Some notes on specefic topics/questions.

  • Perverse sheaves & intersection homology : Notes made for the seminar by the same name that ran during Jan-May ‘24 at ISIB by Charanya Ravi. More details here.
  • Strongly local constructions : A common generalization of normalization of an integral scheme and reducing a scheme to its reduced scheme. Both constructions can be done on rings to yield a new ring and they behave well with localization to assist in gluing. We generalize this procedure by only using the essential fact that the said construction commutes with localization.
  • The uniqueness of \(+\)-construction : Classically, higher \(K\)-groups were defined by taking the homotopy groups of \(BGLR^+\) for a ring \(R\). A crucial step is to show that the homotopy type \(BGLR^+\) is unique. While the existence is well-known and can be found easily in standard references, the uniqueness requires basic facts from obstruction theory and follows from its construction. In this note, we show the existence and uniqueness of \(+\)-construction.
  • Main theorems on length : In this note, we prove the main theorem on length of rings (finite length iff artinian, Theorem 6) and modules (finite length iff finite support, Corollary 9), apart from some other often useful results. This is part of my master’s thesis on intersection theory.
  • A generalization of Swan’s theorem : We generalize Swan’s theorem from compact-Hausdorff spaces to Tychonoff spaces (completely regular or \(T_{3\frac{1}{2}}\)). Consequently, we show that on Tychonoff spaces soft bundles and stably trivial bundles are equivalent.
  • Chow rings and applications in enumerative geometry : We develop and study Chow rings for smooth quasi-projective varieties over an algebraically closed field. The text ends with a discussion on geometry of Grassmannian and computation of its Chow ring and uses it to solve an enumerative problem. Something of interest here is an appendix where we develop an analogue of gluable functor in algebraic setting, which allows us to do algebraic operations on geometric vector bundles directly (like \(\hom, \otimes, \wedge, \text{\rm Sym}\), etc.), without invoking equivalence with locally free sheaves. This is my Master’s Project-1 at ISI Bangalore, conducted under Dr. Suresh Nayak during Aug-Nov ‘24.
  • Simplicial sets, realization & the bar-cobar constructions : My ongoing Master’s Project-2 on simplicial sets under Dr. Anita Naolekar at ISIB. Goal is to study homotopy theory of simplicial sets and study some applications in the form of bar-cobar constructions, together with its uses in computing (co)homology of \(\Omega X\) and \(K(G,n)\). This is also part of my presentation in the eCHT Kan Seminar Winter 2025, conducted by Dr. Jack Carlisle. Many thanks to the eCHT group for this oppurtunity.

Expository

Here are some more expository notes which cover larger areas. They are not complete and I update them regularly. I make these so that I can convince myself that I understand the given topic. Please let me know if you find any mistake!

As most of these notes refer to each other, I have combined all the files below in one file here for proper indexing across documents (\(\sim 850\) pages).

  • Algebraic Geometry : Notes made over the years while reading, solving and discussing theory and problems from Hartshorne and other resources.

  • Commutative Algebra : Made mostly simultaneously with the above to understand the algebraic side. Draws mostly from Atiyah-MacDonald and Eisenbud.

  • Foundational Geometry : These notes contain some formal topics about locally ringed spaces, sheaves and atlases, algebra of \(\mathscr{O}_X\)-modules, category of \(\mathscr{O}_X\)-modules, etc. These are useful while discussing schemes and modules over them. Most of these are informed from Wedhorn.

  • Homotopy Theory : Combines course notes from many courses taken at IISERK and ISIB, apart from self study.

  • Algebraic \(K\)-theory of rings : Some notes on lower \(K\)-theory and \(BGL(R)^+\) made during the summer ‘24 visit to IMSc Chennai under Dr. Rahul Gupta. Something of interest here might be a proof of universal property of \(+\)-constructions using obstruction theory, which helps in proving the uniqueness of the homotopy type of \(BGL(R)^+\).

  • Grothendieck Topologies & Topos Theory : These notes were made while studying this topic under Dr. Amit Kuber during the summer of ‘21.

  • Sheaf Theory : Notes on sheaves and their cohomology. These are fairly complete (except the Cěch-to-derived functor spectral sequence) and are used regularly in other parts.

  • Vector Bundles & Characteristic Classes : Notes made while attending the special topics course on the same topic taught by Dr. Manish Kumar. I will continue completing it in the next semester’s seminar on algebraic topology.

  • Homological Methods : Notes on abelian categories and derived functors. I will soon add a section on derived categories as learned from perverse sheaves seminar.

  • Analysis on \(\mathbb{C}\) : Some typical notes for a standard complex analysis course.

  • Riemann Surfaces : Some notes I made for Riemann surfaces seminar at ISIB which ran around the end of complex analysis course. Goal is to reach Riemann-Roch theorem. These are far from complete.

  • Abstract Analysis : Notes on measure theory and functional analysis which mostly just contains solutions to many exercises solved during the course. Made while taking the courses under the same name at ISIB.

  • Nine Weeks of Fields & Galois Theory : Solutions to assignment problems for advanced group theory, fields and Galois theory for the Algebra-II course at ISIB.

  • Problems on Vector Bundles & Characteristic Classes : Solutions to weekly assignment problems for the special topics course on this topic. Most problems are from Milnor & Stasheff.

  • Ordinary Differential Equations : Notes for the course by the same name at IISERK. Contains an exposition of initial value problems and linearization of autonomous systems.