I want to write an essay about two things for a PhD admission in Mathematics:
First, about all my courses that I have taken in my master’s degree ( I have attached a file includes all my courses names with some details), and I want to mention that I am more interested in Complex Analysis, and Real Analysis with specialty in Metric Space, Compactness, Analytic Function, Singularities, Cauchy’s Theorem, Residues Theorem, and The Riemann Mapping Theorem.
Second, about the Dynamical System and what areas of dynamical system do I think are more important?
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MATH 5110 - Introduction to Modern Analysis
Metric spaces, sequences and
series, continuity, differentiation, the Riemann-Stielties integral,
functions of several variables.
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MATH 5210 - Abstract Algebra I
Group theory, ring theory and
modules, and universal mapping properties.
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MATH 5211 - Abstract Algebra II
Linear and multilinear algebra,
Galois theory, category theory, and commutative algebra.
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MATH 5046 - Introduction to Complex Variables
Functions of a complex variable, integration in the complex
plane, conformal mapping.
MATH 5800 - Investigation of Special Topics
Students who have well defined
mathematical problems worthy of investigation and advanced reading should
submit to the department a semester work plan.
(I have learned in this class how
to use the MatLab and LaTeX)
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MATH 5510 - Numerical Analysis and Approximation
Theory I
The study of convergence, numerical stability, roundoff
error, and discretization error arising from the approximation of differential
and integral operators.
MATH 5120 - Complex Function Theory I
An introduction to the theory of
analytic functions, with emphasis on modern points of view.
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MATH 5070 - Topics in Scientific Computation
(Indefinite Inner Product Spaces)
Topics in indefinite inner products.
Topics in matrix perturbation theory
Topics in structured matrices.
Topics in rational interpolation.
Topics in matrix analysis
- Chapters on Indefinite Inner Products and Orthogonal
polynomials
- The Gohberg-Semencul formula
- Note on Two Krein's theorems via dispacement and the
Gohberg-Semencul formula
- Generalized Schur algorithm for LDL^* factorization
of matrices satisfying displacement equations
- Generalized Schur algorithm for solving the
Hermite-Fejer interpolation problem
- Sketch of the Jordan canonical form
- Proof of the Jordan canonical form
- Canonical form for H-selfadjoint matrices
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MATH 5410 - Introduction to
Applied Mathematics I
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Banach spaces, linear operator theory and application to
differential equations, nonlinear operators, compact sets on Banach spaces, the
adjoint operator on Hilbert space, linear compact operators, Fredholm
alternative, fixed point theorems and application to differential equations,
spectral theory, distributions.
MATH 5411 - Introduction to Applied Mathematics II
MATH 5250 - Modern Matrix Theory and Linear Algebra
The LU, QR, symmetric, polar, and
singular value matrix decompositions. Schur and Jordan normal forms.
Symmetric, positive-definite, normal and unitary matrices. Perron-Frobenius
theory and graph criteria in the theory of non-negative matrices.
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