Applied Mathematics and Modeling (AMM)
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1 Applied Mathematics and Modeling (AMM) Applications of mathematics to the applied sciences Department of Mathematics University of Nebraska-Lincoln 4th March 2006
2 Outline Introductions 1 Introductions Faculty Students 2
3 Faculty Steve Cohn partial differential equations, Schrodinger equations, engineering applications (Courant Institute) Steve Dunbar differential equations, financial mathematics, biology, stochastic analysis (Minnesota) Glenn Ledder asymptotic analysis and perturbations, ecology, hydrogeology (Rensselaer Polytechnic Institute) David Logan differential equations, mathematical ecology, effects of global climate change on ecosystems and eco-physiology (Ohio State) Irakli Loladze differential equations, environmental stoichiometry (Arizona State) Tom Shores numerical analysis, inverse theory, scientific computation (Kansas) Brigitte TenHumberg invasion ecology, optimal decision in insect behavior and life history (Göttingen) ; joint appointment with Mathematics and School of Biological Sciences
4 Eclectic Faculty Bo Deng dynamical systems, chaos, neuroscience, ecology (Michigan State) Wendy Hines dynamical systems, gene propagation models (Georgia Institute of Technology) Richard Rebarber distributed parameter control theory and population ecology (Wisconsin)
5 Students Introductions Faculty Students Undergraduate: We involve numerous undergraduate students in our research programs via programs such as: Summer REUs UNL UCARE program Undergraduate honors theses The RUTE project directed by Glenn Ledder
6 Students Introductions Faculty Students Undergraduate: We involve numerous undergraduate students in our research programs via programs such as: Summer REUs UNL UCARE program Undergraduate honors theses The RUTE project directed by Glenn Ledder
7 Students Introductions Faculty Students Undergraduate: We involve numerous undergraduate students in our research programs via programs such as: Summer REUs UNL UCARE program Undergraduate honors theses The RUTE project directed by Glenn Ledder
8 Students Introductions Faculty Students Undergraduate: We involve numerous undergraduate students in our research programs via programs such as: Summer REUs UNL UCARE program Undergraduate honors theses The RUTE project directed by Glenn Ledder
9 Students Introductions Faculty Students Undergraduate: We involve numerous undergraduate students in our research programs via programs such as: Summer REUs UNL UCARE program Undergraduate honors theses The RUTE project directed by Glenn Ledder
10 Make Titles Informative. Faculty Students Graduate Doctoral Students: Amy Frederick (David Logan) Joan Lubben (Richard Rebarber) Anastasia Mshvidobadze (David Logan) Brian Bockelman joint CS/Math program (Tom Shores) Dan Buettner joing CS/Math program (Tom Shores) In addition, we have a number of pre-doctoral students.
11 What We Want our Students to Acquire/Learn Basic real analysis ( ) and complex variables (823) Applied mathematics ( ) Linear algebra (817) and numerical linear algebra (847) Linear and nonlinear optimization ( ) Mathematical programming skills in MATLAB, Maple, Mathematica and/or other computing platforms Probability and statistics Differential equations (ODE, PDE, difference equations,control theory) Specialty courses outside the department Collaboration with other students and science faculty
12 What We Want our Students to Acquire/Learn Basic real analysis ( ) and complex variables (823) Applied mathematics ( ) Linear algebra (817) and numerical linear algebra (847) Linear and nonlinear optimization ( ) Mathematical programming skills in MATLAB, Maple, Mathematica and/or other computing platforms Probability and statistics Differential equations (ODE, PDE, difference equations,control theory) Specialty courses outside the department Collaboration with other students and science faculty
13 What We Want our Students to Acquire/Learn Basic real analysis ( ) and complex variables (823) Applied mathematics ( ) Linear algebra (817) and numerical linear algebra (847) Linear and nonlinear optimization ( ) Mathematical programming skills in MATLAB, Maple, Mathematica and/or other computing platforms Probability and statistics Differential equations (ODE, PDE, difference equations,control theory) Specialty courses outside the department Collaboration with other students and science faculty
14 What We Want our Students to Acquire/Learn Basic real analysis ( ) and complex variables (823) Applied mathematics ( ) Linear algebra (817) and numerical linear algebra (847) Linear and nonlinear optimization ( ) Mathematical programming skills in MATLAB, Maple, Mathematica and/or other computing platforms Probability and statistics Differential equations (ODE, PDE, difference equations,control theory) Specialty courses outside the department Collaboration with other students and science faculty
15 What We Want our Students to Acquire/Learn Basic real analysis ( ) and complex variables (823) Applied mathematics ( ) Linear algebra (817) and numerical linear algebra (847) Linear and nonlinear optimization ( ) Mathematical programming skills in MATLAB, Maple, Mathematica and/or other computing platforms Probability and statistics Differential equations (ODE, PDE, difference equations,control theory) Specialty courses outside the department Collaboration with other students and science faculty
16 What We Want our Students to Acquire/Learn Basic real analysis ( ) and complex variables (823) Applied mathematics ( ) Linear algebra (817) and numerical linear algebra (847) Linear and nonlinear optimization ( ) Mathematical programming skills in MATLAB, Maple, Mathematica and/or other computing platforms Probability and statistics Differential equations (ODE, PDE, difference equations,control theory) Specialty courses outside the department Collaboration with other students and science faculty
17 What We Want our Students to Acquire/Learn Basic real analysis ( ) and complex variables (823) Applied mathematics ( ) Linear algebra (817) and numerical linear algebra (847) Linear and nonlinear optimization ( ) Mathematical programming skills in MATLAB, Maple, Mathematica and/or other computing platforms Probability and statistics Differential equations (ODE, PDE, difference equations,control theory) Specialty courses outside the department Collaboration with other students and science faculty
18 What We Want our Students to Acquire/Learn Basic real analysis ( ) and complex variables (823) Applied mathematics ( ) Linear algebra (817) and numerical linear algebra (847) Linear and nonlinear optimization ( ) Mathematical programming skills in MATLAB, Maple, Mathematica and/or other computing platforms Probability and statistics Differential equations (ODE, PDE, difference equations,control theory) Specialty courses outside the department Collaboration with other students and science faculty
19 What We Want our Students to Acquire/Learn Basic real analysis ( ) and complex variables (823) Applied mathematics ( ) Linear algebra (817) and numerical linear algebra (847) Linear and nonlinear optimization ( ) Mathematical programming skills in MATLAB, Maple, Mathematica and/or other computing platforms Probability and statistics Differential equations (ODE, PDE, difference equations,control theory) Specialty courses outside the department Collaboration with other students and science faculty
20 What We Want our Students to Acquire/Learn Basic real analysis ( ) and complex variables (823) Applied mathematics ( ) Linear algebra (817) and numerical linear algebra (847) Linear and nonlinear optimization ( ) Mathematical programming skills in MATLAB, Maple, Mathematica and/or other computing platforms Probability and statistics Differential equations (ODE, PDE, difference equations,control theory) Specialty courses outside the department Collaboration with other students and science faculty
21 What we read Where we Publish Journal of Mathematical Biology Bulletin of Mathematical Biology Journal of Theoretical Biology Ecological Modelling Inverse Theory Journal of Differential Equations Applicable Analysis Applied Mathematics and Computation Mathematical and Computer Modelling Water Resources Journal of Hydrology SIAM Journal of Applied Mathematics SIAM Journal of Mathematical Analysis International Journal of Bifurcation and Chaos Mathematical Biosciences
22 Problem 1: Temperature Dependent Arthropod Interactions How does increased CO 2 levels and temperature changes associated with global climate change affect predator-prey interactions? Herbivore-plant interactions? Do these levels cause shifts in their phenologies (development) Use differential and difference equations to model the interactions, predict populations, include stochastic effects, stability (?), etc.
23 Problem 1: Temperature Dependent Arthropod Interactions How does increased CO 2 levels and temperature changes associated with global climate change affect predator-prey interactions? Herbivore-plant interactions? Do these levels cause shifts in their phenologies (development) Use differential and difference equations to model the interactions, predict populations, include stochastic effects, stability (?), etc.
24 Problem 1: Temperature Dependent Arthropod Interactions How does increased CO 2 levels and temperature changes associated with global climate change affect predator-prey interactions? Herbivore-plant interactions? Do these levels cause shifts in their phenologies (development) Use differential and difference equations to model the interactions, predict populations, include stochastic effects, stability (?), etc.
25 Problem 1: Temperature Dependent Arthropod Interactions How does increased CO 2 levels and temperature changes associated with global climate change affect predator-prey interactions? Herbivore-plant interactions? Do these levels cause shifts in their phenologies (development) Use differential and difference equations to model the interactions, predict populations, include stochastic effects, stability (?), etc.
26 Problem 2: Applied Nonlinear Analysis in Ecological Models Mechanistic understanding of chaos generation, cycles, equilibrium Requires global and geometrical multi-timescale analysis from dynamical systems and bifurcation theory Computation and visualization are essential
27 Problem 2: Applied Nonlinear Analysis in Ecological Models Mechanistic understanding of chaos generation, cycles, equilibrium Requires global and geometrical multi-timescale analysis from dynamical systems and bifurcation theory Computation and visualization are essential
28 Problem 2: Applied Nonlinear Analysis in Ecological Models Mechanistic understanding of chaos generation, cycles, equilibrium Requires global and geometrical multi-timescale analysis from dynamical systems and bifurcation theory Computation and visualization are essential
29 Problem 2: Applied Nonlinear Analysis in Ecological Models Mechanistic understanding of chaos generation, cycles, equilibrium Requires global and geometrical multi-timescale analysis from dynamical systems and bifurcation theory Computation and visualization are essential
30 Problem 3: Invasion Ecology and Thistle Spread in Nebraska Model the spread of a particular species of thistle (monocarpic Eurasian Cirsium vulgare). Determine the reliability of the model given that data measurements are uncertain. Requires knowledge of linear algebra (eigenvalue theory), sensitivity and elasticity analysis, statistics
31 Problem 3: Invasion Ecology and Thistle Spread in Nebraska Model the spread of a particular species of thistle (monocarpic Eurasian Cirsium vulgare). Determine the reliability of the model given that data measurements are uncertain. Requires knowledge of linear algebra (eigenvalue theory), sensitivity and elasticity analysis, statistics
32 Problem 3: Invasion Ecology and Thistle Spread in Nebraska Model the spread of a particular species of thistle (monocarpic Eurasian Cirsium vulgare). Determine the reliability of the model given that data measurements are uncertain. Requires knowledge of linear algebra (eigenvalue theory), sensitivity and elasticity analysis, statistics
33 Problem 3: Invasion Ecology and Thistle Spread in Nebraska Model the spread of a particular species of thistle (monocarpic Eurasian Cirsium vulgare). Determine the reliability of the model given that data measurements are uncertain. Requires knowledge of linear algebra (eigenvalue theory), sensitivity and elasticity analysis, statistics
34 Problem 4: Determine Basic Properties of a Nonresonant Schrodinger Equation Is the system completely integrable? Does it have a Hamiltonian structure? What is the long time behavior of solutions?
35 Problem 4: Determine Basic Properties of a Nonresonant Schrodinger Equation Is the system completely integrable? Does it have a Hamiltonian structure? What is the long time behavior of solutions?
36 Problem 4: Determine Basic Properties of a Nonresonant Schrodinger Equation Is the system completely integrable? Does it have a Hamiltonian structure? What is the long time behavior of solutions?
37 Problem 4: Determine Basic Properties of a Nonresonant Schrodinger Equation Is the system completely integrable? Does it have a Hamiltonian structure? What is the long time behavior of solutions?
38 Problem 5: Detecting Diffusive Behavior in Criminal Statistics Certain criminal activities have been observed to have diffusive characteristics. Can such data be fitted to a predictive mathematical model? Requires knowledge of inverse theory, GIS knowledge, PDEs, numerical methods for solving PDEs Requires collaboration with computer science and criminology specialists.
39 Problem 5: Detecting Diffusive Behavior in Criminal Statistics Certain criminal activities have been observed to have diffusive characteristics. Can such data be fitted to a predictive mathematical model? Requires knowledge of inverse theory, GIS knowledge, PDEs, numerical methods for solving PDEs Requires collaboration with computer science and criminology specialists.
40 Problem 5: Detecting Diffusive Behavior in Criminal Statistics Certain criminal activities have been observed to have diffusive characteristics. Can such data be fitted to a predictive mathematical model? Requires knowledge of inverse theory, GIS knowledge, PDEs, numerical methods for solving PDEs Requires collaboration with computer science and criminology specialists.
41 Problem 5: Detecting Diffusive Behavior in Criminal Statistics Certain criminal activities have been observed to have diffusive characteristics. Can such data be fitted to a predictive mathematical model? Requires knowledge of inverse theory, GIS knowledge, PDEs, numerical methods for solving PDEs Requires collaboration with computer science and criminology specialists.
42 Conclusion Introductions No matter what area you choose, Nebraska is a great place to learn and do mathematics!
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