Abstract: Given a vector bundle over an algebraic curve, the space of quotient sheaves of fixed rank and degree carries a scheme structure, called a quot scheme.
In particular, we can discuss the irreducibility and dimension of that space.
The behaviour of quot schemes is wild in general, but we can observe a uniformity in some special instances.
In this talk, I will review some important known results on quot schemes. Also I will briefly explain their generalization to symplectic and orthogonal bundles, based on a series of joint works with George H. Hitching, which were launched from our meeting in 2007 at KIAS.

Abstract: In this talk, we introduce a new combinatorial model of irreducible characters of simple Lie algebras of classical type BCD, which is based on a spin representation of a Clifford algebra. This model turns out to be compatible with type A combinatorics very well and have interesting and important applications in representation theory. As one of its applications, we explain how classical branching rules of irreducible characters in a stable range can be extended to arbitrary highest weights in a bijective way.

Abstract: Entropy is an isomorphism invariant which measures the chaoticity of a dynamical system. Positive entropy systems have been studied for several decades and many of their properties are well understood at least in the case of Z-actions. In the study of general group actions, entropy zero systems arise more naturally and they exhibit diverse dynamics. To understand these systems we begin with the examples of entropy zero Z-actions and investigate their properties including their complexities.

Abstract: Given a domain or a compact submanifold in a Riemannian manifold, its interior volume and boundary volume are associated with geometric inequalities, which are influenced by the geometry of underlying space. In this talk, we discuss various isoperimetric inequalities for minimal submanifolds in a Riemannian manifold. Moreover, submanifolds with variable mean curvature in a Riemannian manifold will be discussed as well. From the analytic point of view, one can associate a submanifold in a Riemannian manifold with the first eigenvalue of the Laplace operator. Recent developments in this direction will be introduced.

Abstract: Abstract: Fano varieties form an important building block of algebraic varieties in view of Minimal Model Program. A two dimensional Fano variety is called a del Pezzo surface. The classification of nonsingular del Pezzo surfaces in term of blowing-up is classical. They are either the projective plane, the quadric surface, and blow-ups of the projective plane in at most $8$ general points.
In this talk, I will briefly mention the minimal model program as a classification device for algebraic varieties. Then, I will talk about the cascades of log del Pezzo surfaces of Picard number one, which can be regarded as a generalization of the classical result on the classification of nonsingular del Pezzo surfaces.

Abstract: Abstract: In the talk, I am going to review previous works on the arithmetic of special values of various twisted L-functions at critical points. Among interesting stories, main emphasis will be put on non-vanishing modulo prime of the special L-values. Ever since the simplest case, namely Dirichlet L-values in Iwasawa theoretic setting, was successfully resolved by Ferrero-Washington in 1979, many researchers have observed that dynamical phenomena play crucial roles in several studies on the topic. Some of the phenomena are roughly generalizations of Kronecker's theorem on equi-distribution statement for (p-adic) normal numbers. In the latter part of the talk, I will introduce relevant works or on-going researches of mine that are joint works with B. Jun, J. Lee, M. Kim, and A. Burungale.

Abstract: Abstract: CR manifold is a generalization of real manifolds in a complex space. In this talk, we briefy investigate local equivalence of CR manifold in view point of Cartan geometry. Then we introduce a differential geometric method for the rigidity of proper holomorphic maps between domains in complex manifolds. We use CR structure as geometric structures preserved by proper holomorphic maps extending smoothly to an open set of the boundary.