MATHEMATICS · TOPOLOGY · DATA · AI
Josh Turner
Assistant Professor of Mathematics
University of Virginia
I study how topology, algebra, and computation reveal structure in mathematical and data-driven systems.
My research spans algebraic topology, category theory, commutative algebra, topological data analysis, machine learning, and large language models.
SELECTED RESEARCH
Mathematics across topology, algebra, data, and AI
Algebraic Topology
Intersection invariants, path spaces, differential forms, diffeology, and homotopy theory.
Categories & Partition Complexes
Topological categories, classifying spaces, tree models, and unitary partition complexes.
Algebraic Structures
Automorphism groups of free Lie, associative, and commutative algebras.
Topological Data Analysis
Persistent homology, time series analysis, and topology applied to complex data.
Artificial Intelligence
Large language models, mathematical reasoning, alignment, and mathematics education.
CURRENT WORK
Trees Associated with Unitary Partition Complexes
With Julie Bergner, Pedro Brunialti Lima de Andrade, and Eleftherios Chatzitheodoridis
In this project, we construct a topological poset of trees whose classifying space is homotopy equivalent to the unitary partition complex. The work connects tree models, topological categories, classifying spaces, and equivariant homotopy theory.
MATHEMATICAL THREADS
Some of the mathematics behind my work
Paths and Topology
\[ PX \longrightarrow X \times X \]
\[ \gamma \longmapsto \bigl(\gamma(0),\gamma(1)\bigr) \]
Persistence
\[ H_k(K_{\epsilon_1}) \longrightarrow H_k(K_{\epsilon_2}) \longrightarrow H_k(K_{\epsilon_3}) \longrightarrow \cdots \]
Algebra
\[ \operatorname{Aut}(L_n) \]
CONTACT
Josh Turner
Assistant Professor of Mathematics
University of Virginia
rbh3vx [at] virginia [dot] edu