Topology and Geometry Seminar at Auburn

The Topology and Geometry Seminar at Auburn University covers various topics in topology, geometry, and their applications, including but not limited to:

• Differential Geometry and Geometric Analysis

• Set-theoretic and Algebraic Topology

• Topological Data Analysis

• Applications of Topology and/or Geometry in Statistics, Physics, Biology, etc

Date: January 30, Friday

Speaker: Ziqin Feng (Auburn)

Title:  Homotopy Types of Vietoris--Rips Complexes from some Structured Graphs

Abstract: We investigate the topological properties of Vietoris–Rips complexes of the hypercube graph $Q_n$​ equipped with the shortest-path metric. The hypercube’s highly symmetric structure make it a natural setting for studying the interaction between combinatorial geometry and topological complexity in Vietoris–Rips filtrations. We first analyze the homotopy types of these complexes at small scale parameters $r=1,2$, and $3$, providing explicit descriptions of their homotopy types.

For larger scale parameters, we relate the appearance of higher-dimensional homology to classical constructions in combinatorial design theory. In particular, we show that the existence of Hadamard matrices governs the emergence of nontrivial homology in high dimensions. More precisely, we prove that the $(2r−1)$th homology group of the Vietoris–Rips complex $VR(Q_n;r)$ nontrivial for all sufficiently large $n$, provided that a Hadamard matrix of order $2r$ exists. This connection reveals a surprising link between the topology of Vietoris–Rips complexes on hypercube graphs and the existence of highly structured combinatorial objects.

Date: February 13, Friday

Speaker: Bei Wang Phillips (University of Utah) 

Zoom: Available upon request

Title: Mapping the Topology of Chemical Latent Spaces

Abstract: Understanding the structure of latent spaces learned by deep neural networks is increasingly central to interpretability, discovery, and scientific insight. In this talk, I will present recent efforts to characterize the topology of latent representations across modalities, spanning image models, word embeddings, and chemical representation learning. In particular, in molecular science, the design of novel drugs and functional materials can be naturally framed as a search over a learned chemical latent space. However, the combinatorial scale of candidate molecules makes exhaustive exploration computationally infeasible. This motivates tools that expose the global organization of these spaces and guide targeted discovery. To address this challenge, we introduce Chemical Mapper, a framework that integrates topological data analysis with geometric deep learning for interactive exploration of chemical latent spaces. At its core, Chemical Mapper employs the mapper construction to summarize the global shape of latent representations as a graph capturing clusters, overlaps, and branching structure. This topological summary reveals how molecules organize into families and how transitions occur between structural and functional regimes. Our results show that Chemical Mapper reveals intrinsic patterns associated with molecular scaffolds, functional groups, and chemical properties, as well as the structural and functional evolutions of the molecules. This talk is based on a joint work with Dhruv Meduri, Chuan-Shen Hu, Cong Shen, and Kelin Xia.

Date: February 20, Friday

Speaker: Xiaolong Li (Auburn)

Title: On Yau’s pinching problem

Abstract: In 1990, Yau asked in his “Open problems in geometry": “The famous pinching problem says that on a compact simply connected manifold if $K_\min > 1/4 K_\max$ (i.e., the minimum of the sectional curvature is bigger than one quarter of its maximum), then the manifold is homeomorphic to a sphere. If we replace $K_\max$ by normalized scalar curvature (equal to the average of sectional curvatures), can we deduce similar pinching theorems?” In this talk, we present a new partial result to Yau’s pinching problem that improves earlier bounds of Gu–Xu, both by sharpening the pinching constants and by weakening the curvature quantities involved. The key observation is a new link between sectional curvature pinching and positive isotropic curvature.

Date: February 26, Thursday (joint with Colloquium)

Speaker: Lei Ni (Zhejiang Normal University & UC San Diego)

Title: Some spectrum estimates and their applications

Abstract: This talk explores spectral estimates for second-order elliptic operators arising from geometric variational problems and their consequences for stability and rigidity. I will focus on Yang–Mills connections and harmonic maps, viewed as critical points of natural energy functionals. Classical results of Simons and Xin show that nontrivial Yang–Mills fields on high-dimensional spheres and stable harmonic maps from spheres exhibit strong instability or triviality. I will describe refinements of these theorems via quantitative estimates on the spectrum of the associated Jacobi operators, including sharp upper bounds for the first eigenvalue and lower bounds on the dimension of negative eigenspaces. These results extend to minimal submanifolds in spheres and provide a unified perspective on Morse index phenomena in these settings. In the second part of the talk, I will discuss a result relating lower bounds of the Laplace spectrum to quasi-conformal diffeomorphisms. This connection leads to a proof of a theorem of Hamilton in full generality concerning convex hypersurfaces and illustrates how spectral information can control global geometric behavior. The overarching theme is that eigenvalue estimates serve as an effective bridge between analysis, topology, and geometric rigidity. 

Date: March 6, Friday

Speaker: Zhe Su (Auburn)

Title: De Rham-Hodge Methods for Data on Manifolds

Abstract: Topological data analysis (TDA) provides powerful tools for understanding the structure of complex, high-dimensional data, yet most existing methods focus on points, graphs, or simplicial complexes. In this talk, I will present our recently developed de Rham-Hodge-based frameworks for analyzing data on manifolds. These methods provide effective and efficient ways to capture both the topological and geometric information of data and are well-suited for integration with machine learning tasks. I will demonstrate their usefulness through applications in mathematical biology, including protein-ligand binding affinity prediction, single-cell RNA velocity analysis, medical image classification, and B-factor analysis.

Date: March 19, Thursday (joint with Colloquium)

Speaker: Mohammad Ghomi (Georgia Tech)

Title: Geometric inequalities in spaces of nonpositive curvature

Abstract: We will survey a number of outstanding problems and recent results in the theory of Cartan-Hadamard manifolds, i.e., complete, simply connected spaces with nonpositive curvature. These are CAT(0) spaces which generalize the Euclidean and hyperbolic spaces and share many of their basic properties. For instance any pair of points may be connected by a unique geodesic, and thus notions of convexity are natural to study. Yet many basic questions concerning isoperimetric inequalities, total curvature of hypersurfaces, or even the structure of the convex hull of 3 points remain open.

Date: March 26, Thursday (joint with Colloquium)

Speaker: Qi S Zhang (UC Riverside)

Title: Laplace comparison theorem and convergence of K\"ahler Ricci flows

Abstract: We will review the classical Laplace and relative volume comparison theorems on Riemannian manifolds with Ricci curvature bounded from below. Then we will show that, after suitable conformal change, versions of these theorems persist on K\"ahler Ricci flows without extra curvature assumptions. Applications to the convergence issue are also discussed. This is a joint work with G. Tian, Z.L. Zhang, M.Zhu and X.H.Zhu.

Date: April 1, Wendesday (joint with SASA seminar)

Speaker: Andrés Contreras (U Chicago)

Title: Weak exponential metrics for log-correlated Gaussian fields

Abstract: Liouville quantum gravity (LQG) is a canonical model for a random surface, and in particular a random metric, which in two dimensions enjoys a rich geometric structure including conformal invariance. In higher dimensions, however, the existence of analogous objects remains open. Currently, only subsequential limits of suitable approximations are known to exist, following work of Ding-Gwynne-Zhuang. We show that any such appropriately normalized subsequential limit must satisfy a canonical set of axioms, analogous to those of the two-dimensional LQG metric. In addition, we establish several quantitative properties of these limits, including sharp moment bounds for distances and optimal Holder comparisons with the Euclidean metric. Joint with Zijie Zhuang.

Date: April 3, Friday

Speaker: Chul Moon (Southern Methodist University)

Title: Statistical Modeling of Topological Features in Medical Imaging: Enhancing Prognostic Precision and Interpretation

Abstract: Tumor morphology provides important insight into growth and metastatic potential. We propose a topological feature obtained by persistent homology to characterize tumor progression in medical images, with particular emphasis on its association with time-to-event data. These features, which are invariant under scale-preserving transformations, capture complex and heterogeneous shape patterns. To incorporate this information, we develop a functional Cox proportional hazards model that represents these topological features in a functional space, utilizing them as functional predictors. This framework enables an interpretable assessment of how topological shape characteristics relate to survival risk.

Date: April 10, Friday

Speaker: Theodora Bourni (University of Tennessee Knoxville)

Title: Ancient solutions to free boundary mean curvature flow

Abstract: We will discuss rigidity results for ancient solutions to the free boundary mean curvature flow in manifolds with convex boundary. In particular, we will show that any free boundary minimal hypersurface of Morse index $I$ admits an $I$-parameter family of ancient solutions that emanate from it. Moreover, among ancient solutions that backward converge exponentially fast to the minimal hypersurface, these exhaust all possibilities. Additionally, we will show how to construct a smooth free boundary mean convex foliation around an unstable free boundary minimal hypersurface. As an application, we use it to provide a more detailed geometric description of mean-convex ancient solutions that backward converge to that minimal surface. This is joint work with Giada Franz.

Date: April 15, Wednesday, 2:00 - 2:50 (joint with the SASA seminar)

Speaker: Xiaodong Cao (Cornell University)

Title: Differential Harnack Estimates: Past and Future

Abstract: In this talk, I will talk about some classical results about Differential Harnack, including Li-Yau’s and Hamilton’s differential Harnack estimates, and Perelman’s Harnack estimate. Then I will talk about Harnack estimates for various parabolic equations, and possible future with Math AI.

Date: April 15, Wednesday, 3:00 - 3:50  (joint with the SASA seminar)

Speaker: Nicolò Forcillo (Michigan State University)

Title: A perturbative approach for flat free boundaries of the one-phase Stefan problem

Abstract: In Stefan-type problems, free boundaries may not regularize instantaneously. In particular, there exist examples in which Lipschitz free boundaries preserve corners. In the two-phase Stefan problem, I. Athanasopoulos,  L. Caffarelli, and S. Salsa showed that Lipschitz free boundaries in space-time become smooth under a nondegeneracy condition,  as well as sufficiently "flat" ones. Their techniques are based on Caffarelli's original work in the elliptic case. In this talk, we present a more recent approach to investigate the regularity of flat free boundaries for the one-phase Stefan problem. It relies on perturbation arguments leading to a linearization of the problem, in the spirit of the elliptic counterpart developed by D. De Silva. This talk is based on a joint work with D. De Silva and O. Savin.

Date: April 17, Friday

Speaker: Elham Matinpour (Notre Dame of Maryland University)

Title: Geometric Stability of Minimal Surfaces with Y-Singularities

Abstract: Singular minimal surfaces arise naturally in geometric variational problems, yet their stability theory remains far less understood than in the smooth setting. In this talk, I will present geometric stability results for minimal surfaces with Y-singularities in $\mathbb{R}^3$. I will describe how instability decomposes into contributions from the smooth faces together with additional spectral effects localized along the singular junctions, governed by a natural Steklov-type eigenvalue problem. This analysis reveals new low-index phenomena that do not occur in the smooth setting. 

Date: August 21, Friday

Speaker: Hal Schenck (Auburn)

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