Matematik Doktora Programı - Sarıyer - İstanbul - Koç Üniversitesi - I475

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Matematik Doktora Programı
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Koç Üniversitesi

Matematik Doktora Programı - Sarıyer - İstanbul

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Matematik Doktora Programı - Sarıyer - İstanbul Matematik Doktora Programı - Sarıyer - İstanbul
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Program Description

The department of Mathematics offers graduate courses leading to Ph.D. degree in Mathematics. The department emphasizes both pure and applied mathematics. Research in the department covers algebra, algebraic geometry, number theory , functional analysis, differential geometry, differential equations, combinatorics, topology, biomathematics, statistics, probability, stochastic analysis and mathematical physics.   In addition to the following courses, students in this program can take any of the courses listed under the “M.S. in Mathematics” program or from other courses not listed here in accordance with their areas of   specialization and subject to the approval of their advisors.  

Degree Requirements

Students can apply to the Ph.D. programs with a B.S. or M.S. degree. The Ph.D. degree requires successful completion of 14 courses beyond the B.S. degree or 7 courses beyond the M.S. degree. All students must pass the Ph.D. Qualifying Examination in the first year after they have been admitted to the Ph.D. program. Students are encouraged to begin research early. Students who have passed the Ph.D. qualifying examination are assisted in matters pertaining to their thesis research by a faculty thesis advisory committee. The research advisor serves as the chair of this committee. The committee meets with the student at least once each semester. Ph.D. students must submit a satisfactory written Ph.D. thesis proposal in their second year of study. At the completion of the Ph.D. research, the students must submit a written Thesis and pass an oral defense to complete the degree requirements.

Research Areas


Algebra and Number Theory

  • Ring Theory and Module Theory, especially Krull dimension, torsion theories, and localization
  • Algebraic Theory of Lattices, especially their dimensions (Krull, Goldie, Gabriel, etc.) with applications to Grothendieck categories and module categories equipped with torsion theories
  • Field Theory, especially Galois Theory, Cogalois Theory, and Galois cohomology
  • Algebraic Number Theory, especially rings of algebraic integers
Combinatorics
  • Combinatorial design theory, in particular metamorphosis of designs, perfect hexagon triple systems
  • Graph theory, in particular number of cycles in 2-factorizations of complete graphs
  • Coding theory, especially relation of designs to codes
  • Random graphs, in particular, random proximity catch graphs and digraphs
Differential Equations
  • Nonlinear ordinary differential equations of molecular dynamics
  • PDE’s of quantum mechanics: time dependent Schrodinger equation
  • Weak, in particular viscosity solutions, of second order equations
  • Asymptotic analysis of reaction diffusion equations
  • Gamma limits of non-convex functionals
  • Geometric flows and level set equations
  • Global behavior of solutions to nonlinear PDE’s
  • Dissipative dynamical systems generated by evolutionary PDE’s
  • PDE’s modeling nonlinear problems of continuum mechanics
Analysis
  • Banach algebras, especially the structure of the second Arens duals of Banach algebras
  • Abstract Harmonic Analysis, especially the Fourier and Fourier-Stieltjes algebras associated to a locally compact group
  • Geometry of Banach spaces, especially vector measures, spaces of vector valued continuous functions, fixed point theory, isomorphic properties of Banach spaces
Mathematical Physics
  • Differential geometric, topologic, and algebraic methods used in quantum mechanics
  • Geometric phases and dynamical invariants
  • Supersymmetry and its generalizations
  • Pseudo-Hermitian quantum mechanics
  • Quantum cosmology
Probability and Stochastic Processes
  • Mathematical finance
  • Stochastic optimal control and dynamic programming
  • Stochastic flows and random velocity fields
  • Lyapunov exponents of flows
  • Unicast and multicast data traffic in telecommunications
  • Probabilistic Inference
Statistics
  • Spatial Statistics, mostly on nearest neighbor methods and multi-species spatial patterns of segregation and association
  • Statistical Pattern Recognition, Classification
  • Statistical Depth
  • Statistics of Medicine concerning morphometric changes in organs and tissues, say, due to a disease
  • Scale, size, and shape comparisons of organs or tissues based on MRI data
  • Linear Models
  • Computationally Intensive Methods: Bootstrap and Randomization
Algebraic Geometry
  • Arithmetical Algebraic Geometry, Arakelov geometry, Mixed Tate motives
  • p-adic methods in arithmetical algebraic geometry, Ramification theory of arithmetic varieties
 Geometry and Topology
  • Topology of low-dimensional manifolds, in particular Lefschetz fibrations, symplectic and contact structures, Stein fillings
  • Symplectic topology and geometry, Seiberg-Witten theory, Floer homology
  • Foliation and Lamination Theory, Minimal Surfaces, and Hyperbolic Geometry
Faculty
  • Attila Askar, Differential Equations
  • Mine Caglar, Probability and Stochastic Processes
  • Emre Alkan, Number Theory 
  • Elvan Ceyhan, Probability and Statistics
  • Tolga Etgu, Topology
  • Varga Kalantarov, Differential Equations
  • Sinan Unver, Algebraic Geometry
  • Selda Kucukcifci, Combinatorics
  • Ali Mostafazadeh, Mathematical Physics
  • Burak Ozbagci, Topology
  • Baris Coskunuzer, Geometric Topology 
  • Ali Ulger, Functional Analysis
  • Emine Sule Yazici, Combinatorics
Curriculum

Students who are admitted with an M.S. degree must complete at least 21 credits of coursework. Students with a B.S. degree must complete an additional 21 credits of coursework by taking courses in the M.S. program. They must also complete the core courses in the “M.S. in Mathematics” program.  
  • In addition, each student has to take a seminar course, MATH 590 Seminar.
  • Students working towards the thesis register for MATH 695 Ph.D. Thesis.
  • Students who have TA assignments must take TEAC 500: Teaching Experience during the semesters of their assignments.
  • Students must also take ENGL 500: Graduate Writing course.

  • MATH 580  Selected Topics in Topology I   
  • MATH 581    Selected Topics in Analysis I  
  • MATH 582    Selected Topics in Analysis II     
  • MATH 583    Selected Topics in Foundations of Mathematics
  • MATH 584    Selected Topics in Algebra and Topology  
  • MATH 585    Selected Topics in Probability and Statistics  
  • MATH 586    Selected Topics in Differential Geometry    
  • MATH 587    Selected Topics in Differential Equations
  • MATH 588    Selected Topics in Applied Mathematics
  • MATH 589    Selected Topics in Combinatorics                        

Course Descriptions

MATH 590 Graduate Seminar
Non-credit presentation of topics of interest in mathematics through seminars offered by faculty, guest speakers and graduate students.   

MATH 695  Ph.D.
Thesis Independent research towards Ph.D. degree. 

TEAC 500 Teaching Experience
Provides hands-on teaching experience to graduate students in undergraduate courses. Reinforces students' understanding of basic concepts and allows them to communicate and apply their knowledge of the subject matter.  

ENGL 500 Graduate Writing
This is a writing course specifically designed to improve academic writing skills as well as critical reading and thinking. The course objectives will be met through extensive reading, writing and discussion both in and out of class. Student performance will be assessed and graded by Satisfactory/Unsatisfactory.
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