Workshop on D-modules and Hypergeometric Functions

Mornings' schedule

    The three courses, two with the duration of about 4h each and one with the duration of about 2h, will take place each morning from Monday, July 11 to Thursday, July 14, in the building C6 of the Faculty of Sciences of the University of Lisbon (see map of the site).

1. Hypergeometric differential equations in dimension one.
    I will review some results of N. Katz in his book"Gauss sums and differential equations"(mainly in Chapters 3 and 5).

• Mellin transform
• The hypergeometric group
• Irreducible hypergeometric DEs
• Determinant formulas
3. Algebraic theory of q-difference equations
• Holonomic q-difference systems
• q-hypergeometric group
(*) Notes of this course are available in http://math.polytechnique.fr/~sabbah/sabbah-lisbonne05.pdf and/or http://math.polytechnique.fr/~sabbah/sabbah-lisbonne05.ps.gz

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COURSE II:   by   Mutsumi Saito (pdf1 and pdf2).

1^st talk (July 11^th).   Introduction to A-hypergeometric systems.
Abstract:
Using Gauss's hypergeometric equation,I explain how its corresponding A-hypergeometricsystem is constructed. In addition, some fundamentalresults are given.

2^nd talk (July 12^th).   Indicial systems of A-hypergeometric systems.
Abstract:
How to find exponents is given.
3^rd talk (July 13^th).   A-hypergeometric series.
Abstract:
How to construct A-hypergeometric series is given.
4^th talk (July 14^th).  A-equivalence classes.
Abstract:
To a parameter vector and a face of the cone generated bythe column vectors of A, a finite set is associated.Such finite sets define an equivalence relation in the parameter space,which classifies the isomorphism classes of A-hypergeometric systems.

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COURSE III:   by   Eduardo Cattani.

Two talks devoted to the joint work with Alicia Dickenstein andBernd Sturmfels on rational hypergeometric functions.

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Afternoons' schedule

In the afternoons, there will be 30mn and 50mn conferences.




	Monday	Tuesday	Wednesday	Thursday
14:00h - 14:50 h	Marius van de Put	Jacques Sauloy	Francisco Castro	Pedro Silva
15:00h - 15:50 h	Orlando Neto	Paul Horja	Ezra Miller	Mutsumi Saito
15:55h - 16:10h	Coffee Break
16:10h - 17:00h	Philippe Maisonobe	Anatoly Koshubei	Uli Walther	Julien Roques (end at 16:40h)
17:10h - 17:40 h	Josep Alvarez	Chris Bremer	Isabel Hartillo
17:50h - 18:30 h	Bruno Fabre	Giovanni Morando	Lisa Nilsson

WORKSHOP DINNER : Wednesday at 20.30 h pm.




Titles and abstracts




Marius Van de Put - Classification, moduli spaces and Galois theory for q-difference equations
Abstract: We present recent work, in cooperation with M. Reversat (Toulouse). q-difference modules over the field of convergent Laurent series at z=0 turn out to have a global character. Split q-difference modules reflect Atiyah's classification of vector bundles on an elliptic curve. More complicated q-difference modules are related to connections with singularities on the elliptic curve. The Galois group of some q-difference modules are direcly related with Stokes matrices for a corresponding irregular differential equation. If time permits a small excursion in positive characteristic will be made.
Orlando Neto - Title to be announced

Philippe Maisonobe - Visite sur les faisceaux pervers d'origine microlocale

Josep Alvarez - Generators of D-modules in positive characteristic
Abstract: Let R = k[x₁,..., x_d] or R = k[[x₁,...,x_d]] be either a polynomial or a formal power series in a finite number of variables over a fields k of characteristic p > 0 and let D_R|k be the ring of k-linear differential operators of R. In this work we prove that if f is a non-zero element of R then Rf, obtained from R by inverting f, is generated as a D_R|k-module by 1/f. This is an amazing fact considering that the corresponding characteristic zero statement is very false. In fact, we prove an analog of this result for a considerably wider class of rings R and a considerably wider class of D_R|k-modules.

Bruno Fabre - Dolbeault Cohomology and Locally Residual Currents
Abstract: Let be X a projective manifold of dimension n. Let be Y₁,...,Y_n n hypersurfaces on X, defining ample line bundles, and intersecting properly.
We show that the Dolbeault cohomology group Hⁱ(\omega ^q) (i < n) of the sheaf of holomorphic q-forms on X can be computed as the i-th cohomology group of some complex of global sections of locally residual currents on X, and that any cohomology class in H^I (\omega^q) contains a \overline\partial-closed locally residual current with support in Y₁\cap ... \cap Y_i .
For q=n, we get a another theorem by restricting to locally residual currents obtained from meromorphic forms with simple poles on the Y_i, obtaining like this a theorem of R. Thomas, B. Khesin, A. Rosly (A polar de Rham theorem, Topology, 43, 2004).

Jacques Sauloy - Function theoretic methods in Galois theory for q-difference equations
Abstract:We describe an analytic approach to the definition and computation ofmonodromy groups and Galois groups for complex linear q-differenceequations: equivalence with flat vector bundles over an elliptic curve,Stokes phenomenon, confluence to differential equations ... A recentprogress features the so-called cosmic Galois group. This is mostly workby J.-P. Ramis, J. Sauloy and C. Zhang.

Paul Horja - Hypergeometric functions, toric Deligne-Mumford stacks andmirror symmetry
Abstract: It is well known that the generalized hypergeometric functions of Gelfand-Kapranov-Zelevinsky (GKZ) play an important role in toric mirror symmetry. I will explain how the classical Mellin-Barnes integral representations of Gamma series are naturally 'mirrored' by Fourier-Mukai integral transforms associated to birational maps of toric stacks. The approach is based on the interplay between the K-theory of toric stacks and the GKZ system of differential equations. This is joint work with Lev Borisov.

Anatoly Kochubei - Holonomic Modules in Positive Characteristic
Abstract: We study modules over the Weyl-Carlitz ring, a counterpart of the Weyl algebra in analysis over local fields of positive characteristic. It is shown that some basic objects of function field arithmetic, like the Carlitz module, Thakur's hypergeometric polynomials, and analogs of binomial coefficients arising in the function field version of umbral calculus, generate holonomicmodules.

Chris Bremer - Local epsilon factors for holonomic D-modules and the Jacquet-Langlands correspondence
Abstract: Beilinson, Bloch, and Esnault have developed a theory of epsilon factors (both local and global) for meromorphic connections on algebraic curves. The local theory is concerned with irreducible, often irregular singular connections E over a laurent series ring K=k((t)), where k is a subfield of C. Epsilon factors are calculated by the determinant of the period matrix associated to the fourier transform of E. The Jacquet-Langlands correspondence suggests that we should be able to construct a theory of epsilon factors for representations of GL(n,K), and that the epsilon factor for E should be the same as the epsilon factor of a suitable representation of GL(n,K). I will discuss my thesis work with Spencer Bloch, which involves developing a theory of epsilon factors on GL(n,K).

Giovanni Morando - Existence theorem for tempered holomorphic solutions of D-modules in dimension 1
Abstract: Let O^t(U) denote the space of tempered holomorphic functions on a relatively compact open subanalytic subset of C. We prove that if P is a differential operator defined on \bar U, regular on U, then there exists a finite subanalytic opencovering U=\cup_{i\in I} U_i such that for any g\in O^t(U) thereexist u_i \in O^t(U_i) (i\in I) such that Pu_i=g|_{Ui}.

Lucia Di Vizio - Monodormy of p-adic q-difference equations (talk in the morning at 11.30h am)
Abstract: I'll speak about a joint work with Y. Andre about monodromy of p-adic q-difference equations. I'll spend some time describing the rank one case, which can be treated in an more elementary way than the general case and making some examples.

Francisco Castro - On the irregularity of hypergeometric D-modules associated witha monomial curve (joint work with N. Takayama (Kobe University)
Abstract: We compute the slopes of the hypergeometric D-module H associated with a smooth monomial curve in Cⁿ. We use induction on the number of variables. We use the fact that the restriction of H to some coordinate hyperplanes are also hypergemetric D-modules of the same type. This last result follows from the restriction algorithm described by Oaku and Takayama.

Ezra Miller - Homological methods for hypergeometric families
Abstract: Hypergeometric systems of partial differential equations, as defined in the 1980s by Gelfand and his coauthors, depend on discrete data and continuous parameters. It has long been an open question how the analytic solutions behave as the continuous parameters vary. In particular, what property of the discrete data determines when the dimensions of the solution spaces remain constant independent of the parameters? A conjectural answer emerged in the late 1990s and was substantially refined in the early 2000s. This talk will describe geometric and homological methods used to prove the conjecture and its refinement in joint work with L. Matusevich and U. Walther.

Uli Walther - Reducibility of the monodromy of GKZ-systems
Abstract: When is HA(\beta) a simple module at a generic point? I show that surprisingly this property is a lattice invariant: it is constant on the coset of \beta mod Z ^d. Moreover, I show that rank-jumping systems are always reducible. I close with recovering a recent difficult theorem of Dickenstein and Sadykov stating that Mellin systems are always reducible. I prove this via the observation that GKZ-systems with integer parametersare necessarily reducible.

Isabel Hartillo - Slopes and hypergeometric systems
Abstract: The classical theory of ordinary differential equations describes regular and singular points, among these, it distinguishes regular and irregular singular points. In the case of ordinarydifferential equation with polynomial coefficients, the theorem of Fuchs gives an algorithmic way of studying singular points andalso gives a rational number which describes (if we have an irregular singular point) the formal power series (not convergent) which is a solution of the equation. So it gives an algorithmic tool to describe regular or singular equations, studying among the singular points if there is any irregular singular point. Slopes are the generalization of this concept in the case of system of partial differential equations with polynomial coefficients, and in the case of holonomic systems a result by Laurent gives a relation of the existence of slopes with the irregular sheafassociated with the system being different from zero.
Let A be a d x n matrix with integer entries and rank(A)=d, such that the columns of the matrix are not in thesame hyperplane. Let \beta \in C^d be a vector, we denote by H_A(\beta) the hypergeometric system defined by A and \beta. If the semigroup generated by the columns of A ispointed and the ring C[\partial]/I_A is a Cohen-Macaulay ring we describe the slopes of the hypergeometricsystem. As application of our result we treat specifically two special cases: the first case is A defined by one row and thesecond A being a (n-1) x n matrix (the codimension one case), these cases allow us to calculate the slopes of suchsystems.

Lisa Nilsson - Convergence of A-hypergeometric series and integrals

Will Traves - Differential Operators and Invariant Theory (ppt) (talk in the morning at 11.30h am)
Abstract: Many mathematicians have studied D-modules associated to quotient spaces under group actions. I will survey some of the results in this area and suggest a new direction based on an old observation due to Cayley. He noticed that the every polynomial invariant of certain group actions satisfies a system of differential equations. I will report on my efforts to use differential operators to create new invariants from old and to study the D-module structure of the ring of invariants under these group actions.

Pedro Silva - Title to be announced

Mutsumi Saito - D-modules on affine toric varieties and A-hypergeometric systems
Abstract: Let A be an integer matrix of size (d, n) with rank (A)=d, and let D(R_A) be the ring of differential operators on theaffine toric variety defined by A. We show that the classification of A-hypergeometric systemsand that of ZA-graded simple D(R_A)-modules are the same. We also give conditions for the algebra D(R_A) being simple.

Julien Roques - Confluence of difference equations to differential equations
Abstract: Let (E[h]) be a family of fuchsian difference equations which is a deformation of an algebraic differential equation (E) on the Riemann sphere, fuchsian at infinity. We will show how the two canonical solutions of (E[h]) are related, when h tends to 0, to the canonical solution of (E) at the infinity which is given by the Frobenius method. We will also show how the connection matrix of (E[h]) is related, when htends to zero, to the local monodromies of the deformed system (E) . The key point is the use of factorial series which satisfy some "(C,\lambda) growth hypothesis"


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