Wednesdays at 3:30 pm in Avery 351

Commutative Algebra Seminar

University of Nebraska ‐ Lincoln

Organizers: Eloísa Grifo and Mark Walker.

The seminar will meet from 3:30 to 5 pm. Talks should aim to be 60 minutes long with ample time after for questions or informal discussion.

2024/2025 academic year

Wednesday, November 6, 2024, 3:30 pm
Sabrina Fowler (UNL)

An Introduction to Gröbner Bases and Their Applications

Abstract: Computations are a powerful tool in commutative algebra, and the foundation of many important computations is Gröbner bases. In this talk I will define the necessary background to understand what Gröbner bases are and go over how to compute them. I will discuss how Gröbner bases can be used to solve fundamental problems, with a focus on The Ideal Membership Problem and The Implicitization Problem.

Wednesday, November 20, 2024, 3:30 pm
Zach Nason (UNL)

Wednesday, December 4, 2024, 3:30 pm
Ben Oltsik (University of Connecticut)

Wednesday, February 5, 2025, 3:30 pm
Rebecca RG (George Mason University)

Wednesday, February 12, 2025, 3:30 pm
Ryo Takahashi (Nagoya University)

Wednesday, March 5, 2025, 3:30 pm
Hal Schenck (Auburn University)

Past talks from the 2024/2025 academic year

Wednesday, August 28, 2024, 3:30 pm
Alexandra Seceleanu (UNL)

Artinian Gorenstein algebras having binomial Macaulay dual generator

Abstract: Every graded artinian Gorenstein ring corresponds via Macaulay-Matlis duality to a homogeneous polynomial, called a Macaulay dual generator. In this way, Macaulay dual generators which are monomials correspond to monomial complete intersection rings. Monomial complete intersections have particularly nice properties: their natural generators form a Gröbner basis, their minimal free resolutions are well understood, and they satisfy the strong Lefschetz property (in characteristic zero).

In this talk we consider Macaulay dual generators which are the difference of two monomials and we seek to understand to what extent the properties listed above still persist for the corresponding artinian Gorenstein algebras. This is joint work with Nasrin Altafi, Rodica Dinu, Sara Faridi, Shreedevi K. Masuti, Rosa M. Miró-Roig, and Nelly Villamizar.

This talk is 95% identical to the one I gave at Notre Dame two weeks before. Anyone who has seen that talk is of course excused.

Wednesday, September 4, 2024, 3:30 pm
Nawaj KC (UNL)

Loewy lengths of modules of finite projective dimension

Abstract: Suppose R is a local Noetherian ring with maximal ideal m. An R-module M has finite length if mⁱ M = 0 for some positive integer i. The minimum i such that mⁱ M = 0 is defined to be the Loewy length of M, denoted ℓℓ(M). Assuming the associated graded ring is Cohen-Macaulay, we will sketch a proof of the following result: if M also has finite projective dimension, then ℓℓ(M) is at least ℓℓ(R/x) where x = x₁, .. , x_d any sufficiently general linear system of parameters on R. This is joint work with Josh Pollitz.

Wednesday, September 11, 2024, 3:30 pm
Sankhaneel Bisui (Arizona State University)

Rational Powers and Summation Formula

Abstract: In commutative algebra, the summation formula is well studied. Mustaţă in 02 proved a summation formula for multiplier ideals, which was generalized by Takagi in 06. In recent work, Hà, Nguyen, Trung, Trung'20, and Hà, Jayanthan, Kumar, and Nguyen proved the summation formula for symbolic powers. Banerjee and Hà proved that the summation formula holds for rational powers of monomial ideals in different polynomial rings. In a recent project, S. Das, T. H. Hà, J. Montaño, and I investigated the summation formula for rational powers. In that project, we introduced the concept of the Rees package, which ensures the summation formula for rational powers. In this talk, I will describe the necessary terms with examples. I will also describe the classes of ideals that satisfy the summation formula for rational powers. I will discuss some results from the joint work with S. Das, T. H. Hà, and J. Montaño.

Wednesday, September 18, 2024, 3:30 pm
Shahriyar Roshan-Zamir (UNL)

Interpolation in weighted projective spaces

There are no pre-requisites for this talk besides some elementary knowledge of algebra, a curious mind and a 55 minute attention span. Everyone is welcome!
Over an algebraically closed field, the double point interpolation problem asks for the vector space dimension of the projective hypersurfaces of degree d singular at a given set of points. After being open for 90 years, a series of papers by J. Alexander and A. Hirschowitz in 1992‐1995 settled this question in what is referred to as the Alexander-Hirschowitz theorem. In this talk, we use commutative algebra to prove analogous statements in the weighted projective space, a natural generalization of the projective space. I will also introduce an inductive procedure for weighted projective planes, similar to that originally due to A. Terracini from 1915, to demonstrate an example of a weighted projective plane where the analogue of the Alexander-Hirschowitz theorem holds without exceptions and prove our example is the only such plane. Furthermore, I will give interpolation bounds for an infinite family of weighted projective planes.

Wednesday, September 25, 2024, 3:30 pm
Jordan Barrett (UNL)

Toward a Zariski-Nagata Theorem for Smooth, Complete Toric Surfaces

Abstract: The Zariski-Nagata theorem is a classical result which expresses the nth symbolic power of a radical ideal I in a polynomial ring over a perfect field in terms of the nth regular powers of the maximal ideals in mSpec(I). In this talk I will discuss a well-known Zariski-Nagata analog for projective varieties. I will also give a brief crash course on toric varieties and their relationship with projective varieties, and I will discuss my recent progress on developing a Zariski-Nagata type theorem for smooth, complete toric surfaces.

Wednesday, October 2, 2024, 3:30 pm
Anna Brosowsky (UNL)

Cartier algebras through the lens of p-families

Abstract: A Cartier subalgebra of a prime characteristic commutative ring R is an associated non-commutative ring of operators on R that play nicely with the Frobenius map. When R is Gorenstein local, its Cartier subalgebras correspond exactly with sequences of ideals called F-graded systems. One special subclass of F-graded system is called a p-family; these appear in numerical applications such as the Hilbert-Kunz multiplicity and the F-signature. In this talk, I will discuss how to characterize some properties of a Cartier subalgebra in terms of its F-graded system. I will further present a way to construct, for an arbitrary F-graded system, a closely related p-family with especially nice properties.

Wednesday, October 9, 2024, 3:30 pm
Krishna Hanumanthu (Chennai Mathematical Institute)

Seshadri constants on blow ups of ruled surfaces.

Abstract: Let X be a projective variety, and let L be an ample line bundle on X. For a point x in X, the Seshadri constant of L at x is defined as the infimum, taken over all curves C passing through x, of the ratios \(\frac{L.C}{m}\), where L.C denotes the intersection product of L and C, and m is the multiplicity of C at x. This concept was introduced by J.-P. Demailly in 1990, inspired by Seshadri's ampleness criterion. Seshadri constants provide insights into both the local behavior of L at x and certain global properties of X. We will give an overview of the current research in this area and discuss some recent results on Seshadri constants of line bundles on blow ups of ruled surfaces.

Wednesday, October 16, 2024, 3:30 pm
Sean Grate (Auburn University)

Betti tables forcing failure of the weak Lefschetz property

Abstract: For most rings, a lot of the data of the ring can be captured via its (minimal) free resolution, in turn summarized by a Betti table. If such a ring is also Artinian, the ring is said to have the weak Lefschetz property (WLP) if multiplication by some linear form is always full rank. Joint with Hal Schenck, we show that if the Betti table of an Artinian algebra has a certain substructure resembling a Koszul complex, then the Artinian algebra cannot have the WLP.

Wednesday, October 23, 2024, 3:30 pm
Lauren Cranton Heller (UNL)

Cellular resolutions of lattice ideals

Abstract: A sublattice of ℤⁿ can be used to construct a binomial ideal in the polynomial ring of n variables. I will discuss a construction of Bayer and Sturmfels which relates the free resolution of such an ideal to the free resolution of a monomial model. In some cases of interest to me the monomial module can then be resolved using a cell complex.

Wednesday, October 30, 2024, 3:30 pm
Adam LaClair (Purdue University)

Koszul Binomial Edge Ideals

Abstract: Koszul algebras are an important family of algebras appearing at the intersection of commutative algebra, combinatorics, and topology, which possess many desirable properties. Given an arbitrary quadratic algebra, it is generally difficult, if not impossible, to determine whether the algebra is Koszul. Thus, finding families of algebras which are Koszul is a challenging and interesting question. In this talk, I will introduce Koszul algebras and present on joint work with Matt Mastroeni, Jason McCullough, and Irena Peeva, where we characterize the Koszul binomial edge ideals in terms of the combinatorics of the underlying graph.

2023/2024 academic year

Note: this is a new page after some issues with the old one; the old seminar announcements will be back online in a few weeks when I manage to restore them to the new format. Thank you for your patience.

Wednesday, December 6, 2023, 3:30 pm
Nikola Kuzmanovski (UNL)

97 years after Macaulay

Abstract: In 1927, Macaulay's paper on lex ideals was published. Macaulay proved that for every homogeneous ideal in a polynomial ring, there exists a nicely structured ideal (called a lex ideal) with the same Hilbert function. This is the fundamental result on Hilbert function and has influenced both algebra and combinatorics throughout the last century. Unfortunately, the fields that Macaulay founded and influenced have developed separately since the publication of the Clements-Lindström Theorem in 1969.
This talk will be about a connection between extremal combinatorics and commutative algebra that will put everything into a single framework. There are several byproducts from this connection. I will deduce old and new results in both fields that give answers to questions posed by Bezrukov, Hoefel, Leck, Mermin and Peeva. Some of these results include other rings in which Macaulay's lex ideal theorem holds, and a classification in this direction.

Wednesday, November 15, 2023, 3:30 pm
Sasha Pevzner (University of Minnesota)

Symmetric group fixed quotients of polynomial rings

Abstract: Let the symmetric group act on the polynomial ring S in n variables via variable permutation. We consider the quotient module M which sets a monomial equal to all of its images under the action. This is a module over the ring of invariants, with relatively little known about its structure. When using integer coefficients, we can embed M as an ideal inside the ring of symmetric polynomials. Doing so gives rise to a family of ideals - one for each n. Localizing at a prime p reveals striking behavior in these ideals, which stay stable (in a sense) as n grows, but jump in complexity each time n equals a multiple of p. In this talk, we will discuss the construction of this family of ideals, as well as a result and conjecture on its structure. The talk will end with a path forward towards solving the conjecture.