Statement of Objectives

Samir ****

I completed my four-year Bachelor of Mathematics degree at Bogazici (Bosporus) University, Istanbul, Turkey in February 2007. In February, 2008, I was selected to Ph.D. program in Mathematics at Yeditepe University, ranking first in the last semester among first year graduate students.

I was introduced to mathematics and physical sciences while at high school, where I considered a career in this area. I noticed that I had a talent for mathematics and was selected for the Mathematics department in the one of top ranked universities of Turkey. My undergraduate education at Bosporus University has not only given me a certain set of skills, but has also helped me understand my fields of interest and my academic strengths and weaknesses.

Through my undergraduate studies, I've experienced some misfortunes. In my first semester, I could not take final exams, due to a hernia operation. I also found that one-hour exams would not allow me to demonstrate my knowledge and potential. Thus, I lost my concentration and motivation. However, I made a new start after graduating from Bogazici University.

In collaboration with Professors R. Agarwal, A. Ashyralyev, V. Shakhmurov I achieved significant results in long term projects and as a result I did some publications that have been submitted to SCI (science citation index) journals, and one has been accepted.

[1]. Maximal B-Regular Integro-Differential Equations.(To appear) Chinese Annals of Mathematics, Ser.B. (Joint with Veli Shakhmurov)

[2]. Operator-Valued (Lq,Lp) Fourier Multipliers. Submitted (Alone)

[3]. A Note On Integral Operators with operator valued kernels. Submitted (Alone)

[4]. Hausdorff-Youngs type Fourier multipliers and applications. Submitted (Alone)

[5]. High-accuracy difference schemes for differential equations of 2n-th order. Submitted (joint with Allaberen Ashyralyev)

[6]. Fourier multipliers on Banach-valued weighted Besov spaces and its applications. Submitted. (Joint with Ravi Agarwal and Veli Shakhmurov).

[7]. A note on the Taylor's decomposition on several points for the odd order differential equations. Submitted (joint with Allaberen Ashyralyev)

[8]. Operator-valued Fourier multipliers in Besov spaces and its applications. Submitted (Joint with Veli Shakhmurov).

I have been studying Differential equations, Difference equations and Harmonic analysis since 2007. The following are some of the results from the above papers:

1

2

3

.....

Presently I am studying the following problems:

• Regularities and Asymptotic behaviors of some important dissipative Nonlinear Equations (Ginzburg-Landau, Whitham and etc. )

• Certain harmonic analysis tools to investigate nonlinear PDE.

• Investigation of Carleman estimates and unique continuation properties for elliptic diff. operator equations

• Utilizing Taylor difference schemes and exact difference schemes for equations

u⁽²ⁿ⁾(t)+Au(t)=f(t), u⁽²ⁿ⁻¹⁾(t)+Au(t)=f(t)

where A is unbounded positive definite self adjoint operator and f∈C^{α,α}((0,T),E) (Banach valued Holder spaces) and to establish stabilities of these schemes.

• Operator-valued averaging method (Krylov-Bogolyubov method) for

((d²u)/(dt²))+Au(t)+µf(u,((du)/(dt)))=0

where A is a positive definite operator µ<<1 small parameter and f∈L₂(R,H).

In February 2008, I was selected for the Ph.D. program at Yeditepe University and I completed my first semster, with rank one among first year graduate students. I completed Functional analysis 2, Qualitative theory of ODE and Differential geometry courses in Spring, 2008. Moreover, I regularly participated in Differential equations and Functional analysis workshops and I presented several talks there.

Workshops (Certified)

1. Partial Differential Equations: Theory and Applications

March 26-June 11 2006 Fatih University,Istanbul Turkey (Organizer Prof.Dr.

A.Ashyralyev)

Talks

a. A Harnack's inequality for second order linear diff. equations

b. Application of Fixed Point Theorems in PDE and Integral equations.

2. Computational Mathematics, November 5, 2006- January 14 2007

Fatih University,Istanbul Turkey (Organizer Prof.Dr. A.Ashyralyev )

Talks

a. High-accuracy difference schemes for differential equations of 2n-th order

b. Coercitivity of Cauchy Problem in Banach valued Holder spaces

3. Numerical Functional Analysis, April 1-June 3 2007

Fatih University,Istanbul Turkey (Organizer Prof.Dr. A.Ashyralyev )

Talks

a. Taylor's decomposition method for the odd order differential equations

b. Some important theorems in Normed spaces. (Open mapping theorem,

Banach-Steinhouse thm, Hanh- Banach thm, Closed mapping theorem)

4. Numerical Functional Analysis 3, April 7-June 9 2008

Fatih University,Istanbul Turkey (Organizer Prof.Dr. A. Ashyralyev )

Talks

a. On Uniform Covexity of Lebesgue spaces.

b. Some Proofs of the Theorem: `'Every Uniform Covex Banach space is Reflexive"

c. Scalar and operator valued Fourier Multipliers in L_{p} spaces and applications.

Recently, I contacted Professor Michael Klibanov. I wrote him about my wish to apply for the Ph.D program at UNCC and he agreed to be my supervisor in Ph.D program. I would like to study ill-posed problems, numerical methods and Carleman estimates during this time. Therefore, I believe that under the supervision of Professor Klibanov, I can achieve significant results and I can gain scientific maturity. Taking everything into consideration, I believe that UNCC is the best place for me to realize my academic plans and goals.

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Note some commas added.

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