Research Seminars: Algebra and Combinatorics

January 15

Title:  Rings with extremal cohomology annihilator

Speaker: Souvik Dey - Charles University, Prague Host: (Dipendranath Mahato, Tai Ha)

Abstract: The cohomology annihilator of Noetherian algebras was defined by Iyengar and Takahashi in their work on strong generation in the module category. For a commutative Noetherian local ring, it can be observed that the cohomology annihilator ideal is the entire ring if and only if the ring is regular. Motivated by this, I will consider the question: When is the cohomology annihilator ideal of a local ring equal to the maximal ideal? I will discuss various ring-theoretic and category-theoretic conditions towards understanding this question and describe applications for understanding when the test ideal of the module closure operation on cyclic surface quotient singularities is the maximal ideal.

Febuary 10

Title:  From Interpolation problems to matroids

Speaker: Paolo Mantero - University of Arkansas - Host: (Alessandra Costantini)

Abstract: Interpolation problems are long-standing problems at the intersection of Algebraic Geometry, Commutative Algebra, Linear Algebra and Numerical Analysis, aiming at understanding the set of all polynomial equations passing through a given finite set X of points with given multiplicities.

In this talk we discuss the problem for matroidal configurations, i.e. sets of points arising from the strong combinatorial structure of a matroid. Starting from the special case of uniform matroids, we will discover how an interplay of commutative algebra and combinatorics allows us to solve the interpolation problem for any matroidal configuration. It is the widest class of points for which the interpolation problem is solved. Along the way, we will touch on several open problems and conjectures.

The talk is based on joint projects with Vinh Nguyen (U. Arkansas).

February 12

Title:  Versions of the circle method
Speaker: Edna Jones - Tulane University

Abstract: The circle method is a useful tool in analytic number theory and combinatorics. The term "circle method" can refer to one of a variety of techniques for using the analytic properties of the generating function of a sequence to obtain an asymptotic formula for the sequence. We will discuss different versions of the circle method and some results that can be obtained by using the circle method.
 

February 21

Title:  Ear decompositions of graphs: an unexpected tool in Combinatorial Commutative Algebra
Speaker: Ngo Viet Trung - Institute of Mathematics, Vietnam Academy of Science and Technology Host: (Tai Ha)

Abstract: Ear decomposition is a classical notion in Graph Theory. It has been shown in [1, 2] that this notion can be used to solve difficult problems on homological properties of edge ideals in Combinatorial Commutative Algebra. This talk presents the main combinatorial ideas behind these results.
[1] H.M. Lam and N.V. Trung, Associated primes of powers of edge ideals and ear decompositions of graphs, Trans. AMS 372 (2019)
[2] H.M. Lam, N.V. Trung, and T.N. Trung, A general formula for the index of depth stability of edge ideals, Trans. AMS, to appear.
 

February 26

Title:  A combinatorial method for the reduction number of an ideal
Speaker: Alessandra Costantini - Tulane University

Abstract: In the study of commutative rings, several algebraic properties are captured by numerical invariants which are defined in terms of ideals and their powers. Among these, of particular relevance are the reduction number and analytic spread of an ideal, which control the growth of the powers of the given ideal for large exponents. Unfortunately, these invariants are usually difficult to calculate for arbitrary ideals, and different methods might be required depending on the specific features of the class of ideals under examination.

In this talk, I will discuss a combinatorial method to calculate the reduction number of an ideal, based on a homological characterization in terms of the regularity of a graded algebra. This is part of ongoing joint work with Louiza Fouli, Kriti Goel, Haydee Lindo, Kuei-Nuan Lin, Whitney Liske, Maral Mostafazadehfard and Gabriel Sosa.

 

March 12

Title:  Algebraic Properties of Invariant ideals. 
Speaker: Sankhaneel Bisui - ASU

Abstract: Let R be a polynomial ring with mn many indeterminate over complex numbers. We can think of the indeterminates as a matrix, X of size m x n.

  
Consider the group G = Gl(m) x Gl(n). Then G acts on R via the group action (A,B)X =AXB^{-1}. In 1980, DeConcini, Eisenbud, and Procesi introduced the ideals that are invariant under this group action.

 
In the same paper, they described various properties of those ideals, e.g., associated primes, primary decomposition, and integral closures.  In recent work with Sudipta Das, Tài Huy Hà, and Jonathan Montaño, we described their rational powers and proved that they satisfy the binomial summation formula. In an ongoing work, Alexandra Seceleanu and I  are formulating symbolic properties of these ideals. In this talk, I will describe these ideals and the properties we are interested in. I will also showcase some results from my collaborations. 

Location:  Gibson Hall, room 310

March 14

Title:  Developments in Interpolation Problem for Projective Spaces
Speaker: Dipendranath Mahato - Tulane University

Abstract: Classical Interpolation problem of estimating new data from a set of known data is well understood under one variable situation. Here we are more interested in higher dimensional Projective Spaces, where we are trying to find the lowest possible degree of the hyper-surface passing through a given set of points with prescribed multiplicity. There are famous conjectures to tackle such problems: Chudnovsky’s Conjecture, Demailly’s Conjecture. I will be discussing those conjectures and some recent developments in this area.

Location:  Gibson Hall, room 310