Singularity Theory

Code:

M4023

Acronym:

M4023

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2015/2016 - 2S

Active?	Yes
Web Page:	https://moodle.up.pt/course/view.php?id=1774
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Master in Mathematics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
M:M	3	Plano de Estudos do M:Matemática	1	-	6	56	162

Teaching language

Suitable for English-speaking students

Objectives

Introduction to results in singularity theory, applications to other sciences.

Use of new mathematical  techniques, and their application in the analysis of models in other sciences.

Learning outcomes and competences

Use of new mathematical  techniques, and their application in the analysis of models in other sciences.

Working method

Presencial

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

Calculus in several variables; algebra: basic notions of algebraic structures, specially rings and ideals; topology of finite dimensional vector spaces; basic notions of differential geometry of curves and surfaces; ordinary differential equations.

Program

Presentation of the subject:
The idea is to generalise to functions of several variables the following information:
We know that for a function f of one real variable, if the first 2n derivatives of f at a point p are zero, and if the (2n+1)-th derivative is non-zero, then the graph of f has an inflection point at p. If the non-zero derivative of f of lowest order has even order, then f has either a local maximum or a local minimum at p.
Moreover, we know that for the function f(x)=x^3, that has an inflection point at x=0, there are perturbations g(x)=x^3+ax^2+bx, of f, with a and b arbitrarily small, for which g(x) has 2 critical points. For f(x)=x^n there are also perturbations g(x) with arbitrarily small coefficients having n-1 critical points.
The generalisation of these results to functions of 2 or more variables uses algebraic structures such as rings and modules.
The results have applications in Physics, Biology, Engineering, Economics, etc.


Program:
Review of differentiable manifolds; maps between manifolds.
Equivalences of maps: germ-equivalence, jet-equivalence, equivalences arising from changes of coordinates.
The Whitney C^k weak and strong topologies in the space of smooth maps and in jet space.
Transversality of submanifolds and of map-germs, Thom's transversality Theorem.
The algebra of germs of smooth functions; its unique maximal ideal; Hadamard's lemma;
the space of jets as a quotient of this algebra by a power of the maximal ideal.
The local algebra of a map-germ, multiplicity of the map-germ.
Determinacy of function-germs: the jacobian ideal, finite determinacy criteria;
Morse's lemma, splitting lemma, Tougeron's theorem for determinacy.
Classification theorems: Arnold's classification of simple function-germs,
Thom classification by codimension.
Unfolding theory: versal unfolding, existence and construction of versal unfoldings of a singularity of finite codimension.
Applications.

Mandatory literature

Poston Tim; Catastrophe theory and its applications. ISBN: 0-273-01029-8
Bruce J. W.; Curves and singularities. ISBN: 0-521-24945-7
Brocker Th.; Differentiable germs and catastrophes. ISBN: 0-521-20681-2
Alexei Davydov; Teoria das Singularidades, CMUP, 2000 (http://cmup.fc.up.pt/cmup/cmup_divulgacao.html)

Complementary Bibliography

Poston Tim; Zeeman.s catastrophe machine

Teaching methods and learning activities

Presentation of topics in lectures.
Exercises presented in the course's homepage, discussed in lectures.

If there are less than 3 students: supervised reading, discussion of exercises.

Evaluation Type

Distributed evaluation without final exam

Assessment Components

designation	Weight (%)
Participação presencial	25,00
Teste	75,00
Total:	100,00

Calculation formula of final grade

For approval the student should obtain a total of 9,5 points adding the marks of the two tests and the oral presentation of an application. The scales are: first test 0 to 5; presentation 0 to 5; second test: 0 to 10.

A minimum of 9.5 in the exam is required for approval.

The final mark is either the sum of the marks obtained in the two tests and presentation or the result of the final exam.

Special assessment (TE, DA, ...)

exam