Literature-reading Seminar
(monthly, joint with
IV. 2012.06.08,
USTC,
14:30-17:30 Zhibin
Zhao (
Title:
Introduction to singularity categories I (1-3)
2012.06.11, USTC,
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)
For background,
click here.
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III.
2011.12.17, GuanLiKeYan
building Room 1518
Program:
9:00-10:30 Ren Wang (
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 (
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 kQ ≡ ModS(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,
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 (
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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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.
