Alexander Wright
University of Michigan, MI·Mathematics & Statistics·Early-career
A surface, like the skin of a donut or a pretzel, can take on many different geometric shapes. A moduli space is a mathematical space in which each point is one such shape.
- Fields
Mathematics & Statistics
- Topics
moduli spaces
Teichmüller theory
Teichmüller space
- Grants
- 1 known grant, through Jun 2027
What they work on
- Build quasi-convex co-bounded planes in Teichmüller spaces (spaces of hyperbolic shapes on a surface) by gluing together pieces obtained from dynamics, toward convex cocompact surface subgroups of mapping class groups.
- Determine whether a non-trivial orbit closure of rank at least 3 exists for translation surfaces (flat surfaces glued from polygons), by classifying special constructions and studying low-genus moduli spaces.
- Show that typical high-genus hyperbolic surfaces are good spectral expanders, where waves and random walks spread quickly, using the trace formula and Mirzakhani's work on random surfaces.
- Develop Patterson–Sullivan theory, a way of measuring how orbits of a group spread out, for mapping class groups of non-orientable surfaces.
- Establish a criterion for unique ergodicity, the property that a flow distributes evenly, on dilation surfaces, a generalization of translation surfaces.
More about the research
These spaces are central to mathematics and theoretical physics, and researchers from many fields work on them. The classical case studies evenly curved (hyperbolic) shapes, which can be approached through geometry, complex analysis, algebraic geometry and matrix groups. More recently, flat geometries with a few sharp cone points have gained attention. They are closely tied to the classical moduli space and to important examples in dynamical systems, which are systems that evolve over time.
Why it matters
This is pure mathematics. It aims to deepen understanding of the geometry of surfaces and higher-dimensional spaces and to open new uses in dynamical systems, the study of how systems change over time. Moduli spaces connect many areas of mathematics and theoretical physics, so progress here could eventually inform work across those fields.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Richard Canary
University of Michigan, MI·Mathematics & Statistics
Study the metric completion of the Hitchin component, a higher Teichmüller space of surface group representations, with respect to a pressure metric that generalizes the Weil-Petersson metric.
Nicholas Salter
University of Notre Dame, IN·Mathematics & Statistics·Early-career
Establish asphericality, the absence of higher-dimensional holes, for special strata of Abelian differentials using the PI's Abel-Jacobi flow method, as progress toward the Kontsevich–Zorich conjecture.
Hannah Larson
University of California-Berkeley, CA·Mathematics & Statistics·Early-career
Develop new intersection-theory methods for moduli spaces of low-genus algebraic curves, spaces whose points each represent a curve, and connect them to closely related moduli spaces.
The school
University of Michigan: 124 labs on PI Pages, $2.1B a year on research, 963 research doctorates a year.
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