Prof. Juan G. Restrepo's research focuses on the analysis of emergent complex behavior in systems of interconnected units using techniques of nonlinear dynamics. Important examples of such systems are collections of network-coupled oscillators and excitable systems such as neurons, and cardiac cells coupled in tissue. The analysis of network-coupled systems requires developing an understanding of quantitative aspects of the structure of the coupling network. Prof. Restrepo is interested in the mathematical modeling of phenomena ranging from opinion dynamics to synchronization of coupled oscillators.
keywords
Nonlinear dynamics of networked and complex systems
APPM 2360 - Introduction to Differential Equations with Linear Algebra
Primary Instructor
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Spring 2018 / Spring 2021 / Spring 2023
Introduces ordinary differential equations, systems of linear equations, matrices, determinants, vector spaces, linear transformations, and systems of linear differential equations. Credit not granted for this course and both MATH 2130 and MATH 3430.
APPM 3010 - Chaos in Dynamical Systems
Primary Instructor
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Fall 2018 / Fall 2020
Introduces undergraduate students to chaotic dynamical systems. Topics include smooth and discrete dynamical systems, bifurcation theory, chaotic attractors, fractals, Lyapunov exponents, synchronization and networks of dynamical systems. Applications to engineering, biology and physics will be discussed.
APPM 3170 - Discrete Applied Mathematics
Primary Instructor
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Fall 2020
Introduces students to ideas and techniques from discrete mathematics that are widely used in science and engineering. Mathematical definitions and proofs are emphasized. Topics include formal logic notation, proof methods; set theory, relations; induction, well-ordering; algorithms, growth of functions and complexity; integer congruences; basic and advanced counting techniques, recurrences and elementary graph theory. Other selected topics may also be covered.
APPM 3570 - Applied Probability
Primary Instructor
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Spring 2019 / Spring 2020 / Spring 2022
Studies axioms, counting formulas, conditional probability, independence, random variables, continuous and discrete distribution, expectation, joint distributions, moment generating functions, law of large numbers and the central limit theorem. Degree credit not granted for this course and ECEN 3810 or MATH 4510. Same as STAT 3100.
APPM 4560 - Markov Processes, Queues, and Monte Carlo Simulations
Primary Instructor
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Spring 2024
Brief review of conditional probability and expectation followed by a study of Markov chains, both discrete and continuous time, including Poisson point processes. Queuing theory, terminology and single queue systems are studied with some introduction to networks of queues. Uses Monte Carlo simulation of random variables throughout the semester to gain insight into the processes under study. Same as APPM 5560 and STAT 4100.
APPM 4720 - Open Topics in Applied Mathematics
Primary Instructor
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Fall 2019
Provides a vehicle for the development and presentation of new topics that may be incorporated into the core courses in applied mathematics. Department enforced prerequisite: variable, depending on the topic, see instructor. May be repeated up to 15 total credit hours. Same as APPM 5720.
APPM 5320 - Introduction to Dynamics on Networks
Primary Instructor
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Fall 2021
Introduces modern approaches to model and analyze dynamical processes on complex networks. Many dynamical processes such as epidemic propagation, opinion formation, synchronization, and cascading processes take place on complex social or technological networks. This course will introduce the tools to understand the interplay between network structure and the outcome of these dynamical processes. Previously offered as a special topics course. Same as APPM 4320.
APPM 5470 - Methods of Applied Mathematics: Partial Differential and Integral Equations
Primary Instructor
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Fall 2022
Studies properties and solutions of partial differential equations. Covers methods of characteristics, well-posedness, wave, heat and Laplace equations, Green's functions and related integral equations. Department enforced prerequisites: APPM 4350 or MATH 4470 and APPM 4360 or MATH 3450.
APPM 5560 - Markov Processes, Queues, and Monte Carlo Simulations
Primary Instructor
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Spring 2024
Brief review of conditional probability and expectation followed by a study of Markov chains, both discrete and continuous time, including Poisson point processes. Queuing theory, terminology and single queue systems are studied with some introduction to networks of queues. Uses Monte Carlo simulation of random variables throughout the semester to gain insight into the processes under study. Same as APPM 4560, STAT 4100 and STAT 5100.
APPM 5720 - Open Topics in Applied Mathematics
Primary Instructor
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Fall 2019
Provides a vehicle for the development and presentation of new topics that may be incorporated into the core courses in applied mathematics. Department enforced prerequisite: variable, depending on the topic, see instructor. May be repeated up to 6 total credit hours. Same as APPM 4720.
APPM 8100 - Seminar in Dynamical Systems
Primary Instructor
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Spring 2018 / Fall 2018 / Spring 2020 / Spring 2021 / Spring 2022 / Spring 2023 / Spring 2024
Introduces advanced topics and research in dynamical systems.
STAT 3100 - Applied Probability
Primary Instructor
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Spring 2019 / Spring 2020 / Spring 2022
Studies axioms, counting formulas, conditional probability, independence, random variables, continuous and discrete distribution, expectation, joint distributions, moment generating functions, law of large numbers and the central limit theorem. Degree credit not granted for this course and ECEN 3810 or MATH 4510. Same as APPM 3570.
STAT 4100 - Markov Processes, Queues, and Monte Carlo Simulations
Primary Instructor
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Spring 2024
Brief review of conditional probability and expectation followed by a study of Markov chains, both discrete and continuous time, including Poisson point processes. Queuing theory, terminology and single queue systems are studied with some introduction to networks of queues. Uses Monte Carlo simulation of random variables throughout the semester to gain insight into the processes under study. Same as APPM 4560 and APPM 5560.
STAT 5100 - Markov Processes, Queues, and Monte Carlo Simulations
Primary Instructor
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Spring 2024
Brief review of conditional probability and expectation followed by a study of Markov chains, both discrete and continuous time, including Poisson point processes. Queuing theory, terminology and single queue systems are studied with some introduction to networks of queues. Uses Monte Carlo simulation of random variables throughout the semester to gain insight into the processes under study. Same as APPM 5560 and APPM 4560.
