CA Seminar for Spring 2024:

In Spring 2024, the Purdue CA seminar will be held in SCHM 307 at 1:30 pm on Wednesdays. Please contact me in case you wish to give a talk in our seminar.

(January 31) Justin Fong, Purdue University.

Title: The F-pure Threshold of Flag Varieties

Abstract: The F-pure threshold (FPT) of an ideal is a numerical invariant of a ring thought of as the positive characteristic counterpart of another numerical invariant, the log canonical threshold, in the setting of characteristic zero. Both provide a measurement of singularities.  In this talk, I will explain my work in determining the values of FPTs for the homogeneous coordinate rings of some classes of flag varieties, with respect to special embeddings into projective space. A flag variety in the most general sense is the homogeneous space G/Q, where G is a reductive group and Q is a parabolic subgroup. A special class of flags, called minuscule flag varieties, have coordinate rings with the structure of an algebra with straightening laws (ASL), which is a type of ring with an underlying poset. I use the properties of an ASL to calculate the fpt of minuscule flags in terms of these posets. For more general classes of flags that are not connected to ASLs, I extend my approach in determining their FPT via the root systems associated to the algebraic group G defining the flag. 

(February 7) Irena Swanson, Purdue University

Title: Primary decomposition in algebraic statistics

Abstract: Conditional independence statements in probability and statistics can be expressed as determinants of certain 2 x 2 matrices.  For certain conditional independence models on n discrete random variables, this independence can be expressed as the ideal generated by 2 x 2 minors along various two-dimensional slices of a corresponding n-dimensional hypermatrix. The primary decomposition structure of these ideals captures various probabilistic properties of the model.

I will describe my old joint work with Amelia Taylor that generalizes the work of Fink and is related to the work of Herzog, Hibi, Hreinsdottir, Kahle, and Rauh; the latter work was presented in the seminar in the fall by Jayanthan A. V.  We relate the structure of minimal primes to certain combinatorial sets, and we interpret these primes via Segre embeddings.

(March 20) Adam LaClair, Purdue University

Title: 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 difficult if not impossible to determine whether the algebra is Koszul. Hence it is a challenging and interesting question to give families of algebras which are Koszul. In this talk I will introduce Koszul algebras and present on joint work with Mastroeni, McCullough, and Peeva where we characterize the Koszul binomial edge ideals in terms of the combinatorics of the underlying graph.

(March 27) Javid Validashti

Title: Combinatorial Interpretations of Multiplicities

Abstract: The classical Hilbert-Samuel multiplicity of an ideal of finite colength in a Noetherian local ring provides an algebraic framework for studying intersection numbers. Two generalizations, namely the $j$-multiplicity and the $\eps$-multiplicity, extend these concepts to ideals without the restriction of finite colength. In this presentation, I will describe how these invariants manifest in various combinatorial structures

(April 3) Taylor Murray, University of Nebraska-Lincoln

Title: Graded Local Cohomology and Graded Bass Numbers

Abstract: Local Cohomology modules have been an indispensable and powerful tool since being introduced by Alexander Grothendieck. However, with great power comes great (big) modules; these modules are rarely non-zero and finitely generated. Therefore, we may ask: when do local cohomology modules have finite Bass numbers? First, we will survey a few results in the literature that address this question. Then, in the case R is a standard graded, finitely generated K-algebra, we investigate how to utilize the graded structure to obtain information about the Bass numbers of Veronese submodules of local cohomology modules.

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CA seminar for Fall 2023:

(August 30) Adam LaClair, Purdue University.

Title: Castelnuovo—Mumford Regularity of Binomial Edge Ideals

Abstract: To any simple graph $G$ is associated the binomial edge ideal $J_{G}$. It is an interesting question to understand commutative algebra invariants of $J_{G}$ in terms of the combinatorics of $G$. In this talk we take up the question of understanding the Castelnuovo—Mumford regularity of $J_{G}$, $\text{reg}(J_{G})$, combinatorially. In this talk we discuss how Conca—Varbaro's breakthrough theorem on preservation of extremal Betti numbers for ideals having a squarefree initial ideal together with combinatorial lower bounds for squarefree monomial ideals can be used to give a new combinatorial lower bound for $\text{reg}(J_{G})$. Time-permitting we also discuss the ideas going into the proof that for various nice classes of graphs equality is attained between this combinatorial invariant and $\text{reg}(J_{G})$.

(September 7, 14, 21) lecture series by A.V.Jayanthan, IIT, Madras.

Title: Depth and regularity of binomial edge ideals in terms of combinatorial invariants.

Abstract: Binomial edge ideals are homogeneous ideals in polynomial rings generated by quadratic binomials which are in one-to-one correspondence with edges of a graph. In this series of three seminars, I will talk about the relation between two important invariants, namely, depth and regularity, of binomial edge ideals and their connection with the combinatorial invariants of the associated graph. I will also talk about interesting open questions in this direction. I will make efforts to make the lectures accessible to graduate students (with some basic commutative algebra background). 

(September 28) Trung Chau, University of Utah.

Title: F-regularity of algebraic sets related to commutator matrices

Abstract: The commutator matrix of two square matrices of the same size X and Y is defined to be XY - YX. In this talk, we will prove that the algebraic sets defined by the anti-diagonal and cross diagonal of the commutator matrices are F-regular, proving a conjecture by Kadyrsizova and Yerlanov. 

(October 4) Cheng Meng, Purdue University.

Title: h-function of local rings of characteristic p

Abstract: For a Noetherian local ring R of characteristic p, we will study a multiplicity-like object called h-function. It is a function of a real variable s that estimates the asymptotic behavior of the sum of ordinary power and Frobenius power. The h-function of a local ring can be viewed as a mixture of the Hilbert-Samuel multiplicity and the Hilbert-Kunz multiplicity. In this talk, we will prove the existence of h-function and mention the properties of h-function, including differentiability, additivity, and behavior under ring maps. We will express Taylor’s s-multiplicity, Trivedi’s Hilbert-Kunz density function, and Mukhopadhyay’s Frobenius-Poincaré function as functionals of h-function and show that they exist in more general settings. We will also indicate how h-function recovers other invariants in characteristic p, including the Hilbert-Kunz multiplicity, the F-signature, and the F-threshold.

(October 18) Takumi Murayama, Purdue University

Title: Uniform bounds on symbolic powers in regular rings via closure theory

Abstract: The containment problem asks: For a fixed ideal I, which symbolic powers of I are contained in an ordinary power of I? We present a closure-theoretic proof of the theorem which says that for ideals I in regular rings R, there is a uniform containment of symbolic powers of I in ordinary powers of I.

(October 25) Cheng Meng, Purdue University

Title: Properties of h-function and invariants in characteristic p

Abstract:

We will talk about properties of h-functions of local rings of characteristic p. There are two kinds of properties of h-functions: one is the pointwise property that is similar to the properties of other multiplicities, including dimension-detecting, additivity, associative formula, and invariant under taking closure. The other is functional property, including m-adic continuity, Lipschitz continuity and smoothness. We will also talk about the asymptotic behavior of h-functions for large s and small s, which shows how the h-function recovers other invariants in characteristic p. We use h-function to prove a weaker version of an inequality appearing in a conjecture by Huneke, Mustata, Takagi and Watanabe.

(November 1) Dan Bath, KU Leuven

Title: Hyperplane Arrangements Satisfy (un)Twisted Logarithmic Comparison Theorems

Abstract: In 1977 Terao conjectured that hyperplane arrangements satisfy the Logarithmic Comparison Theorem (LCT). This says that the logarithmic de Rham complex along the arrangement computes the cohomology of the complement of the arrangement. By the comparison Theorems of Grothendieck and Deligne, this amounts to saying: there is a quasi-isomorphism between the rational de Rham complex with poles of arbitrary order along the arrangement and a subcomplex with poles of order at most one.

We prove this conjecture, as well as a twisted version. The twisted version is the same except the constant local system is replaced by an arbitrary rank one local system, and the logarithmic de Rham complex is given a twisted differential.​ Whereas resonance varieties and the Orlik--Solomon algebra are blind to some of these local systems, our twisted Logarithmic Comparison Theorem sees all local systems. Moreover, any computation is finite-dimensional linear algebra.

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We will sketch our first proof of these results, which is mostly from the world of commutative algebra. Time permitting we may discuss some D-module applications as well as gesture at a second D-module theoretic proof.

(November 15) Arvind Kumar, New Mexico State University

Title: F-threshold of filtration of ideals

Abstract: The F-threshold of I-adic filtration was introduced by Mustata, Takagi, and Watanabe in F-regular rings. Since then, this invariant has been explored by many authors. In this talk, I will define the F-threshold of filtration of ideals. Then, I will concentrate more on the F-threshold of symbolic power filtration and its connection with symbolic F-split. This talk is based on joint work with Dr. Mitra Koley.

(November 29) Jiamin Li, University of Chicago-Illinois

Title: Generic linkage and generalized Frobenius powers

Abstract: Singularity problem of pairs is an important topic in the study of commutative algebra and algebraic geometry. The theory of generic linkage allows us to produce new pairs from the existing ones, and it is natural to ask how the singularities behave under this operation. In this talk we will discuss the application of generalized Frobenius powers by Hernandez, Teixeira and Witt in positive characteristic to this setting, from which we deduce the relation of log canonical thresholds of a pair and its generic linkage, which was previously proved by Niu.