Andrew Marks
University of California-Berkeley, CA·Mathematics & Statistics
A common goal in mathematics is to classify a family of objects completely, by attaching to each one a simple label, or invariant, that tells exactly when two objects are the same. Some families can be classified this way, while for others a complete classification can be proven impossible.
- Fields
Mathematics & Statistics
- Topics
descriptive set theory
computability theory
mathematical logic
- Grants
- 1 known grant, through Jul 2027
What they work on
- Investigate Weiss's question on whether amenability implies hyperfiniteness, meaning whether actions of orderly groups can be built from finite pieces, using Gromov's theory of asymptotic dimension, with applications to topological dynamics and operator algebras.
- Study classical geometric paradoxes, including the Banach-Tarski paradox and Tarski's circle squaring problem, using recent advances in measurable combinatorics.
- Prove dichotomy theorems for definable sigma-ideals, collections of small sets, using Montalbán's true stages framework from computability theory.
More about the research
Descriptive set theory, a branch of mathematical logic, offers a general framework for studying these classification problems. It can compare how hard different problems are and decide when no classification can exist. This project applies those tools to classification problems in computability theory, operator algebras, topological dynamics and ergodic theory, the study of long-term average behavior in dynamical systems.
Why it matters
This is pure mathematics. It aims to show which classification problems across mathematics can be solved and how difficult they are, and when complete classification is provably impossible. The results could eventually inform research in operator algebras, dynamical systems and the theory of computation by clarifying the limits of classification there.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Uri Andrews
University of Wisconsin-Madison, WI·Mathematics & Statistics
Reduce the spectrum problem for strongly minimal theories, which asks which dimensions of their models are computable, to the locally modular case, which is geometrically tame and closely related to groups.
Damir Dzhafarov
University of Connecticut, CT·Mathematics & Statistics
Determine the computability-theoretic and reverse-mathematical strength of Milliken's tree theorem, a generalization of Ramsey's theorem from structural Ramsey theory, by resolving fundamental open questions about it.
Efren Ruiz
University of Hawaii at Manoa, HI·Mathematics & Statistics
Develop new invariants for subshifts of finite type, symbol-sequence dynamical systems, by studying the graph algebras attached to them and how these algebras interact.
The school
University of California-Berkeley: 46 labs on PI Pages, $1.1B a year on research, 825 research doctorates a year.
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