Abstract: The notion of matrix factorizations was introduced by Eisenbud'80 as an explicit description of Maximal Cohen-Macauley modules over a quotient ring defining a hypersurface. On the other hand, when a hypersurface is the defining equation of an ADE singularity, the corresponding root system is constructed by Brieskorn'70 from vanishing cycles of the Milnor fiber. Elliptic root systems were also obtained in this way by K.Saito, but it is hard to do this for more general hypersurface isolated singularities.
Physically, these vanishing cycles should form a Fukaya category, a D-brane category in an A-model. On the other hand, (graded) matrix factorizations form a triangulated category, which is regarded as the D-brane category in a B-model side (Kontsevich, Kapustin-Li'03, ...). This implies we may construct a generalization of root systems from triangulated categories of graded matrix factorizations via a version of homological mirror symmetry conjecture by Kontsevich'94 (see A.Takahashi math.AG/0506347). (recall A-model <-> symplectic geometry, B-model <--> complex geometry, mirror symmetry = symmetry between an A-model and a B-model).A regular system of weights (K.Saito'86) is a quadruple of positive integers $W:=(a,b,c;h)$ satisfying some conditions. Geometrically, each $W$ corresponds to an isolated singularity such as ADE or elliptic one mentioned above. Thus, our aim is to construct a root system and the corresponding Lie theory from the triangulated category of graded matrix factorizations associated to a regular system $W$ of weights, which, we believe, will also be a first step for constructing the primitive form and flat structure.
In this talk, I first review briefly on regular systems of weights and recall the definition of the category of graded matrix factorizations. A natural way of obtaining a root system from a triangulated category of graded matrix factorizations is to show triangulated equivalence of the triangulated category with the derived category of a path algebra of a quiver (with relations, in general). I report the result in the case of type ADE, and another class including those corresponding to what are called unimodular exceptional singularities. In the latter case, the corresponding quivers are "the first" examples of what are called wild quivers in representation theory of finite dimensional algebras. Finally, I comment on a natural stability condition on such a triangulated category.
This talk is based on joint work with Kyoji Saito and Atsushi Takahashi (math.AG/0511155 and a work in preparation).
Abstract: I will try to explain how certain representations $M$ of a quiver $Q$ can be used to construct Calabi-Yau categories $C_M$ of dimension 2. The Calabi-Yau category $C_M$ can be seen as a categorification of a cluster algebra $A_M$. Using the semicanonical basis of the universal enveloping algebra of the positive part of the Kac-Moody Lie algebra associated to Q, one hopes to obtain a basis of $A_M$.
Abstract: The goal of this survey talk is to explain how the representation theory of semi-simple Lie algebras over complex numbers is related to $t$-structures of the derived category of constructible sheaves on flag varieties. Precisely, we give a brief survey about 1) the Riemann-Hilbert correspondence (which relates $D$-modules to constructible sheaves), 2) the Beilinson-Bernstein correspondence (which relates representation theory to $D$-modules), and 3) the Kazhdan-Lusztig polynomials (which relates different $t$-structures). We will not try to prove any of the statements but exhibits examples about $\mathbb P ^1$ and $\mathfrak{sl} ( 2, \mathbb C )$.
Abstract: The first lecture will be an elementary introduction into Beilinson-Bernstein Localization Theorem for crystalline differential operators. In the second lecture we discuss some consequences for the derived categories of coherent sheaves.
Abstract: Let $C$ be a projective irreducible non-singular curve over an algebraic closed field $k$ of characterstic zero. We consider the Jacobian $J(C)$ of $C$ that is a projective abelian variety parametrizing topological trivial line bundles on $C$. We consider its Brill-Noether loci that corresponds to the varieties of special divisors. The Torelli theorem allows us to recover the curve from its Jacobian as a polarized abelian variety. We do a similar study for the Quot scheme $Q_{d,r,n}(C)$ of degree $d$ quotients of a trivial vector bundle on $C$, defining Brill-Noether loci, maps of Abel-Jacobi type and an analogous of the Torelli theorem by applying Fourier-Mukai transforms.