Dr. Clelland's research uses methods from geometry to study a variety of problems in differential equations, including differential equations that arise naturally in various contexts in differential geometry. Her recent and current research in this area includes the study of dynamic equivalence and dynamic feedback linearization for control systems and isometric immersion of Riemannian manifolds, among other topics. Dr. Clelland also studies the mathematics of redistricting, and in particular, mathematical methods for identifying and quantifying partisan bias in district plans. Since 2021, she has consulted with the Colorado Independent Legislative Redistricting Commission and served as an expert witness for lawsuits challenging proposed redistricting plans in Ohio, Wisconsin, and New York. In her 2022 expert witness report for Nichols v. Hochul, regarding redistricting for the New York State Assembly, she developed new methods for evaluating incumbency protection in redistricting; these methods are the subject of ongoing academic work. In Fall 2023, Dr. Clelland held the position of Research Professor at the Simons-Laufer Mathematical Sciences Institute for a semester-long program on Algorithms, Fairness, and Equity. Also in 2023, she developed a new algorithm to repair gaps and overlaps in a noisy polygonal tiling in a way that preserves the intended adjacency relations as closely as possible. This algorithm has important applications to redistricting, where maps of voting precincts often exhibit such problems that must be repaired prior to analyzing districting plans based on the map. Dr. Clelland has written a Python implementation of this algorithm, which is now included in the MGGG Redistricting Lab's open source Maup package, available at https://github.com/mggg/maup.
keywords
Geometry of differential equations, Exterior differential systems, Cartan's method of moving frames, Backlund transformations, Geometry of control systems, Dynamic equivalence for control systems, Isometric embeddings of Riemannian manifolds, Darboux integrability, Mathematics of redistricting and elections, Ensemble analysis
MATH 2001 - Introduction to Discrete Mathematics
Primary Instructor
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Spring 2019 / Fall 2019
Introduces the ideas of rigor and proof through an examination of basic set theory, existential and universal quantifiers, elementary counting, discrete probability, and additional topics. Credit not granted for this course and MATH 2002.
MATH 3001 - Analysis 1
Primary Instructor
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Spring 2024
Provides a rigorous treatment of the basic results from elementary Calculus. Topics include the topology of the real line, sequences of numbers, continuous functions, differentiable functions and the Riemann integral.
MATH 3430 - Ordinary Differential Equations
Primary Instructor
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Spring 2018 / Fall 2019
Involves an elementary systematic introduction to first-order scalar differential equations, nth order linear differential equations, and n-dimensional linear systems of first-order differential equations. Additional topics are chosen from equations with regular singular points, Laplace transforms, phase plane techniques, basic existence and uniqueness and numerical solutions. Formerly MATH 4430.
MATH 4230 - Differential Geometry of Curves and Surfaces
Primary Instructor
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Fall 2018 / Fall 2020 / Fall 2022
Introduces the modern differential geometry of plane curves, space curves, and surfaces in 3-dimensional space. Topics include the Frenet frame, curvature and torsion for space curves; Gauss and mean curvature for surfaces; Gauss and Codazzi equations, and the Gauss-Bonnet theorem. Same as MATH 5230.
MATH 4470 - Partial Differential Equations
Primary Instructor
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Spring 2020 / Spring 2021 / Spring 2024
Studies initial, boundary, and eigenvalue problems for the wave, heat, and potential equations. Solution by separation of variables, Green's function, and variational methods. Same as MATH 5470.
MATH 4810 - Special Topics in Mathematics
Primary Instructor
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Fall 2020
Covers various topics not normally covered in the curriculum. Offered intermittently depending on student demand and availability of instructors. May be repeated up to 7 total credit hours. Same as MATH 5810.
MATH 5230 - Differential Geometry of Curves and Surfaces
Primary Instructor
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Fall 2018 / Fall 2020
Introduces the modern differential geometry of plane curves, space curves, and surfaces in 3-dimensional space. Topics include the Frenet frame, curvature and torsion for space curves; Gauss and mean curvature for surfaces; Gauss and Codazzi equations, and the Gauss-Bonnet theorem. Same as MATH 4230.
MATH 5470 - Partial Differential Equations
Primary Instructor
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Spring 2020 / Spring 2021 / Spring 2024
Studies initial boundary and eigenvalue problems for the wave, heat and potential equations. Solution by separation of variables, Green's function, and variational methods. Department enforced prerequisite: MATH 3430 or MATH 5430. Instructor consent required for undergraduates. Same as MATH 4470.
MATH 5810 - Special Topics in Mathematics
Primary Instructor
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Fall 2020
Covers various topics not normally covered in the curriculum. Offered intermittently depending on student demand and availability of instructors. May be repeated up to 7 total credit hours. Same as MATH 4810.
MATH 6230 - Introduction to Differential Geometry 1
Primary Instructor
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Spring 2018 / Spring 2019 / Spring 2023
Introduces topological and differential manifolds, vector bundles, differential forms, de Rham cohomology, integration, Riemannian metrics, connections and curvature. Department enforced prerequisites: MATH 2130 and MATH 4001. Instructor consent required for undergraduates.
