Title    name   Basic Notions Seminar (Affine) Scheme for Everybody Marco Antei (KAIST) 2011-05-13   The aim of this talk is to give a first definition of   Basic Notions Seminar Introduction to Algebraic Quantum Field Theory Park, Jaesuk (Yonsei University) 2011-04-15   Algebraic quantum field theory aims at providing a mathematical description of the structure of quantum field theories. In this talk I will introduce the basic notions of quantum field theory and QFT algebra, sketch some relevant classical ideas - such as homological and homotopy algebras, basic deformation theory, etc, for a general audience.   Basic Notions Seminar Introduction to Quantum singularity theory I Fan Huijun (Beijing University) 2011-04-05    The study of singularities (such as ADE type singularities) has a long history in mathematics, and has also appeared in physics. In this talk, I will explain the basic concepts in quantum singularity theory. For example, W-curves, Witten equation, moduli spaces and the relation with r-spin curve theory, and integrable hierarchy.   Basic Notions Seminar Infinite dimensional Lie algebras and Hilbert schemes Wei-Ping Li (Hong Kong University of Science and Technology) 2010-12-09   Since the work of Gottsche on Betti numbers of Hilbert schemes of points on surfaces and Nakajima and Grojnowski's independent discovery of Heisenberg algebra actions on the cohomology ring of the Hilbert schemes, there have been a lot of research on the cohomology ring structure via the infinite dimensional Lie algebras. In this talk, I will explain how some infinite dimensional Lie algebras come to play a fundamental role in the seemly totally unrelated research area of topology and geometry of Hilbert schemes.   Basic Notions Seminar The Geography Problems of 4-Manifolds, Part I Park, Jongil (Seoul National University) 2010-10-28   One of the fundamental problems in 4-manifolds is to classify simply connected smooth (symplectic, complex) 4-manifolds. The classical invariants of a simply connected 4-manifold X are encoded by its intersection form Q_X, a unimodular symmetric bilinear pairing on H_2(X:Z). M. Freedman showed that a simply connected closed 4-manifold is determined up to homeomorphism by Q_X. But the situation in the smooth (symplectic, complex) category is strikingly different. Hence it is an important question to know which unimodular, symmetric, bilinear integral forms are realized as the intersection form of a simply connected smooth (symplectic, complex) 4-manifold, and which simply connected smooth (symplectic, complex) 4-manifolds admit more than one smooth (symplectic, complex) structure. We call these geography problems of 4-manifolds. Gauge theory - Donaldson theory and Seiberg-Witten theory - has been very successful in the geography problems. For example, S. Donaldson proved that the intersection form of a simply connected, definite, smooth 4-manifold is diagonalizable, and it has been known that most known simply connected irreducible smooth 4-manifolds with b_2^+ odd > 1admit infinitely many distinct smooth structures. Recently, there has also been a big progress in the case of b_2^+ = 1. In this talks I’d like to review recent known results on the geography problems of 4-manifolds, in particular in the case of b_2^+ = 1, in three levels - smooth, symplectic and complex category.   Basic Notions Seminar An invitation to Kahler-Einstein metrics and Monge-Ampere equations Philippe Eyssidieux (University of Grenoble) 2010-10-08

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