Models of Applied Mathematics 1000-135MMS
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)
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Term 2024Z:
In the academic year 2024/2025, the following models will be discussed: 1. Phase transitions in ferromagnetic models, spontaneous symmetry breaking in the Ising model (discrete probability) |
Term 2025Z:
In the academic year 2024/2025, the following models will be discussed: 1. Phase transitions in ferromagnetic models, spontaneous symmetry breaking in the Ising model (discrete probability) |
Course coordinators
Type of course
Prerequisites (description)
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 criteria
Grade based on homeworks 20%, midterm 20% and final exam 60%
Bibliography
Reading material will be posted on the internet and/or given in the form of hand-outs during the course.
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes: