Program in Applied Mathematics




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CENTRAL EUROPEAN UNIVERSITY

Department of Mathematics and its Applications


Master of Science (M.S.)

Program in Applied Mathematics


2009-2010 MS Student Handbook


Zrinyi u. 14, Third Floor

H-1051 Budapest

Hungary


Telephone: (+36 1) 327 3053

Fax: (+36 1) 327 3166

Email: Mathematics@ceu.hu

Internet: http://web.ceu.hu/math


Table of Contents



Introduction

3

Program Objectives

5

Admissions

6

Program Structure and Graduation Requirements

7

Facilities and Infrastructure

47

Faculty

47

MS Teaching Program

48

Annex

68




Introduction



Degree offered:

Master of Science (MS) in Applied Mathematics

Length of study:

2 years

Graduation requirements:

45 course credits + 5 research credits + thesis (worth 10 credits)

Language of instruction:

English



The Mathematics Department offers a two-year Master of Science (M.S.) program in Applied Mathematics, accredited in the US. It is carried out in cooperation with the Alfréd Rényi Institute of the Hungarian Academy of Sciences.

This program follows the lines of the Bologna process regarding structures, while keeping the US standards and characteristics. At this moment, there is a unanimously accepted subject classification in mathematics. This classification is clearly reflected in the curriculum of graduate programs. In general, the basic courses are the same everywhere. Of course, different programs choose different subjects for elective courses. It is worth mentioning that our M.S. program is strongly oriented to modern topics in Applied Mathematics, including analytical, statistical, numerical and computational methods, which are known to be strong tools in tackling real world applications. Concerning the overall workload in credits, our M.S. program is similar to European programs and stronger than US programs.

Our M.S. program is unique since:


•    It is an international graduate program in the region. The language of instruction is English. There is growing demand in the region for well-trained mathematicians, in particular applied mathematicians. Industrial companies, banks, research institutes, governmental and EU organizations need such kind of specialists. There are specific positions for M.S. graduates, such as, for instance: mathematician, statistician, system analyst, business analyst, optimization analyst, modeler, high school mathematics teacher, mathematics editor, manager. These specialists are supposed to play an important role in the region, as well as in the EU. In fact, we are sure they will be highly appreciated everywhere in the world.


•    Outstanding scholars are involved in both teaching and supervision.


•   This program contributes to the development of CEU as a research focused university, through close cooperation with other CEU departments (Economics, Environmental Sciences and Policy, International Relations and European Studies (IRES), Political Science, Sociology and Social Anthropology) in a joint effort to study regional phenomena, including those related to transition and globalization. Differential equations, graph theory, statistical and numerical methods, calculus of variations, computer simulations are widely used nowadays to develop various models associated with biological, ecological, economic, political, social phenomena. This kind of cooperation is specific to CEU. It is supposed that our graduates will be using their interdisciplinary experience when they hold various positions in the region or elsewhere. 


•   The program can benefit a lot from the opportunities offered by the rich local academic environment, including the Alfréd Rényi Institute of the Hungarian Academy of Sciences, the Eötvös University (ELTE) and the Budapest University of Technology and Economics (BME). Indeed, they can support our program through their staffing and library resources, conferences, seminars, and long experience in running high quality mathematics programs.


Note: In designing this program, we have examined similar existing programs of different universities, such as: The Australian National University, Brown University (Providence, RI, USA), California Institute of Technology (USA), Cambridge University (UK), Columbia University (USA), Illinois Institute of Technology (USA), ELTE (Budapest), Kyoto University (Japan), University of  Calgary (Canada), University of Leeds (UK), University of New South Wales (Sydney, Australia), University of Texas at Austin (USA), University of Western Ontario (Canada), University of Sheffield (UK), Washington University.


Program Objectives


The purpose of our MS program in Applied Mathematics is to offer students who hold a BA or BSc degree with a major in mathematics or a neighboring field: 



  • an opportunity to expand their knowledge in the 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 and research institutions



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



Admissions


Students seeking admission to our MS program have to meet both the general CEU application requirements and the requirements of the Mathematics Department, as follows.


General CEU Admissions Requirements


are available at http://www.ceu.hu/admissions/requirements/general


Specific Requirements of the Mathematics Department


Applicants are required to submit a one-page statement of purpose describing their interest in mathematics, achievements and future goals. In addition, they have to prove familiarity with fundamental undergraduate material, by taking either a Mathematics Exam (for a list of subjects to be covered on the exam see the Annex below) or the GRE Subject Test in Mathematics. Alternatively, candidates will be interviewed.


For more details, please visit: http://www.ceu.hu/math


Program Structure and Graduation Requirements


This is a two-year program in Applied Mathematics, including coursework, research and thesis components.

The M.S. regular courses are delivered during the two main terms, Fall and Winter Term, that start end of September and early January, respectively (and each of them has a duration of 12 weeks). We also schedule special courses, invited lecture series or seminars beyond these two terms, especially during the Spring Term.


Graduation Requirements

M.S. students are required to earn a total of 45 credits in mandatory and elective courses (see Section Schedule of Coursework below). In addition, 5 research credits should be earned by participation in seminars.

All M.S. students are required to attend the Departmental Seminar, starting the Winter Term of their first year. In addition, there is a required Spring MS Seminar for first year students as well as a Winter MS Seminar for second year students. MS seminars aim to introduce students to various topics in mathematics and its applications and facilitate their research work toward the MS Thesis (see Section Thesis below).

First year M.S. students are required to write a preliminary thesis proposal of 3-5 pages and present it in a session of the Spring MS Seminar. This proposal should eventually be improved and extended and then re-discussed during their second year.

As a final condition for graduation, M.S. students are required to write a thesis (worth 10 credits) which is normally defended by the end of the program. The thesis must be submitted 3 weeks in advance. The defense comprises the presentation of the thesis as well as a comprehensive exam on the area of the thesis, covering the material of 3 courses, including at least 2 electives (i.e., either 3 electives or 2 electives + 1mandatory).


The distribution of credits required for graduation is shown in the following table:

Total (a + b) 

60 credits

a. Coursework

45 credits

  • Mandatory Courses

21 credits

  • Elective Courses

24 credits

b. Research + Thesis

5 + 10 = 15 credits


The minimum passing grade for every exam, including thesis, is C+ (worth 2.33). Course grades count towards the final Grade Point Average (GPA) as per credit and constitute 75% of the final GPA. More precisely, the

Final GPA = 0.75 ×nigi ni+ 0.25 × thesis grade,

where ni represents the number of credits attributed to course ci and gi is the grade obtained in the corresponding exam.  The minimum final GPA required for the M.S. degree is 2.67. Note that our university uses the US credit system.


Therefore, the requirements to qualify for the M.S. degree are:


•    passing all mandatory courses;
•    earning at least 45 credits for the overall coursework;
•    earning 5 credits for research;
•    successful thesis;
•    final GPA = 2.67 or higher.

Additional details are available at http://www.ceu.hu/studentlife/current/student-policies, Section Student Rights, Rules, and Academic Regulations. This includes in particular standards and assessment techniques. In addition, specific assessment methods, including specific homework assignments, computer simulations, presentations on various topics, etc. may be required by each instructor, depending on the nature of the course.

Remark. It should be pointed out that our M.S. program is totally in line with the European standards, taking into account that 1 CEU credit = 2 ECTS credits. Recall that the so-called European Credit Transfer and Accumulation System (ECTS) is a standard for comparing the study attainment and performance of students of higher education across the European Union and other collaborating European countries.

  


Course List

Below is a list of courses we are offering. We assume flexibility in choosing different courses from the list of elective courses (see below), depending on the interests of the students. However, students will be assisted in identifying sets of courses which correspond to specific areas of specialization, such as Applied Differential Equations, Applied Statistics, Financial Mathematics, Mathematical Biology, or various combinations of them.
We should also take into account the needs of the job market.
Our course offer exceeds the demand and not all the courses are taken by a given class. On the other hand, further elective courses may be added to the list, if necessary.
Occasionally, we offer non-credit introductory bridge courses to help first year M.S. students meet the level required by the program.


Mandatory Courses 

M1. Basic Algebra 1 (Pal Hegedus)

M2. Basic Algebra 2 (Pal Hegedus)

M3. Real Analysis (Laszlo Csirmaz)

M4. Complex Function Theory (Robert Szoke)

M5. Functional Analysis and Differential Equations (Gabor Elek or Gheorghe Morosanu)

M6. Introduction to Computer Science (Istvan Miklos)

M7. Probability and Statistics (Marianna Bolla)


Elective Courses

  • Matrix Computations with Applications (Robert Horvath)

  • Computations in Algebra (Pal Hegedus)

  • Cryptology (Laszlo Csirmaz)

  • Differential Geometry (Balazs Csikos)

  • Introduction to Discrete Mathematics (Ervin Gyori)

  • Graph Theory and Applications (Balazs Patkos)

  • Non-standard Analysis (Laszlo Csirmaz)

  • Difference Equations and Applications (Gheorghe Morosanu)

  • Evolution Equations and Applications (Gheorghe Morosanu)

  • Applied Partial Differential Equations (Gheorghe Morosanu)

  • Difference Methods for Partial Differential Equations (Gheorghe Morosanu)

  • Control of Dynamic Systems (Gheorghe Morosanu)

  • Combinatorial Optimization (Ervin Gyori)                                             

  • Optimization in Economics (Alexandru Kristaly)

  • Quantitative Financial Risk Analysis (Balazs Janecsko and Imre Kondor)

  • Nonlinear Optimization (Sandor Bozoki)

  • Topics in Financial Mathematics (Alexandru Kristaly)

  • Approximation Theory (Jozsef Szabados)

  • Applied Numerical Analysis (Robert Horvath)

  • Mathematical Models in Biology and Ecology (Paul Georgescu)

  • The Mathematical Theory of Infectious Disease Propagation (Paul Georgescu)

  • Evolutionary Games and Population Dynamics (Paul Georgescu)

  • Bioinformatics (Istvan Miklos)

  • Computational Neuroscience (Tamas Kiss, Krisztina Szalisznyo and Laszlo Zalanyi)

  • Probabilistic Models of the Brain and the Mind (Máté Lengyel, József Fiser)

  • Computational Number Theory (Laszlo Csirmaz)

  • Protocols (Laszlo Csirmaz)

  • Mathematical Methods in Natural Language Processing (Andras Sereny)

  • Stochastic Processes and Applications (Balazs Szekely)

  • Statistics of Stochastic Processes (Balazs Szekely)

  • Multivariate Statistical Inference (Marianna Bolla or Tamas Rudas)

  • Survey Methodology (Tamas Rudas)

  • Topics in Actuarial Statistics (TBA)


For short descriptions, see Section Syllabi below.


Schedule of Coursework

Students are required to earn a total of 45 course credits, out of which (at least) 27 credits should be taken in the first year, including all the mandatory courses M1-M7 (see the list above). Note that each M.S. regular course has 3 credits (1 credit = 12 x 50 minutes = 600 teaching minutes). The remaining necessary course credits are taken in the second year.
To acquire enough general information, among the 45 course credits, students are encouraged to earn some credits in courses from other CEU departments. They are also allowed to take some Ph.D. courses, subject to the course instructor's agreement. Every Ph.D. course is counted as an M.S. course, with the same number of credits.


Thesis

Students are required to write a thesis on a topic related to the specific area of specialization, under close supervision. The thesis should not necessarily include original results. However, since our M.S. students are supposed to become solvers of real world problems, their dissertations should address significant applications related to biological, ecological, economic, industrial, political, social phenomena. They should be able to handle mathematical and computational methods in solving specific problems. Students may also address more theoretical topics in their theses. In such a case, a good survey of a certain modern area in mathematics is expected as well as possible original contributions.
 
Normally, an M.S. student completes her/his coursework in two years, prepares the thesis during the second year, and defends it by the end of the second year. However, according to the general CEU rules, the thesis may be submitted within a maximum of two years of finishing the coursework of the program, with the director of the program’s prior agreement if this has not been done in due course. Additional courses may be offered to some of the students, beyond the first two years of studies, in order to help them write a good thesis and get better prepared in their area of interest.

Basically, our permanent faculty members serve as advisers. In the second year, when the interests of the students get more specific, they are encouraged to choose advisers who are closer to their specific area of interest among the available faculty, including adjunct professors, if they accept and have enough time for consultation.

Besides close and specialized supervision, students can benefit from workshops, seminars, summer schools. 

Normally, the thesis is defended by the end of the student’s second year. The thesis must be submitted 3 weeks in advance. The defense comprises the presentation of the thesis as well as a comprehensive exam on the area of the thesis, covering the material of 3 courses, including at least 2 electives (i.e., either 3 electives or 2 electives + 1 mandatory).


Syllabi

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