강연제목과 초록
① Toru Uemeda (Kyoto Univ.) [3 lectures]
Title : The First Fundamental Theorem and Its Applications
Abstract : In the classical invariant theory, the most basic result is the first
fundamental theorem of vector invariants, which describes explicitly the generators of invariants.Also this kind of results are the main ingredients in the famous book "The Classical Groups" by Herman Weyl. He treats the first fundamental theorems for the classical groups, i.e., the general linear groups, the orthogonal groups, and the symplectic groups. The importance of these results goes beyond the invariant theory. Forty years after Weyl's book, Roger Howe found the phenomena of dual pairs, and recognized its theoretical background in the first fundamental theorems. The theory of dual pairs explains many duality theorems from number theory to physics. In the lecture, I will explain this basic results together with some of its applications. I also hope to mention the problem on invariant differential operators that Professor Yang raised in relation to the Jacobi groups.
② Hiroyuki Ochiai (Kyushu Univ.) [2 lectures]
Title : Polynomial differential operators acting on automorphic forms
Abstract: Differential operators acting on holomorphic Siegel modular forms on the Siegel upper half space can be applied to the explicit computation of special values of automorphic L-functions. We here are interested in the explicit expression ofthese differential operators in the following special setting : the product of two Siegel upper half spaces sits inside the Siegel upper half space with a double size. We can understand these polynomials in the context of the classical reductive dual pairs. The polynomials are considered to be a generalization of Gegenbauer polyno- mials with several variables. This talk is based on the joint work with T. Ibukiyama and T. Kuzumaki.
③ Akihito Wachi (Hokkaido Univ. of Education) [2 lectures]
Title : (1) Capelli identities and b-functions of prehomogeneous vector spaces
Abstract : For a multivariate polynomial there exists a polynomial and a differential operator with polynomial coefficients such that.
The monic generator of the ideal consisting of such is called the b-function of . If is a relative invariant of a prehomogeneous vector space, then is given by replacing variables in with corresponding partial differential operators, which is independent of .In particular, when is the determinant, its -function is easily seen by using the Capelli identity. In this talk, we consider prehomoge-neous vector spaces associated to equioriented quivers of type A, and give an analogue of Capelli identities and the -functions of the relative invariants.
(2) Symmetric pairs and Capelli identities
Abstract : Classical Capelli identities are considered to be associated to dual pairs, which consist of two Lie groups. We consider an analogous setting ofsee-saw pairs, which consist of four Lie groups, and give several identities of differential operators in determinantal form. These identities are naturally arising from the generators of the rings of invariant differential operators over symmetric spaces.
④ Minoru Itoh (Kagoshima Univ.) [3 lectures]
Title : Extensions of the tensor algebra and their applications
Abstract : We introduce a natural extension of the tensor algebra. This extended algebra is based on a vector space as the ordinary tensor algebra is. In addition to "left multiplications" by vectors, we can consider "derivations" by covectors as basic operators on this algebra. These two types of operators satisfy an analogue of the canonical commutation relations, and we can regard the algebra generated by these operators as an analogue of the Weyl algebra and the Clifford algebra (actually this operator algebra contains these algebras naturally as quotient algebras). These extensions of the tensor algebra have some applications : (i) applications to invariant theory related to tensor products, and (ii) applications to immanants. The latter one includes a new method to study the quantum immanants in the universal enveloping algebras of the general linear Lie algebrasand their Capelli type identities (the higher Capelli identities).
⑤ Jae-Hyun Yang (Inha Univ.) [2 lectures]
Title : Invariant differential operators on Siegel-Jacobi space
Abstract : We have a natural action of the Jacobi group (the semidirect product of the symplectic group and the Heisenberg group) on the Siegel-Jacobi space.The Siegel-Jacobi space is a space which is important geometrically andarithmetically. But it is not a reductive symmetric space. First I will discuss differential operators on the Siegel space which are invariant under the natural action of the symplectic group on the Siegel space. I review some works of Hans Maass and Goro Shimura about invariant differential operators on the Siegel space. Secondly I will give a lecture on invariant differential operators on the Siegel-Jacobi space and present some interesting important problems that are natural.
⑥ Young-Hoon Kiem (Seoul National Univ.) [2 lectures]
Title : Elementary construction of moduli spaces of curves and maps via GIT
Abstract :I will explain a construction of the moduli space of pointed stable curves of genus 0, which is often called the Mumford-Knudsen space, by a sequence of explicit blow-ups from the elementary GIT quotient of by thediagonal action of SL(2) with respect to the symmetric linearization O(1,...,1). It is often the case in moduli theory that adding an additional structure makes the problem easier. The problem of parameterizing pointed nodal curves together with a morphism of degree 1 has been solved by Fulton and MacPherson and the moduli space is constructed from by a sequence of explicit blow-ups. Recently it was shown by A. Mustata and M. Mustata that the sequence of blow-ups can be reorganized into a shorter sequence so that the intermediate blown-up spaces have natural moduli theoretic meaning. I will show that by taking the SL(2) quotient of the sequence we obtain a sequence of blow-ups starting from //SL(2) ending with the Mumford-Knudsen space. Furthermore, we show that the intermediate blown-up spaces are Hassett's moduli spaces of weighted pointed stable curves. One key point of the construction is the principle that (GIT) quotient and blow-up commute. If time permits, I will explain an analogous construction for Kontsevich's stable map space to projective space. This talk is based on joint work with D. Hyeon and H. Moon.
⑦ David Hyeon (POSTECH) [2 lectures]
Title : Geometric invariant theory and birational geometry of moduli
space of curves
Abstract : Recently there has been a lot of activity on the so called Hassett-Keel program which studies the relation between log canonical models of various moduli spaces. In the case of moduli space of stable curves, GIT has so far played a central role : I will explain the general idea of the program and present a conjectural critical value formula that suggests a strong interplay between GIT and birational geometry.
⑧ Lin Han (Inha Univ.) [1 lecture]
Title : Non-vanishing for values of modular L-functions modulo and extension groups
Abstract : Let be a prime and an elliptic curve over of odd square free conductor. Denote by the -quadratic twist of for a fundamental discriminant and by the Tate-Shafarevich group of . This presentation shows an instance of application of Galois representation to the studyof algebraic parts of central critical values of twisted L-series modulo primes . This also gives us information on . Under the assumption of BSD conjecture, if the order of is not divisible by a prime for some negative fundamental discriminant , then there are infinitely many negativefundamental discriminant such that ∤.