The 8th HU and SNU Symposium on Mathematics
- Recent developments of Geometry and Topology -
as a part of The 15th HU-SNU Joint Symposium

Contents

Abstract

Seok-Jin Kang
Khovanov-Lauda-Rouquier algebras and cyclotomic categorification theorem
In this talk, we will give an account of Khovanov-Lauda's cyclotomic categorification conjecture that cyclotomic Khovanov-Lauda-Rouquier algebras provide a categorification of highest weight modules for all symmetrizable quantum Kac-Moody algebras. We will also discuss Kang-Kashiwara's proof of the conjecture.
Jaehyun Hong
Rigidities, geometric structures, and Lie algebra cohomologies
We say that a quasi-homogeneous subvariety L/L' of a projective space P^n is rigid if it is determined by the second fundamental form in the following sense: any complex manifold of P^n whose fundamental form is isomorphic to the fundamental form of L/L' is projectively equivalent to an open subset of L/L'. I will talk about how to solve this kind of rigidity problem by transforming it into an equivalence problem of geometric structures.
Dano Kim
Multiplier ideal sheaves and tensor powers of a line bundle
In complex algebraic geometry, the notion of a multiplier ideal sheaf has played an important role in the study of algebraic varieties of general dimension. A multiplier ideal sheaf measures the singularity of a 'singular pole' given by the zeros of a finite set of holomorphic functions. Also more generally, it is natural to define the multiplier ideal sheaf of a plurisubharmonic function. In this talk, we will discuss applications of the fundamental subadditivity property of multiplier ideal sheaves and also of a more recent superadditivity type result due to Popovici. We will also discuss related results stemming from work of Lindholm and Berndtsson on Bergman kernels.
Seonhee Lim
Ford circles and Farey maps for function fields
We will introduce Gauss map analogue for function fields and explain its relation to the geodesic flow on trees. Farey map and its invariant measure will be explored. This is a joint work with D. Kim, H. Nakada and R. Natsui.
Toru Ohmoto
Chern class for singular varieties and enumerative geometry
For a possibly singular complex algebraic variety, the Chern-Schwartz-MacPherson (CSM) class is defined in the homology or the Chow group as a generalization of the integration based on the Euler characteristics measure. In this talk I will talk about some related topics, e.g., we present generating function formulae of pushforward of the CSM class of Hilbert schemes of points on a non-singular variety in relation with the zero dimensional MNOP conjecture.
Tadayuki Watanabe
Invariants of 3-manifolds via Morse homotopy
We give a higher order generalization of Fukaya's Morse homotopy approach to 2-loop Chern-Simons perturbation theory for homology 3-spheres. It is expected that our invariant can be computed explicitly for some examples whereas the original integral expression for CS can hardly be computed. We will also discuss extension of our method to 3-manifolds with positive first Betti numbers.
Masao Jinzenji
Mirror Map as Generating Function of Intersection Numbers
In this talk, we discuss geometric construction of the mirror map used in the mirror computation of Gromov-Witten invariants. We reconstruct the mirror map as a generating function of intersection numbers of the moduli space of holomorphic maps compactified by chains of quasi maps. We also apply this formalism to compute some open Gromov-Witten invariants.
Hitoshi Furuhata
Statistical immersions of spaces of constant Hessian curvature
A statistical manifold is a space with a Riemannian metric and a torsion-free affine connection satisfying the Codazzi equation, which appears in information geometry, affine differential geometry and Hessian geometry. We study immersions between statistical manifolds preserving both structures. In this talk, some new classification theorems will be presented after basic notions will be reviewed.