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Models of Applied Mathematics

General data

Course ID:	1000-135MMS
Erasmus code / ISCED:	11.913 Kod klasyfikacyjny przedmiotu składa się z trzech do pięciu cyfr, przy czym trzy pierwsze oznaczają klasyfikację dziedziny wg. Listy kodów dziedzin obowiązującej w programie Socrates/Erasmus, czwarta (dotąd na ogół 0) – ewentualne uszczegółowienie informacji o dyscyplinie, piąta – stopień zaawansowania przedmiotu ustalony na podstawie roku studiów, dla którego przedmiot jest przeznaczony. / (0619) Information and Communication Technologies (ICTs), not elsewhere classified The ISCED (International Standard Classification of Education) code has been designed by UNESCO.
Course title:	Models of Applied Mathematics
Name in Polish:	Modele matematyki stosowanej
Organizational unit:	Faculty of Mathematics, Informatics, and Mechanics
Course groups:	Elective courses for 1st degree studies in mathematics
Course homepage:	https://www.mimuw.edu.pl/~miekisz/mms.html
ECTS credit allocation (and other scores):	6.00 Basic information on ECTS credits allocation principles: the annual hourly workload of the student’s work required to achieve the expected learning outcomes for a given stage is 1500-1800h, corresponding to 60 ECTS; the student’s weekly hourly workload is 45 h; 1 ECTS point corresponds to 25-30 hours of student work needed to achieve the assumed learning outcomes; weekly student workload necessary to achieve the assumed learning outcomes allows to obtain 1.5 ECTS; work required to pass the course, which has been assigned 3 ECTS, constitutes 10% of the semester student load. view allocation of credits
Language:	Polish
Type of course:	elective courses
Prerequisites (description):	elementary probability theory, ordinary differential equations
Short description:	The aim of this course is to describe various models in applied mathematics in order to help students to plan their master studies and to choose a subject of a future master thesis. We will discuss several mathematical models in physics, biology, economy and social sciences.
Full description:	The aim of this course is to describe various models in applied mathematics in order to help students to plan their master studies and to choose a subject of a master thesis. We will discuss several mathematical models in physics, biology, economy and social sciences (see below). Each example will begin by a brief presentation of a concrete problem stated in a language of a given scientific discipline (we do not assume a previous knowledge of physics, biology, economy, etc.) An appropriate mathematical model (a recurrence equation, a system of ordinary differential equations, a Markov chain) will be constructed. We will then analyze the model. We will end by a discussion of obtained results and a criticism of the model. Possible generalizations and open problems will be presented. Models 1. Fluctuations of the number of protein molecules produced in living cells (birth and death stochastic processes) 2. Valuation of European call options in the binomial model (a present value of money, conditional expected value) 3. Prisoner's Dilemma, Tragedy of Commons - Nash equilibria in game theory 4. Phase transitions in ferromagnetic models, spontaneous symmetry breaking in the Ising model (discrete probability) 5. Systems of ordinary differential equations in ecology (qualitative theory of ordinary differential equations, limit cycles)
Bibliography:	Reading material will be posted on the internet and/or given in the form of hand-outs during the course.
Learning outcomes:	Knowledge and Competence 1. Student knows basic mathematical models of gene expression, he/she is able to compute variance of the number of of protein molecules in the stationary state. 2. Student knows ferromagnetic Ising model, he/she is able to compute magnetization in simple lattice models. 3. Student is able to find Nash equilibria in matrix games and games with continuous strategy spaces. 4. Student knows how to construct mathematical models based on physical, biological and social texts. Social competence Student is able to talk with biologists, physicists, and economists.
Assessment methods and assessment criteria:	Grade based on homeworks 20%, midterm 20% and final exam 60%

Classes in period "Winter semester 2023/24" (past)

Time span:	2023-10-01 - 2024-01-28	Selected timetable range: weekly course term Navigate to timetable MO TU WYK CW W TH FR
Type of class:	Classes, 30 hours more information Lecture, 30 hours more information
Coordinators:	Jacek Miękisz
Group instructors:	Jacek Miękisz
Students list:	(inaccessible to you)
Examination:	Examination
Full description:	In the academic year 2023/2024, the following models will be discussed: 1. Phase transitions in ferromagnetic models, spontaneous symmetry breaking in the Ising model (discrete probability) 2. Fluctuations of the number of protein molecules produced in living cells (birth and death stochastic processes) 3. Prisoner's Dilemma, Tragedy of Commons - Nash equilibria in game theory

Classes in period "Winter semester 2024/25" (future)

Time span:	2024-10-01 - 2025-01-26	Selected timetable range: weekly course term Navigate to timetable
Type of class:	Classes, 30 hours more information Lecture, 30 hours more information
Coordinators:	Jacek Miękisz
Group instructors:	Jacek Miękisz
Students list:	(inaccessible to you)
Examination:	Examination
Full description:	In the academic year 2023/2024, the following models will be discussed: 1. Phase transitions in ferromagnetic models, spontaneous symmetry breaking in the Ising model (discrete probability) 2. Fluctuations of the number of protein molecules produced in living cells (birth and death stochastic processes) 3. Prisoner's Dilemma, Tragedy of Commons - Nash equilibria in game theory

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