Friday, October 21, 2022, 15:30 – 17:00, Darmstadt & Zoom
Speaker: Ryosuke Shimada (TU Darmstadt)
Thursday, November 3, 2022, 16:00 – 17:00, Frankfurt& Zoom
Speaker: Gabriele Bogo (TU Darmstadt)
Abstract: Define E-series and G-series [4, Definition 1.2], and present the basic theory of E-functions following [5, Sec 2.1]; in particular, prove that hypergeometric functions are E-functions (example (4) in [5, Sec 2.1]) and discuss the Siegel-Shidlovsky theorem. State Siegel’s question [4, pag 904]. Introduce the differential algebras E and G [4, Section 1.4], G-operators and E-operators [4, Def 1.6]. Present their basic properties as given in [4, Theorem 1.8] and [4, Theorem 1.10], but do not discuss singularities of the operators. Introduce the Fourier transform [4, Section 1.1], the Laplace transform [4, Sections 1.11,1.12,1.13] and give a complete description of their action on the algebras E and G [4, Prop 1.14]. Conclude by presenting the example in [4, Section 1.15] in order to shed light on the definition of E-operators.
Thursday, November 3, 2022, 16:00 – 17:00, Frankfurt& Zoom
Speaker: Christina Röhrig (TU Darmstadt)
Abstract: Introduce Picard-Fuchs differential equations on curves by following [6, pag 71-72]. State (without proof) that the solution of a Picard-Fuchs differential equation is a G-function [1, pag 110] and the conjectural converse statement (Bombieri-Dwork conjecture [1, pag 11]). As an example, present the Legendre family and its relation with hypergeometric functions as discussed in [9, Sections 2.1-2.2] (the reading of the introduction and of Sections 1,2 of [9] can be helpful in the preparation of the talk.) Introduce and describe the sets G [3, Def 2] and H [3, Section 2.2], and state the main theorem [3, Theorem 1], which relates Siegel’s question to periods of algebraic varieties. Prove the theorem (see [3, Section 6]) by discussing the main ingredients: the appearance of arbitrary elements of G in the asymptotic expansion of E-functions [3, Theorem 3] and the asymptotic expansion of hypergeometric E-functions [3, Theorem 4].
Friday, November 4, 2022, 15:30 – 17:00, Darmstadt & Zoom
Speaker: P. Daniels (University of Michigan)
Abstract: Introduce Picard-Fuchs differential equations on curves by following [6, pag 71-72]. State (without proof) that the solution of a Picard-Fuchs differential equation is a G-function [1, pag 110] and the conjectural converse statement (Bombieri-Dwork conjecture [1, pag 11]). As an example, present the Legendre family and its relation with hypergeometric functions as discussed in [9, Sections 2.1-2.2] (the reading of the introduction and of Sections 1,2 of [9] can be helpful in the preparation of the talk.) Introduce and describe the sets G [3, Def 2] and H [3, Section 2.2], and state the main theorem [3, Theorem 1], which relates Siegel’s question to periods of algebraic varieties. Prove the theorem (see [3, Section 6]) by discussing the main ingredients: the appearance of arbitrary elements of G in the asymptotic expansion of E-functions [3, Theorem 3] and the asymptotic expansion of hypergeometric E-functions [3, Theorem 4].
Friday, November 11, 2022, 15:30 – 17:00, Darmstadt & Zoom
Speaker:M. Lara (Goethe Universität Frankfurt)
Abstract: Introduce Picard-Fuchs differential equations on curves by following [6, pag 71-72]. State (without proof) that the solution of a Picard-Fuchs differential equation is a G-function [1, pag 110] and the conjectural converse statement (Bombieri-Dwork conjecture [1, pag 11]). As an example, present the Legendre family and its relation with hypergeometric functions as discussed in [9, Sections 2.1-2.2] (the reading of the introduction and of Sections 1,2 of [9] can be helpful in the preparation of the talk.) Introduce and describe the sets G [3, Def 2] and H [3, Section 2.2], and state the main theorem [3, Theorem 1], which relates Siegel’s question to periods of algebraic varieties. Prove the theorem (see [3, Section 6]) by discussing the main ingredients: the appearance of arbitrary elements of G in the asymptotic expansion of E-functions [3, Theorem 3] and the asymptotic expansion of hypergeometric E-functions [3, Theorem 4].
Thursday, November 24, 2022, 16:00 – 17:00, Darmstadt& Zoom
Speaker: Rizacan Ciloglu (TU Darmstadt)
Abstract: The goal of the talk is to give an introduction to differential Galois groups and Galois correspondence. The main reference is the book [13]. Introduce differential modules over differential fields (with algebraically closed fields of constants), [13, Definition 1.6.]. Pick your favorite running example to illustrate all the concepts. Introduce Picard-Vessiot extensions [13, §1.3.] and define the differential Galois group [13, Definition 1.25.]. Explain that it is an algebraic group [13, Theorem 1.27.] and prove that the maximal spectrum of the Picard-Vessiot ring is a torsor for 3the differential Galois group [13, Theorem 1.28.]. State the Galois correspondence [13, Theorem 1.34.] and sketch its proof.
Thursday, November 24, 2022, 16:00 – 17:00, Darmstadt& Zoom
Speaker: Leonie Scherer (Goethe Universität Frankfurt)
Abstract: TThe goal of the talk is to introduce Tannakian categories and relate the Tannaka group of a differential module to its differential Galois group. We mostly follow the article [2], specifically Chapter 2. Define affine group schemes and explain how to recover them from their category of finite-dimensional representations, [2, Chapter 2, Proposition 2.8.]. Define neutral Tannakian categories [2, Chapter 2, Definition 2.19.] and give some examples (such as the category of differential modules or algebraic vector bundles with integrable connection). Sketch the proof of the main theorem [2, Chapter 2, Theorem 2.11.]. Explain how to recover the differential Galois group of a differential module through Tannakian formalism, [13, Theorem 2.33.]. Prove [2, Chapter 2, Proposition 2.20.]. This gives an alternate way to see that the differential Galois group is algebraic
Friday, November 25, 2022, 15:30 – 17:00, Darmstadt & Zoom
Speaker: F. Tanania (LMU München)
Abstract: Introduce Picard-Fuchs differential equations on curves by following [6, pag 71-72]. State (without proof) that the solution of a Picard-Fuchs differential equation is a G-function [1, pag 110] and the conjectural converse statement (Bombieri-Dwork conjecture [1, pag 11]). As an example, present the Legendre family and its relation with hypergeometric functions as discussed in [9, Sections 2.1-2.2] (the reading of the introduction and of Sections 1,2 of [9] can be helpful in the preparation of the talk.) Introduce and describe the sets G [3, Def 2] and H [3, Section 2.2], and state the main theorem [3, Theorem 1], which relates Siegel’s question to periods of algebraic varieties. Prove the theorem (see [3, Section 6]) by discussing the main ingredients: the appearance of arbitrary elements of G in the asymptotic expansion of E-functions [3, Theorem 3] and the asymptotic expansion of hypergeometric E-functions [3, Theorem 4].
Thursday, December 1, 2022, 16:00 – 17:00, Frankfurt& Zoom
Speaker: Manuel Müller (TU Darmstadt)
Abstract: In this talk and the next we follow Claude Sabbah’s notes [11] to introduce the basic notions of D-module theory on the affine line. Define the Weyl algebra A1(C) and the algebras D, Db of local differential operators. Talk about their basic properties [11, I, §1.3]. Introduce the notion of good filtration of a D-module [11, I, Definition 3.2.1.]. Define the characteristic variety [11, I, Definition 3.2.5.] and give an example. Define holonomic D-modules and explain that they are precisely the D-modules of the form D/I for some non-zero left ideal I ⊂ D [11, I, Corollary 3.3.5]. Finally globalize the story following [11, III, §1.1.] to introduce holonomic modules over the Weyl algebra A1(C). In particular, we need [11, III, Proposition 1.1.5.] whose key input is [11, I, Lemma 2.3.3].
Thursday, December 1, 2022, 16:00 – 17:00, Frankfurt& Zoom
Speaker: Felix Pennig (TU Darmstadt)
Abstract: Introduce the notion of formal meromorphic connection [11, Definition 4.3.1.] (replacing K by Kb). This is the same as a differential module over Kb. Prove that any holonomic A1(C)- or D-module determines a formal meromorphic connection by extension of scalars (this is one direction of [11, Theorem 4.3.2.]). Prove [11, Proposition 4.3.3], also known as the cyclic vector lemma. Introduce the Newton polygon of a formal meromorphic connection, define the slopes and regular meromorphic connections. Spend the rest of the talk explaining [11, Theorem 5.4.7.], also known as the Levelt-Turrittin decomposition. One of the main ingredients is [11, Theorem 5.3.1.] which gives a splitting of a formal meromorphic connection according to its slopes
Friday, December 9, 2022, 15:30 – 17:00, Darmstadt & Zoom
Speaker: A. Eteve (Cornell University)
Abstract: Introduce Picard-Fuchs differential equations on curves by following [6, pag 71-72]. State (without proof) that the solution of a Picard-Fuchs differential equation is a G-function [1, pag 110] and the conjectural converse statement (Bombieri-Dwork conjecture [1, pag 11]). As an example, present the Legendre family and its relation with hypergeometric functions as discussed in [9, Sections 2.1-2.2] (the reading of the introduction and of Sections 1,2 of [9] can be helpful in the preparation of the talk.) Introduce and describe the sets G [3, Def 2] and H [3, Section 2.2], and state the main theorem [3, Theorem 1], which relates Siegel’s question to periods of algebraic varieties. Prove the theorem (see [3, Section 6]) by discussing the main ingredients: the appearance of arbitrary elements of G in the asymptotic expansion of E-functions [3, Theorem 3] and the asymptotic expansion of hypergeometric E-functions [3, Theorem 4].
Monday December 12, 2022, 13:30 – 15:10, Darmstadt & Zoom
Speaker:Ian Gleason (Universität Bonn)
Abstract: Introduce Picard-Fuchs differential equations on curves by following [6, pag 71-72]. State (without proof) that the solution of a Picard-Fuchs differential equation is a G-function [1, pag 110] and the conjectural converse statement (Bombieri-Dwork conjecture [1, pag 11]). As an example, present the Legendre family and its relation with hypergeometric functions as discussed in [9, Sections 2.1-2.2] (the reading of the introduction and of Sections 1,2 of [9] can be helpful in the preparation of the talk.) Introduce and describe the sets G [3, Def 2] and H [3, Section 2.2], and state the main theorem [3, Theorem 1], which relates Siegel’s question to periods of algebraic varieties. Prove the theorem (see [3, Section 6]) by discussing the main ingredients: the appearance of arbitrary elements of G in the asymptotic expansion of E-functions [3, Theorem 3] and the asymptotic expansion of hypergeometric E-functions [3, Theorem 4].
Tuesday December 13, 2022, 09:50-11:30, Darmstadt & Zoom
Speaker:Ian Gleason (Universität Bonn)
Abstract: Introduce Picard-Fuchs differential equations on curves by following [6, pag 71-72]. State (without proof) that the solution of a Picard-Fuchs differential equation is a G-function [1, pag 110] and the conjectural converse statement (Bombieri-Dwork conjecture [1, pag 11]). As an example, present the Legendre family and its relation with hypergeometric functions as discussed in [9, Sections 2.1-2.2] (the reading of the introduction and of Sections 1,2 of [9] can be helpful in the preparation of the talk.) Introduce and describe the sets G [3, Def 2] and H [3, Section 2.2], and state the main theorem [3, Theorem 1], which relates Siegel’s question to periods of algebraic varieties. Prove the theorem (see [3, Section 6]) by discussing the main ingredients: the appearance of arbitrary elements of G in the asymptotic expansion of E-functions [3, Theorem 3] and the asymptotic expansion of hypergeometric E-functions [3, Theorem 4].
Thursday December 15, 2022, 13:30-15:10, Darmstadt & Zoom
Speaker:Ian Gleason (Universität Bonn)
Abstract: Introduce Picard-Fuchs differential equations on curves by following [6, pag 71-72]. State (without proof) that the solution of a Picard-Fuchs differential equation is a G-function [1, pag 110] and the conjectural converse statement (Bombieri-Dwork conjecture [1, pag 11]). As an example, present the Legendre family and its relation with hypergeometric functions as discussed in [9, Sections 2.1-2.2] (the reading of the introduction and of Sections 1,2 of [9] can be helpful in the preparation of the talk.) Introduce and describe the sets G [3, Def 2] and H [3, Section 2.2], and state the main theorem [3, Theorem 1], which relates Siegel’s question to periods of algebraic varieties. Prove the theorem (see [3, Section 6]) by discussing the main ingredients: the appearance of arbitrary elements of G in the asymptotic expansion of E-functions [3, Theorem 3] and the asymptotic expansion of hypergeometric E-functions [3, Theorem 4].
Friday December 16, 2022, 14:00-17:00, Darmstadt & Zoom
Speaker:Ian Gleason (Universität Bonn)
Abstract: Introduce Picard-Fuchs differential equations on curves by following [6, pag 71-72]. State (without proof) that the solution of a Picard-Fuchs differential equation is a G-function [1, pag 110] and the conjectural converse statement (Bombieri-Dwork conjecture [1, pag 11]). As an example, present the Legendre family and its relation with hypergeometric functions as discussed in [9, Sections 2.1-2.2] (the reading of the introduction and of Sections 1,2 of [9] can be helpful in the preparation of the talk.) Introduce and describe the sets G [3, Def 2] and H [3, Section 2.2], and state the main theorem [3, Theorem 1], which relates Siegel’s question to periods of algebraic varieties. Prove the theorem (see [3, Section 6]) by discussing the main ingredients: the appearance of arbitrary elements of G in the asymptotic expansion of E-functions [3, Theorem 3] and the asymptotic expansion of hypergeometric E-functions [3, Theorem 4].
Thursday, December 22, 2022, 16:00 – 17:00, Darmstadt& Zoom
Speaker: Konstantin Jakob(TU Darmstadt)
Abstract: Define Fourier transform of D-modules and introduce the set of solutions of a D-module in a differential algebra A, [4, §2.1 & 2.2]. Introduce the Tannakian categories Conn0(Gm) and RS0(A1) and explain their relation via Fourier transform of D-modules [4, Proposition 2.3., §2.4]. Define modules of type E and G [4, Def 2.9] and the associated categories E and G. Discuss [4, Theorem 2.12 (1)], which relates D-modules to G-functions, and the analogous statement for E [4, Theorem 2.13]. Construct the monoidal functor Ψ : Conn0(Gm) → {Qℓ − graded vector spaces} 4and conclude that the Galois group of M contains an explicit algebraic torus. Define the divisor div(M) of a module M ∈ Conn0(Gm) [4, §2.6., Lemma 2.7., 2.8]. Finally, introduce hypergeometric D-modules [4, Definition 3.6] and prove [4, Theorem 3.7.], computing their divisor.
Thursday, December 22, 2022, 16:00 – 17:00, Darmstadt& Zoom
Speaker: Yiu Man Wong (Goethe Universität Frankfurt)
Abstract: The goal of the talk is to prove that E is a Tannakian subcategory of Conn0(Gm) [4, Theorem 2.14]. The main tool is Andr´e’s theorem [4, Theorem 2.12 (2)] whose proof uses Bombieri’s characterization of G-functions in terms of their p-adic radius of convergence. the presentation is based on Andr´e’s book [1]. Introduce the notion of global radius of a D-module [1, IV, 3.3]. This requires some basic concepts in p-adic differential equations tat can be found in [1, IV, 1-2]. State that G-modules are precisely the modules with finite global radius [4, pag 76] (see also [4, Sec 1.7,1.8]). Prove the first two points of Lemma 2 in (Andr´e, IV,3). Complete the proof for the category E by following the proof of [4, Theorem 2.14].
Thursday, January 12, 2023, 16:00 – 17:00, Frankfurt& Zoom
Speaker: Jiaming Chen (Goethe Universität Frankfurt)
Abstract: Recall the definition of generalized hypergeometric equations following [8, Chapter 3, §1]. Compute the slope and state [8, Theorem 3.6.]. Define Lie-irreducibility [8, §2.7.] and explain the relation to Kummer induction. The rest of the talk will focus on the computation of differential Galois groups. Following [7, §2.5. & 2.6.] introduce the local differential Galois group, the upper numbering filtration and the unique index N-subgroup. Explain the relation to the slopes. Prove Katz’s Main D.E. Theorem [8, 2.8.1.] using Gabber’s torus trick [8, Theorem 1.0]. Use the various recognition results for semisimple Lie algebras as a black box.
Thursday, January 12, 2023, 16:00 – 17:00, Frankfurt& Zoom
Speaker: Yingkun Li (TU Darmstadt)
Abstract: The talk follows closely [4, Sec 4]. Define Lie-generated objects of a Tannakian category [4, Def 4.3] and motivate this definition as in [4, Sec 4.4]. Prove [4, Theorem 4.7], which shows that an object of E with Galois group SL3 which is Lie-generated by objects of H is Lie-generated by only one specific object. It is not necessary to prove all the special cases in the proof. Finally, prove [4, Theorem 4.9], which describes the symmetry constraint on the Fourier transform of a three-dimensional object of E with Galois group containing SL3
Friday January 20, 2023, Darmstadt & Zoom
Speaker:Thibaud van den Hove (TU Darmstadt)
Abstract: Introduce Picard-Fuchs differential equations on curves by following [6, pag 71-72]. State (without proof) that the solution of a Picard-Fuchs differential equation is a G-function [1, pag 110] and the conjectural converse statement (Bombieri-Dwork conjecture [1, pag 11]). As an example, present the Legendre family and its relation with hypergeometric functions as discussed in [9, Sections 2.1-2.2] (the reading of the introduction and of Sections 1,2 of [9] can be helpful in the preparation of the talk.) Introduce and describe the sets G [3, Def 2] and H [3, Section 2.2], and state the main theorem [3, Theorem 1], which relates Siegel’s question to periods of algebraic varieties. Prove the theorem (see [3, Section 6]) by discussing the main ingredients: the appearance of arbitrary elements of G in the asymptotic expansion of E-functions [3, Theorem 3] and the asymptotic expansion of hypergeometric E-functions [3, Theorem 4].
Thursday, January 26, 2023, 16:00 – 17:00, Darmstadt& Zoom
Speaker: Javier Fresán (Ècole Polytechnique)
Abstract: TAbstract: I will explain why every exponential period function of the form $\int_\sigma e^{-zf} \omega$, where $f$ is a regular function on an algebraic variety $X$ defined over the field of algebraic numbers, $\omega$ is an algebraic differential form on $X$, and $\sigma$ is a rapid decay cycle on $X(C)$, is a linear combination of E-functions "with monodromy” with coefficients in the field generated by usual periods, special values of the gamma function and Euler’s constant.This is how E-functions arise from geometry and gives some intuition of why a positive answer to Siegel’s question about hypergeometric E-functions was extremely unlikely. (Joint work with Peter Jossen).
Friday, January 27, 2023, Darmstadt & Zoom
Speaker:Simon Pepin Lehalleur (Radboud-Universiteit Nijmegen)
Abstract: Introduce Picard-Fuchs differential equations on curves by following [6, pag 71-72]. State (without proof) that the solution of a Picard-Fuchs differential equation is a G-function [1, pag 110] and the conjectural converse statement (Bombieri-Dwork conjecture [1, pag 11]). As an example, present the Legendre family and its relation with hypergeometric functions as discussed in [9, Sections 2.1-2.2] (the reading of the introduction and of Sections 1,2 of [9] can be helpful in the preparation of the talk.) Introduce and describe the sets G [3, Def 2] and H [3, Section 2.2], and state the main theorem [3, Theorem 1], which relates Siegel’s question to periods of algebraic varieties. Prove the theorem (see [3, Section 6]) by discussing the main ingredients: the appearance of arbitrary elements of G in the asymptotic expansion of E-functions [3, Theorem 3] and the asymptotic expansion of hypergeometric E-functions [3, Theorem 4].
Friday, February, 10, 2023, Darmstadt & Zoom
Speaker:Zhouhang Mao (Xiamen University)
Abstract: Introduce Picard-Fuchs differential equations on curves by following [6, pag 71-72]. State (without proof) that the solution of a Picard-Fuchs differential equation is a G-function [1, pag 110] and the conjectural converse statement (Bombieri-Dwork conjecture [1, pag 11]). As an example, present the Legendre family and its relation with hypergeometric functions as discussed in [9, Sections 2.1-2.2] (the reading of the introduction and of Sections 1,2 of [9] can be helpful in the preparation of the talk.) Introduce and describe the sets G [3, Def 2] and H [3, Section 2.2], and state the main theorem [3, Theorem 1], which relates Siegel’s question to periods of algebraic varieties. Prove the theorem (see [3, Section 6]) by discussing the main ingredients: the appearance of arbitrary elements of G in the asymptotic expansion of E-functions [3, Theorem 3] and the asymptotic expansion of hypergeometric E-functions [3, Theorem 4].
Friday, February, 17, 2023, Darmstadt & Zoom
Speaker:Bogdan Zavyalov (Institute for Advanced Study)
Abstract: Introduce Picard-Fuchs differential equations on curves by following [6, pag 71-72]. State (without proof) that the solution of a Picard-Fuchs differential equation is a G-function [1, pag 110] and the conjectural converse statement (Bombieri-Dwork conjecture [1, pag 11]). As an example, present the Legendre family and its relation with hypergeometric functions as discussed in [9, Sections 2.1-2.2] (the reading of the introduction and of Sections 1,2 of [9] can be helpful in the preparation of the talk.) Introduce and describe the sets G [3, Def 2] and H [3, Section 2.2], and state the main theorem [3, Theorem 1], which relates Siegel’s question to periods of algebraic varieties. Prove the theorem (see [3, Section 6]) by discussing the main ingredients: the appearance of arbitrary elements of G in the asymptotic expansion of E-functions [3, Theorem 3] and the asymptotic expansion of hypergeometric E-functions [3, Theorem 4].
