Teaching

MONSOON-WINTER 2023, MTH 640: SEVERAL COMPLEX VARIABLES

The course content and other logistics are in the first-course-hand-out. Click here for a copy. 

Aug 8, 2023: Homework 1 is out. Check out the above link. 

Aug 24, 2023: Homework 2 is out. Check out the above link. 

Sep 11, 2023: Homework 3 is out. Check out the above link. 

Oct 3, 2023: Homework 4 is out. Check out the above link. 

Nov 5, 2023: Homework 5 is out. Check out the above link. 

The final exam for the course is scheduled on November 24th, from 8 to 11 a.m. in the morning. 

PAST TEACHING:

SPRING 2023, MTH305A: SEVERAL VARIABLE CALCULUS AND DIFFERENTIAL GEOMETRY.

The course content and other logistics are in the first-course-hand-out. Click here for a copy. 

We shall spent enough time on Part A of this course, namely the concept of differentiation in more than one variable. Mostly I shall be following Chapter 9 of Rudin's book in the list of references given in FCH. I may also consult Spivak's book.  

HOMEWORK PROBLEM 1: I shall keep on adding the problems in this set until the set-2 of homework problems is out. The set is accessible via the same link as above. 

Office Hours: Fridays: 2:30 p.m. to 3:30 p.m. and 5:30 p.m. to 6:30 p.m. 

HOMEWORK PROBLEM 2 is out. Check out the above link. 

Reading Assignment 1: Study the contraction principle from Rudin's book for a complete metric space. It says that a self-map of a complete metric space that is a contraction has a unique fixed point. 

HOMEWORK PROBLEM 3 is out. Check out the above link. 

Reading Assignment 2: The assignment is clearly stated in HOMEWORK PROBLEM 3. It is about the equality of the mixed partial derivatives for a twice-continuously differentiable function. Part of it, we shall also discuss in our tutorial sessions. 

Test-1: The first test of the course is scheduled on February 4, 2023 from 3:00 p.m. to 4:00 p.m. in L-2. I have not decided the duration of the test but it will be within the stated time-interval. 

The solutions of the problems in Test-1 could be found here. 

HOMEWORK PROBLEM 4 is out. Check out the above link. 

HOMEWORK PROBLEM 5 is out. Check out the above link. 

The mid-semester  exam. for the course is scheduled on February 25, 2023 from 18:00 to 20:00 in L-16. Problems in the mid-sem. will be asked whatever I have covered upto February 15th, 2023. In the upcoming exam. week, I will be on leave on Monday i.e. February 20th and possibly on February 21st too.

The solutions for the problems in midsem. exam paper is available at the above link.

March 7: HOMEWORK PROBLEM 6 is out. Check out the above link. I shall add more problems in this problem set (definitely by the end of this weekend). In the upcoming test (and the final one), problems will be of the same nature as in this problem set.

April 1: HOMEWORK PROBLEM 7 is out. Check out the above link. I shall add more problems in this problem set. 

April 21: HOMEWORK PROBLEM 8 is out. I could add a few more problems in this set. 

The final exam. for the course is scheduled on May 1st, 2023. 

Fall 2022; MTH670A: Potential Theory in the complex plane

I am teaching an elective course in the fall 2022. The course material for this course is taken from the book by Thomas Ransford. The title of the book and the course name are same. The objective of this course is to study subharmonic functions in great detail and see applications in the function theory of holomorphic functions in the complex plane.

The course begins with a quick introduction of harmonic functions and then moving to subharmonic functions. After studying general properties of subharmonic functions, the course explores logarithmic potentials and corresponding energy functional associated to finite, compactly supported, Borel measures. Potentials are subharmonic functions and in a way all subharmonic functions are actually potentials (Riesz Decomposition Theorem). 

One of the most important notion in the potential theory is that of polarity. A set E is called polar if it does not support any measure (with compact support contained in E) that has nontrivial energy (i.e. other that -infinity). The fundamental theorem of potential theory (Frostman's Theorem) then says that for nonpolar compact sets there is a probablity measure (called Equilibrium measure) that has maximal energy and moreover the potential corresponding to it is bounded from below by the energy and is equal to the energy at almost all points except on a polar set contained in the boundary of the compact set. 

Frostman's theorem is a tool for constructing positive, non-constant subharmonic function with approprite regularity which is a very useful tool, e.g. it can be applied to prove the Brelot--Cartan Theorem that says that the upper semicontinuous regularization of supremum of a family of subharmonic functions is subharmonic and is equal  to supremum of the family nearly everywhere (i.e. except on a polar set). This has important consequences, e.g. the set where a nontrivial subharmonic function on a domain admits values -infinity could be shown to be a polar set. Another useful application of Frostman's theorem is: every closed polar set can be realized precisely as the set where a nontrivial subharmonic function takes values -infinity. This has far-reaching consequences on the extension of subharmonic function through a closed polar set. 

Equipped with all the above tools, finally we address the Dirichlet problem on a general planar domain. Following the method by Oskar Perron, we first show that the maximal subharmonic function that is dominated by a given bounded function on the boundary of a planar domain is harmonic and in the case the function is continuous and the boundary points are regular, Bouligand's lemma paves the way towards the solution of the Dirichlet problem. 

The course will end at this juncture and later our plan is to study the rest of the topics present in Ransford's book on potential theory.