Teaching plan for the course unit

 

 

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General information

 

Course unit name: Advanced Mathematics for Scientific Challenges

Course unit code: 573764

Academic year: 2021-2022

Coordinator: Carles Casacuberta Vergés

Department: Department of Mathematics and Computer Science

Credits: 6

Single program: S

 

 

Estimated learning time

Total number of hours 150

 

Face-to-face and/or online activities

60

 

-  Lecture

Face-to-face and online

 

30

 

-  Practical exercises

Face-to-face and online

 

30

Supervised project

30

Independent learning

60

 

 

Competences to be gained during study

 

  • Capacity to understand mathematical concepts involved in problems coming from other disciplines
  • Capacity to construct mathematical models to answer questions coming from science and humanities
  • Capacity to check whether theoretical tools meet the solutions of real problems
  • Capacity to interpret results and test theoretical models through numerical methods 

 

The development of the course will include a gender perspective as much as possible

 

 

 

 

Learning objectives

 

Referring to knowledge

  • To know efficient methods for optimization problems, including linear programming and convex optimization
  • To know the basics of persistent homology for topological data analysis
  • To be able to select suitable methods for each problem depending on the constraints posed to the solution

 

 

Teaching blocks

 

1. Optimization

1.1. Elements of convex analysis

1.2. Linear optimization. The simplex method

1.3. One-dimensional optimization

1.4. Nonlinear unconstrained optimization

1.5. Nonlinear constrained optimization

2. Persistent homology

2.1. Persistent homology of a filtered simplicial complex

2.2. Barcodes and persistence diagrams

2.3. Stability theorems

2.4. Algorithms and software for topological data analysis

2.5. Dimensionality reduction

 

 

Teaching methods and general organization

 

Four one-hour classes will be given every week, including theoretical lectures and problem sessions. In some cases, working with developed software or easy programming will be required. Additional independent learning is assumed. In case of restricted attendance due to health regulations, lectures will be transmitted in streaming. Course materials and complements will be made available through Campus Virtual. 

 

 

Official assessment of learning outcomes

 

Course marks will be based on assignments and resolution of exercises. In case of online teaching due to health regulations, assessment will be based on material delivered through Campus Virtual, possibly complemented with online interviews.

 

Examination-based assessment

The single examination assessment is optional and consists of a written exam.

Reassessment also consists of a written exam. All students are entitled to reassessment. The final grade will be the highest score between course marks and examination marks, if any.

 

 

Reading and study resources

Consulteu la disponibilitat a CERCABIB

Book

S. Boyd, L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004.

http://stanford.edu/~boyd/cvxbook/  EnllaƧ

M. Minoux, Mathematical Programming: Theory and Algorithms, John Wiley & Sons, 1986.

Article

H. Edelsbrunner and J. Harer, Persistent homology: A survey, Surveys on Discrete and Computational Geometry. Twenty Years Later, 257–282 (J. E. Goodman, J. Pach, and R. Pollack, eds.), Contemporary Mathematics 453, Amer. Math. Soc., Providence, Rhode Island, 2008.

R. Ghrist, Barcodes: The persistent topology of data, Bull. Amer. Math. Soc. 45 (2008), 61–75.

https://www.ams.org/journals/bull/all_issues.html  EnllaƧ

F. Chazal and B. Michel, An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists, arXiv:1710.04019, 2017.

https://arxiv.org/pdf/1710.04019.pdf  EnllaƧ

N. Otter, M. A. Porter, U. Tillmann et al., A roadmap for the computation of persistent homology, EPJ Data Science 6:17 (2017).

https://doi.org/10.1140/epjds/s13688-017-0109-5  EnllaƧ