Hang Xue
University of Arizona, AZ·Mathematics & Statistics
Number theory, the study of whole numbers and their patterns, is one of the oldest branches of mathematics. Representation theory studies symmetry by describing how symmetry groups act on spaces.
- Fields
Mathematics & Statistics
- Topics
number theory
representation theory
arithmetic geometry
- Grants
- 1 known grant, through Jul 2027
What they work on
- Study the local Gan-Gross-Prasad conjecture for real orthogonal groups (symmetry groups that preserve lengths) using theta correspondences together with a new relative version of Shahidi's local coefficients.
- Use the same method to give an inductive construction of L-packets for orthogonal groups, the groupings of representations that are central objects in the Langlands conjectures.
- Establish the archimedean counterpart of the arithmetic fundamental lemma by exploiting an inductive structure of orbital integrals and spherical characters, reducing the problem to explicit calculations with doubling zeta integrals.
More about the research
In the 1960s Langlands proposed a web of deep conjectures linking these two fields, and those conjectures still guide much of modern research. In the 1990s Gross and Prasad, following Langlands' ideas, made conjectures about orthogonal groups, which are symmetry groups that preserve lengths, like rotations. Later, with Gan, they extended these conjectures to all classical groups. These Gan-Gross-Prasad conjectures sit where number theory, representation theory and arithmetic geometry meet.
Why it matters
This is pure mathematics. Progress on the Gan-Gross-Prasad conjectures would deepen understanding of how number theory and the mathematics of symmetry are connected, as predicted by the Langlands program. New tools from the project, such as constructions of L-packets for orthogonal groups, could inform further work across number theory, representation theory and arithmetic geometry.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Ken Ono
University of Virginia, VA·Mathematics & Statistics
Develop algebraic, analytic and combinatorial tools for harmonic Maass forms, generalizations of modular forms that underlie Ramanujan's mock theta functions, for attacking open problems in number theory.
Ananth Shankar
Northwestern University, IL·Mathematics & Statistics·Early-career
Construct and study integral models of exceptional Shimura varieties, that is, descriptions by polynomial equations with whole-number coefficients for the varieties attached to exceptional symmetry groups.
Laura DeMarco
Harvard University, MA·Mathematics & Statistics
Characterize the special subvarieties — geometric pieces preserved by the map — that arise in families of dynamical systems built from endomorphisms of algebraic varieties, aiming at a general classification within a unifying conjectural framework.
The school
University of Arizona: 36 labs on PI Pages, $1.0B a year on research, 487 research doctorates a year.
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