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.
November 8: Manav Batavia (Purdue University)
November 15: Muktai Desai (University of Missouri)
November 22: Anusree Vellathum Padath (University of Kansas)
November 29: Amit Phogat (IIT Gandhinagar)
December 6: Mifron Fernandes (University of Missouri)
December 13: Aditya Dwivedi (IIT Bombay)
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.