Eleni Panagiotou

March 27, 2026

Title: Novel metrics of entanglement of open curves in 3-space and their applications to proteins

Abstract: Filamentous materials may exhibit structure-dependent material properties and function that depend on their entanglement. Even though intuitively entanglement is often understood in terms of knotting or linking, many of the filamentous systems in the natural world are not mathematical knots or links. In this talk, we will introduce a novel and general framework in knot theory that can characterize the complexity of open curves in 3-space. This leads to new metrics of entanglement of open curves in 3-space that generalize classical topological invariants, like for example, the Jones polynomial and Vassiliev invariants. For open curves, these are continuous functions of the curve coordinates and converge to topological invariants of classical knots and links when the endpoints of the curves tend to coincide. These methods provide an innovative approach to advance important questions in knot theory. As an example, we will see how the theory of linkoids enables the first, to our knowledge, parallel algorithm for computing the Jones polynomial.

Importantly, this approach opens exciting applications to systems that can be modeled as open curves in 3-space, such as polymers and proteins, for which new quantitative relationships between their structure and material properties become evident. As an example, we apply our methods to proteins to understand the interplay between their structures and functions. We show that our proposed quantitative topological metrics based on static protein structures alone correlate with protein dynamics and protein function. The methods and results represent a new framework for advancing knot theory, as well as its applications to filamentous materials, which can be validated by experimental data and integrated into machine-learning algorithms.

Keywords: entanglement, open curves in 3-space, proteins, polymers, Jones polynomial, Vassiliev invariants, linkoids.

Nikolas Schonsheck

April 10, 2026

Title: Identifying, tracking, and learning the grid cell circular coordinate systems

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. We also observe during this learning process the emergence of geometrically non-local experimentally observed receptive field types: bimodally-tuned head-direction cells and cells with spatially periodic, band-like receptive fields.

Keywords: topological data analysis, persistent homology, neuroscience, neural manifold.