Research
My research spans algebraic topology, category theory, algebra, topological data analysis, and machine learning. A common theme is the use of geometric, homotopical, and computational structure to study problems arising in both pure mathematics and data science.
Algebraic Topology and Intersection Theory
One direction of my research concerns the computation and geometric interpretation of intersection invariants.
I am particularly interested in approaches involving differential forms, diffeological spaces, and ideas originating in the work of Kuo-Tsai Chen on iterated integrals and path spaces.
These methods provide a bridge between classical homotopy-theoretic invariants and differential-geometric constructions.
A related project with Vladimir Chernyak and John R. Klein studies the Hatcher–Quinn intersection invariant using differential forms.
Categories, Trees, and Unitary Partition Complexes
A current direction of my research concerns topological categories, classifying spaces, and unitary partition complexes.
In joint work with Julie Bergner, Pedro Brunialti Lima de Andrade, and Eleftherios Chatzitheodoridis, we study a unitary analogue of the relationship between tree posets and ordinary partition complexes.
Heuts and Moerdijk constructed a poset of trees whose classifying space models the classical partition complex. Our work develops a topological poset of trees whose classifying space is homotopy equivalent to the (n)-th unitary partition complex.
The unitary partition complex is built from decompositions
\[ \mathbb{C}^n = V_1 \perp V_2 \perp \cdots \perp V_k \]
of complex (n)-space into pairwise orthogonal nonzero subspaces.
This project brings together several themes:
- topological categories and topological posets,
- classifying spaces,
- tree models,
- equivariant homotopy theory,
- unitary group actions, and
- partition complexes.
In addition to the main equivalence with the unitary partition complex, the project develops results concerning topological categories that may be useful independently.
Automorphisms of Free Algebras
I am interested in determining generators and structural properties of automorphism groups of several classes of free algebras, including
- free Lie algebras,
- free associative algebras, and
- free commutative algebras.
These questions connect algebraic structure with combinatorial and homotopical techniques.
Topological Data Analysis
Another part of my work concerns topological data analysis (TDA) and its applications to data science.
I am particularly interested in persistent homology and related techniques for studying complex data, including time series data.
A filtration
\[ K_{\epsilon_1} \subseteq K_{\epsilon_2} \subseteq K_{\epsilon_3} \subseteq \cdots \]
induces maps
\[ H_k(K_{\epsilon_1}) \longrightarrow H_k(K_{\epsilon_2}) \longrightarrow H_k(K_{\epsilon_3}) \longrightarrow \cdots, \]
allowing topological features to be tracked across scale.
Topological methods can reveal qualitative geometric structure that may be difficult to identify using conventional statistical summaries alone.
Machine Learning and Data Science
My applied research includes machine learning, numerical methods, and scientific data analysis.
I am interested both in conventional machine-learning methods and in understanding how ideas from topology and geometry can be incorporated into machine-learning pipelines.
This includes work involving machine-learning methods for scientific modeling and parameter calibration.
Large Language Models
I am also interested in large language models (LLMs) and their applications to mathematics, education, and scientific research.
Current interests include
- mathematical reasoning in language models,
- representation structure,
- alignment and AI safety,
- topology-inspired methods for studying model behavior, and
- the use of LLMs in mathematics education.