Spring 2023
The Geometry and Geometric Analysis Seminar at Purdue in Spring 2023 is usually on Mondays 2:30-3:30pm 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 Sai-Kee Yeung are organizing this seminar in Spring 2023. If you have any questions or would to suggest speakers, please contact one of us. If you would like to be on the mailing list, please email Lvzhou.
January 16, 2023 (No Talk due to MLK Day)
January 30, 2023 (zoom: https://purdue-edu.zoom.us/j/91615764975)
Title: The Kaplansky conjectures
Abstract: There is a series of four fundamental and long-standing conjectures on group rings attributed to Kaplansky. For example, the zero divisor conjecture states that the group ring of a torsion-free group with field coefficients has no zero divisors. I will discuss these conjectures, their connections to other open questions in various areas of mathematics, and my recent disproof of the unit conjecture.
February 6, 2023 (zoom https://purdue-edu.zoom.us/j/91615764975)
Title: Hierarchically hyperbolic spaces: What they are and how to make new ones
Abstract: A hierarchically hyperbolic structure is some kind of coordinate system on a given metric spaces where the coordinates take values in hyperbolic spaces, and it gives a very good understanding of the coarse geometry of the space. I will give a brief introduction to this notion and its consequences, discuss a simple criterion to show that a space or group is hierarchically hyperbolic, and illustrate applications of the criterion.
February 13, 2023 (zoom: https://purdue-edu.zoom.us/j/91615764975)
Title: Marked length pattern rigidity
Abstract: Given a closed Riemannian manifold M, the length of the shortest geodesic for each free homotopy class of loops on M is called the (minimal) length of the class. This gives a map called marked length spectrum. It is conjectured that the fundamental group and marked length spectrum together determine the isometric type of negatively curved manifolds. This conjecture has been verified for surfaces and locally symmetric spaces. In this talk, we show that for negatively curved arithmetic manifolds, the fundamental group with all pairs of different equal length classes, i.e., marked length pattern, is enough to recover the metric up to scaling.
February 27, 2023
Title: Geodesic Rays in the Donaldson-Uhlenbeck-Yau Theorem
Abstract: The theorem of Donaldson-Uhlenbeck-Yau says that a holomorphic vector bundle E over a compact Kahler manifold admits a Hermite-Einstein (HE) metric iff E is stable. Historically, this was the first example of a general program linking solvability of certain geometric PDE (the HE metric) with a stability condition, and is something of a spiritual predecessor to the Yau-Tian-Donaldson conjecture. Work on this subsequent conjecture has revealed an important link with a third object, namely, geodesic rays of ``weak" metrics. In joint work with Jonsson, Shivaprasad, we return to the DUY theorem and, by focusing on the analogous geodesic rays in this setup, find a new proof of this celebrated result.
March 3, 2023 (Friday 2:30 pm, exceptional date and location, in person at UNIV 103)
Title: Weil-Petersson curves, traveling salesman theorems, and minimal surfaces
Abstract: Weil-Petersson curves are a class of rectifiable closed curves in the plane, defined as the closure of the smooth curves with respect to the Weil-Petersson metric defined by Takhtajan and Teo in 2006. Their work solved a problem from string theory by making the space of closed loops into a Hilbert manifold, but the same class of curves also arises naturally in complex analysis, geometric measure theory, probability theory, knot theory, computer vision, and other areas. No geometric description of Weil-Petersson curves was known until 2019, but there are now more than twenty equivalent conditions. One involves inscribed polygons and can be explained to a calculus student. Another is a strengthening of Peter Jones's traveling salesman condition characterizing rectifiable curves. A third says a curve is Weil-Petersson iff it bounds a minimal surface in hyperbolic 3-space that has finite total curvature. I will discuss these and several other characterizations and sketch why they are all equivalent to each other. The lecture will contain many pictures, several definitions, but not too many proofs or technical details.
March 7, 2023 (Tuesday 2 pm, exceptional date and time, in person at MATH 731)
Title: The automorphism group of a free group is not virtually a Kahler group
Abstract: Bridson proved that the automorphism group Aut(F_n) of a free group F_n is not a Kahler group. In particular, it is not the fundamental group of a smooth projective variety. I will explain how to extend this to show that no finite-index subgroup of Aut(F_n) is a Kahler group.
March 13, 2023 (No Talk due to Spring Break)
March 20, 2023 (in person)
Title: First order theory of homeomorphism groups of compact manifolds
Abstract: I will describe some recent work on the first order theory of homeomorphism groups of manifolds. I will discuss a new result which shows that the homeomorphism groups of two compact manifolds are elementarily equivalent if and only if the two manifolds are homeomorphic, which resolves an old conjecture of Rubin. I will then describe some of the expressive power of the language of groups in the theory of homeomorphism groups, with implications for the subgroup structure of homeomorphism groups, and for the descriptive set theory of these groups.
March 27, 2023 (in person)
Title: Riemannian metrics on hyperbolic components
Abstract: We introduce a Riemannian metric on certain hyperbolic components of the moduli space of rational maps which is an analogue of the Weil-Petersson metric on Teichmuller spaces. We also show that our metric on the space of quasi-Blaschke products is an extension of McMullen’s Weil-Petersson type metric on the space of Blaschke products and is degenerate along the purely imaginary directions. This is an analogue of Bridgeman’s extension of Weil-Petersson metric to quasi-Fuchsian spaces. The talk is based on joint works with Hongming Nie, Homin Lee and Insung Park.
April 3, 2023 (9:30 am, exceptional time, zoom)
(zoom link: https://purdue-edu.zoom.us/j/91615764975)
(zoom link: https://purdue-edu.zoom.us/j/91615764975)
Title: Are right-angled Artin groups really right-angled angled?
Abstract: The main goal of the talk will be to motivate the extension of reflection groups to rotation groups, a rotation being thought of as fixing pointwise a separating subset and permuting freely-transitively the (possibly infinitely many) components. This approach offers a common point of view on Coxeter groups and graph products of groups such as right-angled Artin groups (which appear as right-angled rotation groups). Geometrically, this suggests the introduction of a new family of graphs, called mediangle graphs, which includes (quasi-)median graphs and Cayley graphs of Coxeter groups.
April 10, 2023 (9:30 am, exceptional time, zoom)
(zoom link: https://purdue-edu.zoom.us/j/91615764975)
(zoom link: https://purdue-edu.zoom.us/j/91615764975)
Title: Spectral gaps of random hyperbolic surfaces
Abstract: We study the low-energy spectrum of the Laplacian on finite-area hyperbolic surfaces. A quantity of particular interest is the spectral gap which provides information about the connectivity of the surface, the rate of mixing of the geodesic flow and error terms in geodesic counting. We shall look at the size of the spectral gap for random hyperbolic surfaces. I will discuss some different constructions of random surfaces and explain recent developments in this area. Based on joint works with Michael Magee and with Joe Thomas.
April 10, 2023 (in person)
Title: Positively curved Finsler metrics on vector bundles
Abstract: While the equivalence between ampleness and positivity holds for vector bundles of rank one, its higher rank counterpart known as Griffiths’ conjecture is still open. There is also a similar but weaker conjecture by Kobayashi who proposed to use Finsler rather than Hermitian metrics to study the equivalence. We will review these two conjectures and state our progress. One of our results is that Kobayashi positivity implies ampleness and convex Kobayashi positivity. We will also discuss how to prove Kobayashi positivity for ample vector bundles with additional curvature assumptions.
April 17, 2023 (9:30 am, exceptional time, zoom)
(zoom link: https://purdue-edu.zoom.us/j/91615764975)
(zoom link: https://purdue-edu.zoom.us/j/91615764975)
Title: Length, scl, and the Hodge Laplacian
Abstract: Geodesic length and stable commutator length give geometric and topological notions of complexity for nullhomologous elements of the fundamental group of a hyperbolic manifold. The ratio of these complexity measures is a sort of geometric-topological isoperimetric ratio called the stable isoperimetric ratio. In this talk, I will discuss this ratio and describe how it relates to different aspects of the geometry and topology of hyperbolic manifolds. In particular, I will highlight a connection to the first eigenvalue of the Hodge Laplacian acting on coexact 1-forms.
April 24, 2023 (in person)
Title: Discrete subgroups of small critical exponents
Abstract: It is conjectured by Kapovich that finitely generated Kleinian groups with critical exponent less than 1 are convex-cocompact. In this talk, we partly answer this in the affirmative by showing that any finitely generated Kleinian groups with sufficiently small critical exponent are convex-cocompact. We also give some geometric properties of hyperbolic manifolds with critical exponent less than 1. This is joint work with Shi Wang.
May 1, 2023 (in person)
Title: The natural flow and homological vanishing in nonpositively curved manifolds
Abstract: We introduce a flow on nonpositively curved manifolds inspired by the natural maps of Besson, Courtois and Gallot for which the Morse theoretic data can be computed in terms of the geometric structure of the manifold. We present several applications of this flow, including conditions for the nonexistence of complex subvarieties and estimates of the Cheeger constant on such manifolds. Most importantly, we show the vanishing of the homology of nonpositively curved manifolds above a certain threshold which is computable from the geometry of its universal cover and the critical exponent of the representation of the fundamental group. Time permitting, we will present some examples including those arising from Anosov representations in higher rank lie groups. This is joint work with Shi Wang and Ben McReynolds.