MathMods :: Joint MSc

Sem3 UGR Biomathematics

granada logo smallApplications  @  UGR  30 ECTS credits

Mathematical modelling. Biomathematics

Description coming soon...

 

Below you can find information about the subjects for this semester.

 

  • Numerical Analysis of PDEs and Approximation [6 credits]

    Numerical Analysis of PDEs and Approximation

    • ECTS credits 6
    • University University of Granada
    • Semester 3
    • Topics

       

      Review of Numerical Methods for Solving Partial Differential Equations and Approximation Theory, according to the following program:
      • Topic 1: Finite difference methods for classical equations and conservation laws. Finite element and finite volume methods. Spectral methods.
      • Topic 2: Approximation theory. Special functions. Orthogonal polynomials.
      • Topic 3: Bézier curves. Spline functions. B-splines. Applications.
      Practice 1: Numerical resolution of boundary value problems.
      Practice 2: Numerical resolution of evolution problems.

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  • Transport PDEs in Kinetic Theory and Fluid Mechanics [6 credits]

    Transport PDEs in Kinetic Theory and Fluid Mechanics

    • ECTS credits 6
    • University University of Granada
    • Semester 3
    • Topics

       

      1 Transport Models:

      Conservation laws (fluids, traffic, etc.) and kinetic models (Liouville, Vlasov, Boltzmann equations).

      2 Linear Transport Equations:

      Initial value problems, first-order equations with regular and singular fields, characteristic equations, and associated dynamical systems.

      3 Introduction to Nonlinear Scalar Conservation Laws:

      Riemann-Hugoniot conditions and admissibility criteria for singularities.

      4 Introduction to Fluid Mechanics Equations:

      Fundamental equations governing fluid dynamics.

      5 The Liouville Equation in Kinetic Theory:

      Derived models: free transport equation, Vlasov-Poisson systems, Vlasov-Maxwell equations, and Boltzmann and Vlasov-Poisson-Fokker-Planck equations.

      6 Generalities on the Vlasov-Poisson System:

      Invariances and conserved quantities, a priori estimates, moment control, weak formulation, moment lemmas, and existence results. Asymptotic behavior in the repulsive case: pseudoconformal law.

      7 Orbital Stability of Galaxies:

      Scattering in gravitational systems and study of polytropes.

      8 Study of Coupled Vlasov-Maxwell Models:

      Analysis of relativistic kinetics and the interaction between charged particles and electromagnetic fields.

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  • Nonlinear Analysis and Differential Equations [6 credits]

    Nonlinear Analysis and Differential Equations

    • ECTS credits 6
    • University University of Granada
    • Semester 3
    • Topics

       

      1 Sobolev Spaces:

      Weak derivatives, duality of Sobolev spaces.Weak topology in Sobolev spaces.Sobolev inequalities and the Rellich theorem.

      2 Eigenvalues and Eigenfunctions of the Laplacian:

      Analysis of spectral properties associated with the Laplace operator.

      3 Differential Calculus in Banach Spaces:

      Fundamental concepts and applications of calculus within the framework of Banach spaces.

      4 Implicit Function Theorem and Bifurcation Theory in Banach Spaces:

      Examination of the implicit function theorem and its implications for bifurcation phenomena in higher-dimensional spaces.

      5 Overdetermined Elliptic Problems:

      Study of elliptic partial differential equations with an excess of boundary conditions.

      6 Moving Plane Method and Symmetry Results:

      Application of the moving plane method to obtain symmetry properties of solutions to differential equations.

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A choice between (a total of 12 ECTS)

  • Mathematical Models in Ecology [6 credits]

    Mathematical Models in Ecology

    • ECTS credits 6
    • University University of Granada
    • Semester 3
    • Topics

       

      Topic I: Introduction to Mathematical Models in Ecology.
      Topic II: Continuous Dynamics of a Single Species.
      Topic III: Species Interactions. Predator-Prey Models.
      Topic IV: Infections. Basic Epidemiological Models.
      Topic V: Discrete Dynamics.

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  • Mobility and Cell Dynamics: Introduction to Tumor Dynamics and Growth [6 credits]

    Mobility and Cell Dynamics: Introduction to Tumor Dynamics and Growth

    • ECTS credits 6
    • University University of Granada
    • Semester 3
    • Topics

       

      Topic 1: Collective Behavior of Species.

      Microscopic Models: Agent-Based.Meso/Macroscopic Models: Kinetic and Hydrodynamic.Micro-Macro Scaling Limits: Mean-Field and Hydrodynamic Limits.Species Dynamics: Reynolds, Visek, Cucker-Smale Models, Aggregation, etc.Synchronization Processes: Kuramoto Model.

      Topic 2: Introduction to Cellular Mobility Processes.

      Mobility and Differentiation.Chemotaxis and Bioconvection Processes. Keller-Segel Model and Variants.Random Walkers and Macroscopic Descriptions.Central Limit Theorems and Anomalous Diffusion.

      Topic 3: Cellular Communication.

      Morphogenesis. Pattern Formation. Cellular Response.Signaling Pathways and Target Genes. Application to the Shh-Gli Pathway.Pattern Formation. Traveling Waves.Critical Analysis of Diffusive Models as Transport Engines.Optimal Mass Transport and Nonlinear Models for Morphogenesis.Synthetic Biology: Biobricks, Creation of Smart Drugs, Gene Therapy, Tissue Regeneration, Protein Design, and Bioplastics.

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  • BIOMAT Course [6 credits]

    BIOMAT Course

    • ECTS credits 6
    • University University of Granada
    • Semester 3
    • Topics

       

      The course is structured around several minicourses and guest seminars, addressing various contemporary issues in the Life Sciences and Social Sciences. The duration of each session will be tailored to the specific topic discussed. Examples of subjects to be covered include: recent findings on complex systems and emergent behaviors in biomedicine and social sciences; nonlinear modeling of interactions among groups of individuals; analysis of collective versus individual behavior; examination of pattern formation in collective processes; cellular communication; application of game theory processes to individual interactions as a driver for modeling cooperation or competition among species; chemotaxis and quorum sensing; economic agent models; cellular movement; inmune system, among others.

      For a historical overview of the topics covered, please refer to the following website:

      Modeling Nature Training

      The minicourses and seminars will be held intensively throughout the semester. Additionally, the course will be complemented with an intensive session in June. The theme for the 2024-2025 course will be anti-aging and cancer, focusing on both cellular modeling and implications for the immune system and inflammatory processes.

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  • Physics of Complex Networks and Interdisciplinary Applications [6 credits]

    Physics of Complex Networks and Interdisciplinary Applications

    • ECTS credits 6
    • University University of Granada
    • Semester 3
    • Topics

       

      Topic 1: Brief Introduction to Complex Systems. Concept of Complex Network.

      Topic 2: Complex Networks as an Example of Complex Systems:

      Concept of Graph, Adjacency Matrix, Random Graphs, Directed and Undirected Networks, Weighted Networks.Node Distribution. Scale-Invariant Networks. Small-World Networks.Node-Node Correlations: Pearson Coefficient, Assortative and Disassortative Networks.Modular Structure. Hierarchical Networks. Multiplex Networks.

      Topic 3: The Brain as a Paradigm of System and Complex Network:

      Structure and Function: Connectivity Matrices (DTI) and Activity (Multielectrodes, EEG, MEG, fMRI).

      Topic 4: Neural Networks:

      Concept of Neural Network, Neuronal Activity Models and Synaptic Transmission, Hodgkin-Huxley Model, Integration and Firing Models, Binary Neuron Models.Synaptic Models: Alpha Function, Excitation and Inhibition, Exponential Models, Dynamic Synapses, Tsodyks-Markram Model.Attracting Neural Networks: Amari-Hopfield Model, Hebbian Learning, Capacity of a Neural Network.Feed-Forward Networks: Perceptron. Firing Rate Models: Wilson-Cowan Model. Fokker-Planck Type Models.Balanced Neural Networks: Homeostatic Balance in Complex Neural Networks, UP/DOWN States in the Cortex.

      Topic 5: Concept of Connectome:

      Construction of Connectomes, Structural Properties of Brain Connectomes, Comparison of Connectomes, Computational Properties.

      Topic 6: Social Networks:

      Statistical Physics Methods in the Context of Social Models.Basic Concepts: Order and Disorder, Ising Model, Importance of Topology (Scale-Free and Bounded Networks), Glauber Dynamics.Social Phenomena as Cooperative/Emergent Phenomena.Social Dynamics Models: Opinion Dynamics and Cultural Dynamics: Axelrod Model.Social Networks and the Internet. Epidemic Models and Virus Propagation in Networks.

      Topic 7: Networks in Ecology:

      Trophic Networks. Stability and May's Paradox. Mutualistic Networks.Nesting Properties and Other Structural Properties.

      Topic 8: Networks in Systems Biology:

      Concept of Gene Regulatory Networks.Cancer and Systemic Diseases. Random Boolean Networks.Genotype vs. Phenotype. Attractors and Phenotypes.

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Home Structure Semester 3 UGR Mathematical modelling. Biomathematics

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