Doctor of Philosophy in Mathematics
Introduction
Mathematics is dynamic and evolving rapidly, especially in the discovery of new knowledge. In-depth, specialized mastery of mathematics is indispensable for development of the society, the nation, and the world.
The Department of Mathematics at Naresuan University occupies the highest ranking and is acknowledged as the leader in the field in Thailand.
Objectives
Desirable characteristics of graduates are as follows:
- Leader of Advanced Mathematics in the academic arenas.
- Advanced Researcher in Mathematics.
- Development of new knowledge in Advanced Mathematics at an up-to-date caliber at all times.
Special Admission Requirements
None
All requirements are in accordance with the Graduate School.
Medium of Instruction
- Thai and English
Research Focus
- Analysis Group
- Algebra Group
Requirement for Graduation
In accordance with the Graduate School requirements.
Credit Requirements*
Requirements | Option 1.1 | Option 2.1 | Option 2.2 |
Coursework | - | 12 | 24 |
Core Courses | - | 3 | 12 |
Electives | - | 9 | 12 |
Required Non-credit Courses | 2 | 2 | 4 |
Dissertation | 48 | 36 | 48 |
Total | 48 | 48 | 72 |
* Minimum credits requirements
Core Courses
Requirements | Option 1.1 | Option 2.1 | Option 2.2 | |||
Course No. | Cr. | Course No. | Cr. | Course No. | Cr. | |
Functional Analysis | - | - | - | - | 252515 | 3 |
Linear Algebra and Matrix Theory | - | - | - | - | 252523 | 3 |
Topology | - | - | - | - | 252561 | 3 |
Special Topics in Advanced Mathematics | - | - | 252681 | 3 | 252681 | 3 |
Total | 0 | 0 | 1 | 3 | 4 | 12 |
Electives
Requirements | Option 1.1 | Option 2.1 | Option 2.2 | |||
Course No. | Cr. | Course No. | Cr. | Course No. | Cr. | |
Measure Theory | - | - | - | - | 252513 | 3 |
Complex Analysis | - | - | - | - | 252514 | 3 |
Set-Valued Analysis | - | - | - | - | 252516 | 3 |
Fixed Point Theory and Applications | - | - | - | - | 252517 | 3 |
Distribution Theory | - | - | - | - | 252518 | 3 |
Equilibrium and Optimization Theory | - | - | - | - | 252519 | 3 |
Advanced Abstract Algebra I | - | - | - | - | 252525 | 3 |
Algebraic Semigroup Theory | - | - | - | - | 252526 | 3 |
Ring and Module Theory I | - | - | - | - | 252527 | 3 |
Ring and Module Theory II | - | - | - | - | 252528 | 3 |
Advanced Abstract Algebra II | - | - | - | - | 252529 | 3 |
Graph Theory and Applications | - | - | - | - | 252534 | 3 |
Fuzzy Theory and Applications | - | - | - | - | 252541 | 3 |
Methods of Applied Mathematics | - | - | - | - | 252572 | 3 |
Advanced Ordinary Differential Equations | - | - | - | - | 252574 | 3 |
Partial Differential Equations | - | - | - | - | 252575 | 3 |
Mathematical Modeling | - | - | - | - | 252576 | 3 |
Applied Linear Algebra | - | - | - | - | 252577 | 3 |
Calculus of Variation | - | - | - | - | 252578 | 3 |
Numerical Analysis | - | - | - | - | 252579 | 3 |
Banach Space Theory | - | - | 252611 | 3 | 252611 | 3 |
Probability and Measure Theory | - | - | 252612 | 3 | 252612 | 3 |
Convex Analysis, Variational Problems and Nonlinear Optimization | - | - | 252613 | 3 | 252613 | 3 |
Geometry of Norm Linear Spaces | - | - | 252614 | 3 | 252614 | 3 |
Noncommutative Ring Theory | - | - | 252621 | 3 | 252621 | 3 |
Homological Algebra | - | - | 252622 | 3 | 252622 | 3 |
Exponential Diophantine Equations | - | - | 252623 | 3 | 252623 | 3 |
Algebraic Number Theory | - | - | 252624 | 3 | 252624 | 3 |
Finite Fields | - | - | 252625 | 3 | 252625 | 3 |
Algebraic Topology | - | - | 252661 | 3 | 252661 | 3 |
Differential Geometry | - | - | 252662 | 3 | 252662 | 3 |
Probability and Stochastic Process | - | - | 252671 | 3 | 252671 | 3 |
Applied Multivariate Analysis | - | - | 252672 | 3 | 252672 | 3 |
Applied Linear Models | - | - | 252673 | 3 | 252673 | 3 |
Partial Differential Equations for Finance | - | - | 252674 | 3 | 252674 | 3 |
Mathematical Quantum Mechanics | - | - | 252675 | 3 | 252675 | 3 |
Advanced Econometric Models | - | - | 252676 | 3 | 252676 | 3 |
Differential Equations and Dynamical Systems | - | - | 252677 | 3 | 252677 | 3 |
Optimization Methods | - | - | 252678 | 3 | 252678 | 3 |
Dynamical Systems for Econometrics | - | - | 252679 | 3 | 252679 | 3 |
Total | 0 | 0 | 20 | >=9 | 40 | >=12 |
Required Non-credit Courses
Requirements | Option 1.1 | Option 2.1 | Option 2.2 | |||
Course No. | Cr. | Course No. | Cr. | Course No. | Cr. | |
Seminar 1 | - | - | - | - | 252683 | 1 |
Seminar 2 | - | - | - | - | 252684 | 1 |
Seminar 3 | 252685 | 1 | 252685 | 1 | 252685 | 1 |
Seminar 4 | 252686 | 1 | 252686 | 1 | 252686 | 1 |
Total | 2 | 2 | 2 | 2 | 4 | 4 |
Dissertation Credit Requirements
Requirements | Option 1.1 | Option 2.1 | Option 2.2 | |||
Course No. | Cr. | Course No. | Cr. | Course No. | Cr. | |
Dissertation 1 Option 1.1 | 252690 | 6 | - | - | - | - |
Dissertation 2 Option 1.1 | 252691 | 6 | - | - | - | - |
Dissertation 3 Option 1.1 | 252692 | 9 | - | - | - | - |
Dissertation 4 Option 1.1 | 252693 | 9 | - | - | - | - |
Dissertation 5 Option 1.1 | 252694 | 9 | - | - | - | - |
Dissertation 6 Option 1.1 | 252695 | 9 | - | - | - | - |
Dissertation 1 Option 2.1 | - | - | 252696 | 6 | - | - |
Dissertation 2 Option 2.1 | - | - | 252697 | 6 | - | - |
Dissertation 3 Option 2.1 | - | - | 252698 | 6 | - | - |
Dissertation 4 Option 2.1 | - | - | 252699 | 9 | - | - |
Dissertation 5 Option 2.1 | - | - | 252790 | 9 | - | - |
Dissertation 1 Option 2.2 | - | - | - | - | 252791 | 3 |
Dissertation 2 Option 2.2 | - | - | - | - | 252792 | 9 |
Dissertation 3 Option 2.2 | - | - | - | - | 252793 | 9 |
Dissertation 4 Option 2.2 | - | - | - | - | 252794 | 9 |
Dissertation 5 Option 2.2 | - | - | - | - | 252795 | 9 |
Dissertation 6 Option 2.2 | - | - | - | - | 252796 | 9 |
Total | 6 | 48 | 5 | 36 | 6 | 48 |
Course Descriptions
25213 Measure Theory | 3(2-2-5) | |
A study of the foundations of real analysis, the Lebesgue outer measure measurable sets and Lebesgue measure, measurable functions, Riemann and Lebesgue integrals, differentiation of functions of bounded variation, measure spaces, convergence in measure, absolute continuity, L spaces, the existence of nonatomic countably additive probabilities, transition probabilities, product measures, convergence in distribution, Skorohod’s theorem, and some applications in econometrics and in economic theory. | ||
252514 Complex Analysis | 3(2-2-5) | |
Foundations of complex analysis, holomorphic and analytic function, Cauchy’s theory, Cauchy’s integral formula and its applications, residue theorem, principles of maximum modulus, singularities. harmonic function and conformal mapping, power series, uniform convergence and Weierstrass convergence theorem, and analytic continuation. | ||
252515 Functional Analysis | 3(2-2-5) | |
Metric spaces, normed spaces and Banach space linear operators, inner products and Hilbert spaces, Hahn-Banach theorem, uniform boundedness theorem, open mapping theorem, and closed graph theorem. | ||
252516 Set-Valued Analysis | 3(2-2-5) | |
Limit of sets, continuity of set-valued functions, closed convex processes, equilibrium and fixed point theorems, constrained inverse function theorem, monotone and maximum monotone functions, derivatives of set-valued functions, and measurability and integration of set-valued functions. | ||
252517 Fixed Point Theory and its Applications | 3(2-2-5) | |
Fixed point theory in metric spaces, fixed point theories for non-expansive mappings in Hilbert spaces, geometry of Banach spaces, fixed point theorems for continuous mappings and non-expansive mappings in Branach spaces, fixed point theorems in topological vector spaces, and iterative approximation of fixed points. | ||
252518 Distribution Theory | 3(2-2-5) | |
A study of the following topics: Dirac-delta function and Delta sequence, Heaviside function and Heaviside sequences, test function and distributions of several variables, linear functionals and distributions, operations on distributions, convergence of distributions, Schwartz-Sobolev theory of distributions, and direct product and convolution of distribution. | ||
252519 Equilibrium and Optimisation Theory | 3(2-2-5) | |
Minimization problem, the proximation theory, conjugate function, Frenchel’s theory, sub-differential of convex functions, tangent and normal cones, generalized gradients of locally Lipschitz functions, Debreu-Gale-Nikaido theory, equilibrium of dynamic economies, and the Leray-Schauder theory. | ||
252523 Linear Algebra and Matrix Theory | 3(2-2-5) | |
Linear treansformations and their matrices, invariant subpace, linear functional diagonalisation, Jordan canonical forms, inner product spaces, unitary and orthogonal matrices, Gram-Schmidt algorithm, and bilinear forms. | ||
52525 Advanced Abstract Algebra 1 | 3(2-2-5) | |
A study of Groups, isomorphism theorems, group actions, Sylow theorems, rings, ideals, polynomial rings, unique factorisation domains, fields and field extensions, and an introduction to Galois theory. | ||
252526 Algebraic Semigroup Theory | 3(2-2-5) | |
Elementary concepts, Green’s relations, simple and O-simple semigroups, inverse semigroups, and transformational semigroups. | ||
252527 Ring and Module Theory | 3(2-2-5) | |
Modules and sub modules, homomorphism of modules, direct summands, direct sums and products of modules, decomposition of rings, generating and cogenerating, semi simple modules, socl and radical chain conditions, modules with composition series, semi simple rings, andlocal rings and artinian rings. | ||
252528 Rings and Module Theory 11 | 3(2-2-5) | |
Functors, projective modules and generators, injective modules and cogenerators, tensor functions and flat modules, Morita dualities, noetherian rings, semiperfect and perfect rings, and quasi-Frobenius rings and serial rings. | ||
252259 Advanced Abstract Algebra 11 | 3(2-2-5) | |
Solvable groups, Jondan-Holder theory, free groups, classification of extension fields, and the fundamentals of Galois’ theory. | ||
252534 Graph Theory and Applications | 3(2-2-5) | |
Graphs, subgraphs, trees, connectivity, paths and cycles, matchings, chromatic numbers, independent sets and cliques, Ramsey’s theory, planar graphs, directed graphs, and algebraic graph theory. | ||
252541 Fuzzy Theory and Applications | 3(2-2-5) | |
Fuzzy set theory, operations on fuzzy sets, fuzzy logic and approximate reasoning, fuzzy relations, fuzzy system modeling and stability analysis, and fuzzy logic control. | ||
252561 Topology | 3(2-2-5) | |
Elementary set theory, functions and relations, partially ordered sets, Zorn’s lemma, abstract topological spaces, metric spaces, bases and sub-bases, convergence, filters and nets, separation axioms, continuity and homeomorphisms, connectedness, separability, and compactness. | ||
252572 Methods of Applied Mathematics | 3(2-2-5) | |
Matrices, equivalence, quadratic and hermitian forms, eigenvalues, invariants, function spaces and Sturm-Liouville problems, calculus of variations, Euler-Lagrange equations, constraints, variable endpoints, Sturm-Liouville theory, integral equations, Green’s functions, Hilbert-Schmidt theory, and singular integral equations. | ||
252574 Advanced Ordinary Differential Equations | 3(2-2-5) | |
Existence theorems, linear and non-linear differential equations, regular and singular boundary value problems, stability theory of linear and non-linear systems, Liapunov’s second method, and geometric theory of differential equations in the plane. | ||
252575 Partial Differential Equations | 3(2-2-5) | |
The Cauchy problem for partial differential equations; the classification of second order linear partial differential equations; the properties of solutions for elliptic, parabolic, and hyperbolic equations; the existence of solutions for elliptic partial differential equations; topics from Fourier and Laplace transforms; potential theory; Green’s functions; integral equations; and Sobolev spaces and Schwartz distributions. | ||
252576 Mathematical Modeling | 3(2-2-5) | |
The fundamental concepts of mathematical modeling, graphic modeling, processes of modeling, modeling using data, adjusting the model, modeling using differential equations, and modeling using difference equations. | ||
252577 Applied Linear Algebra | 3(2-2-5) | |
A study of matrices and linear algebraic systems, inner products and norms, minimization and least squares approximation, orthogonality, equilibrium, eigenvalues and eigenvectors, linear dynamical systems, and iteration of linear systems. | ||
252578 Calculus of Variations | 3(2-2-5) | |
The variance of functional depending on the function of one variable, the variance of functional depending on unknown functions, the variation of functional depending on the function of several variables, direct method in variational problems, such as Euler’s finite difference methods, the Ritz method, and the Kantorovich method. | ||
252579 Numerical Analysis | 3(2-2-5) | |
Numerical solutions of linear systems, numerical solutions of non-linear equations, numerical solutions of ordinary equations, numerical solutions of partial differential equations, finite differences and applications to interpolation and differentiation, and integration and summation of series and finite element methods. | ||
252611 Banach Space Theory | 3(2-2-5) | |
Reflexivit, weak and weak star topology, convexity and smoothness, modules of convexity and smoothness, duality mappings, Banach limit, metric projection, application of contractions, operator on normed spaces, and existence theories of fixed points in Banach spaces. | ||
252612 Measure and Probability Theory | 3(2-2-5) | |
Measure and integration, convergence concepts, random variables, independence and conditional expectation, laws of large numbers, central limit theorems, and martingale theorems. | ||
252613 Convex Analysis, Variational Problems and Non-Linear Optimisation | 3(2-2-5) | |
Convex functions, minimization of convex functions and variational inequalities, duality in convex optimization, duality by the minimax theory, non-smooth optimization, and Karush-Kuhn-Tucker theory. | ||
252614 Geometry of Norm Linear Spaces | 3(2-2-5) | |
Convexivity and smoothness, generalisations of uniform convexity and uniform smoothness, modulus of convexity and smoothness, duality mappings, differentiability of norms, and applications of geometric properties on norm linear spaces. | ||
252621 Noncommutative Ring Theory | 3(2-2-5) | |
Noetherion rings, rings with descending chain condition, Wedderburn structure theory for semi-simple rings, rings of quotients, and rings characterized by their modules. | ||
252622 Homological Algebra | 3(2-2-5) | |
Modules over a ring, tensor products and groups of homorphisms, categories and functors, homology functors, projective and injective modules, and derived functors. | ||
252623 Exponential Diophantine Equations | 3(2-2-5) | |
Reviews of algebraic number theory; estimates of linear forms in logarithms; purely exponential equations; binary recurrence sequences of orders 2, 3 and 4; Thue equation; super-elliptic equation; Thue-Mahler equation; and perfect powers in binary recurrence sequences. | ||
252624 Algebraic Number Theory | 3(2-2-5) | |
Algebraic numbers and integers, relations among rational functions, coefficients of analytic functions of one complex variable and recurrence relations, Pisot numbers and their basic properties, and p-adic analysis. | ||
252625 Finite Fields | 3(2-2-5) | |
The structure of finite fields, polynomials over finite fields, and factorization of finite fields. | ||
252661 Algebraic Topology | 3(2-2-5) | |
Covering spaces, homotopy theory, homotopy groups, homology and cohomology. | ||
252662 Differential Geometry | 3(2-2-5) | |
Differentiable manifolds, fibre bundles, Riemannian geometry, affine connections, integration on manifolds, vector fields, and introductory Lie groups and Lie algebras. | ||
252671 Probability and Stochastic Process | 3(2-2-5) | |
Basic properties of probability models, independence, random variables and their distributions, conditional probability, conditional probability law of large numbers, central limit theory, random walk, Markov chains and Brownian motion, and selected applications. | ||
252672 Applied Multivariate Analysis | 3(2-2-5) | |
Multivariate normal distribution, multivariate data plots, multivariate analysis of variance: discriminant analysis, logistic regression methods, canonical correlation analysis, principle component analysis, factor analysis, cluster analysis, multidimensional scaling, statistical packages, and applications. | ||
252673 Applied Linear Models | 3(2-2-5) | |
Multiple linear regression, analysis of variance, transformation and weighting in linear models, logistic regression for binary response data, generalised linear models, estimation and inference using iterative reweighted least square (IRLS) Poisson regression, log-linear models for contingency tables, models for survival data and non- linear regression techniques, and applications. | ||
252674 Partial Differential Equations for Finance | 3(2-2-5) | |
Deterministic optimal control, dynamic programming, Hamilton-Jacobi equations, viscosity solutions, stochastic optimal control, discrete and continuous time, the Hamilton-Jacobi-Bellman equation, verification of arguments, stochastic differential equations, Ito’s lemma, backward and forward Kolmogorov equations, the Feynman-Kac formula, stopping times, Girsanov’s theory, parabolic equations, fundamental solution, boundary value problems, maximum principle, and applications to finance including portfolio optimisation and option pricing. | ||
252675 Mathematical Quantum Mechanics | 3(2-2-5) | |
Formalisation of quantum mechanics including mathematical methods of quantum theory, basic postulates of quantum mechanics and equation of motion, approximation methods and scattering theory, and introductory relativistic quantum mechanics. | ||
252676 Advanced Econometric Models | 3(2-2-5) | |
The nature of econometrics and data, regression analysis with cross sectional data, and regression analysis with time-series data. | ||
252677 Differential Equations and Dynamical Systems | 3(2-2-5) | |
General theory of linear systems, local theory of non-linear systems, stable manifolds and Hartman-Grobman theories, global theory of non-linear systems, Poincare-Bendixon theory, limit cycles, Poincare maps, and bifurcations. | ||
252678 Optimisation Methods | 3(2-2-5) | |
Static and comparative static models, convex sets and concave functions, static optimization, and applications for microeconomics. | ||
252679 Dynamical Systems for Econometrics | 3(2-2-5) | |
The basic concepts and scalar systems, the dynamical system of higher dimensions and some applications, an introduction to dynamic optimizations, and some applications to dynamic optimisations. | ||
252681 Special Topics in Advanced Mathematics | 3(2-2-5) | |
Study and analysis of topics in advanced mathematics that are of special interest. | ||
252683 Seminar 1 | 1(0-2-1) | |
Practice in how to search, analysis of data, and an oral presentation of research or an article of current interest in mathematics. | ||
252684 Seminar 2 | 1(0-2-1) | |
Conducting presentation and discussion of a research topic of interest in theoretical or applied mathematics. | ||
252685 Seminar 3 | 1(0-2-1) | |
Conducting a presentation and discussion of current research in the field of mathematics relevant to their proposed dissertation research. | ||
252686 Seminar 4 | 1(0-2-1) | |
Practice in how to write and present research in mathematics. | ||
252690 Dissertation 1, Option 1.1 | 6 Credits | |
A literature review of basic knowledge and research relevant to the topic, creating guidelines and hypotheses for the proposed research, and submitting a progress report to the program committee. | ||
252691 Dissertation 2 Option1.1 | 6 Credits | |
A further literature review, intensive study of basic knowledge and research relevant to the topic, a review of frameworks and guidelines, and submission of a summary report of the research and a progress report to the program committee. | ||
252692 Dissertation 3, Option 1.1 | 9 Credits | |
Setting up research hypotheses, conducting research within the frameworks and guidelines, and submitting a summary report and a progress report to the program committee. | ||
252693 Dissertation 4, Option 1.1 | 9 Credits | |
Preparing a dissertation proposal, conducting the research following the defined methodology, frameworks, and guidelines, and submitting a progress report to the program committee. | ||
252694 Dissertation 5, Type 1.1 | 9 Credits | |
Reviewing and summarising the preliminary research results, preparing an article(s) suitable for publication in national or international mathematical journals, and submitting a summary research report and a progress report to the program committee. | ||
252695 Dissertation 6, Option 1.1 | 9 Credits | |
Summarising the research results, writing up the complete dissertation, preparing for the dissertation defense, and submitting a summary of the dissertation results to the program committee. | ||
252696 Dissertation 1, Option 2.1 | 6 Credits | |
Conducting a literature survey, reviewing basic knowledge and research relevant to the topic, exploring the direction for a dissertation research study, preparing frameworks and guidelines, and submitting a progress report to the program committee. | ||
252697 Dissertation 2, Option 2.1 | 6 Credits | |
Researching and compiling further information relevant to the research and within the frameworks and guidelines, and submitting a summary research report and progress report to the program committee. | ||
2522698 Dissertation 3, Option 2.1 | 6 Credits | |
Establishing research assumptions, frameworks, and guidelines; preparing a dissertation proposal; and submitting a summary report of the research and progress to the program committee. | ||
252699 Dissertation 4, Option 2.1 | 9 Credits | |
Reviewing the research making any improvements or modifications necessary based on expert opinions, writing a research article for publication in a national or international journal in mathematics, and presenting a summary report of the research and a progress report to the program committee. | ||
252790 Dissertation 5, Option 2.1 | 9 Credits | |
Writing a final dissertation, preparing for the dissertation defense, and submitting a summary of the dissertation results to the program committee. | ||
252791 Dissertation 1, Option 2.2 | 3 Credits | |
Conducting a literature review in various databases and research articles on fundamental knowledge on topics of interest and writing a progress report for presentation to the program committee. | ||
252792 Dissertation 2, Option 2.2 | 9 Credits | |
Setting up research hypotheses within established guidelines and frameworks, writing a dissertation proposal, and submitting a summary report of the proposed research and dissertation progress to the program committee. | ||
252793 Dissertation 3, Option 2.2 | 9 Credits | |
Establishing research hypotheses, conducting research within allocated guidelines and framework. Summary report of research and dissertation progress report to present to committee of this program | ||
252794 Dissertation 4, Option 2.2 | 9 Credits | |
Conducting research within allocated guidelines and frameworks, preparing a dissertation proposal, and submitting a summary report of the research findings for presentation to the program committee. | ||
252795 Dissertation 5, Option 2.2 | 9 Credits | |
Conducting a review of the research, writing an article for publication in a national or international journal of mathematics, making any improvements or modifications to the research results based on expert opinions, and presenting a summary of the dissertation results to the program committee. | ||
252796 Dissertation 6, Option 2.2 | 9 Credits | |
Writing a complete dissertation, preparing for a dissertation defense, and presenting a summary of the dissertation to the program committee. |
Course Organisers
Dean | Plubtieng, Somyot, Prof., Ph.D. |
Program Director | Plubtieng, Somyot, Prof., Ph.D. |
Graduate Faculty Full Members | Namnak, Chaiwat, Asst. Prof., Ph.D.Petrot, Narin, Asst. Prof., Ph.D.
Siripitukdet, Manoj, Assoc.Prof., Ph.D. Wanakeeree, Rabian, Assoc.Prof., Ph.D. |