Literature-reading Seminar (monthly, joint with Anhui University):

IV.  2012.06.08, USTC, GuanLiKeYan Building, 1308

14:30-17:30   Zhibin Zhao (Anhui Univ.)

Title: Introduction to singularity categories I (1-3)

2012.06.11, USTC, GuanLiKeYan Building, 1308

9:00-11:00  Longgang Sun (USTC)

Title: Introduction to singularity categories II (1-2)

14:00-16:00  Xiao-Wu Chen (USTC)

Title: Introduction to singularity categories III (1-2)

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III. 2011.12.17,  GuanLiKeYan building Room 1518

Program:

9:00-10:30       Ren Wang (Hubei Univ./USTC)

Title: Coherent algebras and noncommutative projective lines (after Piontkovski)

Abstract:  A well-known conjecture says that every one-relator group is coherent. We state and partly

prove a similar statement for graded associative algebras. In particular, we show that every Gorenstein

algebra A of global dimension 2 is graded coherent. This allows us to define a noncommutative analogue

of the projective line \$\mathbb P^1\$ as a noncomutative scheme based on the coherent  noncommutative

spectrum qgr A of such an algebra A, that is, the category of coherent A-modules modulo the torsion ones.

This category is always abelian Ext-finite hereditary with Serre duality, like the category of coherent

sheaves on \mathbb P^1. In this way we obtain a sequence \$\mathbbP_n^1(n\geq 2 ) \$ of pairwise

non-isomorphic noncommutative  schemes which generalize the scheme \$ \mathbb P^1=\mathbb P^2\$.

10:30-12:00      Yanhong Bao (Anhui Univ./USTC)

Title: Connections on central bimodules--Part II (after Dubois-Violette and Michor)

Abstract:  We will discuss the duality between \$A\$-\$A\$-central bimodules and \$Z(A)\$- modules, which

may induce the notion of connections for \$Z(A)\$- modules by the dual notion for central bimodules. We

will introduce hermitian structures and noncommutative (pseudo-) Riemann structures of a unitial *-algebra,

and give the definition of Levi-Civita connections for \$Der(A)\$ as a \$Z(A)\$-module.

Lunch break

14:00-15:30     Xiujian Wang (Anhui Univ.)

Title: Rank functions on rooted tree quivers (after Ryan-Kinser)

Abstract: For a quiver Q, the free abelian group R(Q) corresponding to the representations of Q has a ring

structure  where the multiplication is given by the tensor product. We will show that if Q is a rooted tree, then

the associated reduced ring  R(Q)_red is a finitely generated Z-module, which can be described explicitly. It is

done by investigating the functors from the category of representations of Q to the category of finite dimensional

vector spaces.

15:30-17:30     Longgang Sun (USTC)

Title:  Category equivalences involving graded modules over path algebras of quivers (after Smith)

Abstract:  Let Q be a finite quiver with arrow set Q_1, k a field, and kQ its path algebra with its standard

grading. Set QGr(kQ):= Gr(kQ)/FdimkQ  (graded kQ-modules modulo those that are unions of finite dimensional

modules).  This paper mainly shows the following categorical equivalences:

QGr kQModS(Q) ≡ GrL(Q^) ModL(Q^)_0 QGr kQ(n).

Here S(Q) is a direct limit of finite dimensional semisimple algebras; Q^0 is the quiver without sources or sinks  that is

obtained by repeatedly removing all sinks and sources from Q; L(Q^) is the Leavitt path algebra of Q; L(Q^)_0 is its

degree zero component; and Q^(n) is the quiver whose incidence matrix is the n-th power of that of Q.

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II. 2011.11.19, Anhui University, New Campus, BoXue NanLou B403

Program:

9:00-10:30       Yanhong Bao (Anhui Univ./USTC)

Title: Connections on central bimodules (after Dubois-Violette and Michor)

Abstract: We will recall several noncommutative generalizations of smooth vector fields and

differential forms on a smooth manifold.  Based on derivations of an algebra, the definition of

a connection on a central bimodule is recalled, and some constructions and special cases are

discussed. Finally, we discuss a duality between bimodules and modules over the center, and

apply the duality to the 1-forms.

10:30-12:00       Huanhuan Li (Hubei Univ./USTC)

Title:  Calabi-Yau algebras viewed as deformations of Poisson algebras (after Berger-Pichereau)

Abstract: Let A=A(f) denote the quadratic algebra defined as the quotient of k<x1, ..., xn> by the

two-sided ideal generated by f. The main purpose of the article is to study the algebras B=B(f)

defined by the potential w=fz (where z is an another generator). For any non-degenerate form f,

the algebra B is Koszul and 3-Calabi-Yau. When n=2, the classification of B leads to three types of

algebras. One of the types is viewed as a deformation of a Poisson algebra S, whose Poisson bracket

is non-diagonalizable quadratic. In the case that the potential of S has non-isolated singularities,

the homology of S is computed. Then the Hochschild homology of B is obtained.

Lunch break

14:30-16:00       Longgang Sun (USTC)

Title: Tilting and cluster tilting for quotient singularities (after Iyama-Takahashi)

Abstract: We will show that the stable categories of graded Cohen-Macaulay modules over quotient

singularities have tilting objects. In particular, these categories are triangle equivalent to derived categories

of finite dimensional algebras. The method is based on higher dimensional A-R theory.

16:00-17:30       Xiujian Wang (Anhui Univ.)

Title: Tensor products on quiver representations (after Herschend)

Abstract: We will recall some definitions related to quivers, and translate the well-known tensor product

of representations of a group given by diagonal actions to the case of representations of a quiver. Then

we provide three different approaches and exhibit their close relationship to the point-wise tensor product.

Finally, we discuss monoidal structures roughly.

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I. 2011.10.15,  GuanLiKeYan building Room 1208

Program:

9:00-10:30       Xiao-Wu Chen (USTC)

Title: Singularity category of a non-commutative resolution of singularities (after Burban-Kalck)

Abstract: We will recall the notion of singularity category of a noncommutative ringed space. For

isolated singularity, it has a nice description via maximal Cohen-Macaulay modules. For the special

case of a curve with nodal singularity, a complete description of the singularity category is available.

10:30-12:00      Dawei Shen (USTC)

Title: Residue and duality for singularity categories of isolated Gorenstein singularities (after Murfet)

Abstract:  We will study Serre duality on the singularity categories of isolated Gorenstein singularities

and give an explicit formula for the Serre duality. We will use generalized fractions, Grothendieck

residue symbols and complete injective resolutions to give the formula. Then we will prove Kapustin-Li

formula by applying the above results to hypersurface singularities.

Lunch break

14:30-16:00     Tao Lu (Shandong Univ./USTC)

Title: Langlands duality for representations of quantum groups (after Frenkel-Hernandez)

Abstract: We establish a correspondence (or duality) between the characters and the crystal bases of

finite-dimensional reprentations of quantum groups associated to Langlands dual semi-simple Lie algebras.

This duality may also be stated purely in terms of semi-simple Lie algebras. To explain this duality, we

introduce an "interpolating quantum group" depending on two parameters which interpolates between

a quantum group and its Langlands dual.

16:00-17:30     Ren Wang (Hubei Univ./USTC)

Title:  Gerasimov's theorem and N-Koszul algebras (after Berger)

Abstract:   Let A be a graded algebra with a single homogeneous relation. We will introduce a criterion for A

to be N-Koszul, where N is the degree of the relation. This criterion uses a theorem of Gerasimov. we will give

an alternative proof of Gerasimov's theorem for N=2, and determine which of the PBW deformations

of a symplectic form are Calabi-Yau.