2016 academic year

No. 1

Date: May 13 (Fri) 16:30 - 17:30

Place: B702, Fac. Sci., Hiroshima University

Speaker: Kazuki HIROE (Josai University)

Title:On irregular spectral curves

Abstract:

    As irregular spectral curves, we consider plane algebraic curves obtained by the classical limit of linear ordinary differential equations with irregular singularities on the Riemann sphere. In this talk, I will discuss the following comparisons between singularities of curves and differential equations. Local invariants: Milnor & intersection number of curve singularities and Komatsu-Malgrange irregularity of differential equations. Global invariants: Euler characteristics of the normalization of spectral curves and Katz' rigidity index of differential equations. Local transformations: blow-up and local Fourier transform. Link of singularity: links of plane curve singularities and links of irregular types of differential equations. Moreover I will give a characterization of formal deformation of local formal differential equations preserving the isotopy of links and show that isomonodromic deformation with irregular singularities by Jimbo-Miwa-Ueno and admissible deformation by Boalch preserve the link isotopy.

No. 2

Date: June 3 (Fri) 16:30 - 18:00

Place: B707, Fac. Sci., Hiroshima University

Speaker: Hajime NAGOYA (Kanazawa University)

Title: Irregular conformal blocks, with applications to Painleve tau functions

Abstract:

    Conformal blocks are the building blocks of the correlation functions of two-dimensional conformal field theories. For a long time, conformal blocks with regular singularities have attracted much attention. Recently, conformal blocks with irregular singularities (irregular conformal blocks, ICB) have also been paid attention. We discuss a recent progress on irregular conformal blocks and give a conjectural formula of the combinatorial expansion of irregular conformal blocks of rank one at an irregular singular point. Note that combinatorial formulas for series expansions of irregular conformal blocks at regular singular points have been already known, which are obtained by easy confluence limits. We also explain how to obtain an expansion of ICB of rank one at an irregular singular point by a confluence limit of regular conformal blocks and show that the expansion of PVI tau function at infinity in terms of CB goes to the previously conjectured expansion of PV tau function at infinity in terms of ICB by some confluence limit.

No. 3

Date: June 24 (Fri) 16:30 - 18:00

Place: B702, Fac. Sci., Hiroshima University

Speaker: Hiroshi YAMAZAWA (Shibaura Institute of Technology)

Title: Singular solutions of $q$-difference-differential equations of the Briot-Bouquet type

Abstract:

    G\'{e}rard and Tahara introduced the Briot-Bouquet type partial differential equations. We will try to q-discrete the equations to q-difference-differential equations. In this talk we will report the existence of solutions in the case of the characteristic exponent $\rho(0)\not=1,2,\dots$, and in another case.

No. 4

Date: July 15 (Fri) 15:00 - 17:50

Place: B707, Fac. Sci., Hiroshima University

15:00 - 16:30

Speaker: Shingo KAMIMOTO (Hiroshima Univeraity)

Title: Exact asymptotics and resurgent analysis

Abstract:

    In 1886, Poincare introduced the notion of ``Asymptotic expansion�f�f to give a meaning to formal solutions of linear ordinary differential equations at an irregular singular point. Afterward, asymptotic analysis at an irregular singular point was developed by contributions of many splendid mathematicians; Hukuhara, Turrittin, Birkhoff, Malgrange, Majima, Sibuya, �c In this talk, we focus on the works by Ramis and Ecalle in 1980�fs. We will review ``Exact asymptotics�h and ``Resurgent analysis�h initiated by them, which are the analytical methods based on the Borel summation method, and the results concerning the resurgence of formal series solutions of nonlinear differential and difference equations will be given.

16:50 - 17:50

Speaker: Hiroyuki USAMI (Gifu University)

Title:On asymptotic properties of positive solutions of a kind of Lanchester-type model

Abstract:

    In 1916, F. W. Lanchester proposed a simple system of ODEs as a model equation describing combats between aircrafts. It has been understood that similar systems are applicable to describe other types of combats as well as many phenomena appearing in logistics, economic and biology. However, it seems that the analysis of Lanchester-type models were mainly based on numerical methods. So, in this talk we give rigorous results concerning asymptotic properties of positive solutions of the system.

No. 5

Date: December 9 (Fri) 16:30 - 17:45

Place: B707, Fac. Sci., Hiroshima University

Speaker: Takao SUZUKI (Kindai University)

Title: A higher order Painlev\'e system in two variables and extensions of the Appell hypergeometric functions $F_1$, $F_2$ and $F_3$

Abstract:

    In this talk we propose an extension of Appell hypergeometric function $F_2$ (or equivalently $F_3$). It is derived from a particular solution of a higher order Painlev\'e system in two variables. On the other hand, an extension of Appell's $F_1$ was introduced by Tsuda. We also show that those two extensions are equivalent at the level of systems of linear partial differential equations.

No. 6

Date: December 23 (Fri) 16:30 - 18:00

Place: B707, Fac. Sci., Hiroshima University

Speaker: Yoko UMETA (Yamaguchi University)

Title:Stokes geometry for a unified family of $P_{\mathrm{J}}$-hierarchies (J=I,II,IV,34)

Abstract:

    In this talk, we introduce a unified family of $P_{\mathrm{J}}$-hierarchies (J=I,II,IV,34) with a large parameter. [Kawai,Koike,Nishikawa,Takei],[Takei] and [Honda] studied that the Stokes geometry of $(P_{\mathrm{J}})_m$ and Noumi-Yamada system have closed relationship with those of their underlying Lax pair. The unified system also has the properties. We consider relations between the Stokes geometry of non-linear equation and that of its underlying Lax pair.

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