The Geometry and Geometric Analysis Seminar at Purdue in Spring 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 and Nicholas McCleerey are organizing this seminar in Spring 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: Rigid spines on geometric armadillo tails
Abstract: In this talk, we will study a particular type of finite-area, infinite translation surface. We will show that these surfaces exhibit a cylinder decomposition which does not manifest on finite translation surfaces. Moreover, we will identify a special curve in the closure of the cylinder decomposition that serves as an obstruction for the existence of a parabolic affine diffeomorphism in the direction of the cylinders. This is joint work with Dami Lee.
Title: Graph manifolds and their Thurston norm
Abstract: A classical approach to studying the topology of a manifold is through the analysis of its submanifolds. The realm of 3-manifolds is particularly rich and diverse, and we aim to explore the complexity of surfaces within a given 3-manifold. After reviewing the fundamental definitions of the Thurston norm, we will present a constructive method for computing it on Seifert fibered manifolds and extend this approach to graph manifolds. Finally, we will outline which norms can be realized as the Thurston norm of some graph manifold and a key property of such norms.
Title: An improved ABP estimate in the complex setting
Abstract: In this talk, we will present an improved ABP estimate in the complex setting and discuss its applications to complex Hessian equations. These include a sharp gradient estimate for complex Monge-Ampere equations and a bound on the sup-slope for a class of Hessian equations. The approach is based on a comparison version of the iteration method developed by Guo, Phong, and Tong.
Title: Generalized McCann’s Theorem with Application to Michael-Simon Inequality
Abstract: In optimal transport theory, McCann's theorem provides the existence and description of optimal transport maps on Riemannian manifolds with the distance squared cost. In this talk, I will show a generalization of McCann's theorem to a submanifold setting. As an application, I will use this generalization to prove Michael-Simon inequality in manifolds with lower bounds on intermediate Ricci curvatures.
Title: Viscosity solution to complex Hessian quotient equation
Abstract: In this paper, we prove the existence of viscosity solutions to complex Hessian equations on compact Hermitian manifolds, assuming the existence of a strict subsolution in the viscosity sense. The results cover the complex Hessian quotient equations. This generalized our previous results where the equation needs to satisfy a determinant domination condition. This is a joint work with Prof. Jingrui Cheng.
Title: Stable invariants of words from random matrices
Abstract: Let w be a word in a free group. A few years ago, Magee and I discovered that the stable commutator length of w, which is a well-studied topological invariant, can also be defined in terms of certain Fourier coefficients of w-random unitary matrices.
But there are very natural ways to tweak the random-matrix side of this story: one may consider, for example, w-random permutations or w-random orthogonal matrices, and apply the same definition to obtain other "stable" invariants of w. Are these invariants interesting? Do they have, too, alternative topological/combinatorial definitions?
In a joint work with Yotam Shomroni, we present a conjectural picture and prove some parts of it. No background is assumed - I will define all notions.
Title: Simplicial volume and isolated, closed totally geodesic submanifolds of codimension one
Abstract: We show that for any closed Riemannian manifold with dimension at least two and with nonpositive curvature, if it admits an isolated, closed totally geodesic submanifold of codimension one, then its simplicial volume is positive. As a direct corollary of this, for any nonpositively curved analytic manifold with dimension at least three, if its universal cover admits a codimension one flat, then either it has nontrivial Euclidean de Rham factors, or it has positive simplicial volume. This is based on a joint work with Chris Connell and Shi Wang, arXiv:2410.19981.
Title: Groups acting on trees with APLA and their bounded cohomology
Abstract: We present a family of groups of automorphisms of a regular tree T that have almost prescribed local action (APLA) on the edges around the vertices of T. Since their introduction by Le Boudec, these groups have provided examples for addressing various group-theoretic questions.
In this talk, we prove a condition for the vanishing of their continuous bounded cohomology. Moreover, we show that when this condition is not satisfied, the continuous bounded cohomology in degree two is infinite-dimensional.
Title: On Symmetric Product of Surfaces
Abstract: We discuss some geometric and topological properties of symmetric products SP^n(M_g) of orientable surfaces. In particular we investigate when they admit a positive scalar curvature metric. In view of Gromov's Positive Scalar Curvature conjecture we compute the macroscopic dimension of universal covers of such products. Also, we compute the Lusternik-Schnirelmann category and the topological complexity of them. (This is a joint work with Luca Di Cerbo and Ekansh Jauhari)
Title: Hessian Equations and special concavity on Riemannian manifolds
Abstract: We will discuss the problem of solvability of the real Hessian quotient equation on Riemannian manifolds. The difficulty concerns obtaining a C^2 estimate for this equation, which is known to be one of the few open problems related to hessian type equations. We will provide the solution in real dimension two and discuss some special properties of quotient operators. This is work in progress in collaboration with Pengfei Guan.
Title: The natural flow and the (co)homology of non-positively curved manifolds
Abstract: Recently with Chris Connell and Shi Wang, we introduced a flow that aimed at being a smooth version of the natural map introduced by Besson-Courtios-Gallot. I will discuss how one can use the natural flow/map to prove (co)homological vanishing theorems for non-positively curved manifolds. One crucial feature of the natural flow is that the necessary k-Jacobian estimates needed for proving (co)homological vanishing become k-trace conditions which are substantially easier to deal with. I will report on some forthcoming vanishing results which imply interesting "gap" results for the possible (co)homological dimensions of the infinite covers of compact locally symmetric spaces of higher rank. Specifically, we show that there is a gap in possible dimensions below the dimension of the locally symmetric manifold that is linear in the real rank. In particular, this gap can be arbitrarily large and these provide the first examples for large gaps.