algebraic solutions of second order linear differential equations with finite primitive Galois group can be gotten from certain hypergeometric equations via a change of variables. Finally, if time permits, I will discuss questions concerning algebraic relations among solutions of linear differential equations.

A brief introduction to this theory can be found in:

"Introduction to the Galois Theory of Linear Differential Equations" in: Algebraic Theory of Differential Equations, M.A.H. MacCallum and A.V. Mikhalov, eds., London Mathematical Society Lecture Notes Series (no. 357), Cambridge University Press, 2009, 1-82. (also item 64 on my list of publications at https://singer.math.ncsu.edu/ms_papers2.html)

More adventurous people can also consult the first four chapters of "Galois Theory of Linear Differential Equations", M. van der Put and M.F. Singer, Grundlehren der mathematischen Wissenschaften, Volume 328, Springer, 2003,

or

"Algebraic Groups and Differential Galois Theory", T. Crespo and Z. Hajto, Graduate Studies in Mathematics 122, American Mathematical Society, 2011.

#### Duco van Straten, Mainz: *"Differential Equations, Geometry and Arithmetic"*

The differential equations that describe the "variation of cohomology" in families of varieties, commonly called "Picard-Fuchs equations", have many fascinating properties that reflect the purely topological, the Hodge-theoretical and the arithmetic properties of the varieties in question. The solutions to these equations are loosely referred to as "periods", which are examples of "G-functions" whose divers properties are related in multiple ways and interwoven in a beautiful intricate pattern.

The aim of the course to introduce the audience to this rich field by way of examples, among which the Legendre differential equation and its generalisations play a prominent role.

In the first lecture we introduce some important data attached to Fuchsian differential equations, like monodromy representation, Riemann symbol and the notion of rigidity.

In the second lecture we will explore the interaction with Hodge-theoretical invariants and monodromy and the significance of points of maximal unipotent monodromy.

In the third lecture we explore the relation of differential equations of Picard-Fuchs type and arithmetic properties of families of varieties by the example of so called Calabi-Yau operators.

Literature: It is always useful to take a dive in classical books on ordinary differential equations, like

E.L. Ince: "Ordinary Differential Equations", or

E.G.C. Poole: "Introduction to the theory of Differential Equations".

More background in algebraic geometry is required for

P. Deligne: "Equations differentielles à points singuliers réguliers" or

Y. André: "G-Functions and Geometry"".

An accessible text for Hodge-theory is

H. Movasati: "A course in Hodge Theory with emphasis on multiple integrals".

I will probably make also use of my paper "Calabi-Yau operators".

#### Daniel Vargas-Montoya, Toulouse, Masha Vlasenko, Warsaw: *"A p-adic Approach to Differential Equations"*

The differential equations in this mini-course will be classical. Namely, we will consider linear ordinary differential equations with coefficients in the field of rational functions Q(t). When such equations arise from the Gauss-Manin connection in algebraic geometry (so called Picard-Fuchs differential equations), their solutions possess nice arithmetic properties captured in the notion of a G-function, where "G" stands for "geometry". We will introduce and study the p-adic Frobenius structure of a differential equation, which enforces the above mentioned nice properties for a given prime number p. Picard-Fuchs differential equations possess such a structure for almost every prime number p.

The main sources we are using to prepare these lectures are books "p-adic Differential Equations" by Kedlaya and "An Introduction to G-functions" by Dwork, Gerotto and Sullivan.

In Lecture 1 we will discuss the definition of the p-adic Frobenius structure, its existence and uniqueness. We will also explain the motivation coming from the deformation theory of local zeta functions of algebraic varieties originating in the work of Bernard Dwork.

Lecture 2 will focus on arithmetic consequences of the existence of Frobenius structure, such as algebraicity of solutions modulo p, p-integrality of expansion coefficients and mirror maps, and other p-adic analytic properties of solutions. Some of these properties will require special assumptions on the Frobenius structure, or can be reformulated in terms of the Frobenius structure's properties.

Lecture 3 will address the question of the existence of the p-adic Frobenius structure for Fuchsian differential operators with rational local exponents. We will present a proof of existence in the rigid case due to Daniel Vargas-Montoya and discuss the connection with G-functions.

Prerequisites: knowledge of p-adic numbers and of ordinary differential equations (Cauchy theorem, monodromy, regular singularities, Fuchsian operators). Excellent references for the latter subject are "Local theory of meromorphic connections in dimension 1 (Fuchs theory)" by Haefliger and "Gauss' hypergeometric function" by Beukers. Familiarity with hypergeometric functions will also be beneficial as we are going to use them in examples.

#### Don Zagier, Bonn: *"Periods, differential equations, and modular forms"*

1. Periods, in two senses: Integrals of algebraic forms over algebraic varieties (closed or
with boundary) defined over Q, or else integrals over the fibres of a family of algebraic
varieties (say, over P^{1}). The former form a countable set including numbers like π or
ζ(3). The latter are functions of one variable (or more, but here just one), always
satisfy a differential equation (Picard-Fuchs equation or Gauss-Manin connection), and
include for instance hypergeometric functions or the generating function of the Apery
numbers.

A very interesting question is to know whether a given linear ODE is of PF type (also called geometric). Here there are various conjectures saying that certain properties of the ODE that always hold in the geometric case are in fact sufficient. We will discuss some of these, with several examples.

2. Modular forms: Modular forms (of which no prior knowledge will be assumed!) are a basic and well-known topic in modern number theory , but strangely, even many experts are not aware that they always satisfy differential equations. In fact, there are three very different types of differential equations satisfied by a modular form f(z):

(i) a linear ODE with algebraic coefficients if f(z) is expressed as a function of a modular function (= modular form of weight 0);

(ii) a non-linear ODE with constant coefficients, always 3rd order, now with respect to z and

(iii) a linear ODE w.r.t. z, but now with quasimodular forms as coefficients.

The first type of ODE is always geometric (of which the Apéry example is an example) and hence related to algebraic geometry; the second type (of which the famous Chazy equation is the simplest and best-known example) is important in the theory of integrable systems, and the third type plays an important role in conformal field theory and the study of vertex operator algebras (which would only be mentioned in passing since I cannot assume knowledge of this). I would describe each of these types briefly, with diverse examples.

### Week 2: Lectures

##### Preliminary list of lectures - still to be completed. Click on title for abstract.

##### Frits Beukers, "Picard-Fuchs equations and modular forms - with some applications"

The relationship between Picard-Fuchs equations and modular forms has been explained in Don Zagier's lectures
of last week and we will briefly recall part of it (see reference below). It is well-known that in
many explicit examples of Picard-Fuchs equations the coefficients a(n) of a Taylor series solution
have interesting congruence properties, the Apery numbers being a well-known case. Examples of such
congruences are the Lucas congruences a(k) = a(k_{0}) * a(k_{1}) * · · · * a(k_{r})
(mod p) where k_{0},k_{1},...
are the digits of k in base p where p is (almost) any prime. Other examples are the so-called
supercongruences which have been discovered in abundance the past 15 years. In many (but not all!)
cases such congruences can be proven directly by exploiting the connection with modular forms.

Background reading: Sections 4 and 5 of Don Zagier's overview paper 'The arithmetic and topology of
differential equations' in: Proc. European Congress of Math, European
Mathematical Society (2018), 717-776.

##### Jean-Benoît Bost, "Arithmetic algebraization theorems and the analogy between number fields and function fields"

In this lecture, I will describe how various classical results of Diophantine geometry, notably concerning algebraicity and transcendence in relation to algebraic foliations, may be seen as counterparts, valid over number fields, of various classical algebraicity results in analytic and formal geometry.
This analogy concerns, not only the statement of these Diophantine results, but also their proofs.

Reading material: J.- B. Bost, Theta invariants of Euclidean lattices and infinite-dimensional Hermitian vector bundles over arithmetic curves, vol 334 of Progress in Mathematics, Birkhäuser 2020

##### Alin Bostan, "TBA"

##### Francis Brown, "Mellin transforms and P-recurrences, point counts over finite fields and invariants of graphs"

Abstract: Coming soon

##### Éric Delaygue, "On Abel's problem and Gauss congruences"

A classical problem due to Abel is to determine if a homogeneous linear differential equation of
order 1, with an algebraic function as coefficient, admits a non-zero algebraic solution.
Given such an equation, Risch designed an algebraic algorithm that determines whether there exists
a non-zero algebraic solution or not.

I will prove an arithmetic characterization of Abel's problem, in terms of Gauss congruences, in
the case where the coefficient of the equation admits a Puiseux expansion with rational coefficients.
I will use this criterion to completely solve the hypergeometric case and prove a prediction Golyshev
made using the theory of motives.

M. F. Singer, Algebraic solutions of n-th order linear differential equations, Proc. Queen’s Number Theory Conf. 1979, Queen’s Pap. Pure Appl. Math. 54 (1980), 379–420.

F. Baldassarri, B. Dwork, On second order linear differential equations with algebraic solutions, Am. J. Math. 101 (1979), 42–76.

D. Zagier, The arithmetic and topology of differential equations, in Proceedings of the European Congress of Mathematics, Berlin, 18-22 July, 2016, Mehrmann, V.; Skutella, M. (Eds.), European Mathematical Society (2018), 717–776.

É. Delaygue, T. Rivoal, On Abel’s problem and Gauss congruences, Int. Math. Res. Not. IMRN (2024), no. 5, 4301–4327.

A. Bostan, An arithmetic characterization of some algebraic functions and a new proof of an algebraicity prediction by Golyshev, preprint (2024), available at https://arxiv.org/abs/2408.13951.

##### Javier Fresán, "On G-functions of differential order 2"

G-functions are power series that solve a linear differential equation and satisfy some growth
conditions of arithmetic nature. Those which are solution of a differential equation of order 1 are
algebraic of a rather particular shape. A rich family of G-functions of differential order 2 is
given by algebraic substitutions of the classical Gauss hypergeometric series. However, I will argue
that G-functions of order 2 are infinitely much richer than those. Indeed, there exist infinitely
many non-equivalent differential equations of order 2 whose solutions include a G-function that is
not a polynomial in algebraic substitutions of hypergeometric series. This solves Siegel's problem
for G-functions, as formulated by Fischler and Rivoal, and answers a question of Krammer in
connection to his counterexample to a conjecture by Dwork. (Joint work with Josh Lam and Yichen Qin)

I. I. Bouw and M. Möller. Differential equations associated with nonarithmetic Fuchsian groups. J. Lond.
Math. Soc., 81(2010), 65–90.

S. Fischler and T. Rivoal. On Siegel’s problem for E-functions. Rend. Semin. Mat. Univ. Padova, 148 (2022), 83–115.

J. Fresán and P. Jossen. A non-hypergeometric E-function. Ann. of Math., 194 (2021), 903–942.

J. Fresán, Y. H. J. Lam, Y. Qin. On Siegel's problem and Dwork's conjecture for G-functions. Coming soon.

D. Krammer. An example of an arithmetic Fuchsian group. J. reine angew. Math. 473 (1996) 69–86.

Y. H. Joshua Lam and D. Litt. Geometric local systems on the projective line minus four points.
Preprint, arXiv:2305.11314, 2023.

##### Gilles Christol, "TBA"

##### Vasily Golyshev, "Beyond Complex Multiplication"

For a class of Picard-Fuchs equations, one can find special arguments with extra polynomial dependencies between solutions by using 'poor man's adeles', i.e., by gluing material modulo different primes using the Chinese remainder theorem.

##### Yoshishige Haraoka, "Higher dimensional Katz theory"

The Katz theory on rigid local systems brought a big progress in the theory of Fuchsian ODE's.
The theory is extended to irregular singular equations. We are also interested in extending it
to the higher dimensional case.

As a typical example, we consider a KZ-type equation. If one chooses an independent variable,
we can define the middle convolution in the direction of the variable, and get another KZ type
equation with possibly different rank. This operation has many applications. For example, we
can show, by operating middle convolutions, that any rigid Fuchsian ODE can be extended to a
KZ-type equation with the positions of the singular points as new variables. Moreover, if we
combine the middle convolution with the restriction operation to a singular locus, we will
obtain several non-rigid Fuchsian ODEs having integral representations of solutions. We can
also define the multiplicative middle convolution for KZ type equations.

##### Charlotte Hardouin, "Extending Hölder's result on the differential transcendence of the Gamma function"

Among the functions of the complex variable, we distinguish the rational and algebraic
functions from the transcendental functions. In this last class, we consider a new
hierarchy depending on the differential algebraic relations satisfied by the function.
We call hypertranscendental or differentially transcendental any function that doesn't satisfy
a polynomial relation together with its derivatives. A celebrated example is given by the Gamma
function, which is hypertranscendental by a result of Hölder.

The Gamma function satisfies the well known discrete functional equation Gamma(x+1) = x Gamma(x).
With Michael Singer, we developed in 2008 a parametrized Galois theory in order to understand
the type of differential algebraic relations satisfied by solutions of linear discrete equations
from a geometric point of view.

In this talk, I will show how one can use this parametrized Galois theory to prove that
Hölder's result extends to any non-rational and not of exponential type solution of a shift
equation. This result is in collaboration with Boris Adamczewski and Thomas Dreyfus.

##### Hiraku Kawanoue, "The exponential function in characteristic p"

In characteristic 0, a solution of a Fuchsian differential equation is a product of a
monomial whose exponent is called a local exponent and a power series possibly combined with logarithms.
F. Fuernsinn and H. Hauser proved the analogous assertion to above in characteristic p, where logarithms
are replaced by the formal variables z_1, z_2,..., which satisfy z_1'=1/x and z_{i+1}'=z_i'/z_i, introduced
by T. Honda and B. Dwork.

Now consider the exponential function exp_p in characteristic p as the solution of the differential equation
y=y', computed by the algorithm given in their paper. Then, the structure of exp_p is not simple. For example,
the algebraicity of its ``constant term'' (exp_p with setting all z_i's to be 0) is not clear (a question
by A. Bostan). Note that the solution of a linear differential equation has ambiguity by multiplication of
p-th power elements. Thus we have choices for the expression of the solution.

In my talk, I will present a beautiful expression of the exponential function, which is quite structured.
I also present the generalization of this observation to any linear differential equations of order 1 in
characteristic p.

This is a joint work with F. Fuernsinn and H. Hauser. The main reference is ``On Abel's Problem about
logarithmic Integrals in positive characteristic'' (arXiv:2401.14154) by F. Fuernsinn, H. Hauser and myself.

##### Maxim Kontsevich, "Holonomic D-modules and positive characteristic & p-Determinants and monodromy of differential operators"

The talk is related to a generalization of Katz' nilpotence conjecture to the irregular case. There are two relevant papers:

Holonomic D-modules and positive characteristic, Japan. J. Math. 4 , 1-25 (2009), e-print 1010.2908 where the conjecture is basically formulated, and its follow-up for some formal deformations of exponential-motivic D-Modules.

p-Determinants and monodromy of differential operators (with A.Odesskii), Selecta Mathematica (2022) 28:5, e-print 2009.12159.

In the second paper we have a very nice and unexpected result relating p-curvature and determinant of the logarithm of monodromy (notice the very unusual order of words: determinant of logarithm instead of logarithm of determinant!). The proof there is very convoluted. I have now a better simple proof which I can present in the talk, and a lot of new (semi)-experimental material, including generalization to q-difference equations.

##### Andrea Pulita, "p-adic differential equations over Berkovich curves"

We propose to discuss (a selection of) the following items:

An introduction to the basic ideas of Berkovich spaces and why they are a powerful tool in non-Archimedean geometry and in particular in the topic of p-adic differential equations.

Discussing the importance of radii of convergence of solutions to p-adic differential equations and their implications.

Presenting some local and global decomposition theorems that apply to p-adic differential equations. This is similar to the decomposition theorem of a differential equation over C((x)) by the slopes of its Newton Polygon. In fact, this is a particular case of a more general theorem.

An overview of the finite-dimensionality results for the de Rham cohomology associated with these equations. The fact that contrary to the algebraic case, they do not always have a finite dimensional de Rham cohomology and the information about the cohomology is encoded in the radii.ns.

##### Yunqing Tang, "The arithmetic of power series and applications to periods"

Borel and Dwork gave conditions on when a nice power series with rational number coefficients comes from a rational function in terms of meromorphic convergence radii at all places. Such a criterion was used in Dwork's proof of the rationality of zeta functions of varieties over finite fields. Later, the work of André, Bost, Charles and many others generalized the rationality criterion of Dwork in general framework of Arakelov theory and deduced many applications in the arithmetic of differential equations and elliptic curves. In this talk, we will discuss some further refinements and generalizations of the criteria of André, Bost, and Charles and their applications to modular forms and irrationality of certain periods. This is joint work with Frank Calegari and Vesselin Dimitrov.

##### Marius van der Put, "First order differential equations in characteristic zero and in positive characteristic"

This concerns classification, Painlevé property, stratification, algebraic relations between solutions, Grothendieck-Katz conjecture and more. The above fits with the theme of the workshop.

##### Masaaki Yoshida, "Shift operators of a Fuchsian equation with an accessory parameter"

For years I worked on the Schwarz maps of the hypergeometric equations. But in these years I got interested in Fuchsian equations with an accessory parameter, and the shift operators for them. Typical example is the Dotsenko-Fateev equation.

##### Masahiko Yoshinaga, "Towards a Kontsevich-Zagier type conjecture for holonomic series"

The Kontsevich-Zagier formulated a conjecture states that any two integrations with the same value are
expected to transformed to each other by a few transformations rules. In this talk we would like to
discuss similar questions for infinite series.

### Participants

#### Organizers & Scientific Committee:

Herwig Hauser, Vienna (local host)

Alin Bostan, Paris

Francis Brown, Oxford

Hiraku Kawanoue, Chubu & Kyoto

Shihoko Ishii, Tokyo

Michael Singer, North Carolina

#### Participants Week 1

- Hiroki Aoki (Tokyo)
- Mariemi Alonso-García (Madrid)
- Alin Bostan (Paris)
- Manfred Buchacher (Linz)
- Christopher Chiu (Leuven)
- Roland Donninger (Vienna)
- Eleonore Faber (Graz)
- Claudia Fevola (Paris)
- Florian Fürnsinn (Vienna)
- Francisco García Cortés (Sevilla)
- Nutsa Gegelia (Mainz)
- Luisa Gietl (Vienna)
- Vasily Golyshev (Trieste)
- Anna Goncharuk (Kharkiv)
- Sebastian Gontarek (Wroclav)
- Charlotte Hardouin (Toulouse)
- Herwig Hauser (Vienna)
- Zhiqiang He (Grenoble)
- Martin Kalck (Graz)
- Manuel Kauers (Linz)
- Hiraku Kawanoue (Chubu/Kyoto)
- Christoph Koutschan (Linz)
- Christian Krattenthaler (Vienna)
- Abhiram Mamandur Kidambi (Leipzig)
- Anton Mellit (Vienna)
- Luis Narváez (Sevilla)
- Hadrien Notarantonio (Paris)
- Éric Pichon-Pharabod (Paris)
- Andrea Pulita (Grenoble)
- Feliks Rączka (Warsaw)
- Armin Rainer (Vienna)
- Harald Rindler (Vienna)
- Julien Roques (Lyon)
- Bruno Salvy (Lyon)
- Josef Schicho (Linz)
- Michael Singer (Raleigh)
- Yunqing Tang (Berkeley/Caltech)
- Gerald Teschl (Vienna)
- Marius van der Put (Groningen)
- Duco van Straten (Mainz)
- Masha Vlasenko (Warsaw)
- Shoji Yokura (Kagoshima)
- Masaaki Yoshida (Fukuoka)
- Masahiko Yoshinaga (Osaka)
- Sergey Yurkevich (Vienna)
- Don Zagier (Bonn)
- Wadim Zudilin (Utrecht)

#### Participants Week 2:

- Hiroki Aoki (Tokyo)
- Matthias Aschenbrenner (Vienna)
- Frits Beukers (Utrecht)
- Gregor Böhm (Vienna)
- Jean-Benoît Bost (Paris)
- Alin Bostan (Paris)
- Francis Brown (Oxford)
- Manfred Buchacher (Linz)
- Antoine Chambert-Loir (Paris)
- Christopher Chiu (Leuven)
- Gilles Christol (Paris)
- Éric Delaygue (Lyon)
- Roland Donninger (Vienna)
- Eleonore Faber (Graz)
- Javier Fresán (Paris)
- Florian Fürnsinn (Vienna)
- Francisco García Cortés (Sevilla)
- Nutsa Gegelia (Mainz)
- Luisa Gietl (Vienna)
- Vasily Golyshev (Trieste)
- Sebastian Gontarek (Wroclav)
- Yoshishige Haraoka (Tokyo)
- Charlotte Hardouin (Toulouse)
- Herwig Hauser (Vienna)
- Zhiqiang He (Grenoble)
- Martin Kalck (Graz)
- Manuel Kauers (Linz)
- Hiraku Kawanoue (Chubu/Kyoto)
- Maxim Kontsevich (Paris)
- Christoph Koutschan (Linz)
- Christian Krattenthaler (Vienna)
- Abhiram Mamandur Kidambi (Leipzig)
- Anton Mellit (Vienna)
- Hadrien Notarantonio (Paris)
- Peter Paule (Linz)
- Andrea Pulita (Grenoble)
- Feliks Rączka (Warsaw)
- Armin Rainer (Vienna)
- Harald Rindler (Vienna)
- Julien Roques (Lyon)
- Josef Schicho (Linz)
- Michael Singer (Raleigh)
- Yunqing Tang (Berkeley/Caltech)
- Gerald Teschl (Vienna)
- Marius van der Put (Groningen)
- Duco van Straten (Mainz)
- Daniel Vargas-Montoya (Toulouse)
- Masha Vlasenko (Warsaw)
- Shoji Yokura (Kagoshima)
- Masaaki Yoshida (Fukuoka)
- Masahiko Yoshinaga (Osaka)
- Don Zagier (Bonn)
- Wadim Zudilin (Utrecht)