WEEKLY ACTIVITIES

Stanley depth of powers of monomial ideals
 Speaker: S. A. Seyed Fakhari (Postdoc Simons)Time: 9h00, Wednesday, February 27, 2019Location: Rom 612, Building A6, Institute of Mathematics Abstract: Let \$S=mathbb{K}[x_1,dots,x_n]\$ be the polynomial ring in \$n\$ variables over a field \$mathbb{K}\$ and suppose that \$M\$ is a nonzero finitely generated \$mathbb{Z}^n\$-graded \$S\$-module. Let \$uin M\$ be a homogeneous element and \$Zsubseteq {x_1,dots,x_n}\$. The \$mathbb {K}\$-subspace \$umathbb{K}[Z]\$ generated by all elements \$uv\$ with \$vin mathbb{K}[Z]\$ is called a {it Stanley space} of dimension \$|Z|\$, if it is a free \$mathbb{K}[Z]\$-module. A decomposition \$mathcal{D}\$ of \$M\$ as a finite direct sum of Stanley spaces is called a {it Stanley decomposition} of \$M\$. The minimum dimension of a Stanley space in \$mathcal{D}\$ is called the {it Stanley depth} of \$mathcal{D}\$ and is denoted by \${rm sdepth} (mathcal {D})\$.The quantity \$\${rm sdepth}(M):=maxbig{{rm sdepth}(mathcal{D})mid mathcal{D} {rm is a Stanleydecomposition of} Mbig}\$\$ is called the {it Stanley depth} of \$M\$. We say that a \$mathbb{Z}^n\$-graded \$S\$-module \$M\$ satisfies the {it Stanley's inequality} if \$\${rm depth}(M) leq {rm sdepth}(M).\$\$ In fact, in 1982 Stanley conjectured that every \$mathbb{Z}^n\$-graded \$S\$-module satisfies the Stanley's inequality.This conjecture has been disproved by Duval, Goeckner, Klivans and Martin. However it is still interesting to find classes of modules which satisfy the Stanley's inequality.It is a general philosophy that high powers of ideals have nice homological behavior. Thus, one would expect that the Stanley's inequality could be true for high powers of an ideal. In this talk we focus on this question and review the recent developments in this regard.

Highlights

 21/02/20, Colloquium Lecture:Bernoulli Numbers in Positive Characteristic 23/02/20, Conference:SEAMS school “Arithmetic, Geometry and Model Theory” 04/03/20, Conference:MIS-IMH research school on Mathematics of Data 09/03/20, Conference:School and Conference on Computational Commutative Algebra 23/04/20, Conference:Hội thảo TỐI ƯU VÀ TÍNH TOÁN KHOA HỌC lần thứ 18 24/07/20, Conference:AMC-2020 Satellite Workshop Deterministic and Stochastic Aspects of Differential Equations 25/07/20, Conference:AMC-2020 Satellite Workshop Arithmetic of automorphic forms and their function field analogues 25/07/20, Conference:AMC-2020 Satellite Workshop on Nonlinear Analysis and Optimization Theory 25/07/20, Conference:AMC-2020 Satellite Workshop Minimal Free Resolution and Related Topics