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The CCAC runs a virtual seminar series on recent results in combinatorial commutative algebra. All are welcome to join. Further information is given below. Join our mailing list to learn about upcoming talks.
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Upcoming Talks
Upcoming Talks
To be Determined
To be Determined
(Contact the organizers if you are interested in giving a talk)
DATE: TBD (Fall 2025)
TIME:
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Past Talks
Past Talks
Assistant Professor, Bryn Mawr College
Assistant Professor, Bryn Mawr College
DATE: APRIL 29, 2025
TIME: 11:00 AM Central Time (CS) [12:00 EST]
TITLE: Degree of h-polynomials of edge ideals
ABSTRACT: This talk will focus on the degree of $h$-polynomials of edge ideals for finite simple graphs, focusing on combinatorial formulas for various fundamental graph classes, including paths, cycles, and bipartite graphs. Moreover, we provide a characterization of all connected graphs where the sum of the Castelnuovo-Mumford regularity and the degree of the $h$-polynomial of an edge ideal attains its maximum, which is the number of vertices in the graph. This is joint work with Jennifer Biermann, Augustine O'Keefe, Joseph Skelton and Gabriel Sosa.
Professor, Okayama University
Professor, Okayama University
DATE: MARCH 18, 2025
TIME: 10:30 AM Central Time (CS) [11:30 EST]
TITLE: The local cohomology modules of Stanley--Reisner rings with low codimension
ABSTRACT: Abstract of talk
Tenured Researcher, Vietnam Academy of Science and Technology
Tenured Researcher, Vietnam Academy of Science and Technology
DATE: FEBRUARY 25, 2025
TIME: 9:30 AM Central Time (CS) [10:30 EST]
TITLE: Asymptotic depth of invariant chains of edge ideals
ABSTRACT: We completely determine the asymptotic depth, equivalently, the asymptotic projective dimension of a chain of edge ideals that is invariant under the action of the monoid Inc of increasing functions on the positive integers. Our results and their proofs also reveal surprising combinatorial and topological properties of corresponding graphs and their independence complexes. In particular, we are able to determine the asymptotic behavior of all reduced homology groups of these independence complexes. This is joint work with T.Q. Hoa, D.T. Hoang, D.V. Le and T.T. Nguyen.
Professor, University of Kurdistan
Professor, University of Kurdistan
DATE: JANUARY 28, 2025
TIME: 9:30 AM Central Time (CS) [10:30 EST]
TITLE: Polymatroidal Ideals
ABSTRACT: Let R = K[x_1, . . . , x_n] be a polynomial ring over a field K and let be a polymatroidal ideal. This presentation is going to be about some interesting properties and results of polymatroidal ideals. In particular, our focus includes properties of astab (associated prime stability) and dstab (depth stability) of polymatroidal ideals. additionally, we explore Cohen-Macaulay and unmixed polymatroidal ideals.
Furthermore, we give some examples such that the astab and dstab are unrelated for such monomial ideals and also we give some known problems and conjectures about this subject.
Associate Professor, Ilam University
Associate Professor, Ilam University
DATE: NOVEMBER 26, 2024
TIME: 11:30AM Central Time (CS) [12:30 EST]
TITLE: Ideals with (componentwise) linear powers
ABSTRACT: Let S be a polynomial ring over a field K, and let A be a finitely generated standard graded S-algebra. We discuss a criterion in terms of the initial ideal of the defining ideal of A, which implies that all graded components of A have linear quotients and with additional assumptions are componentwise linear. A typical example of such algebra is the Rees algebra of a graded ideal. Applying this criterion to the Rees algebras of cover ideals of graphs and sortable monomial ideals, we obtain ideals whose powers are componentwise linear.
Post-Doc Fellow, Chennai Mathematical Institute
Post-Doc Fellow, Chennai Mathematical Institute
DATE: OCTOBER 29, 2024
TIME: 11:30 Central Time (CST)
TITLE: CURRENT TRENDS ON THE V-NUMBER
ABSTRACT: In this talk, we will discuss a newly defined invariant of graded ideals known as the v-number. We will explore results regarding the relationship between the v-number and Castelnuovo-Mumford regularity for several classes of graded ideals. We will also see how the v-number can be extended in the case of graded modules. Then, we will analyze the asymptotic behaviour of the v-number corresponding to a Noetherian filtration. Finally, we will conclude with some interesting problems and conjectures on the v-number.
SLIDES: Click here for a copy of the slide.
Professor, Texas State University
Professor, Texas State University
DATE: SEPTEMBER 24, 2024
TIME: 11:30 Central Time (CST)
TITLE: Hilbert Series of Edge Ideals, Property P, and f-vectors
ABSTRACT: Given a square-free monomial ideal I in a polynomial ring R, there is a well-known formula that connecting the h-vector formed by the coefficients of the Hilbert Series of R/I to the f-vector, which counts faces of the Stanley-Reisner complex associated to the ideal. This formula involves an alternating sum and binomial coefficients. This talk focuses on edge ideals of graphs. For such ideals, there are conditions under which the Hilbert series can be more directly computed from the f-vector corresponding to a smaller graph. For instance, if G is the suspension of a graph H, meaning G is formed by adding a “whisker” to each vertex of H, then the h-vector of G is precisely the f-vector of the Stanley-Reisner complex corresponding to H. For more general graphs, edges satisfying Property P from graph theory correspond to regular elements. Under mild conditions, these regular elements can be combined to form a regular sequence. Working modulo this regular sequence allows one to use the f-vector corresponding to a smaller graph to more directly compute the h-vector. When R/I is Cohen-Macaulay and there is a maximum regular sequence of edges satisfying the conditions above, then the h-vector is precisely the f-vector corresponding to this new graph.
Associate Professor, Texas State University
Associate Professor, Texas State University
DATE: April 9, 2024
TIME: 11:30 EST
TITLE: Tropical Type Ideals with an Application to Toric Edge Rings of Bipartite Graphs
ABSTRACT: A tropical hyperplane can be thought of as an affine shift of the normal fan of a simplex in R^d. A collection of such objects decomposes the ambient space into regions that have natural monomial labelings given by "type" data. We study the resulting type ideals, in the process providing minimal cellular resolutions, and also relating certain initial ideals to Stanley-Reisner ideals of root polytopes. We focus especially on applications to toric edge rings of bipartite graphs, where we establish new results regarding regularity, as well as provide new "tropical" proofs of known results. This is joint work with Ayah Almousa and Ben Smith
Assistant Professor, Universidad de los Andes
Assistant Professor, Universidad de los Andes
DATE: March 4, 2024
TIME: 11:30 EST
TITLE: Castelnuovo-Mumford Regularity of Symbolic Powers of Cover Ideals of Graphs
ABSTRACT: Let G be a graph. The cover ideal J(G) of G is the ideal generated by squarefree monomials corresponding to vertex covers of G. One can also define this ideal as the Alexander dual of the edge ideal of G. In this talk we review the recent results about the Castelnuovo-Mumford regularity of 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 >= 2. Here J(G)^(k) denotes the k-th symbolic power of J(G).
Associate Professor, Gebze Technical Univ.
Associate Professor, Gebze Technical Univ.
DATE: February 13, 2024
TIME: 11:30 EST
TITLE: Squarefree Powers of Edge Ideals
ABSTRACT: Let I be a monomial ideal. The k'th squarefree power of I is generated by the squarefree monomials in I^k. In this talk, we discuss some results and conjectures about the regularity and the depth function of squarefree powers of edge ideals.
SLIDES: Click here for a copy of the slides
Professor, Tulane University
Professor, Tulane University
DATE: January 23, 2024
TIME: 11:30 EST
TITLE: Binomial Expansion for Rational Powers and Integral Closures of Sums of Ideals
ABSTRACT: Let A and B be algebras over a field k. Let I and J be nonzero proper ideals in A and B, respectively. We investigate the question of when a binomial expansion for rational powers of I+J (considered as an ideal in A (x)_k B) exists. This question is motivated from the facts that similar expansions were known to exist for symbolic powers and adjoints of I +J.
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