Vrije Universiteit Amsterdam, April 8, 2025.
Please register by clicking the link to the right.
NU-building, 9th floor (Math department), seminar room next to the common area.
Tentative schedule
10:00 – 10:50 Coffee, arriving
10:50 – 11:00 Welcome
11:00 – 11:45 Francesca Arici (Leiden)
11:45 – 12:30 Karthik Viswanathan (UvA)
12:30 – 14:00 Lunch
14:00 – 14:45 Ioannis Diamantis (Maastricht)
14:45 – 15:30 Rui Dong (VU)
15:30 – 15:45 Coffee
15:45 – 16:30 Wout Moltmaker (UvA)
16:30 – 18:00 Borrel
Organisers:
Abstracts:
Francesca Arici
Toeplitz extensions in mathematical physics and K-theory
Motivated by the role played by Toeplitz-type extensions in the study of solid-state physics, we review the theory of Toeplitz algebras, their central role in operator K-theory and operator algebras, and applications to computation of invariants of solid-state systems. We will discuss also some open problems in K-theory related to the Douglas—Arveson conjecture in index theory.
Ioannis Diamantis
Bonded Knots and Braids: From Topology to Algebra and Protein Structure
In biological systems, proteins are intricate molecular structures formed by long chains of amino acids. These chains fold into specific three-dimensional shapes, stabilized by intramolecular bonds between certain amino acid residues. These bonds are crucial for maintaining the protein’s structure and ensuring its proper function. From a mathematical perspective, proteins can be modeled as bonded knots, where the backbone of the protein is represented as a knot or an open curve, and the stabilizing interactions correspond to connections between different segments of the structure.
In this talk, I will introduce the theory of bonded knots and its natural extension to bonded braids, exploring their structural and algebraic properties. Bonded knots generalize classical knots by incorporating additional constraints in the form of bonds, which can be classified into three main categories: long bonds (both topological and rigid-vertex), regular bonds (where bonds are unknotted), and tight bonds (modeled as line segments with no crossings). For each category, I will present Reidemeister-type theorems, both in the topological and rigid setting, describing the corresponding moves, and introduce fundamental invariants.
Building upon this framework, I will discuss bonded braids, presenting an Alexander theorem that establishes a connection between bonded knots and bonded braids in the topological setting. I will introduce the bonded braid monoid, describe its generators and defining relations, and state a Markov theorem that characterizes equivalent bonded braids. Finally, I will show how the bonded braid monoid embeds into a group.
If time permits, I will conclude with an exploration of bonded knots on the torus and their connection to doubly periodic bonded tangles, discussing their infinite covers and potential applications.
This talk aims to provide a comprehensive introduction to the emerging theory of bonded knots and braids, highlighting its interplay between topology, algebra, and biological structures.
Rui Dong
Fast Computation of the Up Persistent Laplacian for Pseudomanifolds and Cubical Complexes
We show that an orthogonal basis for the kernel of a non-branching matrix can be computed in quadratic time. Non-branching matrices are real matrices with entries in $\{-1,0,1\}$, where each row contains at most two non-zero entries. Such matrices naturally arise in the study of Laplacians of pseudomanifolds and cubical complexes. Building on this result, we show that the up persistent Laplacian can be computed in quadratic time for pairs of such spaces. Furthermore, we show that the up persistent Laplacian of $q$-non-branching simplicial complexes can be represented as the Laplacian of an associated hypergraph, thus providing a higher-dimensional generalization of the Kron reduction, as well as a Cheeger-type inequality. Finally, we highlight the efficiency of our method on image data.
Wout Moltmaker
Thistlethwaite theorems for knotoids and linkoids
Knotoids are a natural generalization of knots, consisting of knot diagrams modeled on the interval rather than the circle. Similarly linkoids are diagrams with multiple knotted components, which may be closed or open-ended. Knotoids and linkoids have found applications in protein- and DNA topology, but are relatively poorly understood compared to classical knots and links. In this talk I will introduce knotoids and linkoids and show how they are used in applications. Then I will define 'twisted' knotoids, which describe knotoids in arbitrary closed surfaces, and define the 'arrow' polynomial for twisted knotoids which is an extension of the Kauffman bracket polynomial invariant. Finally I will give a Thistlethwaite-style theorem for these, which states that the arrow polynomial of a twisted knotoid can be obtained as an evaluation of the Bollobás-Riordan polynomial of a ribbon graph associated with the knotoid diagram.
Karthik Viswanathan
Persistent Topological Features in Large Language Models
Recent advances in large language models (LLMs) have triggered growing interest in understanding their internal decision-making processes. In this talk, I will introduce a topological framework for analyzing LLMs using zigzag persistence, a powerful tool from topological data analysis, to capture how topological features such as n-dimensional “holes” appear, vanish, and reappear as data representations evolve layer by layer. Our method tracks these features directly across their full evolutionary paths, enabling a more holistic view of how prompts and their relative positions change in the representation space. I will present the topological descriptors that reveal unique dynamical signatures in both datasets and model architectures, while also illustrating how these descriptors can provide a principled criterion for layer pruning, preserving overall system performance and offering a global perspective on layer interactions.