Academics / Courses / DescriptionsES_APPM 495: Selected Topics
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Description
Geophysical and astrophysical fluid dynamics
Fluid dynamics plays a central role in many natural systems ranging from the Earth’s atmosphere and oceans, rocky and liquid metal planetary interiors, to the gaseous interiors of stars and giant planets. In each of these systems, flows are driven, and affected, by the competing action of buoyancy, rotation, gravity, radiation, and magnetic fields. This course will present the typical assumptions made when modeling these systems, followed by an investigation of some of the important waves, instabilities, and turbulence present in geophysical and astrophysical fluids. These topics will be investigated with a combination of theoretical and numerical techniques, but no background in numerical analysis will be assumed or required. Specific topics may include: Navier—Stokes equations and the Boussinesq & Anelastic approximations; internal gravity waves; inertial waves; convection; baroclinic instability; shear instability; tides & parametric subharmonic instability; homogeneous isotropic turbulence; rapidly rotating turbulence; and, wave turbulence.
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Active matter: Modeling of active fluids
Flocks of birds and schools of fish are familiar examples of emergent collective behavior, where interactions between self-driven (active) individuals lead to coherent motion (flow) on a scale much larger than the isolated unit. In recent years, similar phenomena have been observed with active micro-units such as bacteria, chemically-activated motile colloids, even microtubule–kinesin bundles (active nematics)
Active matter is a HUGE field. This course will focus on the hydrodynamics of microswimmers (bacteria, motile colloids). Topics include Stokes equation, scallop theorem, multipole expansion, rheology of active suspensions, active turbulence.
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Introduction to Calculus of Variations
Euler-Lagrange equation, extremals, fixed and variable endpoints, constrained problems, multidimensional problems, applications.
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Introduction to Integral Equations
Fredholm and Volterra equations, Hilbert-Schmidt kernel, Neumann series, eigenvalue problems, adjoint operators, Fredholm alternative theory, connections to differential equations.
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Introduction to Network Science
This five week module will provide a brief introduction to the study of complex networks, with applications drawn from physics, biology, and the social sciences. Topics to be covered include weights, degree distributions, shortest paths, centrality, clustering, community structure, small-world networks, random networks, and dynamics (as time permits). No required text
Prerequisites: Vector calculus. Linear algebra and programming experience helpful, but not required.
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Modeling Biological Oscillations
This 0.5-credit class discusses various types of oscillations in biological systems, focusing on
how to develop mathematical models for such systems. Topics include oscillations in animal
populations, genetic oscillators, circadian rhythms, and cellular Calcium oscillations.
The class is aimed at advanced undergraduates (juniors/seniors) and first-year graduate students. No textbook required; lecture notes are provided.
Prerequisites: Differential equations and some experience in programming.
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Modeling Metabolic Networks
The goal of this course is for students to understand and use the mathematical tools needed to model and analyze metabolic networks. Students will learn the physical and chemical assumptions which lead to different mathematical descriptions including, thermodynamics, the chemical master equations, and mass-action kinetics. We will study standard and state of the art techniques for analyzing systems of kinetic equations and their mass-conservation and thermodynamic constraints including optimization and model reduction techniques for network simplification. We will also touch on asymptotic analysis, reaction-diffusion systems, dynamic simulation, data-based methods, and identifiability analysis. Class will be a mix of lecture and discussion of primary literature with project-based assignments as the main evaluation. No textbook required; lecture notes will be provided.
Prerequisites: An undergraduate level understanding of analytical analysis and numerical simulation of ordinary differential equations is required. Experience in biological systems, thermodynamics, and nonlinear dynamics may be helpful but is not required.
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Modeling Social Systems
This five-week module will present a survey of mathematical models for social systems. Those systems may describe human or animal behavior, evolutionary dynamics, or engineered systems, among other things; the common thread is that coupling between individuals (social effects) is key to system behavior. No required text
Prerequisites: Vector calculus and ordinary differential equations. Programming experience helpful, but not required.
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Parameter Estimation for Dynamical Systems
Dynamical systems have specific parameter identifiability challenges which make parameter estimation from experimental data challenging. In this module we will discuss theory for parameter identifiability and information geometry, and apply methods for nonlinear parameter estimation to dynamical systems with different structural characteristics. We will focus on a variety of dynamical systems including classical oscillators, stiff ODEs with fixed points (which characterize biological regulatory and metabolic networks), and chaotic systems.
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Soft Matter: Modeling of cell membranes
Cells and cellular organelles are enveloped by membranes made of lipid bilayers. In this course, we will discuss the mechanical properties of the lipid bilayer (large lateral stability, fluidity, small resistance to bending) and model their role in shaping cells (e.g., why is the red blood cell a biconcave disk?) and in processes such as membrane remodeling and mechanosensing.
Topics include geometry of surfaces, Monge parametrization, membrane curvature elasticity, Helfrich theory, the shape equation of vesicles, thermal shape fluctuations-------------------------------------------------------------------------------------------------------------------
Topics in Computational Neuroscience
This 0.5-credit class investigates a range of aspects of neuronal networks. They include sensory
processing and its understanding within the frameworks of efficient, sparse, and predictive
coding and the roles of top-down (centrifugal) inputs; cortical networks, their irregular dynamics
arising from excitation-inhibition balance, and the emergence of rhythmic activity; place cells
and grid cells in the navigation system. The class is aimed at advanced undergraduates (juniors/seniors) and first/second-year graduate students. No textbook required; lecture notes are provided.
Prerequisites: Differential equations and some experience in programming.