The commutative algebra group of the Institute for Research in Fundamental Sciences (IPM) is organizing the 17th seminar on Commutative Algebra and Related Topics. The goal is to bring together people who are working on commutative algebra and related fields, introduce recent developments to the young researchers and PhD students, and to acquaint the young researchers with the new trends in the subject.

Because of the pandemic situation of Covid 19, the seminar will be held virtually. Contributions to this event are highly appreciated.

# Seminar (January 6-8, 2021)

## Invited Speakers

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The question of finding or even effectively bounding the betti numbers of an ideal in a commutative ring is a difficult one. Even more complicated is using the structure of an ideal $I$ to find information about its powers $I^r$.
Taylor's thesis described a free resolution of any ideal generated by $q$ monomials as the simplicial chain complex of a simplex with $q$ vertices. Taylor's resolution, though often far from minimal, works for every monomial ideal $I$, giving upper bounds for the betti numbers of $\beta_{i}(I) \leq {q \choose i }$ where $q \choose i$ is the number of $i$-faces of a $q$-simplex.
If $I$ is generated by $q$ monomials and $r$ is a positive integer, then $I^r$ can be generated by $q^r$ monomials, and therefore its betti numbers are bounded by $q^r \choose i$, a number that grows exponentially.
The question that we address in this talk is: can we find a subcomplex of the $q^r$-simplex whose simplicial chain complex is a free resolution of $I^r$ for any given monomial ideal $I$ generated by $q$ monomials?
We will explore this question by considering "redundant" faces of the Taylor complex of $I^r$, which will lead us to a (much smaller) subcomplex of the Taylor complex to resolve $I^r$.
This talk is based on joint work with Susan Cooper, Sabine El Khoury, Sarah Mayes-Tang, Liana M., and Sandra Spiroff.

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Let $S=\mathbb{K}[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables over a field $\mathbb{K}$. To every simple graph $G$ with vertex set $V(G)=\{x_1, \ldots, x_n\}$ and edge set $E(G)$, one associates its {\it edge ideal} $I=I(G)$ defined by $$I(G)=\big(x_ix_j: \{x_i,x_j\}\in E(G)\big)\subseteq S.$$ The focus of this talk is on the Alexander dual of edge ideals. Namely, the ideal$$J(G)=\bigcap_{\{x_i, x_j\}\in E(G)}(x_i, x_j),$$which is called the {\it cover ideal} of $G$. The reason for this naming is that $J(G)$ is minimally generated by squarefree monomials corresponding to the minimal vertex covers of $G$.
We review the recent results about the symbolic powers of cover ideals. In particular, we characterize all graphs $G$ with the property that $J(G)^{(k)}$ has a linear resolution for some (equivalently, for all) integer $k\geq 2$. Also, we determine an upper bound for the regularity of symbolic powers of certain classes of graphs including bipartite graphs, unmixed graphs and claw-free graphs. Furthermore, we compute the largest degree of minimal generators of $J(G)^{(k)}$ when $G$ is either an unmixed of a claw-free graph. Moreover, we study the asymptotic behavior of depth of symbolic powers of cover ideals.

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## Registration

Everybody is welcome to participate in this event. Please note that attending in this virtual seminar is free of charge and there is no need for a registration fee but you need to have the link code which will be available by your request through the email ipmcommalg17.sem@gmail.com.

## Call for Papers

Papers will be accepted for presentation at the seminar subject to approval by the Organizing committee. Please send your submissions (extended abstract or full paper) electronically (in PDF and Tex formats) to ipmcommalg17.sem@gmail.com with the subject: "17th_commalg_abstract". .

## Important Dates

Deadline for registration and paper submission: December 8, 2020 (هجده آذر ماه 1399)
Decision on acceptance of attendance and talks: December 29, 2020 (نهم دی ماه 1399)

### IPMInstitute for Research in Fundamental Sciences

Niavaran

P.O. Box 19395-5746, Tehran - Iran

• Tel: +98 21 222 90 928, Fax: +98 21 222 90 648
• ipmcommalg@ipm.ir
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