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 3
2025/26
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 12
2025/26
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 5
2025/26
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 15
2025/26
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 8
2025/26
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 & 22
2025/26
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 18
2025/26
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 22
2025/26
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 8
2025/26
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 1
2025/26
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 18
2025/26
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 6
2025/26
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 30
2025/26
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.
