# Prospective MS and PhD Students

Welcome at the CEU Department of Mathematics and its Applications. **You can apply using the online application form which will shortly become available.**

Institutional TOEFL test is provided at the CEU Open House, November 17

ttps://www.ceu.edu/institutional-toefl-registration

**I. The MS Programs**

The objectives of this program are to offer students:

- an opportunity to expand their knowledge in several fields of mathematics and its applications like financial mathematics, cognitive science or network science 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.

Application fee is EUR 30 (payable only once, non-refundable) “The fee should be paid by bank card via the CEU online application form (secure payment)”.(https://www.ceu.edu/admissions/how-to-apply/checklist#step-7 )

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.edu/admissions/how-to-apply/checklist), 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 March 2, 2017 (the time and venue will be specified later). For details of this math exam, please visit Section CEU Mathematics Entrance Examination below.

For details of this math exam, please visit Section CEU Mathematics Entrance Examination below. *With questions please contact the Mathematics Department at mathematics@ceu.edu . *

Details here

https://www.ceu.edu/admissions/after-you-apply/program-specific-exams

Instead of taking the CEU Mathematics Entrance Examination, candidates may choose to take the GRE Subject Test in Mathematics.

**Details: http://www.ceu.edu/admissions/after-you-apply/admissions-process**

**II. The Doctoral (PhD) Program**

offers an innovative curriculum encompassing both pure mathematics and cutting-edge contemporary applications in a cooperation with the Renyi Institute of Mathematics. 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 https://www.ceu.edu/node/13694 , 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. MS and PhD in Mathematics: Mathematics Entrance Exam or GRE:**

In order to give evidence of proficiency in mathematics, applicants should take the CEU mathematics entrance exam. It is a written examination covering the basic material in algebra and analysis.

The exam will be held on **March 2, 2017 **at CEU. For those candidates not residing close to Budapest the Department of Mathematics will attempt to arrange for a mathematician at the candidate's home university to administer it on **March 2, 2017**. In that case please contact the Mathematics Department at mathematics@ceu.edu.

Instead of taking the CEU Mathematics Entrance Examination, candidates may choose to take the GRE Subject Test in Mathematics (a scanned copy of the test score sheet needs to be submitted with the online application).

Useful links: https://www.ceu.edu/apply

application forms: https://www.ceu.edu/admissions/how-to-apply/application-forms

**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: https://www.ceu.edu/apply

**Department Specific Documents**

A one-page statement of purpose describing the candidate's interest in mathematics, achievements and future goals;

GRE Subject Test in Mathematics score report (as a substitute for Math Entrance Exam), if any

are required, depending on the case.

The fee should be paid by bank card via the CEU online application form (secure payment).(https://www.ceu.edu/admissions/how-to-apply/checklist#step-7 )

**Important Dates:**

**https://www.ceu.edu/admissions/how-to-apply/deadlines**

**also visit: **

https://www.ceu.edu/admissions/how-to-apply

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 Rk. Connected and convex subsets of Rk.
- 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.