威尼斯人

“2018年苏州大学交换代数研讨会”会议日程安排

（2018年11月23日--11月25日,威尼斯人-威尼斯人娱乐场 ）

交换代数会议报告摘要

报告题目：Diagonal matrix reduction over Hermite rings

报告人：陈焕艮教授（杭州师范大学）

摘要：A commutative ring $R$ is an elementary divisor ring if every matrix over $R$ admits a diagonal reduction. Every elementary divisor ring is a Harmite ring. We explore when a Hermite ring is an elementary divisor ring. We shall define the termZabvasky subsetof a ring to study diagonal matrix reduction. Let $S$ be a Zabavsky subset of a Hermite ring $R$. We prove that $R$ is an elementary divisor ring if and only if $aR+bR+cR=R$ with $a,b,c\in R$ implies that there exist $p,q\in R$ such that $((ap+bq)R+cqR)\cap S=\emptyset$. If $aR+bR=R$ with $a,b\in R$ implies that there exists a $y\in R$ such that $a+by\in S$, then$R$ is an elementary divisor ring. Many known results are thereby generalized to much wider class of rings.

报告题目：Ideal approximation theory and its application

报告人：扶先辉副教授（东北师范大学）

摘要：In this talk, I will present an overview of our work on ideal approximation theory. References:

(1) X.H. Fu, P. A. Guil Asensio, I. Herzog, B. Torrecillas, Ideal approximation theory, Adv. Math. (2013)244:750–790;

(2) X. H. Fu, I. Herzog, Powers of the phantom ideal, Proc. London Math. Soc. (2016) 112(4):714-752;

(3) S. Estrada, X. H. Fu, I. Herzog, S. Odabasi, The ideal cotorison pair generated by a set of objects, preprint;

(4) H,Y, Zhu, J.S. Hu, X.H. Fu, The intersection of ideals, preprint.

报告题目：On the regularity of product of irreducible monomial ideals

报告人：高玉彬副教授（陕西师范大学）

摘要：Let $I_1,I_2, \cdots, I_t$ be graded ideals in a polynomial ring $K[x_1,x_2, \cdots, x_n]$ over a field $K$. If these ideals are complete intersections, A. Conca and J. Herzog conjectured that the Castelnuovo-Mumford regularity of the product $I_1I_2 \cdots I_t$ satisfies ${\text{reg}}(I_1I_2 \cdots I_t)\leq {\text{reg}}(I_1)+ {\text{reg}}(I_2)+ \cdots {\text{reg}}(I_t)$. In this talk, when $t$ equals 3 and 4, we show that the above inequality holds for irreducible monomial ideals.

报告题目：Temperley-Lieb algebras and their generalizations

报告人：李彦博副教授（东北大学）

摘要：The Temperley-Lieb algebra first arose in the context of Potts models and ice models in statistical mechanics.It is a good tool to compute the Jones polynomial of a link. Graham generalized it to seven families.In this talk, we first recall the definitions of these algebras. Then we review the presentations of type A and type D T-L algebras given by means of diagrammatic. Finally, we study the Forked Temperley-Lieb algebra,which can be viewed as a quotient of the type D T-L algebra.

报告题目：Blowup rings of three dimensional Ferrers diagram

报告人：申伊塃副教授（中国科技大学大学）

摘要：In this talk, we report our recent investigation of the Rees algebra and the toric ring of the squarefree monomial ideal associated to the three-dimensional Ferrers diagram. Under the projection property condition, we describe explicitly the presentation ideals of the Rees algebra and the toric ring. We show that the toric ring is a Koszul Cohen–Macaulay normal domain, while the Rees algebra is Koszul and the defining ideal is of fiber type. In addition, we investigate the Castelnuovo--Mumford regularity and the multiplicity of the toric rings. In particular, in the rectangular case, we are able to provide direct formulas for these two important invariants. Then, we compare these invariants for an accompanied pair of Ferrers diagrams under some mild conditions, and bound the Castelnuovo--Mumford regularity for more general cases. This is a joint work with Kuei-Nuan Lin (Penn State Greater Allegheny).

报告题目：Boolean graphs-A survey

报告人：武同锁教授（上海交通大学）

摘要：A Boolean graph is the zero divisor graph of a Boolean ring. For a positive integer $n$, let $[n]=\{1,2,\cdots, n\}$, and $2^{[n]}$ the power set of $[n]$. A finite Boolean graph $B_n$ is isomorphic to a graph defined on the vertex set$2^{[n]}\setminus \{ {[n]}, \varnothing\}$, where two vertices are adjacent if and only if their meet is empty. In this paper, we give a survey of some works done in the area of research related to Boolean graphs, in both graph theoretic and algebraic aspects. We also introduce some most recent unpublished works by the author and others.

报告题目：The invariants for quantized symplectic groups

报告人：肖占魁副教授（华侨大学）

摘要：In this talk we present a categorical version of the first and second fundamental theorems of the invariant theory for the quantized symplectic groups. Our methods depend on the theory of braided strict monoidal categories which are pivotal, more explicitly the diagram category of framed tangles. This is joint work with Yuping Yang and Yinhuo Zhang.

报告题目：A note on the structure of finite free resolutions

报告人：周才军教授（上海师范大学）

摘要：In this talk, we present a prime avoidance lemma for matrices, and make use it to study the structure of finite free resolutions。