The Geometry and Geometric Analysis Seminar at Purdue in Spring 2026 is usually on Mondays 11:30am-12:20pm Eastern Time in MATH 731 if we meet in person. Some talks will be on zoom and the link will be included in the email announcement.
Lvzhou Chen, Mathew George, and Nicholas McCleerey are organizing this seminar in Spring 2026. If you have any questions or would like to suggest speakers, please contact one of us. If you would like to be on the mailing list, please email Lvzhou.
Title: Asymptotically large free semigroups in Zariski dense discrete subgroups of Lie groups
Abstract: An important quantity in the study of discrete groups of isometries of Riemannian manifolds, Gromov hyperbolic spaces, and other interesting geometric objects is the critical exponent. For a discrete subgroup of isometries of the quaternionic hyperbolic space or octonionic projective plane, Kevin Corlette established in 1990 that the critical exponent detects whether a discrete subgroup is a lattice or has infinite covolume. Precisely, either the critical exponent equals the volume entropy, in which case the discrete subgroup is a lattice, or the critical exponent is less than the volume entropy by some definite amount, in which case the discrete subgroup has infinite covolume. In 2003, Leuzinger extended this gap theorem for the critical exponent to any discrete subgroup of a Lie group having Kazhdan’s property (T) (for instance, a discrete subgroup of SL(n,R), where n is at least 3).
In this talk, I will present a result which shows that no such gap phenomenon holds for discrete semigroups of Lie groups. More precisely, for any Zariski dense discrete subgroup of a Lie group, there exist free, finitely generated, Zariski dense subsemigroups whose critical exponents are arbitrarily close to that of the ambient discrete subgroup.
As an application, we show that the critical exponent is lower semicontinuous in the Chabauty topology whenever the Chabauty limit of a sequence of Zariski dense discrete subgroups is itself a Zariski dense discrete subgroup.
Title: On the Yau–Tian–Donaldson conjecture
Abstract: Any compact Riemann surface is topologically determined by its genus, i.e. the number of "holes", and by the Uniformization Theorem, it admits a unique metric of constant curvature 1, 0 or -1. In higher (complex) dimension, the situation is more complicated, but the Yau-Tian-Donaldson conjecture states that the existence of a metric of constant scalar curvature---which is an analytic object---is governed by a purely algebro-geometric condition. I will present joint work with S. Boucksom, where we prove a version of this conjecture.
Title: Anabelian geometry for 3-manifolds
Abstract: In the 70s, Mazur observed that knots in the 3-sphere and primes in number fields share some remarkable similarities. This has developed in a field called arithmetic topology, which pursues analogous results between number fields and 3-manifolds. In this talk, we will highlight a new analogy between anabelian geometry and 3-manifolds. In particular, we present an analogue of the foundational Neukirch-Uchida theorem, which states that the isomorphism class of a number field is determined by its absolute Galois group. Our analogy will involve infinite-component links which satisfy a Chebotarev-type property introduced by McMullen, which can be viewed as analogous objects to the prime numbers in the integers. This is joint work with Nadav Gropper and Jun Ueki.
Title: Rigidity and classification of holomorphic mappings among moduli space of curves and locally Hermitian symmetric spaces
Abstract: We will explain known results for the problem between two locally symmetric spaces and the problem from a locally symmetric space to a moduli space in a geometric setting of harmonic maps and Bochner formula. We will also explain a classification of holomorphic mappings from a moduli space of curves of genus at least $2$ to a locally Hermitian symmetric space under some natural assumptions on its corresponding representation of the Torelli group.