You can find the titles and abstracts of the talks below.
Title: Identifying, matching, and learning in vivo nonlinear coordinate systems
Speaker: Niko Shoncheck
Abstract: Brains use a variety of coordinate systems to encode information. Sometimes these coordinate systems are linear and can be recovered from population activity using standard techniques. Often, however, they are not: many coordinate systems exhibit nonlinear global topology for which such tools can be less effective. Notably, grid cells in the entorhinal cortex comprise two linearly independent circular coordinate systems that, together, exhibit toroidal topology. Recent recordings using high-density probes confirm this toroidal topology persists during spatial and non-spatial behavior, and can be quantified and decoded with persistent (co)homology.
We ask a next natural question: is the propagation of circular coordinate systems through neural circuits a generic feature of biological neural networks, or must this be learned? If learning is necessary, how does it occur? We apply methods from topological data analysis developed to quantitatively measure propagation of such nonlinear manifolds across populations to address these problems. We identify a collection of connectivity and parameter regimes for feed-forward networks in which learning is required, and demonstrate that simple Hebbian spike-timing dependent plasticity reorganizes such networks to correctly propagate circular coordinate systems.
Title: Crystallographic Phase Retreival, Multi-Reference Alignment and Second Moments
Speaker: Arun Suresh
Abstract: X-ray crystallography is a widely used technology for reconstructing the lattice structure of crystallized protein molecules from the diffraction patterns produced when the crystal is bombarded with X-rays. This technology plays a critical role in drug discovery and medicine. At the heart of this method lies the so-called "phase problem": if we treat the protein's electron density as the signal to be reconstructed, the diffraction pattern fails to captures any phase information of this signal. The crystallographic phase retrieval problem therefore seeks to recover an unknown signal from its crystallographic phase-less measurements. In this talk, we will introduce the crystallographic phase retrieval problem, discuss its well-posed-ness, and outline a few results that determine if and when recovery is possible. To establish these recovery results, we will outline how the crystallographic phase retrieval problem is a special instance of a problem seeking to invert the second moment measurements of a more general framework known as the Multi-Reference Alignment (MRA) model.
This talk is based on joint works with Dan Edidin, Nadav Dym and Tamir Bendory.
Title: Equivalence of Landscape and Erosion Distances for Persistence Diagrams
Speaker: Cagatay Ayhan
Abstract: This talk is based on our recent work (arXiv:2506.21488), which establishes connections between three of the most prominent metrics on persistence diagrams in topological data analysis: the bottleneck distance, Patel’s erosion distance, and Bubenik’s landscape distance. Our main result shows that the erosion and landscape distances are equal, thereby bridging the former's natural category-theoretic interpretation with the latter's computationally convenient structure. The proof utilizes the category with a flow framework of de Silva et al., and leads to additional insights into the structure of persistence landscapes. Our equivalence result is applied to prove several results on the geometry of the erosion distance. We show that the erosion distance is not a length metric, and that its intrinsic metric is the bottleneck distance. We also show that the erosion distance does not coarsely embed into any Hilbert space, even when restricted to persistence diagrams arising from degree-0 persistent homology. Moreover, we show that erosion distance agrees with bottleneck distance on this subspace, so that our non-embeddability theorem generalizes several results in the recent literature.
Title: Proximal Transport Divergences for Robust Generative Modeling
Speaker: Benjamin Zhang
Abstract: Generative modeling involves finding mappings from simple to sample reference distributions to complex target distributions. In practical application, these target distributions may be quite singular or may not have a density at all, making finding generative flows for them difficult. Proximal optimal transport (OT) divergences are a novel class of discrepancy measures that interpolate between information divergences and OT distances. These divergences provide a principled foundation for the proximal optimization methods commonly used in generative modeling. We establish that these divergences possess key mathematical properties: smoothness, boundedness, and computational tractability. We then demonstrate that generative flows trained with these divergences are well-posed with desirable regularity properties, making them particularly effective for learning distributions on low-dimensional manifolds while maintaining stable and robust training dynamics. Finally, we apply these robust generative models to likelihood-free inference for generating samples from Bayesian posteriors supported on low-dimensional manifolds.