Operator Theory and It’s Applications

This course had a significant influence on my point of view on lot’s of different problems, from study of different ODE/PDE’s to even various kinds of “Optimization” problems, because technically all of those concepts (i.e. ODE/PDE’s or Optimization problems) are certain operators from one space to another. Below is details of what we covered in that course:

During this course we first had a review of Hilbert and Banach Spaces, then we fully studied different kinds of operators such as “Linear”, “Bounded”, “Compact”, “Hermitian”, “Self-Adjoint” and “Unitary” Operators. In the second part of the course we reviewed several versions of “Spectral Theorem” with proofs for it.

In the last part of the course we first studied several kinds of unbounded operators such as “Closed and Closeable, Symmetric and Self-Adjoint” unbounded operators and after that we learned Spectral and Bloch Theory for unbounded operators and the spectrum of periodic Schrodinger operators.

Functional Analysis

I made a strong foundation for my mathematical analysis through this course together with “Real Analysis and Measure Theory”. In the first part of the course we learned “Banach Spaces” , “Strong, Weak and Weak* Topologies”.

After that, we went through three important theorems which were “Hahn-Banach, Open Mapping and Closed Graph” Theorems. Hilbert spaces was the next topic we covered and at the end we covered two important theorems “Spectral Theory of Bounded Operators” and “Theory of Distributions”.

Complex Analysis

The importance of the course “Complex Analysis” for me was due to the fact that later on, I was able to apply it’s concepts in order to develop my theorem in “Fluid Dynamics” where there, I explained “The Behavior of Newtonian Fluids in Multiply Connected Domains”.

In this course we first reviewed “Continuous and Holomorphic Functions” on the complex plane and integration along the curves. In the second part we reviewed “Cauchy’s Theorem and it’s applications” especially on evaluation of some integrals. In the third part of the course we reviewed “Meromorphic Functions and the Logarithm” included “Homotopies and Simply Connected Domains”, “The Complex Logarithm” and “Fourier Series and Harmonic Functions”.

Finally in the last part of the course, we learned conformal mappings included “The Schwarz Lemma”, “The Riemann Mapping Theorem” and “Conformal Mappings onto Polygons”.

Real Analysis (Measure Theory & Integration)

I built a strong foundation for my mathematical analysis through this course together with the functional analysis course. At first we defined “Sigma Algebra and Measures”, then we defined “Integration” based on those definitions. In the next part we defined “Convergence of Functions”. After that we learned “Radon-Nikodym Theorem” and “Introduction to Lp Spaces”.

Linear Optimization

During the course “Linear Optimization” I got familiar with different methods of optimizing systems of linear inequalities with multiple variables. In particular, I got familiar with “Fundamental Theorem of Linear Programming”, “Weak Duality Theorem”, “Strong Duality Theorem”, “Complimentary Slackness Theorem”, “Economic Interpretation of Dual Variables” and “Karush-Kuhn-Tucker Conditions”.

Background in Science and Engineering

I believe my multiple background in science and engineering can be extremely useful for any team project and I would able to use my mathematical background to look at the project from a general point of view, find and create unique and specific solutions for it using my background in engineering and bring the project to the next level.

During my undergrad studies, I had the experience of looking at the techniques we learned in our engineering courses from an abstract point of view and even sometimes find a mathematical proof for them. Through practicing that method and by getting experienced in it, finally in my master thesis in mathematics, I was able to develop a mathematical theorem which was explaining the behavior of Newtonian’s flow in certain situation.

I believe that result, can totally explain my ability in looking at any project from a abstract point of view, generalize the problem and finally predict the solution for any problem in similar situations.

Complex Fluid (Math 559)

In this course we had an overview of Non-Newtonian Fluid Dynamics, and discussed two approaches for building constitutive models for complex fluids: continuum modeling and kinetic-microstructural modeling. We also learned about “multiphase complex fluids” and “numerical models and algorithms for computing complex fluid flows”. Through this course, I got a deep understanding of the fact that how we can apply different mathematical models for certain types of physical problems.

 

Master Thesis

Important Result:

Deriving a theorem in Fluid Dynamics which explains how boundary conditions would affect behavior of Newtonian fluids and obtaining numerical results using accurate numerical schemes.

Numerical Schemes Used in Thesis:

Spectral Methods, Compact finite differences (The most accurate scheme applicable to the problem)

Advanced Mathematical Techniques Used for Modeling the Problem:

Dynamical Systems, Linear Algebra, Complex and Real Analysis, Inversion Theory and Conformal Mapping, Topology.

Math Thesis Link