Professor Miller's research centers around problems
in geometry, algebra, topology, combinatorics, and
computation originating in mathematics and the
sciences, including biology, chemistry, computer
science, medical imaging, and statistics.
The techniques range, for example, from abstract
algebraic geometry of varieties to concrete metric
or discrete geometry of polyhedral spaces; from
deep topological constructions such as equivariant
K-theory and stratified Morse theory to elementary
simplicial homology; from functorial perspectives
on homological algebra in the derived category to
constructions of complexes based on combinatorics
of cell decompositions; or from central limit
theorems via geodesic contraction on stratified
spaces to dynamics of explicit polynomial vector
fields on polyhedra.
Beyond motivations from within mathematics, the
sources of these problems lie in, for example,
graphs and trees in evolutionary biology and
medical imaging; mass-action kinetics of chemical
reactions; computational geometry, symbolic
computation, and combinatorial game theory; and
geometric statistics of data sampled from highly
non-Euclidean spaces. Current datasets under
consideration include MRI images of blood vessels
in human brains and mouse lungs, vein structures in
fruit fly wings for developmental morphological
studies, and fMRI time-course images of human
brains for classification of schizophrenia patients.
Professor Miller's research centers around problems in geometry, algebra, topology, combinatorics, statistics, probability, and computation originating in mathematics and the sciences, including biology, chemistry, computer science, and imaging.
The techniques range, for example, from abstract algebraic geometry or commutative algebra of ideals and varieties to concrete metric or discrete geometry of polyhedral spaces; from deep topological constructions such as equivariant K-theory and stratified Morse theory to elementary simplicial and persistent homology; from functorial perspectives on homological algebra in the derived category to specific constructions of complexes based on combinatorics of cell decompositions; from geodesic contraction applied to central limit theorems for samples from stratified spaces to dynamics of explicit polynomial vector fields on polyhedra.
Beyond motivations from within mathematics, the sources of these problems lie in, for example, graphs and trees in evolutionary biology and medical imaging; mass-action kinetics of chemical reactions; computational geometry, symbolic computation, and combinatorial game theory; Lie theory; and geometric statistics of data sampled from highly non-Euclidean spaces. Examples of datasets under consideration include MRI images of blood vessels in human brains and lungs, 3D folded protein structures, and photographs of fruit fly wings for developmental morphological studies.