Purdue Geometry and
Geometric Analysis
Seminar
The Geometry and Geometric Analysis Seminar at Purdue in Spring 2024 is usually on Mondays 3:30pm-4:30pm 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 and Nicholas McCleerey are organizing this seminar in Spring 2024. 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.
Spring 2024
Next talk
April 22, 2024 (in person)
Title: Almost-Kahler Ricci Solitons
Abstract: In this talk, we introduce almost-Kahler Ricci solitons. They are obstructions to the existence of almost-Kahler metrics of constant Chern scalar curvature on symplectic Fano manifolds. We study deformations of such metrics and we give some examples. We also study the Lie algebra of real holomorphic vector fields on manifolds admitting such solitons. In particular, we partially extend Matsushima theorem to the almost-Kahler setting.
All talks in Spring 2024
January 8, 2024 (in person)
Title: Residual finiteness growth functions of surface groups with respect to characteristic quotients
Abstract: Residual finiteness growth functions of groups have attracted much interest in recent years. These are functions that roughly measure the complexity of the finite quotients needed to separate particular group elements from the identity in terms of word length. In this talk, we study the growth rate of these functions adapted to finite characteristic quotients. One potential application of this result is towards linearity of the mapping class group.
January 15, 2024 (No talk due to MLK day)
January 22, 2024 (in person)
Title: Plurisubharmonicity of the Dirichlet energy
Abstract: Harmonic mappings of Riemannian manifolds are characterized as critical points of the Dirichlet energy $E$. Given a smooth family of harmonic mappings $\{f_t\}_{t\in S}$, it is natural to view the energy $E$ as a function on the parameter space $S$. In other words, we consider the function $E: S\to \mathbb{R}$, $t\mapsto E(f_t)$. In this talk, we will discuss a subharmonicity property of the energy function $E$ and study its connection with rigidity and moduli theories.
January 29, 2024 (zoom https://purdue-edu.zoom.us/j/91967506594)
Title: A new vanishing criterion for bounded cohomology
Abstract: Bounded cohomology is a powerful and by now well-established tool to study properties of groups and spaces. However, it is often difficult to compute and, beyond the case of amenable groups, many questions remain widely open.
In joint work with Francesco Fournier-Facio, Yash Lodha and Marco Moraschini, we present a new algebraic condition that implies the vanishing of the bounded cohomology of a given group for a big family of coefficient modules. This condition is satisfied by many non-amenable groups of topological, geometric or algebraic origin.
February 19, 2024 (zoom https://purdue-edu.zoom.us/j/91967506594)
Title: Singular cscK metrics on smoothable varieties
Abstract: Searching for canonical metrics in Kähler classes has been a central theme in Kähler geometry for decades. This talk aims to explain a method for investigating canonical metrics in families of singular varieties, employing relative versions of pluripotential theory and variational approach in families. We shall start by reviewing fundamental concepts and important properties within the variational picture of constant scalar curvature Kähler (cscK) metrics. I will then introduce notions of weak and strong topologies of quasi-plurisubharmonic functions in families, and explain several properties of entropy and Mabuchi functional extended to the family framework. Finally, I will demonstrate how these properties contribute to obtaining the stability of coercivity of Mabuchi functional for the family parameter and the construction of cscK metrics on smoothable varieties. This is joint work with T. D. Tô and A. Trusiani.
February 26, 2024 (in person)
Title: A variational problem in Kahler geometry
Abstract: Consider the following problem: Given a convex body in some real Euclidean space, among all ellipsoids inscribed in it find/characterize the one with greatest volume. This problem was posed and solved in a 1948 paper by Fritz John. A rather straightforward complex variant can be viewed as a question concerning hermitian metrics on line bundles over complex projective space. The talk will be about a certain generalization of this latter problem from projective spaces to general compact Kahler manifolds.
March 4, 2024 (in person)
Title: On the solvability of general inverse $\sigma_k$ equations
Abstract: In this talk, first, I will introduce general inverse $\sigma_k$ equations in Kähler geometry. Some classical examples are the complex Monge–Ampère equation, the J-equation, the complex Hessian equation, and the deformed Hermitian–Yang–Mills equation. Second, by introducing some new real algebraic geometry techniques, we can consider more complicated general inverse $\sigma_k$ equations. Last, analytically, we study the solvability of these complicated general inverse $\sigma_k$ equations.
March 11, 2024 (No talk due to Spring break)
March 18, 2024 (in person)
Title: Stir fry Homeo_+(S^1) representations from pseudo-Anosov flows
Abstract: A total linear order on a non-trivial group G is a left-order if it’s invariant under group left-multiplication. A result of Boyer, Rolfsen and Wiest shows that a 3-manifold group has a left-order if and only if it admits a non-trivial representation into Homeo_+(S^1) with zero Euler class. Foliations, laminations and flows on 3-manifolds often give rise to natural non-trivial Homeo_+(S^1)-representations of the fundamental groups, which have proven to be extremely useful in studying the left-orderability of 3-manifold groups.
In this talk, we will present a recipe of stir frying these Homeo_+(S^1)-representations. Our operation generalizes a previously known ``flipping'' operation introduced by Calegari and Dunfield. As a consequence, we constructed a surprisingly large number of new Homeo_+(S^1)-representations of the link groups. We then use these newly obtained representations to prove the left-orderablity of cyclic branched covers of links associated with any epimorphisms to Z_n. This is joint work with Steve Boyer and Cameron Gordon.
March 25, 2024 Postponed to Fall Semester
Title: Higher rank lattice actions with positive entropy
Abstract: In this talk, I will discuss about smooth higher rank lattice actions on manifolds with positive entropy. For instance, when lattices in SL(n,R) (with n>2) act on an n-dimensional manifold with positive entropy, we can see that the lattice is commensurable with SL(n,Z) and some information about manifolds. This is joint work with Aaron Brown.
April 1, 2024 (in person)
Title: A Flowing Construction to Generalize Poincaré-Type Metrics
Abstract: Poincaré-type metrics are a flavor of cusp metrics on the complement of a divisor in a compact Kähler manifold exhibiting many friendly geometric properties. Yet, as shown by H. Auvray in the context of Calabi's extremal metrics, they have certain limitations owing to their sensitivity to the geometry of the ends. We introduce a construction called gnarling which augments Poincaré-type metrics by incorporating certain holomorphic flows along the divisor. After outlining several aspects of these gnarled metrics, including a key growth estimate, we show their utility in perturbing classes of cscK Poincaré-type metrics.
April 8, 2024 (special time: 1:30-2:30pm; zoom https://purdue-edu.zoom.us/j/91967506594)
Title: Sharp spectral gaps for scl from negative curvature
Abstract: Stable commutator length is a measure of homological complexity of group elements, which is known to take large values in the presence of various notions of negative curvature. We will present a new geometric proof of a theorem of Heuer on sharp lower bounds for scl in right-angled Artin groups. Our proof relates letter-quasimorphisms (which are analogues of real-valued quasimorphisms with image in free groups) to negatively curved angle structures for surfaces estimating scl.
Talk Recording (Passcode: B3fyZd%3)
April 15, 2024 (in person)
Title: Entire holomorphic functions on Fermat surface in $\mathbb{C}^3$
Abstract: Let $S_n=\{x_1^n+x_2^n+x_3^n=1\}$ be the Fermat surface of degree $n$ in $\mathbb{C}^3$. A natural question is whether $S_n$ contains some `non-trivial' solution $x_i=f_i$ in entire holomorphic functions $f_i$ from $\mathbb{C}$. In this talk we will explain the approach of a recent rejoint work with Tuen-Wai Ng in showing that there is no non-trivial solution for $n=6$, the only case unknown. The question is usually attributed to Hayman who showed that there is no solution for $n\geq 7$ in 80's.
April 22, 2024 (in person)
Title: Almost-Kahler Ricci Solitons
Abstract: In this talk, we introduce almost-Kahler Ricci solitons. They are obstructions to the existence of almost-Kahler metrics of constant Chern scalar curvature on symplectic Fano manifolds. We study deformations of such metrics and we give some examples. We also study the Lie algebra of real holomorphic vector fields on manifolds admitting such solitons. In particular, we partially extend Matsushima theorem to the almost-Kahler setting.