# Applied mathematics and mathematical statistics

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1 Applied mathematics and mathematical statistics The graduate school is organised within the Department of Mathematical Sciences.. Deputy head of department: Aila Särkkä Director of Graduate Studies: Marija Cvijovic Syllabus (approved by the vice-rector for research education on October 12, Reference number C ) 1. Description of subject and goals The purpose of the graduate school in applied mathematics and mathematical statistics is to give the student fundamental knowledge within applied mathematics, orientation about current problems and applications, a deeper insight into one or several parts of the subject, and the ability to independently carry out research work. The aim of the program until the licentiate degree is to give the student the ability to independently take part in research and development work. The aim of the program until the doctoral degree is to give the student the ability to critically and independently plan, lead, carry through, and present research and development work. The studies within the school are typically performed in connection to research topics that are actively pursued at the faculty. Example of such topics are: Computational mathematics Computational mathematics is a field that studies problems both in pure and applied mathematics, using methods based on a synthesis of mathematical analysis and numerical/symbolical computation. Computational mathematics treats the whole process from mathematical model to computer implementation. Development and analysis of computational algorithms are crucial components. Questions considered include stability, convergence, and efficiency of computational methods. An important area in computational mathematics is the modelling and numeric of partial differential equations. This area includes the construction and analysis of efficient computational algorithms in numerical linear algebra, methods of discretization such as the finite element method, as well as different aspects of high performance computing such as adaptivity and effective use of parallel computer architectures. Common applications in computational mathematics are structural mechanics, fluid mechanics, biomedicine, architecture, and mathematical physics. Modelling of kinetics and dynamical system Kinetic and dynamical models are tools for studying systems whose state change over time. A prime example of this is the Boltzmann equation that describes the position and velocity of molecules in a gas. Research on this topic consists both of formulating models

2 based on reasonable assumptions, and analyzing the behavior of such models. The models usually take the form of ordinary or partial differential equations, but discrete models also exist. In order to study these, a combination of computer simulation and mathematical theory is employed, e.g. functional analysis, stochastic analysis and measure theory. The questions investigated are often motivated by the domain of application and include stability of solutions, stationary (time-independent) solutions and asymptotic behavior. Examples of applications are flocking behavior among animals, bacterial movement and the growth of tumors. Mathematical statistics Mathematical statistics is used to describe, analyze and predict random events and uncertainty in data. The subject comprises two main subparts: probability theory, that provides the theoretical foundation and deals with the underlying stochastic processes and mathematical models of random phenomena, and statistical inference, that treats all aspects of data management, such as the collection, organization, analysis and presentation of data. The access to new technologies for data collection and the rapidly increasing possibilities of virtual data sharing has led to a massive increase in available data within a number of fields, something that places an ever increasing demand on the access to efficient and robust analysis tools. Thereby, mathematical statistics constitutes a fundamental component within most sciences. Many of the researchers within the department work interdisciplinary, examples being the identification of functional elements in DNA, the movement of particles in particle systems, spatial processes in climate and material sciences, genetic epidemiology of inheritable diseases, characteristics of the growth and extinction of populations, analysis and prediction of traffic data, and models of cell signaling systems. Optimization Mathematical optimization is an applied mathematical field comprising modelling, theory, and solution methodology for decision problems. Since the subject is closely connected with applications, also the modelling of real problems in a mathematical form is central. The development of theory as well as methodology has over the years walked hand in hand with the development of computers and software. The goal of the postgraduate education is to give a broad overview of the research field and an advanced scientific training in a specific research area within the field. The research in the research group of mathematical optimization comprises theory, modelling, and methodology development for large-scale structured linear, non-linear, integer, and combinatorial optimization, as well as graph and polyhedral theory. The research spans a wide field from fundamental mathematical research to more applied research in cooperation with industrial companies. Applications include scheduling of industrial production and maintenance activities, simulation based design optimization, planning of electricity production and distribution, safety in electrical networks, and traffic and transport planning.

4 4.1. Course requirements For the doctoral degree, the course work should amount 90 credits of which 30 credits should come from common courses and 22.5 credits should come from specialized courses. Additionally, 15 credits should be from courses in Generic and Transferable Skills (GTS). The remaining credits (22.5) should be from elective courses. For the licentiate degree, the course work should amount 45 credits of which 15 should come from common courses and 7.5 from specialized courses. Additionally, 9 credits should be from courses in Generic and Transferable Skills (GTS). The remaining credits (13.5) should be from elective courses Common courses The common courses can be selected from three main topics. For the doctoral degree, at least 30 credits should come from common courses (i.e. four complete courses) where at least one course comes from each of the three main topics. For the licentiate degree, at least 15 credits should come from common courses (i.e. two complete courses) from at least two of the three main topics. Computational mathematics Ordinary and partial differential equations Numerical linear algebra Mathematical statistics Statistical inference Stochastic processes Optimization Linear and non-linear optimization Combinatorial optimization 4.3 Specialized courses The specialized courses aim to give the student a deeper understanding within specific topics. The specialized courses are selected by the student in consultation with the examiner and supervisor and can partially consist of self-study courses. Example of topics for the specialized courses include Computational mathematics The finite element method and its implementation Stochastic partial differential equations Computational geometry Multiscale methods Wavelet analysis

5 Integration theory Functional analysis Geometric integration Mathematical statistics Statistical inference Linear regression models Bayesian inference Analysis of time series Experimental design Stochastic processes, Markov theory and queueing theory Integration theory Weak convergence Martingales Optimization Linear and non-linear optimization Convex analysis Discrete/integer optimization Combinatorial optimization Optimization on graphs and networks Simulation based optimization Optimization under uncertainty Multiple objective optimization Large-scale optimization 4.4 Generic and Transferable Skills Generic and Transferable skills (GTS) aims to give doctoral students at Chalmers professional and individual development, and is a program of activities/courses not directly linked to the respective areas of research. For the doctoral degree, 15 credits should be from courses in Generic and Transferable Skills (GTS) of which 9 credits should be obtained for the licentiate degree. In addition to the courses within Generic and Transferable Skills, the student is also required to participate in the introduction day for doctoral students (before the licentiate examination, at latest). Further requirements are an oral popular science presentation to be performed prior to the PhD thesis defence and a written popular science presentation to be published on the back of the PhD thesis.

6 Get more information: General Introduction for Doctoral Students Graduate Courses - Generic and Transferable Skills 5. Thesis 5.1. Licentiate thesis A licentiate thesis requires that the scientific work is presented in the form of a report corresponding to at least 75 credits. The thesis should be presented at a public seminar. The thesis is graded with pass or fail Doctoral thesis A doctoral thesis requires that the scientific work corresponding to 150 credits is presented in the form of a report and defended publically. The thesis should be of such quality that it fulfils the standard requirements for publication, either in its entirety or in abridged form, in a scientific journal of good quality. The dissertation is graded with pass or fail. 6. Requirement for the degree 6.1. Licentiate degree A licentiate degree consists of 120 credits. The requirements for the licentiate degree include 1. completion of a licentiate thesis corresponding to 75 credits (as described above), 2. completion of a study course corresponding to 45 credits distributed over courses according to Section Doctoral degree A doctoral degree consists of 240 credits. The requirements for the doctoral degree include 1. completion of the doctoral thesis corresponding to 150 credits (as described above), 2. completion of a study course corresponding to 90 credits distributed over courses according to Section Degree names The names of the degrees within the graduate school in applied mathematics and mathematical statistics are

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