Teaching

Approximately two thirds of this course is on Commutative Algebra and one third on Homological Algebra. The goal of the Commutative Algebra part is to sere as an introduction to Algebraic Geometry.

The picture depicts the functoriality of the connecting homomorphism (if one is chasing elements).

functoriality of connecting maps

Syllabus
Lectures (Refresh often to get the latest version of a Lecture)

Homework Assignments

Homework 1 (Solutions here)
Homework 2 (Solutions here)
Homework 3 (Solutions here)
Homework 4 (Solutions here)
Homework 5 (Solutions here)
Presentation Topics

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MATH300 Introduction to Complex Analysis (Winter 2022 at University of British Columbia)

A first course in complex analysis for undergraduates

A picture for the residue theorem

functoriality of connecting maps

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MGE023 Algebraic Geometry-I (Sep-Dec 2021)

The course covered basic scheme theory, cohomology of quasi-coherent sheaves, Serre duality for proper schemes, Riemann-Roch for curves and surfaces, and applications of these techniques.

The desingularisation of a lemniscate obtained by looking at the proper transfrom of the lemniscate by blowing up the plane at the singularity. The red curve is the proper transform of the lemniscate. The z-axis is the exceptional divisor.

integral curve of compactified logistic equation

Prerequisites and co-requisites
Lectures and Tutorials Lectures 4, 5, and from Lecture 17 onwards were by Rupam Karmakar
Supplementary Notes
Homework Assignments

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MGE050 Abelian Varieties (Jan-April 2021)

The course covered the basic theory of abelian varieties via scheme theory. Topics included: Embedding of abelian varieties in projective space, Isogenies, Riemann-Roch for abelian varieties, the index of a line bundle, criterion for ampleness of a line bundle via its Hilbert polynomial, the construction of the dual abelian variety of a given abelian variety.

The picture of an abelian variety of dimension one over the complex numbers

integral curve of compactified logistic equation

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MU2202 Differential Equations

The course covers basic techniques of solving ordinary differential equations (separable, linear with constant coefficients, variation of parameters, etc) and theorems on the existence, uniquenss of solutions, their variation with initial conditions and parameters, rectification of vector fields, one-parameter groups of diffeomorphisms and the associated autonomous differential equations, the equation of variations associated to differential equation whose vector field is C¹, linearisation, and more.

Here is a picture of an integral curve of the logistic equation y'=y(1-y) when the phase space is compactified by a one point compactification of the y-axis. The compactification is the red circle.The blue curve is an integral curve whose initial phase is greater than 1, the thin black line on the right is the line y=∞, the red line the usual time axis y=0, and (faint) the green line on top the line y=1. The blue dot is where the inegral curve crosses y=∞. See Lectures 17 and 18 below.

integral curve of compactified logistic equation

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MU1202 Analysis II (undergraduate)

Topology of Rⁿ: Euclidean, l¹ and l^∞ norms on Rⁿ and the equivalence of convergence of sequences in Rⁿ, open and closed sets, sequential compactness, continuous functions defined on subsets of Rⁿ, boundedness of continuous functions defined on compact sets. Characterisation of open sets in Rⁿ.

Differentiation: Review of inner product spaces and linear maps on inner product spaces, Derivative as a linear map, Chain rule, Matrix representation and and partial derivatives, Comparison of real and complex derivatives, sufficient condition for differentiability, equality of mixed partial derivatives. Taylor's formula and its application to maxima and minima, Inverse function theorem, Implicit function theorem, Tangent space of level sets and gradient, Lagrange multiplier method, diagonalisation of symmetric operators.

Here is a picture of the surface x²+y³ -y²+z²=0 used in Lecture 10 below to illustrate the Implicit Function Theorem.

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MGE094 Rigid Analytic Geometry (Aug-Nov 2019 )

Non-archimedean fields, Tate algebras, affinoid algebras, rigid spaces, GAGA for rigid spaces, Tate's uniformisation of elliptic curves, Mumford's uniformisation theorem.

Here is how ultra-metrics can be visualisd. This is Figure 2 in Jan E. Holly's very readable article Pictures of Ultrametric Spaces, the p-adic Numbers, and Valued Fields, American Mathematical Monthly, October 2001.

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MU2202 Differential Equations (Jan-April 2019)

Please visit the site for this course ( Jan-April 2021) on Differential Equations for files. The files for the 2019 course are no longer available.

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Graduate Analysis I (Aug-Nov 2018)

Measure Theory, basic functional analysis, L^p spaces, Fourier series, Fourier integrals, Fourier inversion formula.

And here are some approximate identities:

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Introduction to Combinatorics MAT344 (Winter 2018 at U of Toronto)

The graph below is crucial to computing Catalan numbers. See Lecture 23.

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Graduate Complex Analysis (Jan-April 2017)

Picture of Schwarz reflections of triangles in the unit disc leading to a proof of the Little Picard Theorem. See Lecture 26 for details.

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Graduate Algebra I (Aug-Nov 2015)

The Notes are divided by topics rather than by lectures. Not everything taught is up on the website.

Below is a picture ocurring in the proof of the Snake Lemma in an abstract abelian category

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Compact Riemann Surfaces (Jan-April 2013)

The course revolved around Homework problems (see the outline), and I haven't scanned my lecture notes for the subject. The HW problems may be of some interest.

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Torsors (Aug-Nov 2012)

The course covered the basics of torsors, regarding them as algebraic geometric analogues of principal bundles. The first few lectures covered the very basics of principal bundles on manifolds. Later the course covered faithful flat descent, fpqc, fpff topologies, schemes as sheaves in the fpqc topology, torsors in this setting, reduction of structure groups in this general setting, as well as fibre bundles associated with a torsor and the action of the underlying group scheme on a scheme.

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Sheaves and Cohomology (Aug-Nov 2010)

The notes are not organised as lectures. But nevertheless the notes cover a reasonable amount of what was taught (the last bits of the course dealt with sheaves on Stein manifolds, and I did not save my hand-written notes).

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Nurture Lectures (1998)

These are eight lectures I gave to the students in the Nurture programme of the National Board of Higher Mathematics (NBHM) in 1998. I am also linking to Homework problems they were asked to do in 1997 and in 1998. The lectures were for 20 talented second year undergraduates in various Indian institutes and universities who had been slected based on their performance as high school students in the Olympiad camp run by the NBHM. This particular cohort (which graudated from high school in 1996) visited the Harish-Chandra Research Institute (then called the Mehta Research Institute) the summers of 1997, 1998, 1999 and 2000. I was asked to teach them analysis. I cannot find my lectures from 1997 or 1999. To keep in touch with them after the summer, they were assigned homework problems, solutions to which they mailed in throughout the year. The students were not necessarily specialising mathematics in the institutes they were affiliated to, and had to find time to work out problems outside the requirements of their official course of study. In any case, they were bright, and eight lectures were enough for them to pick up measure theory (well enough for them to do the homework problems).

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