Seminar on Global Analysis

Japanese version here

### 10 November 2008, 16پF30 room 301, Graduate School of Science and Technology, Kumamoto University Donald A. Lutz (San Diego State University) Asymptotic Integration of some classes of linear differential equations

Abstract: This talk will involve theorems which have been used for obtaining asymptotic representations for solutions of several different kinds of second order linear differential equations. The methods used to obtain the results originally were quite ad hoc and depended heavily on the special structure of the equations. It will be shown how the results can also be obtained from a more unified perspective as part of a general theory for systems of linear differential equations originated by N. Levinson. In doing so, the results can in many cases also be improved and extended.

### 25 June 2009, 16پF30 room 301, Graduate School of Science and Technology, Kumamoto University Vladimir P. Kostov (Universite de Nice-Sophia Antipolis) On the Schur-Szegö composition of polynomials

Abstract: The Schur-Szegö composition (SSC) of the degree n polynomials P:=∑j=0najxj and Q:=∑j=0nbjxj is the polynomial P*Q:=∑j=0najbjxj/Cnj. We recall first some classical results. When the polynomials are hyperbolic, i.e. with all roots real, and when all roots of P have the same sign, then the multiplicity vector of P*Q is completely defined by the multiplicity vectors of P and Q. When both P and Q have all their roots negative, then the SSC defines a semi-group action on the set of multiplicity vectors considered as ordered partitions of n.
If a (complex) polynomial P has one of its roots at -1, then it is representable as an SSC of n-1 polynomials of the form (x+1)n-1(x+ai) where the numbers ai are uniquely defined up to permutation. We shall discuss some properties of the mapping sending the symmetric polynomials of the roots of P into the ones of the numbers ai.

### 27 October 2009, 16پF30 room 401, 3rd building of Faculty of Science, Kumamoto University Marius van der Put (University of Groningen) Classification of meromorphic differential equations

Abstract: The classification of linear differential equations over the field K=C({z}) of the meromorphic functions at z=0 (i.e., the field of the convergent Laurent series) is a highlight of the theory of asymptotics. Starting with simple examples we will give a survey of this and show how this leads to explicit monodromy spaces. The relation with a theorem of Sibuya and the fundamental paper of Jimbo-Miwa-Ueno will be discussed.

### 10 December 2009, 16پF30 room 401, 3rd building of Faculty of Science, Kumamoto University Mitsuo Kato (University of the Ryukyus) Reflection subgroups of Appell's F4 monodromy groups

Abstract: Assume the system of differential equations E4(a,b,c,c';X,Y) satisfied by Appell's hypergeometric function F4(a,b,c,c';X,Y) has a finite irreducible monodromy group M4(a,b,c,c'). The monodromy matrix Γ3* derived from a loop Γ3 once surrounding the irreducible component C={(X,Y)|(X-Y)2-2(X+Y)+1=0} of the singular locus of E4 is a complex reflection. The minimal normal subgroup NC of M4 containing Γ3* is, by definition, a finite complex reflection group of rank four. Let P(G) be the projective monodromy group of the Gauss hypergeometric differential equation 2E1(a,b,c). It is known that NC is reducible if ε:=c+c'-a-b-1∉Z or if ε∈Z and P(G) is a dihedral group. We prove that if ε&isinZ, then NC is the (irreducible) Coxeter group W(D4), W(F4), W(H4) according as P(G) is a tetrahedral, octahedral, icosahedral group, respectively.

### 8 March 2010, 17پF00 room 301, Graduate School of Science and Technology, Kumamoto University Timur Sadykov (Siberian Federal University) Bases in the solution space of the Mellin system

Abstract: I will present a joint work with Alicia Dickenstein. We consider algebraic functions z satisfying equations of the form
a0 zm + a1zm1 + a2 zm2 + … + an zmn + an+1 =0.
Here m > m1 >… > mn>0, m,mi N, and z=z(a0,…,an+1) is a function of the complex variables a0, …, an+1. Solutions to such equations are classically known to satisfy holonomic systems of linear partial differential equations with polynomial coefficients. In the talk I will investigate one of such systems of differential equations which was introduced by Mellin. We compute the holonomic rank of the Mellin system as well as the dimension of the space of its algebraic solutions. Moreover, we construct explicit bases of solutions in terms of the roots of initial algebraic equation and their logarithms. We show that the monodromy of the Mellin system is always reducible and give a formula for the holonomic rank of a generic bivariate hypergeometric system.

### 27 May 2010, 16پF30 room 301, Graduate School of Science and Technology, Kumamoto University Raimundas Vidunas (Kobe University) Transformations between Heun and hypergeometric equations

Abstract: It is known that Heun's functions (or differential equations) can be reduced to Gauss hypergeometric functions by rational changes of its independent variable only if its parameters, including the fourth singular point location parameter t and the accessory parameter, take special values. The talk will present a classification of Heun functions reducible to Gauss hypergeometric functions via such transformations. Some arithmetic properties of the parameter t will be noted.

### 2 July 2018, 16پF30 room 301, Graduate School of Science and Technology, Kumamoto University Raimundas Vidunas (Osaka University) Hypergeometric expressions for modular functions

Abstract: The well-known Rogers-Ramanujan series of modular level 5 can be expressed in terms of 2F1-hypergeometric functions with the icosahedral projective monodromy. We show that similar series of level 7 can be expressed as 3F2-hypergeometric functions with the PSL(2,7) projective monodromy of 168 elements.

### 30 January 2019, 16پF30 room 401, 3rd building of Faculty of Science, Kumamoto University Davide Guzzetti (SISSA) Non-generic isomonodromy deformations at an irregular singularity and Frobenius manifolds

Abstract: Some of the main results of  and  (see also  for a synthetic exposition with examples), concerning non-generic isomonodromy deformations of a certain linear differential system with irregular singularity and coalescing eigenvalues, are discussed. The results are the analytic part of a joint work with G. Cotti and B. Dubrovin. We were motivated by the problem of extending to coalescent structures the analytic theory of Frobenius manifolds, in view of the computation of the monodromy data of the quantum cohomology of Grassmannians , , . Analytically, this problem translates to the problem of extending the isomonodromic deformation theory of Jimbo-Miwa-Ueno to certain non generic cases.
References
 G. Cotti, B. Dubrovin. D. Guzzetti: Isomonodromy Deformations at an Irregular Singularity with Coalescing Eigenvalues. arXiv:1706.04808 (2017). To appear in Duke Math. J.
 G. Cotti, B. Dubrovin. D. Guzzetti: Local Moduli of Semisimple Frobenius Coalescent Structures. arXiv:1712.08575 (2017).
 G. Cotti, D. Guzzetti: Analytic geometry of semisimple coalescent Frobenius structures. Random Matrices Theory Appl. 6 (2017), no. 4, 1740004, 36 pp.
 G. Cotti, D. Guzzetti: Results on the Extension of Isomonodromy Deformations to the case of a Resonant Irregular Singularity. Random Matrices Theory Appl. (2018).
 D. Guzzetti: Notes on non-generic Isomonodromy Deformations. SIGMA 14 (2018), 087, 34 pages.
 G. Cotti, B. Dubrovin, D. Guzzetti. Helix Structures in Quantum Cohomology of Fano Varieties. arXiv:1811.09235 (2018).

### 28 November 2019, 16پF30 room 401, 3rd building of Faculty of Science, Kumamoto University Jiro Sekiguchi (Tokyo University of Agriculture and Technology) On construction of algebraic potentials

Organizers

پ@Yoshishige HARAOKA (Kumamoto Univ.) haraoka -at- kumamoto-u.ac.jp
پ@Hironobu KIMURA (Kumamoto Univ.) hiro -at- aster.sci.kumamoto-u.ac.jp
پ