Overview
Professor Miller's research centers around problems in geometry,
algebra, topology, probability, statistics, 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 collapse 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.
algebra, topology, probability, statistics, 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 collapse 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.
Office Hours
Office hours: Tuesday, 13:00 – 14:15 in Physics 209 or outside
Thursday, 13:00 – 14:15 in Physics 209 or outside
Thursday, 13:00 – 14:15 in Physics 209 or outside
Current Appointments & Affiliations
Professor of Mathematics
·
2009 - Present
Mathematics,
Trinity College of Arts & Sciences
Recent Publications
RNAprecis: Prediction of full-detail RNA conformation from the experimentally best-observed sparse parameters.
Preprint · March 5, 2025 Full text Link to item CiteMinimal resolutions of lattice ideals
Journal Article Journal of Pure and Applied Algebra · March 1, 2025 A canonical minimal free resolution of an arbitrary co-artinian lattice ideal over the polynomial ring is constructed over any field whose characteristic is 0 or any but finitely many positive primes. The differential has a closed-form combinatorial descri ... Full text CiteMINIMAL FREE RESOLUTIONS OF NUMERICAL SEMIGROUP ALGEBRAS VIA APÉRY SPECIALIZATION
Journal Article Pacific Journal of Mathematics · January 1, 2025 Numerical semigroups with multiplicity m are parametrized by integer points in a polyhedral cone Cm, according to Kunz. For the toric ideal of any such semigroup, the main result here constructs a free resolution whose overall structure is identical for al ... Full text CiteRecent Grants
RTG: Linked via L-functions: training versatile researchers across number theory
Inst. Training Prgm or CMEKey Faculty · Awarded by National Science Foundation · 2023 - 2028R2 [Reciprocal Relationships]: Mentorships to Strengthen and Sustain STEM Teachers
Inst. Training Prgm or CMECo-Principal Investigator · Awarded by National Science Foundation · 2020 - 2025HDR TRIPODS: Innovations in Data Science: Integrating Stochastic Modeling, Data Representation, and Algorithms
ResearchSenior Investigator · Awarded by National Science Foundation · 2019 - 2023View All Grants
Education, Training & Certifications
University of California, Berkeley ·
2000
Ph.D.
Brown University ·
1995
B.S.