The Geometry and Geometric Analysis Seminar at Purdue in Fall 2025 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 Fall 2025. 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: Geometric Properties of Similarity Structure Groups
Abstract: Countable Similarity Structure (CSS) groups are a certain type of discrete homeomorphism group of a compact ultrametric space that act by local similarity. They are essentially due to Farley and Hughes and are a generalized class of Thompson groups. I will introduce CSS* groups, a certain subclass that includes the Higman-Thompson groups $V_{d,r}$ and prove that CSS* groups are non-inner amenable and not acylindrically hyperbolic. Throughout the talk I will provide background on both properties and, time permitting, discuss their connections to operator algebras. This is joint work with Eli Bashwinger.
Title: Calabi-Yau metrics and optimal transport
Abstract: Recent development in the study of Calabi-Yau metrics have revealed an intriguing connection with the theory of optimal transport, namely that Calabi-Yau metrics in certain degenerate and asymptotic regimes are often described by solutions to optimal transport problems. In this talk, I will discuss some recent advances in the regularity theory of optimal transport maps and its relationship with problems in Kahler geometry. Based on joint works with T. Collins and S.-T. Yau.
Title: Complex Monge-Ampere equations from geometric flows
Abstract: We present various setups where natural geometric flows lead to Monge-Ampere flows when the flow is started with suitable initial data. Examples include the G2 Laplacian flow on 7-manifolds, the hypersymplectic flow on 4-manifolds, the anomaly flow on Calabi-Yau 3-folds, and Type IIA flow on symplectic 6-manifolds.
Title: The second rational homology group of the Torelli group
Abstract: After giving an introduction to finiteness properties of the mapping class group and its subgroups, I will discuss a recent theorem I proved with Minahan calculating the second homology group of the Torelli group.
Title: Non Vanishing of the Fourth Bounded Cohomology of Free Groups and Codimension 2 Subspaces
Abstract: Bounded cohomology is a powerful albeit very hard to compute invariant. Nothing encapsulates that more than the as of yet mysterious bounded cohomology of free groups. During this talk I will give a very brief introduction to bounded cohomology, further motivate why one should care about the bounded cohomology of free groups and then explain how to show that it is non-zero in degrees two, three and four.
Title: Norms of spherical averaging operators on hyperbolic groups
Abstract: I will discuss how operator norms for spherical averaging operators on Gromov hyperbolic groups grow with respect to the radius. I will also mention an intriguing application towards a combinatorial notion of expansion.
Title: Coupled Kähler-Einstein metrics and coupled Ding stability
Abstract: A foundational theorem in Kähler geometry states that a Kähler-Einstein metric exists on a Fano manifold (with discrete automorphisms) if and only if it is uniformly Ding stable. When Kähler-Einstein metrics do not exist, we can seek coupled Kähler-Einstein metrics, introduced by Hultgren and Witt-Nyström, defined in terms of decompositions of the anticanonical bundle. The main result of this talk is the equivalence between the coupled uniform Ding stability (as appropriately defined) and the existence of coupled Kähler-Einstein metrics. Time permitting, we also discuss another equivalent condition involving the stability threshold and its coupled version. This is a joint work with Kento Fujita.
Title: Asymptotic analysis of stability thresholds
Abstract: Stability thresholds, particularly the \alpha_k and \delta_k invariants, are a fundamental topic in the theory of K-stability, with connections to various different fields such as algebraic geometry, convex geometry, and geometric analysis. In this talk, we investigate their asymptotic behavior, revealing new phenomena in both toric and non-toric settings.
In the toric setting, Ehrhart theory precisely describes the asymptotics via lattice point approximations of the moment polytope. We establish the stabilization of \alpha_k and derive an asymptotic expansion for \delta_k. In the general setting, we demonstrate that \alpha_k may fail to stabilize. To study their asymptotics we analyze the Okounkov body and its discrete approximation, which are a generalization of moment polytopes on toric varieties. Using tools from convex geometry and lattice point enumeration techniques, we prove the first asymptotic result for \delta_k. Based on joint work with Y. Rubinstein and G. Tian.
Title: Simplicity of the kernel of the flux homomorphism for non-orientable open surfaces
Abstract: For an orientable open surface with an area form, the group of area-preserving diffeomorphisms admits a non-trivial homomorphism called the flux homomorphism, and the kernel of the flux coincides with the group of Hamiltonian diffeomorphisms. If the surface is not closed, the Hamiltonian diffeomorphism group admits a surjective homomorphism to the real line, called the Calabi homomorphism. Hence, this group is not simple, and Banyaga proved that the kernel of the Calabi homomorphism is simple. In this talk, we discuss the case of non-orientable open surfaces. For non-orientable surfaces, the notion of area density plays the role of area forms, and we can formulate the flux homomorphism for the group of area-density-preserving diffeomorphisms. I will explain that, in contrast to the orientable case, the kernel of the flux homomorphism for non-orientable open surfaces are simple. This talk is based on joint work with KyeongRo Kim (KIAS).
Title: Generalized Torsion in Amalgams
Abstract: Recently, generalized torsion has been studied in connection with group orderability, fundamental groups of 3-manifolds, and purely group-theoretic questions. In this talk, we explore generalized torsion in amalgams of groups. I will present a sufficient condition ensuring that an amalgam of two groups is generalized torsion-free, and discuss applications to the construction of generalized torsion-free groups satisfying certain non-orderability properties, including 3-manifold groups and one-relator groups. We will also cover an analogs of Bergman’s theorem and a Freiheitssatz-type result, and conclude with a discussion of some questions that arose during our study. This is a joint work with Adam Clay.
Title: On Hofer's geometry of autonomous flows on the two-sphere
Abstract: Hofer's geometry on the group of Hamiltonian diffeomorphisms is an actively studied yet still elusive notion in symplectic topology. For example, the growth of an autonomous Hamiltonian flow in Hofer's metric is not yet well understood. Polterovich and Rosen have shown that generically this growth is asymptotically linear, and in all known cases where it is not, it appears to be bounded. This dichotomy is known for open connected surfaces of infinite area by a result of Polterovich and Siburg from 2000. I will discuss a new approach to this question, which establishes a strong version of such a dichotomy for the two-sphere. This talk is based on a joint work with Lev Buhovsky, Ben Feuerstein, and Leonid Polterovich.