Mathematical Methods in the Sciences

Code:

M3012

Acronym:

M3012

Level:

300

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2018/2019 - 1S

Active?	Yes
Web Page:	http://cmup.fc.up.pt/cmup/complexity
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Mathematics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:B	0	Official Study Plan	3	-	6	56	162
L:CC	0	Plano de estudos a partir de 2014	2	-	6	56	162
			3
L:F	5	Official Study Plan	2	-	6	56	162
			3
L:G	0	study plan from 2017/18	3	-	6	56	162
L:M	40	Official Study Plan	2	-	6	56	162
			3
L:Q	0	study plan from 2016/17	3	-	6	56	162

Teaching language

Suitable for English-speaking students

Objectives

In the academic year 2018/19, the Mathematical Models in Science course will be devoted to mathematical models to understand complex systems, that is, systems consisting of an enormous number of "individuals" that interact with each other, giving rise to emerging phenomena, not explainable only by the individual characteristics of each. In other words systems in which "the whole is much greater than the sum of the parts". Systems of this type are very frequent in Physics, Biology, Sociology, Ecology, Epidemiology and other areas of knowledge.

The aim is therefore to construct a conceptual (and formal) framework to explain how interactions between the (microscopic) elements of a system can lead to cooperative phenomena, and emerging properties of process dynamics. This strategy, which allows us to move from microscopic interaction to emergent collective phenomena characteristic of all Complex Systems, is strongly inspired by the methodology of Statistical Physics. It is seen as a general paradigm of the passage from the site to the large-scale global properties of complex systems, and has served as a motivation for many areas of mathematics (dynamical systems, nodes theory, enumerative geometry, and others).

The mathematical models used are vast. From information theory, entropy, random fields, Gibbs measurements, statistical physics models, percolation, cellular automata, agent modeling, and many others, all using "classical" mathematical methods, which will be reviewed during class.

Several applications will be addressed to Mathematics and Natural Sciences described above. The course does not presuppose any background in Physics, Biology or other sciences.

Learning outcomes and competences

It is intended that the examples selected for the program of this curricular unit allow to appreciate the use of mathematics in other contexts, namely, Mathematics and Natural Sciences, described above. Students should acquire autonomy and critical sense in the use of models and resources in the various applications of mathematics.

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

Students are expected to reveal consolidated knowledge in the various areas that are studied in a degree in Mathematics.

Program

1. Information and Entropy
2. Models of Statistical Physics. A formal framework for analyzing complex systems
3. The Ising Model. Phase transition.
4. Cellular automata. Percolation. Phase transitions.
5. Complex Networks
6. Agent modeling
7. Criticality, self-organization and universality
8. Applications to Physics and Biology and Ecology

Mandatory literature

Barrat Alain; Dynamical processes on complex networks. ISBN: 978-0-521-87950-7
Grimmett Geoffrey; Percolation. ISBN: 3-540-64902-6
Lesne Annick; Renormalization methods. ISBN: 0-471-96689-4

Comments from the literature

Detailed notes will be provided on the classes, written by the teacher responsible

Teaching methods and learning activities

The classes will have a theoretical part explained by the teacher and a part of solving exercises and analyzing examples of application of the theory to various concrete situations in the areas of Physics, Biology, Ecology and others. The course does not presuppose any prior background in these areas.

keywords

Physical sciences > Mathematics > Applied mathematics
Physical sciences > Mathematics > Chaos theory
Physical sciences > Physics > Statistical physics

Evaluation Type

Distributed evaluation with final exam

Assessment Components

designation	Weight (%)
Apresentação/discussão de um trabalho científico	60,00
Exame	40,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	100,00
Frequência das aulas	52,00
Trabalho escrito	10,00
Total:	162,00

Eligibility for exams

The approval of the course requires a higher grade than 10 values. The final classification will be obtained according to the following weights:
Presentation / discussion of a group work with final presentation: 60%; Final written exam: 40%.

Calculation formula of final grade

The final classification will be obtained according to the following weights:
Presentation / discussion of a group work with final presentation: 60%; Final written exam: 40%.

Classification improvement

The improvement of classification is made in final exam at the time of appeal