30758 - ALGEBRIC AND TOPOLOGICAL METHODS
Department of Decision Sciences
Course taught in English
BAI (6 credits - I sem. - OP | MAT/03) - BEMACS (6 credits - I sem. - OP | MAT/03) - BESS-CLES (6 credits - I sem. - OP | MAT/03) - BGL (6 credits - I sem. - OP | MAT/03) - BIEF (6 credits - I sem. - OP | MAT/03) - BIEM (6 credits - I sem. - OP | MAT/03) - BIG (6 credits - I sem. - OP | MAT/03) - CLEACC (6 credits - I sem. - OP | MAT/03) - CLEAM (6 credits - I sem. - OP | MAT/03) - CLEF (6 credits - I sem. - OP | MAT/03) - WBB (6 credits - I sem. - OP | MAT/03)
Course Director:
GIUSEPPE SAVARE'
GIUSEPPE SAVARE'
Suggested background knowledge
A first course in Mathematics (basic knowledge of univariate and multivariate calculus, linear algebra)
Mission & Content Summary
MISSION
Many relevant achievements in the analysis of data and in the theory of economic and computational models have their origins in topological and geometric results involving algebraic and combinatorial techniques.
Topology is that branch of mathematics which deals with qualitative geometric information and studies the essential properties of shapes, in particular those that are independent of any particular representation in coordinates and are invariant for continuous deformations. In order to deal with shapes, simplified combinatorial representations are constructed through processes such as triangulation. The intuitive idea is that one can try to distinguish or perhaps even characterise shapes based on the occurrences of patterns within a space: they are described by algebraic objects that allow for computations.
Particularly prominent applications are fixed point theorems, game theory equilibria, persistent homology for the study of forms and data.
This course provides an introduction to the basic concepts and aims at making elementary topological and algebraic methods more easily accessible. It presents a number of relevant results in algebra, combinatorics and geometry, and provides the necessary background in algebraic topology, explaining their main applications.
CONTENT SUMMARY
Fundamentals of Topology
- Topological spaces and basic operations
- Continuity
- Connectedness and compactness
Towards Algebraic Topology: Group Theory
- Group structures
- Group actions and quotients
- Topological groups
Foundations of Algebraic Topology
- Homotopy and retractions
- The fundamental group
- Covering spaces
- Basics of homology theory: Loop spaces on graphs, Euler’s formula, simplicial complexes
Applications
- Sperner Lemma and Brouwer’s Fixed Point Theorem
- KKM theory and fixed-point theorems for multivalued maps. Applications to general equilibrium and game theory.
- The Borsuk–Ulam Theorem and Tucker’s Lemma (discrete version)
- Persistent homology in data analysis
Intended Learning Outcomes (ILO)
KNOWLEDGE AND UNDERSTANDING
At the end of the course student will be able to...
- Recognize and understand the main algebraic structures.
- Understand the topological perspective and become familiar with basic topological concepts.
- Understand the fundamental algebraic structures associated with topological spaces.
- Know the main results related to continuous maps, their topological properties, and applications to fixed-point theory.
APPLYING KNOWLEDGE AND UNDERSTANDING
At the end of the course student will be able to...
- Compute algebraic invariants such as homology groups and the fundamental group in simple cases.
- Analyze the geometric and topological structure of data.
- Apply homology theory techniques to explore and understand data structures.
- Use fixed-point theorems and multivalued maps to formulate and solve mathematical problems in applied contexts.
Teaching methods
- Lectures
- Practical Exercises
- Individual works / Assignments
DETAILS
Some exercises will be solved during the lectures, with the aim of applying the main theoretical results to different problems.
Assessment methods
| Continuous assessment | Partial exams | General exam | |
|---|---|---|---|
|
x | x |
ATTENDING AND NOT ATTENDING STUDENTS
Partial Exams: Students may opt to take two written exams—one at the midpoint of the course and one at the end. The final grade will be the average of the two scores.
General Exam: written exam at the end of the course that contributes to the overall assessment.
Each written exam includes both simple problem-solving exercises and open-ended questions related to the theoretical results discussed in class.
Teaching materials
ATTENDING AND NOT ATTENDING STUDENTS
- Lecture notes of the course
- M.A. Amstrong: Basic topology. Springer, 1983.
- M. Manetti: Topology. Springer, Cham, 2015.
- K. Parthasarathy: Topology. An introduction. Springer 2022.
Further references:
- J. Matousek: Using the Borsuk-Ulam Theorem, Lectures on Topological Methods in Combinatorics and Geometry. Springer, 2008.
- H.Edelsbrunner, J.L. Harer: Computational Topology. An Introduction. American Mathematical Society, Providence, 2010.
Last change 02/06/2025 19:44