Abstract:
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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Abstract:
Among the current trends of commutative algebra, the role of combinatorics is distinguished. Especially the combinatorics of finite simple graphs has created fascinating research projects in commutative algebra. Let $G$ be a finite simple graph on the vertex set $[n] = \{1,...,n\}$ and $E(G)$ the set of edges of $G$. Recall that a finite simple graph is a finite graph which possesses no loop and no multiple edge together with no isolated vertex. Let $S = K[x_1,...,x_n]$ denote the polynomial ring in $n$ variables over a field $K$. The {\em edge ideal} of $G$ is the ideal $I(G)$ of $S$ which is generated by those monomials $x_ix_j$ with $\{i, j\} \in E(G)$. Since projdim $(S/I(G))$ and reg $(S/I(G))$ determine the size of the Betti table of the graded minimal free resolution of $S/I(G)$, the question of finding the possible pairs of projdim $(S/I(G))$ and reg $(S/I(G))$, where $G$ ranges among all finite simple graphs on $[n]$, is attractive and reasonable. My talk will be a quick survey of [arXiv:2007.14176] with Adam Van Tuyl, {\em et al.}, and of [arXiv:2002.02523] with Huy T\`ai H\`a. No special knowledge will be required to understand my talk.

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Abstract:
We investigate algebraic properties of an exponential counterpart of numerical semigroups from a relative point of view. This approach follows Grothendieck's philosophy on algebraic geometry and commutative algebra, which emphasizes homomorphisms rather than rings. The classical study of numerical semigroup rings is a special case of our relative situation. Indeed, for numerical semigroup $S$ there naturally arise two numerical semigroup algebras: (1)~The ring $\kappa[\![\mathbf{u}^S]\!]$ is an algebra over a Noether normalization $\kappa[\![\mathbf{u}^s]\!]$, where $s$ is a non-zero element of $S$. (2)~The ring $\kappa[\![\mathbf{u}^S]\!]$ serves also as a coefficient ring for the algebra $\kappa[\![\mathbf{u}]\!]$.
Singularities such as Cohen-Macaulayness, Gorensteiness and complete intersection of the ring $\kappa[\![\mathbf{u}^S]\!]$ are in fact properties of the algebra $\kappa[\![\mathbf{u}^S]\!]/\kappa[\![\mathbf{u}^s]\!]$. We may replace the power series ring $\kappa[\![\mathbf{u}^s]\!]$ by an arbitrary numerical semigroup ring $R$ and consider singularities of a numerical semigroup algebra $\kappa[\![\mathbf{u}^S]\!]/R$. See [2,3] for investigations emphasizing algebras over a fixed coefficient ring The purpose of this talk is to clarify notions of the ring $\kappa[\![\mathbf{u}^S]\!]$ that are in fact notions of the algebra $\kappa[\![\mathbf{u}]\!]/\kappa[\![\mathbf{u}^S]\!]$. We will replace the power series ring $\kappa[\![\mathbf{u}]\!]$ by an arbitrary numerical semigroup ring and regard it as an algebra over various coefficient rings [1].
The talk is based on joint work with I-Chiau Huang.

References:
[1] I-C. Huang and R. Jafari, Coefficient rings of numerical semigroup algebras, preprint, 2020.
[2] I-C. Huang and R. Jafari, Factorizations in numerical semigroup algebras, J. Pure Appl. Algebra 223 (2019), no.~5, 2258-2272.
[3] I-C. Huang and M.-K. Kim, Numerical semigroup algebras, Comm. Algebra 48 (2020) no.~3, 1079-1088.

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Abstract:
For a reduced hypersurface $V(f)\subset\Bbb{P}^n$ of degree $d$, the Castelnuovo-Mumford regularity of the Milnor algebra $M(f)$ is well understood when $V(f)$ is smooth, as well as when $V(f)$ has isolated singularities. We study the regularity of $M(f)$ when $V(f)$ has a positive dimensional singular locus. In certain situations, we prove that the regularity is bounded by $(d-2)(n+1)$, which is the degree of the Hessian polynomial of $f$. However, this is not always the case, and we prove that in $\Bbb{P}^3$ the regularity of the Milnor algebra can grow quadratically in d. The talk bases on joint work with Laurent Buse, Alexandru Dimca, and Gabriel Sticlaru.

Reference:
[1] Laurent Buse, Alexandru Dimca, and Hal Schenck, Gabriel Sticlaru, \emph{The Hessian polynomial and the Jacobian ideal of a reduced}, https://arxiv.org/abs/1910.09195.

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Abstract:
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. Interested audiences may look at [1,2] and their references.

References:
[1] S. A. Seyed Fakhari, Homological and combinatorial properties of powers of cover ideals of graphs, in Combinatorial Structures in Algebra and Geometry (D. Stamate, T. Szemberg, Eds), Springer Proceedings in Mathematics and Statistics 331 (2020), 143--159.
[2] S. A. Seyed Fakhari, On the minimal free resolution of symbolic powers of cover ideals of graphs, preprint 2020.

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Abstract:
This talk is about associated primes of powers of an ideal in Noetherian commutative rings. Brodmann proved that the set of associated primes stabilizes for large powers. In general, the number of associated primes can go up or down as the exponent increases. This talk is about ways of computing associated primes and about sequences $\{a_n\}$ for which there exists an ideal $I$ in a Noetherian commutative ring $R$ such that the number of associated primes of $R/I^n$ is $a_n$.
This is a report on four separate projects with Sarah Weinstein, Jesse Kim, Robert Walker, and ongoing work with Roswitha Rissner.

References:
[1] Sarah Weinstein and Irena Swanson, Predicted Decay Ideal, arXiv:1808.09030.
[2] Irena Swanson and Robert M. Walker, Tensor-Multinomial Sums of Ideals: Primary Decompositions and Persistence of Associated Primes, arXiv:1806.03545.
[3] Jesse Kim and Irena Swanson, Many associated primes of powers of primes, arXiv:1803.05456.

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The talk deals with multiplicity based criteria for the integral dependence of two arbitrary ideals $I \subset J$ of an equidimensional and universally catenary Noetherian local ring. We focus on a criterion that uses the multiplicity sequence introduced by Achilles and Manaresi. For the basics, Sections 1.1-1.3, 8.1-8.3 and 11.1-11.3 of the book by Swanson-Huneke, Integral Closure of Ideals, Rings, and Modules could be useful.
This is a report on joint work with Claudia Polini, Ngo Viet Trung, and Javid Validashti [1].

Reference:
C. Polini, N.V. Trung, B. Ulrich, and J. Validashti, Multiplicity sequence and integral dependence, Math.Ann. 378 (2020), 951-969.

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Abstract:
Let $(R,\mathfrak{m}_R)$ be a local ring. An $R$-module (algebra) $B$ is said to be a balanced big Cohen-Macaulay module (algebra) provided every system of parameters of $R$ is a regular sequence on $B$ and $B/\mathfrak{\mathfrak{m}}_{R}B\neq0$. When such a module $B$ is a finitely generated module we say that $B$ is a maximal Cohen-Macaulay module. Although the celebrated conjecture on the existence of balanced big Cohen-Macaulay modules (algebras) has been settled affirmatively since 2016, the conjecture that every complete local ring admits a maximal Cohen-Macaulay module, so-called the Small Cohen-Macaulay Conjecture, is still widely open even in dimension $3$ and even for rings containing a field. In this talk, we report the results of our investigation on the Small Cohen-Macaulay Conjecture in a joint work with Kazuma Shimomoto. For example, we present (and we discuss) the following reduction of the conjecture: Let $(A,\mathfrak{m}_A)$ be a complete regular ring and $X_1,\ldots,X_n$ indeterminates over $A$ and consider the excellent regular local ring $A':=A[X_1,\ldots,X_n]_{(\mathfrak{m}_A,X_1,\ldots,X_n)}$. Let $\mathfrak{p}$ be a prime almost complete intersection ideal of $A'$. If the factorial extended Rees algebra of $A'$ w.r.t. $\mathfrak{p}$ (or its ordinary Rees algebra) admits a graded maximal Cohen-Macaulay module then the Small Cohen-Macaulay Conjecture holds. In other words, roughly speaking, one can say that the Small-Cohen-Macaulay Conjecture reduces to the graded modules over (factorial) blow-up algebras of excellent regular local rings.

References:
[1] Yves Andre, Weak functoriality of Cohen-Macaulay algebras, J. Am. Math. Soc., 33 (2020) no. 2, 363–380.
[2] B. Bhatt, On the non-existence of small Cohen-Macaulay algebras, J. Algebra 411 (2014), 1–11.
[3] W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge, (1998).
[4] M. Hochster, Current state of the homological conjectures, Tech. Report, University of Utah, http://www.math.utah.edu/vigre/minicourses/algebra/hochster.pdf, (2004).
[5] C. Huneke and G. Lyubeznik, Absolute integral closure in positive characteristic, Adv. Math. 210 (2007), 498–504.
[6] H. Schoutens, A differential-algebraic criterion for obtaining a small maximal Cohen-Macaulay module,To appear in Proc. Amer. Math. Soc.
[7] H. Schoutens, Hochster’s small MCM conjecture for three-dimensional weakly F-split rings, Commun. Algebra, 45 (2017), 262-274.
[8] E. Tavanfar, Reduction of the Small Cohen-Macaulay Conjecture to excellent unique factorization domains, Arch. Math. 109, (2017), 429-439.

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Abstract:
Tilting theory is a topic in representation theory of algebras and its fundamental idea is to relate the module categories to their derived categories of two algebras. One of the important results in this theory, the tilting theorem, was proved by Brenner and Butler [1] that states a tilting module $T$ over an algebra $\Lambda$ induces two torsion pairs, one in the category of $\Lambda$-modules and the other one in the category of $End_{\Lambda}(T)$-modules in conjunction with a pair of crosswise equivalences between the torsion and torsion-free classes. The notion of tilting modules was introduced by Happel and Ringel in [3]. Then Rickard [5] introduced the concept of tilting complexes and developed Morita theory for derived categories of module categories. The concept of tilting complexes has been generalized in different directions. One of them is due to Keller and Vossieck [4] and involves the notion of silting complexes. In [2], Buan and Zhou generalized the tilting theorem to a silting theorem for a $2$-term silting complexes over a finite dimensional algebra $\Lambda$. In this talk, we give a relative version of a silting theorem for any abelian category which is a finite $R$-variety for some commutative Artinian ring $R$.

References:
[1] S. Brenner, M.C.R. Butler, Generalizations of the BernsteinGelfandPonomarev reflection functors, in: Representation Theory, II, Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont.,1979, in: Lecture Notes in Math., vol.832, Springer, BerlinNew York, 1980, pp.103-169.
[2] A. B. Buan, Y. Zhou, A silting theorem, J. Pure Appl. Algebra 220 (2016), 2748-2770.
[3] D. Happel, C.M. Ringel, Tilted algebras, Trans. Am. Math. Soc. 274(2) (1982), 399-443.
[4] B. Keller, D. Vossieck, Aisles in derived categories, Bull. Soc. Math. Belg., Ser. A 40 (2) (1988),239-253.
[5] J. Rickard, Morita theory for derived category, J. Lond. Math. Soc. (2) 39 (2) (1989), 436-456.

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