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

March 19

Title:  Geometric vertex decomposition
Speaker: Thai Nguyen - University of Dayton

Abstract: Geometric vertex decomposition is a useful technique in various algebro-geometric contexts such as liaison theory and Groebner bases theory. It is a degeneration technique that was first used in the work of Knutson-Miller-Yong to study Schubert determinantal ideals. It can also be thought of as an ideal-theoretic generalization of vertex decomposition of simplicial complexes. It was shown in the work of Klein-Rajchgot that geometrically vertex decomposable (gvd) ideals possess various nice algebraic properties as those of the Stanley-Reisner ideal of vertex decomposable simplicial complexes. In this talk, I shall survey some results using this technique. I shall also discuss some homological invariants of gvd ideals, with emphasis on toric ideals of graphs. The talk will include results from my project with Jenna Rajchgot and Adam Van Tuyl.


Location:  Gibson Hall, room 310

March 26

Title:  Asymptotic counts of number fields generated by plane curves
Speaker: Michael Allen - LSU Host: (Olivia Beckwith)

Abstract: Every irreducible polynomial f(x) with integer coefficients corresponds uniquely to a field extension of the rational numbers which consists of Q, a root x of f, and all combinations thereof under the standard arithmetic operations. For example, f(x) = x^2-2 produces the field Q(sqrt{2}) = {a + bsqrt{2} : a, b \in Q}. If f is a polynomial in two or more variables, we can produce infinitely many such fields corresponding to solutions to f=0. For f(x,y) = y^2-x^3-x-1, we have solutions (1, \sqrt{3}), (2, \sqrt{11}), (3, \sqrt{31}) and so on, and so the curve defined by f(x,y)=0 ``generates" the fields Q(\sqrt{3}), Q(\sqrt{11}), and Q(\sqrt{31}).

Recently, Mazur and Rubin suggested using this algebraic information as a means to study the geometric properties of a curve. One can easily ask the reverse question: ``If we know something about a curve C, what can we say about the fields that it generates?" We approach this question through the lens of arithmetic statistics by counting the number of such fields with bounded size---under some appropriate notion of size---for an arbitrary fixed plane curve C. This is joint work in progress with Renee Bell, Robert Lemke Oliver, Allechar Serrano L\'{o}pez, and Tian An Wong.


Location:  Gibson Hall, room 310

April 9

Title:  Tropical Methods in Motivic Enumerative Geometry
Speaker: Andrés Jaramillo Puentes - University of Tübingen

Abstract: Over the complex numbers the solutions to enumerative problems are invariant: the number of solutions of a polynomial equation or polynomial system, the number of lines or curves in a surface, etc. Over the real numbers such invariance fails. However, the signed count of solutions may lead to numerical invariants: Descartes' rule of signs, Poincaré-Hopf theorem, real curve-counting invariants. Since many of this problems have a geometric nature, one may ask the same problems for arbitrary fields.

Motivic homotopy theory allows to do enumerative geometry over an arbitrary base, leading to additional arithmetic and geometric information. The goal of this talk is to illustrate a generalized notion of sign that allows us to state a movitic version of classical theorems like the Bézout theorem, the tropical correspondence theorem and a wall-crossing formula for curve counting invariants for points in quadratic field extensions.

Location:  Gibson Hall, room 310

April 23

Title:  A modular framework for generalized Hurwitz class numbers
Speaker: Andreas Mono - Vanderbilt University Host: (Olivia Beckwith)

Abstract: We discover a neat linear relation between the mock modular generating functions of the level 1 and level N Hurwitz class numbers.  This relation gives rise to a holomorphic modular form of weight 32 and level 4N for N > 1 odd and square-free. We extend this observation to a non-holomorphic framework and obtain a higher level analog of Zagier’s Eisenstein series as well as a preimage under the ξ-operator. All of these observations are deduced from a more general inspection of the weight 12 Maass–Eisenstein series of level 4N at its spectral point s = 34 . This idea goes back to Duke, Imamo¯glu and Tóth in level 4 and relies on the theory of so-called sesquiharmonic Maass forms. Furthermore, we connect the aforementioned results to a regularized Siegel theta lift as well as a regularized Kudla–Millson theta lift for odd prime levels, which builds on earlier work by Bruinier, Funke and Imamo¯glu. 

This is joint work with Olivia Beckwith.

Location:  Gibson Hall, room 310

April 30

Title:  Integral Riemann-Roch for Q
Speaker: Katia Consani - Johns Hopkins University

Abstract: I will present a joint result with A. Connes showing the existence of an integral, Riemann-Roch formula for the number field Q. At the root of this result is the characterization of the integers as a polynomial ring $\mathbb{F}_1[X]$, where the `variable' X fulfills the relation $1+1=X+X^2$.

Location:  Gibson Hall, room 310