Seminars and Presentations

These notes arose as a part of my preparation for a miniseries of two lectures at ISI, Kolkata. A list of references is given at the beginning of the first set of notes. Broadly, I followed the books Cohen-Macaulay Rings by Bruns and Herzog, Commutative Ring Theory by H. Matsumura, Commutative Algebra with a View Towards Algebraic Geometry by D. Eisenbud, and Basic Commutative Algebra by B. Singh.

Lecture 1: Introduction to Gorenstein Rings and Matlis Duality

Elaborated notes on Gorenstein Rings, Injective Hulls and Matlis Duality

Lecture 2: Dualising Modules

The following are lecture notes that arose from the weekly presentations on this topic as a part of my coursework under the supervision of Prof. Mahuya Dutta at ISI, Kolkata. The references I followed to prepare the lectures are Differential Forms in Algebraic Topology by R. Bott and L. W. Tu, From Calculus to Cohomology by I. Madsen and J. Tornehave, Differential Topology by V. Guillemin and A. Pollack, Connections, Curvature and Cohomology (Vol-1) by W. Greub, S. Halperin and R. Vanstone and Foundations in Differentiable Manifolds and Lie Groups by F. W. Warner. The links to the lectures are given below.

Lecture 1: Introduction to de Rham cohomology and de Rham cohomology with compact supports on Euclidean spaces

Lecture 2: Geometric realisation, pullbacks of forms, functoriality, and Poincare lemma for star shaped open euclidean subsets

Lecture 3: The Mayer-Vietoris sequence and its consequences, smooth approximation theorem and homotopy invariance, computations, the Brouwer's fixed point theorem, and the Hairy Ball theorem

Lecture 4: The Jordan-Brouwer separation theorem, Invariance of domain and dimension of manifolds, some computations, introduction to de Rham cohomology on Manifolds, Mayer-Vietoris sequence, corresponding theory for compactly supported cohomology on manifolds, and some computations

Lecture 5: Orientability, Integration on Manifolds, generalised Stokes's theorem, and generalised Poincare lemma

Lecture 6: Homotopy invariance of de Rham cohomology on Manifolds, Poincare lemma for compact cohomology, degree of a proper map, good covers and introduction to Mayer-Vietoris argument, finite dimensionality of de Rham cohomology and compact cohomology

Lecture 7: Poincare Duality (for orientable Manifolds of finite type and of general type), Betti numbers, signature and Euler characteristic, and properties of the degree of maps between compact orientable manifolds

Lecture 8: Introduction to fiber bundles, Kunneth formula, and Leray-Hirsch theorem

Lecture 9: Kunneth formula for compact cohomology, closed and compact Poincare dual of a closed oriented submanifold, and their examples

Here are some additional discussions on Multilinear Algebra which act as a prerequisite to the discussions from Lecture 2 onwards.

Lecture 1 

Lecture 2 

Lecture 3 

Lecture 4 

The lecture notes on Classifying Spaces arose out of a seminar held at ISI, Kolkata, on December, 2022, under the supervision of Prof. Samik Basu. This notes consists of some background material on Principal G-bundles and explains the Milnor join construction, establishing the existence of classifying spaces corresponding to any topological group G. I have followed Stephen A. Mitchell's notes on principal bundles and classifying spaces, Bachelor thesis of Bouke Jansen on Principal bundles, J. Milnor's papers named Construction of Universal Bundles I and II, Tammo tom Dieck's book on Algebraic Topology as a part of the preparation of the lecture notes. The link to the lecture is here.