February 8 & 15: Abhiram Subramanian (Chennai Mathematical Institute)
Codimension-two Varieties 1: Cohomology of Vector Bundles and Modules
Abstract: 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 (also slightly overkill).
February 22: Mifron Fernandes (University of Missouri)
March 1: Shanea Sebastian (University of Missouri)
March 8: Suhas Vadan Gondi (University of California San Diego)
March 22: Dr. Manohar Kumar (IIT Madras)
April 5: Manav Batavia (Purdue University)
May 3: Pooja Joshi (Texas A&M University)
May 10: Agilan Amirthalingam (University of Kansas)
February 1: Srishti Singh (University of Missouri)
Numerical Semigroups of Sally Type
Abstract: 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 (named after Judith Sally, 1937-2024). 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.
January 25: Aryaman Maithani (University of Utah)
On invariant rings of permutation groups
Abstract: 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 also independent: the a-invariant, the Hilbert series of S^G, the Hilbert series of H^n(S)^G and H^n(S^G). If you would like the answers to these questions, this talk is for you.
January 18: Aditya Dwivedi (IIT Bombay)
Equivalence of strong and weak F-regularity of positively graded rings.
Abstract: 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: Anusree Vellathum Padath (University of Kansas)
Sheaf Cohomology via Derived Functors and Čech Cohomology
Abstract: 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.
November 15: Amit Phogat (IIT Gandhinagar)
Algebra Structures and Grade 3 Gorenstein Ideals
Abstract: 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: Manav Batavia (Purdue University)
The arithmetic rank of residual intersections of a complete intersection ideal
Abstract: 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 upto 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: Lisa Mandal (IISER Kolkata)
Set-theoretic Complete Intersection for Curves in Affine 3-folds
Abstract: 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 it's class in the grothendieck group is torsion.
October 18: Dr. Omkar Javadekar (Chennai Mathematical Institute)
Integral closure of ideals and their Castelnuovo--Mumford regularity
Abstract: A conjecture due to Kuronya--Pintye states that if I is a homogeneous ideal of a k[x₁,...,xₙ] and J is its integral closure, then reg(J) ≤ reg(I). In this talk, we will prove the conjucture for certain classes of monomial ideals.The talk is based on the following recent work: A comparison of the regularity of certain classes of monomial ideals and their integral closures
October 11: Kesavan Mohana Sundaram (University of Nebraska)
The Derived Category of a DG Algebra
Abstract: 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).
October 4: Ganapathy K (IIT Madras)
DG-algebras and some applications
Abstract: Differential graded(dg) 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.
September 27: Aniketh Sivakumar (Tulane University)
Algebraic invariants and Convex bodies
Abstract: 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.
September 20: Sreehari Suresh Babu (University of Kansas)
Componentwise linear ideals from sums
Abstract: 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.
September 13: Agilan Amirthalingam (University of Kansas)
Syzygies of Projective varieties
Abstract: This will be an introductory talk on the connections between the syzygies of projective varieties (of their coordinate rings) 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: Soumyadeep Misra (University of Kansas)
Symbolic Powers of Edge Ideals and Minh’s Conjecture
Abstract: 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: Aryaman Maithani (University of Utah)
Polynomial invariants of GL₂: Conjugation over finite fields
Abstract: 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.
August 23: Aryaman Maithani (University of Utah)
Invariant Theory of Commutative Rings
Abstract: 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's done a first course in module theory.