MS in Applied Mathematics, PhD in Mathematics and its Applications

Degrees offered	Master of Science (MS) in Applied Mathematics	PhD in Mathematics and its Applications
Length of study	2 years	minimum 3 years
Graduation requirements	50 credits + thesis (worth 10 credits)	54 credits + thesis
Language of instruction	English	English

I. The MS Program 

This is an international two-year MS degree program in Applied Mathematics, accredited in the US.

The objectives of this program are to offer students:

• an opportunity to expand their knowledge in several fields of mathematics and its applications by providing courses at graduate level

• a unique academic experience via a high-quality international program taught in English

• preparation for a professional career in education, or in business, industry or research institutions

• for those wishing to continue their studies in the PhD track, enough knowledge in mathematics and its applications, as well as an opportunity to decide whether they are willing to become mathematicians.

Financial aid awards are determined at the end of the admissions process when the department completes its academic merit-ranking list. Due to the fact that the number of applicants eligible for admission is normally higher than the number of students to whom CEU can offer support, financial aid offers depend on an applicant's place in the merit-ranking list.
Financial Aid Packages for Master's Students: full fellowship, partial fellowship, full tuition waiver, partial tuition waiver.

Candidates who submit complete application packages and provide documentation that they are able to support themselves may be admitted for an additional number of places, if they satisfy the same entry requirements (see below).

Entry Requirements for the MS Program

Applicants are expected to hold a (three- or four-year) first degree (i.e., BA or BSc), with a major in mathematics or a neighboring field (e.g., computer science, economics, engineering, physics), or provide documentation indicating that they will earn this degree by the time of enrolling in our MS program.

In addition to meeting the general CEU Admissions Requirements (see http://www.ceu.hu/admissions), applicants are also required to submit a one-page statement of purpose describing their interest in mathematics, their achievements, and future goals. In addition, they must prove familiarity with the fundamental undergraduate material by taking the

Mathematics Entrance Exam. This is free of charge. It is a written examination covering the basic material in algebra and analysis. The examination will be held on February 11, 2012 at the locations listed here. Detailed list is also available in the application form. If a student has difficulties arranging to take the math exam because of traveling impediments (or there is no nearby CEU coordinator), the Department will attempt to arrange that a mathematician in her/his own university administer it (either on February 11, 2012 or on another day). Please contact us: mathematics@ceu.hu. For details on this math exam, please visit Section CEU Mathematics Entrance Examination below.

Instead of taking the CEU Mathematics Entrance Examination, candidates may choose to take the GRE Subject Test in Mathematics. The GRE score report must be received before February 29, 2012. Otherwise, these applications may only be considered as late applications (see below). The GRE option is offered to applicants who do not have access to the CEU examination.

Alternatively, in exceptional cases, candidates will be interviewed.

Remarks:

1. The language of instruction at CEU is English. Candidates may either take the TOEFL test (or other equivalent English test accepted by CEU) on their own expenses (the score report of which must be received before February 29, 2011), or take the TOEFL examination organized by CEU at the locations listed here, on February 11, 2012. Note that the Math Entrance Exam takes place on the same day at the same location (and both these examinations are free of charge). Of course, candidates who have already taken the English test before, with a score accepted by CEU, need to take only the Mathematics Entrance Exam on February 11, 2012. For further issues, including exemptions from the language requirements, see http://www.ceu.hu/admissions/requirements.

2. Candidates who are offered acceptance or are placed on the waiting list will be notified after March 20, 2012, via the online application status tracking feature.

3. Late applications will be considered on a case-by-case basis, if they reach CEU before April 30, 2012. Of course, in such cases, TOEFL (or equivalent) if applicable + GRE score reports should be sent to CEU. Instead of taking the GRE candidates may choose to take a Mathematics exam at their home universities (in this case, please contact us: mathematics@ceu.hu). Alternatively, in exceptional cases, candidates will be interviewed.

Application DocumentsAll applicants must provide CEU with the documents listed below. All application materials are submitted electronically to CEU. For information on how to apply see http://www.ceu.hu/admissions/apply.

Application Documents Checklist

• Completed online CEU application form;
• A one-page statement of purpose describing the candidate's interest in mathematics, achievements and future goals;
• CV (resume), including a list of publications, if any;
• Academic records (originals and notarized copies will be submitted by the time of enrollment by the admitted students only);
• 2 letters of recommendation.

In addition,

• GRE Subject Test in Mathematics score report (as a substitute for Math Entrance Exam), and 
• Proof of English proficiency

are required, depending on the case.

Important Dates:

January 25, 2012, 24:00 Pacific Standard Time (PST): application deadline for applicants who wish to take any CEU-administered admissions examinations (TOEFL and/or the Mathematics Entrance Exam), and/or are requesting exemption from the English language requirement (see http://www.ceu.hu/admissions/requirements#language); for other applicants the deadline is February 29, 2012 (see below);

February 11, 2012: CEU-administered admissions examinations (TOEFL and Mathematics Entrance Exam);

Febryary 29, 2012: application deadline for candidates submitting complete application packages, including language and/or GRE scores;

March 20, 2012: starting date for admissions decisions (which are made available to candidates via the online application system).

II. The Doctoral (PhD) Program

offers an innovative curriculum encompassing both traditional mathematics and cutting-edge contemporary applications. It is designed to ensure that students acquire rigorous and state of the art knowledge and to offer research opportunities under expert supervision.

Our Doctoral (PhD) program is registered to grant the PhD in Mathematics and its Applications by the Board of Regents of the University of the State of New York (U.S.A.) for, and on behalf of, the New York State Education Department. So our PhD is recognized throughout the world. Our PhD students are prepared especially for careers in academic and research institutions, but there are many other job opportunities for them. For some examples of positions held by our PhD alumni, click here.

Doctoral enrollment may continue up to a maximum of six years. Students that are admitted into our PhD program are eligible to receive a full CEU Doctoral fellowship for at least three years. The CEU fellowship award covers tuition, the student activity fee, medical insurance, and provides a full living scholarship.

Strong candidates who submit complete application packages and provide documentation that they are able to support themselves may be admitted for an additional number of places, if they satisfy the same entry requirements (see below). Transfer from other institutions' doctoral programs to our PhD program is also possible in exceptional cases.

Entry Requirements for the PhD Program

There are general CEU application requirements, which can be found at the CEU web site http://www.ceu.hu/admissions, where information is also available on how to apply, deadlines, application documents, etc.

In addition, there are some specific application requirements of the Department of Mathematics and its Applications, as follows:

1. Eligibility Requirements:

Students from any country may apply. Applicants are expected to have earned a master's degree (or equivalent) in mathematics or a related field (physics, engineering, computer science, etc.) from a recognized university or institution of higher education, or provide documentation indicating that they will earn this degree by the time of enrolling in our PhD program. We also accept excellent candidates with a bachelor's degree only, if they have completed at least four years of college by the time of enrolling, and have a strong mathematical background.

2. Statement of Purpose:

Applicants are required to submit a one-page statement describing their interest in mathematics, their achievements and future goals.

3. Mathematics Entrance Exam:

In order to give evidence of proficiency in mathematics, applicants should take the CEU Mathematics Entrance Exam. This is free of charge. It is a written examination covering the basic material in algebra and analysis.

The exam will be held on February 11, 2012 at the locations listed here. Detailed list is also available in the application form. If a student has difficulties arranging to take the math exam because of traveling impediments (or there is no nearby CEU coordinator), the Department will attempt to arrange that a mathematician in his/her own university administer it (either on February 11, 2012, or on another day). Please contact us: mathematics@ceu.hu. For details on this exam, please visit Section CEU Mathematics Entrance Examination below.

Instead of taking the CEU Mathematics Entrance Examination, candidates may choose to take the GRE Subject Test in Mathematics. The GRE score report must be received before February 29, 2012. Otherwise, such applications may only be considered as late applications (see below). The GRE option is offered to applicants who do not have access to the CEU examination.

Alternatively, in exceptional cases, candidates will be interviewed.

Remarks:

1. The language of instruction at CEU is English. Candidates may either take the TOEFL test (or another equivalent English test accepted by CEU) on their own expenses (the score report of which must be received before February 29, 2012), or take the TOEFL examination organized by CEU at the locations listed here, on February 11, 2012. Note that the Math Entrance Exam takes place on the same day at the same location (and both these examinations are free of charge). Of course, candidates who have already taken the English test before, with a score accepted by CEU, need to take only the Mathematics Entrance Exam on February 11, 2012. For further issues, including exemptions from the language requirements, see http://www.ceu.hu/admissions/requirements.

2. Candidates who are offered acceptance or are placed on the waiting list will be notified after March 20, 2012, via the online application status tracking feature.

3. Late applications will be considered on a case-by-case basis, if they reach CEU before April 30, 2012. Of course, in such cases, TOEFL (or equivalent) if applicable + GRE score reports should be sent to CEU. Instead of taking the GRE candidates may choose to take a Mathematics exam at their home universities (in this case, please contact us: mathematics@ceu.hu). Alternatively, in exceptional cases, candidates will be interviewed.

4. Candidates for the PhD program may also apply, at the same time, to our MS program (see above).

Application Documents

All applicants must provide CEU with the documents listed below. All application materials are submitted electronically to CEU. For information on how to apply see http://www.ceu.hu/admissions.

Application Documents Checklist

• Completed online CEU application form;
• A one-page statement of purpose describing the candidate's interest in mathematics, achievements and future goals;
• CV (resume), including a list of publications, if any;
• Academic records (originals and notarized copies will be submitted at the time of enrollment by the admitted students only);
• 2 letters of recommendation.

In addition,

• GRE Subject Test in Mathematics score report (as a substitute for the Math Entrance Exam), and
• Proof of English proficiency

are required, depending on the case.

Important Dates:

January 25, 2012, 24:00 Pacific Standard Time (PST): application deadline for applicants who wish to take the CEU-administered admissions examinations (TOEFL and/or the Mathematics Entrance Exam), and/or are requesting exemption from the English language requirement (see http://www.ceu.hu/admissions/requirements#language); for other applicants the deadline is February 29, 2012 (see below);

February 11, 2012: CEU-administered admissions examinations (TOEFL and Mathematics Entrance Exam);

February 29, 2012: application deadline for candidates submitting complete application packages, including language and/or GRE scores;

March 20, 2012: starting date for admissions decisions (which are made avalable to candidates via the online application system).

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CEU Mathematics Entrance Examination

The exam takes 3 hours and consists of problems in algebra and analysis. Of course, problems may involve a mixture of analysis and algebra. Some problems are computational, some ask for proofs, and some ask for examples or counterexamples.

Here is a list of subjects which are required for the entrance exam:

Algebra

    Linear Algebra:

• Vector spaces over R, C, and other fields: subspaces, linear independence, basis and dimension.
• Linear transformations and matrices: constructing matrices of abstract linear transformations, similarity, change of basis, trace, determinants, kernel, image, dimension theorems, rank; application to systems of linear equations.
• Eigenvalues and eigenvectors: computation, diagonalization, characteristic and minimal polynomials, invariance of trace and determinant.
• Inner product spaces: real and Hermitian inner products, orthonormal bases, Gram-Schmidt orthogonalization, orthogonal and unitary transformations, symmetric and Hermitian matrices, quadratic forms.

    Abstract Algebra:

• Groups: finite groups, matrix groups, symmetry groups, examples of groups (symmetric, alternating, dihedral), normal subgroups and quotient groups, homomorphisms, Sylow theorems.
• Rings: ring of integers, induction and well ordering, polynomial rings, roots and irreducibility, unique factorization of integers and polynomials, homomorphisms, ideals, principal ideals, Euclidean domains, prime and maximal ideals, quotients, fraction fields, finite fields.

Analysis

• Real numbers as a complete ordered field. Extended real number system. Topological concepts: neighborhood, interior point, accumulation point, etc. 
• Sequences of real numbers. Convergent sequences. Subsequences. Fundamental results. 
• Numerical series. Standard tests for convergence and divergence. 
• Real functions of one real variable. Limits, continuity, uniform continuity, differentiation, Riemann integration,  fundamental theorem of calculus, mean value theorem, L'Hopital's rule, Taylor's theorem, etc. 
• Sequences and series of functions. Pointwise and uniform convergence. Fundamental results. Power series and radii of convergence. 

• The topology of  R^k. Connected and convex subsets of  R^k.  
• Functions of several real variables. Limits, continuity, uniform continuity. Continuous functions on compact or  connected sets. Partial derivatives. Differentiable functions. Taylor's theorem. Maxima and minima. Implicit and inverse function theorems. 
• Multiple integrals. Integrals in various coordinate systems. Vector fields in Euclidean space (divergence, curl, conservative fields), line and surface integrals, vector calculus (Green's theorem in the plane, the divergence theorem in 3-space).  

• Ordinary differential equations. Elementary techniques for solving special differential equations (separable, homogeneous, first order linear, Bernoulli's, exact, etc.). Existence and uniqueness of solutions to initial value problems (Picard's theorem). Linear differential equations and systems. Fundamental results. 

Moreinfo: https://mathematics.ceu.hu/node/19256