We meet weekly on Wednesday at 3:00 PM (Central time), GI 310
Organizers: Alessandra Constantini and Kalina Mincheva.
Spring 2025
January 15
Souvik Dey, Charles University, Prague. Rings with extremal cohomology annihilator.
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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.
February 10 (note the special date and place, DW 103)
Paolo Mantero, University of Arkansas. From Interpolation problems to matroids.
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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
Edna Jones, Tulane University. Versions of the circle method.
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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 (note the special date)
Ngo Viet Trung, Institute of Math, Vietnam Academy of Science and Technology. Ear decompositions of graphs: an unexpected tool in Combinatorial Commutative Algebra.
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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
Alessandra Constantini, Tulane University A combinatorial method for the reduction number of an ideal.
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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
Sankhaneel Bisui, Arizona State University. Algebraic Properties of Invariant ideals.
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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 $m \times n$. Consider the group $G = Gl(m) \times Gl(n)$. 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.
March 14 (note the special date)
Dipendranath Mahato, Tulane University. Developments in Interpolation Problem for Projective Spaces.
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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.
March 19
Thai Nguyen, University of Dayton. Geometric vertex decomposition.
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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.
March 26
Michael Allen, LSU. Asymptotic counts of number fields generated by plane curves.
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Every irreducible polynomial $f(x)$ with integer coefficients corresponds uniquely to a field extension of the rational numbers which
consists of $\mathbb{Q}$, a root $\alpha$ of $f$, and all combinations thereof under the standard arithmetic operations.
For example, $f(x) = x^2-2$ produces the field $\mathbb{Q}(\sqrt{2}) = \left\{a + b\sqrt{2} : a, b \in \mathbb{Q}\right\}$.
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 $\mathbb{Q}(\sqrt{3}), \mathbb{Q}(\sqrt{11})$, and $\mathbb{Q}(\sqrt{31})$.
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 Lòpez, and Tian An Wong.
April 9
Andrés Jaramillo Puentes, University of Tübingen. Tropical Methods in Motivic Enumerative Geometry.
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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.