Abstracts

Suprajo Das, Matlis duality, February 2, 9 & 16

Abstract: We plan to give a series of three lectures on topics that are fundamental to the study of local cohomology. The first lecture will focus on injective modules, essential extensions, and injective hulls. The second talk will address injective modules over Noetherian rings and the theory of injective hulls of the residue field of a Noetherian local ring. The final talk will address Matlis Duality over complete local rings. If time permits, we shall also discuss graded Matlis Duality.

Arnab Saha, Introduction to p-adic Hodge theory, February 6

Abstract: We will discuss some of the basics of the theory such as the comparison theorems between Galois representations and filtered isocrystals via the Fontaine functor.  

Akhil Surendran, Prismatic cohomology and some comparison theorems, February 13

Abstract: We will discuss the theory of prismatic cohomology due to Bhargav Bhatt and Peter Scholze. Beginning with the notion of p-derivations, we will go through the construction of prisms and prismatic site, and finally showing some comparison theorems with the known cohomology theories such as de Rham and crystalline cohomology.  

Jugal K. Verma, Hilbert's 14 th problem and Rees' solution to Zariski's conjecture, February 20 

Abstract: I will present Rees’s solution of Zariski conjecture. This conjecture is a generalization of Hilbert’s 14 th problem. Rees used the existence of non torsion points on elliptic curves to construct a non Noetherian symbolic Rees algebra. This algebra turns out to be a counterexample to Zariski's conjecture. 

                This solution gave rise to many developments in commutative algebra. In particular RC Cowsik connected the Noetherian property of the Symbolic Rees algebra to the still unresolved open problem of the set theoretic complete intersection property of algebraic curves in three dimensional affine space.

Jugal K. Verma, An introduction to Gorenstein rings, March 09, 16, 23 & 30

Abstract: Three approaches to Gorenstein rings by Northcott-Rees, Hyman Bass and A. Grothendieck will be introduced. It will be proved that these three approaches are equivalent. Many examples of Gorenstein rings related to algebraic curves, rings of invariants of finite groups and semigroup rings will be discussed. 

Abstract (March 30): I will describe methods of construction of Artinian Gorenstein rings using bilinear forms. I will prove a theorem of F. S. Macaulay about the symmetry of the $h$-vector of the Hilbert series of a standard graded Gorenstein ring. If time permits, I will also prove a result of Daniel Gorenstein about the conductor of the coordinate ring of an algebraic curve. 

Siddhi Balu Ambhore, Binomial Ideals: Structure, Associated Primes, and Primary Decomposition, March 13 & 18

Abstract: We present the main results from the foundational paper "Binomial Ideals" by D. Eisenbud and B. Sturmfels (Duke Math. J., 1996). A binomial ideal is an ideal in a polynomial ring generated by polynomials with at most two terms. We prove that over an algebraically closed field, the associated primes, the radical, and the primary components of any binomial ideal are again binomial.

Sandipan Das, The Atiyah-Weil Criterion for Holomorphic Vector Bundles over Compact Riemann Surfaces, March 20

Abstract:  Unlike smooth vector bundles, a holomorphic vector bundle over a compact complex manifold does not, in general, admit a holomorphic connection. This is fundamentally due to the absence of holomorphic partitions of unity. In this talk, we discuss the necessary and sufficient conditions for the existence of such connections on compact Riemann surfaces. This result, originally established by Atiyah and Weil, is presented here using the modern formulation of Biswas and Raghavendra. 

Pavan Hasmukhbhai Adroja, Introduction to Toric Varieties, March 27

Abstract:  Toric varieties form an important class of algebraic varieties characterised by their connections with combinatorial structures such as fans and convex polytopes. In this talk, we will discuss their construction from combinatorial data and explore some of their geometric and algebraic properties. 

Riddhi Patil, Serre’s Conjecture for Finitely Generated Projective Modules, April 03

Abstract: In this talk, we will discuss the proof of a well-known result in commutative algebra, Serre's Conjecture, also known as the Quillen-Suslin Theorem.