Scheduled speakers for the 2026/27 Virtual CAAGS seminar.
Additional talk titles, abstracts, and lecture materials
will be added as they become available.
October 182026/27
The arithmetic rank and cohomological dimension
of some residual intersections
The arithmetic rank of an affine variety is the minimal
number of equations needed to define it set-theoretically,
that is, the smallest number of polynomials generating
the defining ideal upto radical. Despite this elementary
definition, it is often difficult to compute: the minimal
generators up to radical often bear little relation to
the given ideal generators, and their optimal number can
depend on the characteristic. Local cohomological dimension
provides a lower bound for arithmetic rank, but is itself
a subtle invariant.
Residual intersections generalize the classical notion
of algebraic linkage. In this talk, we determine the
arithmetic rank and cohomological dimension of generic
residual intersections of generic Gorenstein ideals of
height three in any characteristic. Our methods also
reveal structural properties of the top local cohomology
modules supported at these ideals. Although these
residual intersections are not set-theoretic complete
intersections, we will demonstrate that they admit
linear specializations which are.
This is joint work with Vaibhav Pandey and Uli Walther.
Given a ring R and an R-module M, the cohomological
support variety of M is a geometric object that encodes
homological information about M. Work of Briggs, Grifo,
and Pollitz considers the case when R is the quotient
of a monomial ideal I, and M = R. Their work classifies
all varieties that arise when I has 5 minimal generators.
In this talk, we will exhibit monomial ideals with
support varieties that obtain arbitrary codimension
(and admit no embedded deformation) as well as support
varieties that are the union of arbitrarily many
hyperplanes. We will also classify all varieties that
arise when I has 6 minimal generators.
This is based on joint work with Fagerstrom, Faur,
Katz, Stern, and Watson.
The classical inequality of Serre gives an upper
bound for the Poincaré series of the residue field.
Rings attaining this upper bound are called Golod
Rings. In this talk, we discuss some classical
results, recent developments, and some open questions.
A classical problem in enumerative geometry is to
count the number of lines in $\mathbb{P}^3$ meeting
four general lines. Schubert calculus provides a way
to answer this and much more general incidence
questions.
In this talk we answer above stated problem by
reinterpreting lines as points of a Grassmannian
and incidence conditions as Schubert varieties.
This translates a geometric counting problem into
a simple combinatorial calculation.
This talk is mainly prepared from the expository
article “Variations on a Theme of Schubert Calculus”
by Maria Gillespie.
September 62026/27
Congruence modules and Wiles defects of determinantal
rings of maximal minors
I will talk about the computations of congruence modules
and Wiles defects for determinantal rings
$\mathcal{O}[X]/I_t(X)$, where $\mathcal{O}$ is a DVR
and $X$ is a matrix of indeterminates of size
$t \times n$.
The objects were introduced by Khare, Iyengar,
and Manning as higher codimension analogues of
some things that are relevant in number theory.
In my talk, I will not focus on the number theory
aspects but rather the commutative algebra behind
these computations.
This is joint work with Kashif Khan.
August 232026/27
Density functions for the epsilon multiplicity
of homogeneous ideals
Let $(R,\mathfrak m)$ be a Noetherian local ring
of dimension $d$, and let $I\subseteq R$ be an
ideal. Ulrich and Validashti defined the
$\varepsilon$-multiplicity of $I$ as
This invariant may be viewed as a generalization
of the classical Hilbert--Samuel multiplicity.
Cutkosky showed that the $\limsup$ in this
definition can be replaced by an actual limit
when $R$ is analytically unramified. A surprising
example due to Cutkosky, Hà, Srinivasan, and
Theodorescu demonstrates that this limit can be
an irrational number even when $R$ is a regular
local ring.
In this talk, we shall focus on the case of
homogeneous ideals in a standard graded domain
over a field. Motivated by Trivedi's approach to
the Hilbert--Kunz multiplicity via density
functions, we introduce a compactly supported
continuous real-valued function, called the
$\varepsilon$-density function, whose integral
recovers the $\varepsilon$-multiplicity. If time
permits, we will present explicit examples and
discuss applications in the context of integral
closures.
This talk is based on joint works with Roy
and Trivedi.
2025/26 Academic Year
2025/26 Academic Year Talks
Speaker information, talk titles, and abstracts
from the 2025/26 Virtual CAAGS seminar.
Binomial edge ideals, introduced independently by
Herzog, Hibi, Hreinsdóttir, Kahle, and Rauh, and by
Ohtani, provide a bridge between graph theory and
commutative algebra by associating a binomial ideal
to a simple graph G. The binomial edge ideal of G,
denoted by J_G, enables the study of combinatorial
properties of graphs through algebraic invariants
and is closely related to determinantal ideals
and conditional independence ideals arising in
algebraic statistics.
In this talk, we begin by discussing some basic
properties of binomial edge ideals and their Gröbner
bases. We then present the irreducible decomposition
of J_G. Subsequently, we study the symbolic powers
of J_G. Finally, we discuss the conjecture of Ene,
Rinaldo, and Terai concerning the equality between
the symbolic and ordinary powers of J_G, that is,
J_G^{(k)} = J_G^k, together with updates related
to this conjecture.
May 32025/26
Bounds on Bass numbers of local cohomology modules
Introduced in the 1960s in algebraic geometry to
study sheaves and their cohomology, local cohomology
has become a fundamental tool in commutative algebra.
Despite extensive study, its structure remains
poorly understood, primarily because these modules
are rarely finitely generated.
Consequently, investigating their finiteness
properties—such as associated primes and Bass
numbers—has become a central problem toward
understanding their structure. Let R be a polynomial
ring over an uncountable algebraically closed field
of characteristic zero. In our talk, we investigate
upper bounds for the Bass numbers of local cohomology
modules of R. Our approach relies on the theory of
D-modules, which I will briefly recall for a
general audience.
April 122025/26
Free Resolutions of Symmetric Algebras of Ideals
with Deviation Two
For an ideal I in a ring R, the deviation of I is
defined as the difference between its minimal number
of generators and its grade. This invariant plays
a central role in measuring how far an ideal is
from being a complete intersection. In polynomial
rings, ideals of deviation zero and one correspond
to complete intersections and almost complete
intersections, respectively, and their algebraic
and homological properties have been extensively
studied.
In this talk, we focus on ideals of higher positive
deviation, where the structure becomes significantly
more complex. Our primary aim is to study certain
classes of ideals that arise naturally in this
context and to understand their role in the
construction of free resolutions of symmetric
algebras. The symmetric algebra of an ideal
encodes information about its generators and
relations and serves as an approximation to the
Rees algebra. Constructing explicit resolutions
for these algebras is a fundamental problem in
commutative algebra.
April 52025/26
A sharp upper bound on cohomological dimension
in unramified mixed characteristic
Given an ideal I in a regular local ring A,
the cohomological dimension of I in A is the
index of the highest non-vanishing local
cohomology of A supported at I. Determining
effective upper bounds on the cohomological
dimension in terms of topological invariants
of Spec(A/I) is a central problem in commutative
algebra: foundational results include the
Hartshorne--Lichtenbaum Vanishing Theorem and
the Second Vanishing Theorem.
In equal characteristic, Faltings established
in 1980 a general bound on the cohomological
dimension of an ideal in terms of its “big height”.
In this talk, we extend Faltings’ results to
the unramified mixed characteristic setting
and show that the resulting bound is sharp.
The notion of divisors serves as an important
tool for studying geometry on a variety. In this
talk, the concept of Weil divisors and their
relation to fractional ideals will be introduced.
We'll also discuss divisor class groups and
linear systems.
If time permits, I would briefly talk about
Kähler differentials on varieties, and the
divisor of a rational n-form on a variety.
Prerequisites: A first course in Algebraic
Geometry using the classical approach and the
concept of coherent sheaves discussed previously
in Mifron's talk.
March 152025/26
Positivity in Algebraic Geometry - nefness of line bundles
One of the central ways to study projective
varieties and their morphisms is through the
positivity properties that they carry.
Positivity can be understood by studying the
line bundles on the varieties - are they nef?
big? ample? pseudo-effective? These questions
help understand the numerical and geometric
properties of our variety.
In this talk, we will define these terms,
particularly focusing on the nef cone. We
will state some important properties in this
context and see why the nef cone is a useful
enlargement of the ample cone. We will focus
on developing an intuitive understanding via
examples of surfaces. The aim is to give an
accessible entry point into the language of
positivity and to show why the nef cone is
such a useful invariant in algebraic geometry.
March 82025/26
Equations of Kalman Varieties and the Geometric Method
The Kalman variety of a linear subspace in a
vector space consists of all endomorphisms
that contain an eigenvector in that subspace.
In this talk, I will give some basic properties
of these varieties including giving set-theoretic
equations for them. I will then introduce a
powerful method for computing syzygies, known
as the Kempf-Lascoux-Weyman geometric technique,
which I will apply in our set-up to obtain a
minimal free resolution of the normalisation
of the Kalman varieties.
Time permitting, I will briefly talk about how
to obtain syzygies of the Kalman varieties from
the syzygies of its normalisation.
In this talk we will first discuss about
sheafification of a module. We will define
quasi-coherent and coherent sheaves and look
at various constructions of sheaves from
quasi-coherent coherent sheaves. In the end
if there is time we will look at some
theorems about quasi-coherent coherent
sheaves by Serre.
Prerequisites: In this talk I will work with
classical affine and projective varieties
(No schemes). I will also assume that
everyone knows definition of presheaves
and sheaves.
February 8 & 222025/26
Codimension-two Varieties 1:
Cohomology of Vector Bundles and Modules
In this two-part series, we will look at
Hartshorne’s conjecture and some sufficient
conditions under which it holds.
In the first session, we will look at the
prerequisite material, namely the cohomology
of vector bundles over projective space and
the relation between sheaf cohomology and
local cohomology. We will also look at Serre’s
construction for codimension-two local
complete intersections, and how this gives
an equivalent formulation for Hartshorne’s
conjecture.
In the second session, we will state and
prove some sufficient conditions for
Hartshorne’s conjecture. This is based
on a joint work with Manoj Kummini
(arXiv:2511.19962).
Prerequisites: some basic knowledge of
projective varieties and vector bundles.
In particular, sections I.1-I.5, II.1-II.5,
and III.1-III.7 of Hartshorne’s Algebraic
Geometry should be sufficient.
A numerical semigroup S is a finitely-generated
submonoid of the natural numbers with a finite
complement. Semigroups appear across many areas
of maths, including algebraic geometry, number
theory, integer programming, and even music theory.
They provide a nice combinatorial framework to
study otherwise hard-to-compute algebraic properties.
One of these properties is the betti sequence
of a semigroup ring, i.e., the sequence of ranks
of free modules in a minimal free resolution.
In this talk, we will compute various such
invariants of a class of numerical semigroups
called Sally type semigroups. In particular,
we will see how Hochster’s combinatorial formula
can be used to get the minimal number of generators
for the defining ideal I of a semigroup ring R/I,
where R is a polynomial ring, by bypassing all
ideal-theoretic complications.
Consider the action of a subgroup G of S_n
acting on the polynomial ring
S = k[x_1, ..., x_n] by permuting the
variables. Noting that this action does not
really refer to the base field k, one may
ask if the ring of invariants S^G is also
suitably independent of k. For example,
the properties of being Cohen–Macaulay,
Gorenstein, a UFD, F-regular, etc.
One may also ask whether suitable numerical
invariants are independent: the a-invariant,
the Hilbert series of S^G, the Hilbert series
of H^n(S)^G and H^n(S^G).
January 182025/26
Equivalence of strong and weak F-regularity
of positively graded rings
In the study of singularities via positive
characteristic methods, several notions of
F-regularity were introduced in the foundational
work of Hochster and Huneke. While these notions
are central to the field, their equivalence
remains an open problem. We will prove the
equivalence of these notions in the case of
positively graded rings following the work
of Lyubeznik and Smith.
The talk assumes basic knowledge of
Noetherian rings and modules.
November 222025/26
Sheaf Cohomology via Derived Functors and Čech Cohomology
Sheaf cohomology can be defined abstractly
as the right derived functors of the global
section functor, but it can also be computed
concretely using Čech cohomology. In this
talk I will introduce both viewpoints and
explain the mechanism that relates them.
After briefly recalling the derived-functor
definition of Hi(X,F), I will define Čech
cohomology using an open cover and then
discuss the Leray condition, under which
the Čech complex gives an acyclic resolution
of F, and prove that the two cohomology
theories coincide.
The main aim of the talk is to describe an algebra
structure on the minimal free resolution and, as an
application, obtain a structure theorem for grade 3
Gorenstein ideals. The talk is based on the 1977
paper of Eisenbud and Buchsbaum titled: Algebra
Structures for Finite Free Resolutions, and Some
Structure Theorems for Ideals of Codimension 3.
November 82025/26
The arithmetic rank of residual intersections
of a complete intersection ideal
The arithmetic rank of a variety is the
minimal number of equations needed to
define it set-theoretically, i.e., the
smallest number of polynomials generating
the defining ideal up to radical. Computing
this invariant is notoriously difficult:
the minimal generators up to radical often
bear little relation to the given ideal
generators and can vary unpredictably
across characteristics.
Residual intersections provide a natural
extension of the classical notion of algebraic
links. We establish a general upper bound for
the arithmetic rank of any residual intersection
of a complete intersection ideal in an arbitrary
Noetherian ring, and we show that this bound is
sharp under specific characteristic assumptions.
This work is joint with Kesavan Mohana Sundaram,
Taylor Murray, and Vaibhav Pandey.
November 12025/26
Set-theoretic Complete Intersection for Curves
in Affine 3-folds
Let A be an affine 3-fold over a C_1
field of characteristic zero. We show
any smooth curve with trivial conormal
bundle is a set theoretic complete
intersection provided its class in
the Grothendieck group is torsion.
October 182025/26
Integral closure of ideals and their
Castelnuovo--Mumford regularity
A conjecture due to Kuronya--Pintye
states that if I is a homogeneous ideal of
k[x₁,...,xₙ] and J is its integral closure,
then reg(J) ≤ reg(I). In this talk, we will
prove the conjecture for certain classes
of monomial ideals.
The talk is based on the recent work:
A comparison of the regularity of certain
classes of monomial ideals and their
integral closures.
Derived category of a DG Q-Algebra A,
denoted D(A), is obtained by formally
inverting quasi-isomorphisms between DG
A-modules. In this talk, we introduce the
notion of thick subcategory T’ of a
triangulated category T. We will see
some examples and its properties.
The smallest thick subcategory generated
by an object X in T is denoted by Thick(X).
By the work of Dwyer-Greenless-Iyengar,
we will also see that the objects of
Thick(R) are exactly the perfect complexes
in D(R).
Differential graded algebra is a chain
complex which is also a graded algebra,
where the differential in the chain
complex is compatible with the product
structure. In this talk, we introduce
dg-algebras and give some examples.
Then we discuss situations where the
existence of dg-algebra structures
are useful.
The study of the growth of ideals has led
to the discovery of rich families such as
symbolic powers and integral closures,
whose behaviour encodes deep algebraic
and geometric information. In recent years,
convex-geometric methods via Newton
polyhedra and Newton–Okounkov bodies
have provided powerful tools for
understanding these objects.
In this talk, I will introduce these
ideas and illustrate how convex bodies
offer a natural framework for studying
graded families of ideals. In particular,
I will explain how the volume of these
convex bodies captures various
multiplicities of graded families.
A componentwise linear ideal in a polynomial
ring S is an ideal I such that the ideal
generated by each component of I has a linear
resolution. Given two componentwise linear
ideals I and J, we study necessary and
sufficient conditions for I+J to be
componentwise linear. We provide a
complete characterization when dim S=2.
As a consequence, we show that any
componentwise linear monomial ideal in
k[x,y] has linear quotients using
generators in non-decreasing degrees.
When dim S is arbitrary, we describe
how one can build a componentwise
linear ideal from a given collection
of componentwise linear monomial ideals,
satisfying some mild compatibility
conditions, using only sum and product
with square-free monomials. This is
a joint work with Prof. Hailong Dao.
This will be an introductory talk on
the connections between the syzygies
of projective varieties and their
geometry. We will see how Koszul
cohomology can be computed using some
special vector bundles on the variety.
I will point out some open questions
at the end.
This talk will be accessible to anyone
who has had a couple of courses in
commutative algebra. I will define
all the geometric terms involved.
September 62025/26
Symbolic Powers of Edge Ideals and Minh’s Conjecture
We will study symbolic powers of edge ideals
and a conjecture of Minh predicting that
their regularity coincides with that of
the ordinary powers. After a brief review
of symbolic powers and their combinatorial
description via vertex covers, I will try
to explain a proof that the conjecture
holds for complete graphs.
The key idea is to show that the difference
between symbolic and ordinary powers is
small enough to force their regularities
to agree. This talk should be accessible
to anyone who has completed a first course
in commutative algebra and has a basic
familiarity with symbolic powers
and regularity.
August 302025/26
Polynomial invariants of GL₂:
Conjugation over finite fields
Consider the conjugation action of GL2(K)
on the polynomial ring K[X2x2]. When K
is an infinite field, the ring of
invariants is a polynomial ring generated
by the trace and the determinant.
We describe the ring of invariants
when K is a finite field, and show
that it is a hypersurface.
Given a group G acting on a ring R,
we consider the subring R^G, the
subring of elements fixed by G.
It's a natural question to ask
what good properties of R are
inherited by R^G. Some of these
questions were considered by
Hilbert and Noether, and were a
motivation to study Noetherian rings.
We will discuss some of these results.
This talk should be accessible to
someone who has done a first
course in modules.
