An introductory lecture following the approach of the book by Herlihy, Kozlov, Rajsbaum Distributed Computing through Combinatorial Topology, Elsevier-Morgan Kaufmann, based on the notions of indistinguishability and perspectives. Illustrating the use of simplicial complexes via simple examples. No preliminary knowledge either about topology or about distributed computing.

A very elementary introduction to the combinatorial topology approach to distributed computing.

Geometric and topological features play an important role in many physical systems, especially in fluid dynamics where forces drive continuous deformation. The benign influence of geometry, which is the muscular actor in topological dynamics, has sometimes a more devilish nature. Here we show how two different types of singularities that may emerge during the evolution of magnetic fields and soap films are responsible for the spontaneous reorganization and energy change of these systems. A first example is given by the inflexional instability of magnetic knots that under conservation of self-linking number brings inflexional magnetic knots to inflexion-free braids through an integrable singularity of torsion. A second example is given by the continuous deformation of a soap film Möbius band to a 2-sided disk by a drastic change of the soap film topology through a twisted fold catastrophe. These are examples of generic behaviors that trigger energy relaxation in a wide variety of different physical contexts. [1] Moffatt, H.K. & Ricca, R.L. (1992) Helicity and the Călugăreanu invariant. Proc. R. Soc. A 439, 411-429. [2] Ricca, R.L. (2005) Inflexional disequilibrium of magnetic flux tubes. FluidDyn. Res. 36, 319-332. [3] Goldstein, R.E., Moffatt, H.K., Pesci, A.I. & Ricca, R.L. (2010) A soapfilm Möbius strip changes topology with a twist singularity. Proc.Natnl. Acad. Sci. USA 107, 21979-21984.

Geometric devils in topological dynamics.

Using the notion of contiguity of simplicial maps, and its relation (via iterated subdivisions) to the notion of homotopy between continuous maps, we adapt Farber's topological complexity to the realm of simplicial complexes. We show that, for a finite simplicial complex K, our discretized concept recovers the topological complexity of the realization of K.

Simplicial complexity: piecewise linear motion planning in robotics.

Large-scale homology computations are rendered tractable in practice by a combinatorial version of Morse theory. Assuming that my internet connection survives, in this talk I will introduce discrete Morse theory, describe how it helps with computing various types of homology, and outline a powerful new higher-categorical extension. The audience will not require prior knowledge of any of the technical terms mentioned above.

Discrete Morse Theory

The world would look very different to us if, like many insects and other animals, our eyes could see the polarization of light. In particular, we would see a distinctive “fingerprint” pattern in the blue sky overhead, as well as many crystalline materials that surround us. In this talk I will describe the geometry of optical polarization patterns from Maxwell’s equations, and in doing so reveal the deep and surprising relationship between the polarization pattern of birefringent crystals and the blue sky, based on a geometric and topological understanding of the tensors involved in the propagation and scattering of light.

Polarization geometry of crystals and the blue sky.

Knots and supercoiling are both introduced in bacterial plasmids by catalytic processes involving DNA strand passages. I will report on a recent study where we used molecular dynamics simulations and a mesoscopic DNA model, to study the simultaneous presence of knots and supercoiling in DNA rings and the kinetic and metric implications which may be relevant for the simplyfing action of topoisomerases. Finally, I will discuss how the same modelling and simulation approach can be used shed light on the complex experimental phenomenology of knotted DNA translocating through solid state nanopores.

Knotted DNA: conformational, dynamical and pore-translocation properties

Discrete Exterior Calculus (DEC) is a relatively new method for solving partial differential equations numerically. The central idea of the method is to develop a discrete version of Exterior Differential Calculus. In this talk, we will briefly review the various operators of Exterior Differential Calculus and their discrete versions. We will then focus our attention in 2D DEC, presenting a geometric interpretation of DEC as well as numerical comparisons with the Finite Element Method in order to show DEC’s competitive performance.

A brief introduction to Discrete Exterior Calculus

Cellular DNA is a long, thread-like molecule with remarkably complex topology. Enzymes that manipulate the geometry and topology of cellular DNA perform many vital cellular processes (including segregation of daughter chromosomes, gene regulation, DNA repair, and generation of antibody diversity). Some enzymes pass DNA through itself via enzyme-bridged transient breaks in the DNA; other enzymes break the DNA apart and reconnect it to different ends. In the topological approach to enzymology, circular DNA is incubated with an enzyme, producing an enzyme signature in the form of DNA knots and links. By observing the changes in DNA geometry (supercoiling) and topology (knotting and linking) due to enzyme action, the enzyme binding and mechanism can often be characterized. This talk will discuss topological models for DNA strand passage and exchange, including the analysis of site-specific recombination experiments on circular DNA and the analysis of packing geometry of DNA in viral capsids.

DNA Topology

One of the paramount challenges in neuroscience is to understand the dynamics of networks of interconnected neurons. In this talk I will describe a novel approach to classifying network dynamics that employs tools from topological data analysis. The efficacy of our method was established by studying simulated activity in three small artificial neural networks in which we varied certain “biological” parameters, giving rise to dynamics that could be classified into four regimes, and showing that a machine learning classifier trained on features extracted from persistent homology could accurately predict the regime of the network it was trained on and also generalize to other networks not presented during training. I will also discuss the application of these methods to networks of actual neurons in mice. This is joint work with Jean-Baptiste Bardin and Gard Spreemann.

Topological exploration of neuronal network dynamics.

After recalling classical topological persistence, I introduce the generalization of the persistent Betti number functions to what we call persistence functions. These are defined axiomatically on a wide class of concrete categories. Then, three types of processes generating persistence functions are shown: coherent samplings and ranging or steady features. Examples of all of them are given on weighted graphs coming from real situations. This is based on a joint work with Mattia G. Bergomi, Pietro Vertechi and Lorenzo Zuffi.

Non-topological persistence.

The Cheeger constant is a measure of the edge expansion of a graph, and as such plays a key role in diverse mathematical areas, in particular in combinatorics and theoretical computer science. In recent years there is an interest in higher dimensional notions of expansion that likewise provide a quantitative measure of cohomological acyclicity of a complex in dimension k. In this talk we will focus on a k-dimensional version of the graphical Cheeger constant called coboundary expansion, and on a related parameter called cosystolic expansion. After introducing these notions and outlining their relevance to random complexes and to topological overlap theorems, we will discuss two results. The first concerns an application of cosystolic expansion to a stability version of the cohomological classification of G-covering spaces (joint work with Irit Dinur). The second is a lower bound on coboundary expansion under certain symmetry assumptions (joint work with Dmitry Kozlov).

Aspects of High Dimensional Expansion.

Very recently, chemists have succeeding in synthesizing purified samples of polymers which are more topologically complicated than the linear or ring polymers which have previously been created. The new polymers have different graph types, including theta-curves, multithetas, K_{3,3}, and K_4, with new types arriving at a rapid rate. In this talk, we develop a theory of Gaussian random graph embeddings in order to model the shapes of these polymers in solution. The theory allows us to connect the expected radius of gyration of the polymers with the Kirchhoff index of the graph. We will prove a theorem that describes the asymptotic behavior of the expected radius of gyration for large graphs of fixed topology in terms of the degree-Kirchhoff index of the graph. Last, we show that our theory provides a good fit to data from the Tezuka lab's experimental measurements of the relative sizes of topological polymers with different graph types. This talk is joint work with Tetsuo Deguchi and Erica Uehara (Ochanomizu University) and Clayton Shokwiler (Colorado State University), with support from the Simons Foundation (#354225 and #52412) and the Japan Science and Technology Agency (CREST Grant Number JPMJCR19T4, Deguchi Lab) and JSPS KAKENHI Grant Number JP17H06463.

Modeling topological polymers with Gaussian random graph embeddings.

DNA replication in bacteria yields two interlinked circular chromosomes. Returning the chromosomes to an unlinked monomeric state is essential to cell survival. Simplification of DNA topology is mediated by enzymes, such as recombinases and topoisomerases. We here focus on site-specific recombinases that recognize two short segments of DNA (the recombination sites), introduce two double stranded breaks and recombine the ends. The local action of site-specific recombinases is a reconnection event which is modeled mathematically as a band surgery. The banding is coherent or non-coherent, depending on the relative orientation of the recombination sites. We use tools from low dimensional topology to investigate the mechanism of action of these enzymes, and we complement the analytical work with computer simulations. The numerical work provides a quantitative measure to distinguish among pathways of topology simplification by recombination, and also informs the search for bandings between specific pairs of link types.

Topological modeling of DNA recombination

There exists a wide class of winding problems, where one or more planar Brownian motions get entangled around each other and obstacles in the plane. These can be seen as models of entanglement of long molecules, or as a general framework for understanding dynamical topological complexity. I will review some classical results, such as Spitzer's law, which gives the angular distribution for a planar Brownian motion around the origin. I will discuss a recent variation (with Huanyu Wen) where we add a drift that promotes entanglement. Finally, I will present ongoing work (with Gage Bonner and Benedek Valko) on the much more challenging problem of winding around several obstacles. In some limits, this can be reduced to computing the escape rate for a random walk on a so-called regular languages.

Random entanglements: Winding of planar Brownian motions

When dealing with complex high-dimensional data sets, several machine learning tasks rely on having appropriate low-dimensional representations. These reductions are often phrased in terms of preserving statistical or metric information and may fail to recover important topological features. We will describe in this talk several schemes to take advantage of the underlying topology of a data set, in order to produce informative low-dimensional coordinates.

Topological Dimensionality Reduction.

Almost everything we encounter in our 3-dimensional world is a surface - the outside of a solid object. Comparing the shapes of surfaces is, not surprisingly, a fundamental problem in both theoretical and applied mathematics. Deep mathematical results are now being used to study objects such as bones, brain cortices, proteins and biomolecules. This talk will discuss joint work with Patrice Koehl that introduces a new metric on the space of genus-zero surfaces and applies it in this context.

Some New Geometric Methods to Distinguish Shapes and Applications in Biology

Indecomposables obtained through decompositions of persistent homology are regarded as topological summary of real data. However, as is well known, there exist pathologically complicated indecomposables in multi-parameter persistent homology in purely algebraic setting, and this fact makes it difficult to build mathematical theory on that setting. Our fundamental question is, how much should we care about such complicated indecomposables in the real data, and what is a suitable framework to study this question? To this aim, we will show several ongoing results, especially, (1) large deviation principle on 1 parameter persistent homology, and (2) law of large numbers on multi-parameter persistent homology. Then we will discuss how these two results (partially) answer to the original question.

Characterizing rare events in persistent homology.

A multitude of knot invariants, including quantum invariants and their categorifications, have been introduced to aid with characterizing and classifying knots and their topological properties. Relations between knot invariants and their relative strengths at distinguishing knots are still mostly elusive. In addition to Ball Mapper, and machine learning techniques, we develop a new approach using filtrations to analyze infinite data sets where representative sampling is impossible or impractical, an essential requirement for working with knots and the data from knot invariants. Although of different origins, these methods confirm and illuminate similar substructures in knot data created for 10 million knots up to 17 crossings.

A landscape of knots: TDA and dimension reduction techniques in theoretical mathematics.

Persistent homology is a well known invariant encoding geometric properties of an underlying metric space. While the nature of this encoding and precise geometric information it contains are not well understood, it is known that geodesic circles (a special class of closed geodesics in a geodesic space) can sometimes be detected with one- or higher-dimensional persistent homology. In this talk we will consider a converse task: we want to use persistent homology to detect as many geodesic circles in a space as possible. To that end we introduce a new construction of complexes called selective Rips complexes. They can be thought of as custom made complexes designed to detect geodesic circles. It turns out that employing selective Rips complexes we can detect and locate a fairly large class of geodesic circles using one- and two-dimensional persistent homology.

Detecting geodesic circles with selective Rips complexes.

COVID-19 is a disease caused by a coronavirus named SARS-CoV-2. As the virus has been spreading around the world, hundred of thousands of viral genomes have been sequenced. Genomes provide an accurate record of variation and evolution and can inform how these viruses emerge and evolve. In this talk I will discuss how mutations and recombinations shape SARS-CoV-2 evolution, and how topological data analysis techniques can help to understand the role of these processes in the emergence of this and other potential future viruses.

COVID-19, Viruses, Recombination and Topology

Magnetic helicity is an ideal MHD invariant; it measures geometric and topological properties of a magnetic field. The talk will begin by reviewing helicity and its mathematical properties. It can be decomposed in several ways (for example, self and mutual helicity, Fourier spectra, field line helicity, linking, twist, and writhe). The talk will also review methods of measuring helicity flux, as well as applications in solar and stellar astrophysics. The poloidal-toroidal decomposition of a vector field will be discussed, with a generalization to arbitrary simply connected compact surfaces. I will then discuss some new developments in measuring localized concentrations of helicity in a well-defined, gauge invariant manner, using wavelets.

Magnetic helicity: decompositions and methods of localization.

Reeb graphs and main other related graphical signatures have extensive use in applications, but only recently has there been intense interest in finding metrics for these objects. In this talk, we introduce an extension of smoothing on Reeb graphs, which we call truncated smoothing; this in turn allows us to define a new family of metrics which generalize the interleaving distance for Reeb graphs. Intuitively, we ``chop off'' parts near local minima and maxima during the course of smoothing. After formalizing truncation as a functor, we show that when applied after the smoothing functor, this prevents extensive expansion of the range of the function, and yields particularly nice properties (such as maintaining connectivity) when combined with smoothing. Further, for certain choices of the truncation parameter, we have additional structure which we can take advantage of to construct a categorical flow for any choice of slope m in [0,1]. Using the infrastructure built for a category with a flow, this then gives an interleaving distance for every m in [0,1], which is a generalization of the original interleaving distance, which is the case m=0. While the resulting metrics are not stable, we show that any pair of these for m,m' in [0,1) are strongly equivalent metrics, which in turn gives stability of each metric up to a multiplicative constant. This talk is on joint work with Erin Chambers and Tim Ophelders.

The Truncated Interleaving Distance for Reeb Graphs

One of the key goals in topological data analysis is find or infer the global structure of a sampled space by correctly determining the homology groups. Given a sample (data) embedded in Euclidean space (or some more general space), the general pipeline is to build a distance based filtration and compute the corresponding persistence diagram. The standard way of proving results about the result is via the bottleneck stability of persistence diagrams. However, it turns out that in many cases this provides a highly pessimistic approximation result. In this talk, I will present recent results which show that the structure of an underlying space appears much earlier than expected and is remarkably stable. I will describe how these results fit into our understanding of the stability of persistence and show how the persistent approach can be useful in new ways. The talk will assume some passing familiarity with persistent homology and TDA but will not assume any previous knowledge about percolation.

Homological Percolation and Persistence.

George Dantzig’s Simplex method is a work horse of modern computational mathematics. Because linear programs (LPs) appear in all areas of application the Simplex methods was elected as one of the top 10 algorithms in the 20th century. Surprisingly, despite its popularity and almost 80 years of research we still do not fully understand its complexity and behavior. To bound the number of iterations of the Simplex method we take a geometric point of view and investigate the possible lengths of monotone paths inside the oriented graphs of polyhedra (oriented by the objective function). We are interested in both the shortest and the longest monotone paths possible and estimate the (monotone) diameter and the height of some famous combinatorial polyhedra (such as TSP, fractional matching polytopes, and others). But how far apart are two monotone paths of an LP? E.g., if I use two pivot rules the Simplex method traces two paths, is there a notion of distance between them?. As we look at all monotone paths together we see a metric space structure which can be used to count how many are there or to generate them randomly. This comes from a very natural graph (the flip-path graph) and CW complex structures (cellular-strings complex) on the set of monotone paths These objects are important in topological combinatorics (Baues complexes and polyhedral subdivisions studied by Billera, Sturmfels, Gelfand, Kapranov, Zelevinsky and others in the early 1990’s) and in discrete optimization (In particular the complexity of the simplex method with work going back to Klee). In my talk I will discuss some natural counting problems associated with the discrete geometric situation. Our main enumerative results include bounds on the number of monotone paths, and on the the diameter of the space of monotone paths (how far are two monotone paths from each other?). The picture is fairly complete in dimension three, but plenty of open problems remain for high dimensional polytopes. The new theorems presented in my talk come from three recent papers (in Arxiv), the first joint work with Moise Blanchard (MIT) and Quentin Louveaux (U. Liege) the second joint work with Christos Athanasiadis (U. Athens) and Zhenyang Zhang (UC Davis), the third with Sean Kafer (U. Waterloo) and Laura Sanita (Eidhoven U).

A CW-complex of Monotone Polyhedral Paths: Geometric/topological meditations about the performance of the Simplex Method.

Configuration spaces of points in the plane are well studied and the topology of such spaces is well understood. But what if you replace points by particles with some positive thickness, and put them in a container with boundaries? It seems like not much is known. To mathematicians, this is a natural generalization of the configuration space of points, perhaps interesting for its own sake. But is also important from the point of view of physics––physicists might call such a space the "phase space" or "energy landscape" for a hard-spheres system. Since hard-spheres systems are observed experimentally to undergo phase transitions (analogous to water changing into ice), it would be quite interesting to understand topological underpinnings of such transitions. We have just started to understand the homology of these configuration spaces, and based on our results so far we suggest working definitions of "homological solid, liquid, and gas". This is joint work with a number of collaborators, including Hannah Alpert, Ulrich Bauer, Robert MacPherson, and Kelly Spendlove.

Configurations spaces of particles: homological solid, liquid, and gas.

Persistent Homology (PH) is a widely used topological descriptor for data. In order to get a systematic understanding of the data science scenarios where PH is successful, it is crucial to know about its discriminative power, i.e. the ability to identify and disambiguate patterns in the data, or in other words it is crucial to know about the information loss and the invariances of PH. Formally these interrogations translate into the following inverse problem: Given an element in the image of PH, a so-called barcode D, what is the fiber (pre-image) of PH over D? There are several ways of defining PH: for point clouds in a metric space, for filter functions on a simplicial complex and for continuous functions on an arbitrary space, to name a few. Hence there are as many inverse problems to address. In this talk I will review the simplicial situation as well as that of Morse functions on a smooth manifold, with the aim of showing some geometrically surprising fibers and transmitting my interest for these intricate inverse problems. 

Inverse Problems for Persistent Homology

We generalize several basic notions in algebraic topology to categories which contain both topological spaces classically treated by classical homotopy theory as well as more discrete and combinatorial spaces of interest in applications, such as graphs and point clouds. The advantage of doing so is that there are now non-trivial 'continuous' maps from paracompact Hausdorff spaces to finite spaces (given the appropriate structure), and one may then compare the resulting topological invariants on each side functorially. We find that there are a number of possible such categories, each with its own particular homotopy theory and associated homologies, and, additionally, that there is a generalization of the coarse category which allows finite sets to be non-trivial (i.e. not 'coarsely' equivalent to a point). We will give an overview of these theories and several applications, show how they are related to familiar objects in applied topology, such as the Vietoris-Rips homology, and discuss the advantages and disadvantages of each. We finish by describing a recent construction of sheaf theory in the category of Čech closure spaces, a strict generalization of the category of topological spaces.

Applied topology from the classical point of view

Topological data analysis is a rapidly developing area of mathematics with applications in data science. In addition to revealing the shape of data we develop tools for visualizing high dimensional scalar and vector valued functions. As an example, we explore relations between various knot invariants, and extrapolate how presented tools may help to compare various, high-dimensional descriptors of fixed datasets. In particular, we show how these ideas can be used to compare different mapper-type graphs of the same dataset. This is a joint work with Davide Gurnari, Anna Jurek, Simon Rudkin and Radmila Sazdanovic.

Data, their relations and shape - topology in action

Reconnection is the process by which two approaching vortices cut and connect to each other. As a topologically changing event, it has been a subject of considerable fundamental interest for decades – not only in (classical) viscous flows but also in quantum fluids, as well as in numerous other fields, such as plasmas, polymers, DNAs, and so on. For viscous fluid flows, reconnection is believed to play a significant role in various important phenomena, such as turbulence cascade, fine-scale mixing, and aerodynamic noise generation. We first delineate the underlying mechanism involved in vortex reconnection and its apparent role in turbulence cascade. Then we address the helicity dynamics involved in viscous reconnection occurring in evolutions of a trefoil knotted vortex and a Hopf-link. For both cases, we find that the global helicity H gradually decreases before reconnection but sharply increases during reconnection – this effect increases with increasing vortex Reynolds number (Re≡circulation/viscousity). This suggests that H for viscous flows is not conserved as Re→∞. Both positive and negative helical structures are simultaneously generated before and during reconnection, and their different decay rates due to asymmetric reconnection appears to cause such an increase of H during reconnection. By examining the topological aspects of the helicity dynamics, we find that different from H, the sum of linking and writhing numbers (i.e., Lk+Wr) continuously drop during reconnection. Our results suggest that the twist, which increases with Re, plays a more important role in helicity dynamics than recognized before, particularly at high Re. Note: the video of this web-seminar combines the contributions by <a href="https://seminargeotop-a.com/previous-talks/55">Fazle Hussain</a> (I part) and Jie Yao (II part).

Dynamics of viscous vortex knots and links

Reconnection is the process by which two approaching vortices cut and connect to each other. As a topologically changing event, it has been a subject of considerable fundamental interest for decades – not only in (classical) viscous flows but also in quantum fluids, as well as in numerous other fields, such as plasmas, polymers, DNAs, and so on. For viscous fluid flows, reconnection is believed to play a significant role in various important phenomena, such as turbulence cascade, fine-scale mixing, and aerodynamic noise generation. We first delineate the underlying mechanism involved in vortex reconnection and its apparent role in turbulence cascade. Then we address the helicity dynamics involved in viscous reconnection occurring in evolutions of a trefoil knotted vortex and a Hopf-link. For both cases, we find that the global helicity H gradually decreases before reconnection but sharply increases during reconnection – this effect increases with increasing vortex Reynolds number (Re≡circulation/viscousity). This suggests that H for viscous flows is not conserved as Re→∞. Both positive and negative helical structures are simultaneously generated before and during reconnection, and their different decay rates due to asymmetric reconnection appears to cause such an increase of H during reconnection. By examining the topological aspects of the helicity dynamics, we find that different from H, the sum of linking and writhing numbers (i.e., Lk+Wr) continuously drop during reconnection. Our results suggest that the twist, which increases with Re, plays a more important role in helicity dynamics than recognized before, particularly at high Re. Note: the video of this web-seminar combines the contributions by Fazle Hussain (I part) and <a href="https://seminargeotop-a.com/previous-talks/56">Jie Yao</a> (II part).

Vortex reconnection and turbulence cascade

This talk will aim to present an overview of the three dimensional structures of proteins. These large and intricate molecules perform the vast majority of the biological functions of life and the structures of over 170,000 proteins have been determined and are stored in the Protein Databank. A detailed understanding of their structures has gradually emerged over the last 50 years. Chirality within protein structures is observed at all 'levels' of structure, starting with the basic stereochemistry of the polypeptide chain, through local chain folding, to the 'tertiary' structure of the whole chain and even to chirality of large complexes.

The Wonderful World of Protein Structures

Jonathan M. Fogg and Lynn Zechiedrich - Baylor College of MedicineDNA in cells is supercoiled and constrained into loops. Despite the ubiquity and importance of supercoiling in regulating nearly every aspect of DNA activity, relatively little is known about how. To determine how supercoiling influenced DNA shape, we determined the 3-D structures of individual 336 bp DNA minicircles over a wide range of supercoiling from s = -0.019 to +0.085 (Irobalieva et al. 2015). Supercoiled DNA forms far more bent and contorted shapes than predicted. We sought to understand how DNA formed these shapes using coarse-grained molecular dynamics simulations (Wang et al. 2017), which predicted that site-specific disruptions to base pairing may explain otherwise energetically unfavorable sharp DNA bends. Likewise, bending strain at the apices of highly writhed DNA circles leads to broken base pairs. Probing for and mapping where base-pair disruptions occur, we discovered that negative supercoiling transmits mechanical stress along the DNA backbone to disrupt base pairing at specific distant sites (Fogg et al. 2021). This unprecedented base-pair disruption cooperativity among distant sites localizes certain sequences to superhelical apices to facilitate DNA writhing and relieve torsional strain, likely preventing more extensive denaturation that can cause genomic instability. We also discovered how cells may exploit DNA looping to position DNA nicks to facilitate repair. Our data explain how DNA can form short loops through base-pair disruption and reveal a complex interplay between looping- and supercoiling-mediated site-specific disruptions to base pairing and the 3-D conformation of DNA, which influence how genomes are stored, replicated, transcribed, repaired, and likely other aspects of DNA activity. We hope to harness these looping- and supercoiling-mediated site-specific denaturation and mechanical correlations to design novel DNA shapes for nanotechnology.Irobalieva, R.N.*, Fogg, J.M.*, Catanese, D.J., Sutthibutpong, T., Chen, M., Barker, A.K., Ludtke, S.J., Harris, S.A., Schmid, M.F., Chiu, W., and Zechiedrich, L. (2015) Structural diversity of supercoiled DNA. Nature Comm. Oct 12;6:8440 PMC4608029 (*equal contribution)Wang, Q., Irobalieva, R. N., Chiu, W., Schmid, M. F., Fogg, J. M., Zechiedrich, L., and Pettitt, B.M. (2017) DNA sequence determines conformational distribution of minicircles under torsional stress. Nucleic Acids Res. 45, 7633–7642 PMC5737869 Fogg, J.M., Judge, A.K., Stricker, E., Chan, H.L., and Zechiedrich, L. Supercoiling and looping promote DNA base accessibility and coordination among distant sites. Nature Comm. in press.

Cooperativity of looping- and supercoiling-mediated base-pair disruption among distant sites modulates the 3-D structure of DNA to control its activity

The process of driven translocation of polymer chains through a narrow pore can be severely hindered by the presence of self and mutual entanglement. In circular chains this entanglement is trapped in the form of knots and links that may act as potential obstruction at the pore affecting the translocation dynamics. Here we present theoretical results mainly based on extensive Langevin simulations on the driven translocation dynamics of topologically linked rings. We highlight the role of link complexity, pore size and driving force field on the translocation process and suggest how to extend nanopore sensing techniques to probe the topological properties of these systems and, for instance, to distinguish knotted from linked states and two component to multicomponent links.

Getting interlocked circular chains through the needle’s eye

The combinatorial graph Laplacian, as an operator on functions defined on the vertex set of a graph, is a fundamental object in the analysis of and optimization on graphs. There is also an algebraic topology view of the graph Laplacian which arises through considering boundary operators and specific inner products defined on simplicial (co)chain groups. This permits extending the graph Laplacian to a more general operator, the q-th combinatorial Laplacian to a given simplicial complex. An extension of this combinatorial Laplacian to the setting of pairs (or more generally, a sequence of) simplicial complexes was recently introduced by (R.) Wang, Nguyen and Wei. In this talk, I will present serveral results (including a persistent version of the Cheeger inequality) from our recent study of the theoretical properties for the persistence Laplacian, as well as efficient algorithms to compute it. This is joint work with Facundo Memoli and Zhengchao Wan.

Persistent Laplacian: properties and algorithms

We investigate a sheaf-theoretic interpretation of stratification learning from geometric and topological perspectives. Our main result is the construction of stratification learning algorithms framed in terms of a sheaf on a partially ordered set with the Alexandroff topology. We prove that the resulting decomposition is the unique minimal stratification for which the strata are homogeneous and the given sheaf is constructible. In particular, when we choose to work with the local homology sheaf, our algorithm gives an alternative to the local homology transfer algorithm given in Bendich et al. (2012), and the cohomology stratification algorithm given in Nanda (2020). Additionally, we give examples of stratifications based on the geometric techniques of Breiding et al. (2018), illustrating how the sheaf-theoretic approach can be used to study stratifications from both topological and geometric perspectives. This approach also points toward future applications of sheaf theory in the study of topological data analysis by illustrating the utility of the language of sheaf theory generalizing existing algorithms. This is joint work with Adam Brown (<a href=https://doi.org/10.1007/s00454-020-00206-y>click here</a>).

Sheaf-Theoretic Stratification Learning From Geometric and Topological Perspectives

The Covid-19 pandemic has highlighted the need for novel antiviral strategies. In this talk, I will demonstrate that insights into the geometric principles underpinning virus architecture provide a key to uncovering the mechanisms by which viruses replicate and infect their hosts. Geometric and topological descriptors of virus architecture, combined with stochastic simulations, reveal how viruses navigate the knife’s edge between stability and instability, guaranteeing protection for their genetic cargo while also enabling its timely release. Models of virus architecture also provide a novel perspective on open problems in virus assembly. This includes the origin and control of polymorphic particle assembly, which arises, amongst others, when virus-derived protein containers are functionalised to present antigens for applications in vaccinology. They moreover play an instrumental role in the discovery of genome- encoded virus assembly instructions. These results shed new light on selective pressures on viral evolution and pave the way for innovation in antiviral therapy and virus nanotechnology.

Mathematical Virology: Geometry in the Fight against Viral Infection

I am going to talk about two problems that Message Passing Neural Networks (MPNNs) have been shown to be struggling from. The first one – known as over-squashing – is unavoidable in the MPNN class and concerns the input graph topology. This relates to how information propagates in a graph. We show that discrete curvature quantities (old and new) could help us understand where messages are being lost and we can provably characterize the over-squashing phenomenon in terms of curvature. The second problem consists in analysing GNNs as multi-particle dynamics using the lens of gradient flows of an energy. We investigate what happens when instead of learning the MPNN equations we learn an energy and then let the equations follow the gradient flow of such energy. This allows us to understand further the role of the channel-mixing matrix that is ubiquitous in standard graph convolutional models as a bilinear potential inducing both attraction and repulsion along edges via its positive and negative eigenvalues respectively.

Over-squashing and over-smoothing through the lenses of curvature and multi-particle dynamics

Vortex lines and streamlines in turbulent flows, visualized in the experiments or in the numerics, appear chaotic, twisted, perhaps linked or knotted. The physical meaning of this complexity and its relation to the dynamics is still obscure. In this lecture I shall address this problem - the geometrical and topological complexity of turbulence - in the arguably simpler context of "quantum fluids". Quantum fluids (superfluid helium, atomic Bose-Einstein condensates, etc) are studied in the laboratory at temperatures close to absolute zero. At these low temperatures the fundamental quantum properties of matter are not masked by thermal disorder. In particular, any rotational motion is constrained by quantum mechanics to individual vortex lines of fixed strength (phase defects of a complex order parameter), unlike what happens in ordinary fluids where vorticity is a continuous field. Quantum turbulence, created by stirring a quantum fluid, is thus conceptually simpler than ordinary turbulence, consisting of a tangle of individual vortex lines rather than a disordered continuous vorticity field. After describing some surprising similarities between quantum turbulence and ordinary turbulence, I shall show how the geometry and the topology of quantum turbulence can be quantified in a relatively simple way, hence demonstrate that quantum turbulence is indeed knotted. Is ordinary turbulence knotted too? References: <ul><li>Barenghi, C.F. and Parker, N.G. 2016 A primer on quantum fluids, Springer.</li> <li>Barenghi, C.F., Skrbek, L., and Sreenivasan, K.R. 2014 Introduction to quantum turbulence, Proc. Nat. Acad. Sci. (USA) 111 (Suppl. 1), 4647. </li> <li>Cooper, R.G., Mesgarnezhad, M., Baggaley, A.W., and Barenghi, C.F. 2019 Knot spectrum of turbulence, Sci. Reports9, 10545.</li> <li>Barenghi, C.F. 2022 Tangled vortex lines: dynamics, geometry and topology of quantum turbulence, arXiv:2207.04260v1, to appear in Knotted fields, ed. Ricca, R.L. and Liu, X., Springer.</li</ul>

Is turbulence knotted?

Knot theory is divided into several subfields. One of these is hyperbolic knot theory, which is focused on the hyperbolic structure that exists on many knot complements. Another branch of knot theory is concerned with invariants that have connections to 4-manifolds, for example the knot signature and Heegaard Floer homology. In my talk, I will describe a new relationship between these two fields that was discovered with the aid of machine learning. Specifically, we show that the knot signature can be estimated surprisingly accurately in terms of hyperbolic invariants. We introduce a new real-valued invariant called the natural slope of a hyperbolic knot in the 3-sphere, which is defined in terms of its cusp geometry. Our main result is that twice the knot signature and the natural slope differ by at most a constant times the hyperbolic volume divided by the cube of the injectivity radius. This theorem has applications to Dehn surgery and to 4-ball genus. We will also present a refined version of the inequality where the upper bound is a linear function of the volume, and the slope is corrected by terms corresponding to short geodesics that have odd linking number with the knot. My talk will outline the proofs of these results, as well as describing the role that machine learning played in their discovery. This is joint work with Alex Davies, Andras Juhasz, and Nenad Tomasev.

Knot theory and machine learning

Circular coordinate systems -- here, cycles -- are ubiquitous in data encoded by the brain. Classical ideas from topology tell us that the structure of the encoded data must be reflected in the activity of the encoding neural populations, and methods from topological data analysis have been highly successful at detecting signatures of such encodings. The next natural question we might ask is how we assign meaning or semantics to observed cycles  Here, we describe a new method for using a measure of cross-similarity to register, or falsify the registration of, cycles across populations. We demonstrate its use in simulated and experimental data, and discuss ongoing work using these tools to investigate how feed-forward networks propagate cycles. This is joint work with Iris Yoon, Niko Schonsheck, and several others.

Tracking cycles in neural codes

I will present the results of two projects applying topological data analysis (TDA) and machine learning (ML) to biological data. In the first, we have developed a new tool, TDAExplore, that combines TDA and ML to both classify biological images and to provide a visualization that is biologically informative. In the second, we use TDA and ML to classify quasi-periodic biological videos and we apply TDA to such a video to produce synthetic periodic videos.

Topological Data Analysis for Biological Images and Video

Multi-agent Epistemic Logic is a modal logic of knowledge. It allows to reason about a finite set of agents who may know facts about the world, and about each other. In this talk, I will present a new semantics for epistemic logic, based on simplicial complexes. In this approach, the knowledge of the agents is modeled by a higher-dimensional space called a simplicial model; and the truth of an epistemic logic formula can be evaluated by inspecting the various possible paths in this space. I will illustrate these ideas using examples from the theory of distributed computing, where the agents correspond to individual processes who can exchange information in order to solve a task. Both topological invariants and logical invariants can be leveraged to prove that some distributed computing tasks are impossible to solve.

Knowledge and topology: a simplicial approach

In this talk I will present applications of General Topology to Epistemic Logic (=the logical aspects of knowledge, knowabiity and belief) and Formal Learning Theory. I show that topological methods can throw light on issues such as the value of simplicity as a learning strategy (cf. Ockham's Razor) and the analysis of epistemic paradoxes (e.g. the connection between the so-called Surprise Exam Paradox and the Cantor-Bendixson process of calculating the perfect core). Time-permitting, I may present some complete and decidable logical axiomatizations of these notions and maybe even give a hint concerning the completeness proofs.

The Topology of Knowing (Or How to Avoid Unexpected Exams)

One goal of applied category theory is to understand open systems: that is, systems with a boundary of some sort, through which matter, energy or information can flow in or out.   We can describe a large class of open systems using the mathematics of decorated cospans, which we explain here.  In various examples these ideas have been implemented in software.  An interesting example comes from stock-flow diagrams, which are widely used in epidemiology to model the dynamics of populations. Although tools already exist for building these diagrams and simulating the systems they describe, we have created a new package called StockFlow which uses decorated cospans to overcome limitations of the existing tools.  This is joint work with Xiaoyan Li, Sophie Libkind, Nathaniel Osgood and Evan Patterson.

Compositional Modeling with Decorated Cospans

Persistent homology, a popular method in TDA, can be used to define feature maps encoding geometrical properties of data. In this talk I will present a method, developed by the TDA group at KTH, which allows to construct feature maps with learnable parameters, stable with respect to distances on persistence modules. The feature maps are in fact defined starting from distances between persistence modules rather than on the barcode decomposition, making the method suitable for generalisations. Particular focus will be on understanding parametrised families of such feature maps, such as those stable with respect to p-Wasserstein distance. The use of Wasserstein stable features will be illustrated on real world and artificial datasets.

Stable and interpretable topological feature maps

In a simplicial complex, a pair of simplices  are a collapsing pair, if one is a unique maximal coface of the other which is then a free face. Such a pair can be collapsed by removal of the two simplices and all simplices between them – think about an edge in a solid tetrahedron; collapsing means removing the edge, the interior of the tetrahedron and the interior of the two faces containing that edge. This leads to a homotopy equivalence. There is a similar notion for cubical complexes. A sequence of collapses leads to a simpler (fewer simplices/cubes) space.
For a directed space, which is a topological space with a selected set of paths, the directed paths, directed homotopy equivalence is a very strong requirement, and not what should be the basis of collapsing.
We study the following setting: A Euclidean Cubical Complex, an ECC, is a subset of R^n which is a union of elementary cubes. An elementary cube is a product of n intervals [ai,ai+e], where e is either 0 or 1. A directed path in an ECC is continuous and non-decreasing in all coordinates.
We define a notion of collapse with the aim of preserving various properties of spaces of directed paths.
This is joint work with the WiT, Women in Topology, group R. Belton, R. Brooks, S.Ebli, L.F., B.T.Fasy, N.Sanderson, E. Vidaurre.

Collapsing in directed topology

Many natural structures such as proteins, climbing vines, and seashells exhibit a well defined chirality, some are left-handed, some are right-handed, some are both. The ultimate origin of chirality is one of Nature's great mystery. However, geometry and mechanics play a fundamental role in assigning chirality and carrying this information from microscopic to macroscopic scales. In this talk, I will discuss the general problem of chirality, chirality measure, and chirality transfer, trace its history, and use examples from chemistry and biology to obtain general principles with some surprising twists.

The geometry and mechanics of chirality: from Maxwell's perversion to Feynman's obsession

For a successful analysis a suitable representation of data by objects amenable for statistical methods is fundamental. There has been an explosion of applications in which homological representations of data played a significant role. I will present one such representation called stable rank and introduce various novel ways of using it to encode geometry, and then analyse, data. I will provide several illustrative examples of how to use stable ranks to find meaningful results. 

Data, geometry, and homology

Frames are overcomplete systems of vectors in Hilbert spaces. They were originally introduced in the 1950s in the context of non-harmonic Fourier series, and came to renewed prominence in the 1980s in signal processing applications. More recently, there has been burgeoning interest in frames in finite-dimensional Hilbert spaces, with applications to signal processing, quantum information, and compressed sensing. In this talk, I will describe some ways in which tools from differential, Riemannian, and symplectic geometry can be applied to problems in frame theory. Some key tools that crop up are Hamiltonian actions, the Cartan decomposition, and geometric invariant theory. This is joint work with Tom Needham and partially with Dustin Mixon and Soledad Villar.

Geometric Approaches to Frame Theory

A theta-curve is a spatial embedding of the unique graph with two vertices joined by three parallel edges. Like knots and links, theta-curves and their mathematical properties are relevant to the mathematical modeling of biopolymers. In this talk, we will investigate unknotting operations and define new invariants of theta-curves. We will also generalize the statement that 'unknotting number one knots are prime' to theta-curves. This is joint work with several sets of authors.

Entanglement and invariants of theta-curves

We will give a brief introduction to the subject. The survey of main ideas and tools will be complemented with applications to specific standard distributed tasks. We will conclude with stating an open problem in combinatorial topology which is related to the complexity of the Weak Symmetry Breaking distributed task.

Simplicial Methods in Distributed Computing

In many situations in physics, the path of light is determined not only by spatial geometry but also by an underlying local density (e.g., mass concentration in general relativity, refractive index in optics). Consider a scenario where a Riemannian manifold in Euclidean space is shaped by a density function, with only a finite sample of points available. How can we infer the original metric and determine the manifold's topology? This talk introduces a density-based method for estimating topological features from data in high-dimensional Euclidean spaces, assuming a manifold structure. The key to our approach lies in the Fermat distance, a sample metric that robustly infers the deformed Riemannian metric. Theoretical convergence results and implications in the homology inference of the manifold will be presented. Additionally, I will show practical applications in time series analysis with examples from real-world data. This talk is based on the article: X. Fernandez, E. Borghini, G. Mindlin, and P. Groisman. "Intrinsic Persistent Homology via Density-Based Metric Learning." Journal of Machine Learning Research 24 (2023) 1-42.

The Fermat principle in Riemannian geometry

Cusp shapes are widely observed in nature, most famously as the bright caustic lines on the inside of a coffee cup. This can be understood from the fact that a cusp arises from the smooth deformation of a parameterized curve. Remarkably, the same generic cusp can be formed on the surface of a viscous fluid with surface tension, as demonstrated by Jeong and Moffatt [J. Fluid Mech. 241, 1, (1992)], using complex mapping techniques. However, their observation is limited to very specific and idealized geometries. Here we demonstrate that cusps are indeed local solutions to the Stokes equation with surface tension, regardless of the global flow.

Geometrical singularities and free surface cusps

We prove using the mathematics of tangles that if a protein terminus passing through a single loop results in a locally knotted protein, then Taylor's twisted hairpin model is the most likely method for creating such knots. In this case the knotted products will all be twist knots. If we assume a right-handed chirality bias, which is common in proteins, then  the majority of these twist knots will be  right-handed trefoils ($+3_1$), followed by left-handed trefoils ($-3_1$), achiral figure eight knots ($4_1$) and right-handed five crossing twist knots ($-5_2$). An alternative pathway has been observed computationally where a terminus passes through two loops.  We use 3-string tangle analysis to model this pathway.  This is joint work with Garrett Jones and Puttipong Pongtanapaisan.

Modelling knotted proteins with tangles

Some twenty years ago Michael Farber defined the topological complexity of robot motion planning as a measure of the difficulty to construct predictable motion plans for mechanical devices (like robots) that are allowed to move in a given work space. More recently, we defined the complexity of a kinematic map that takes into account the kinematic relation between the internal states of a serial mechanism and its spatial poses. In our talk we will discuss a more general motion planning for parallel mechanisms. In particular, we will consider mechanisms that are redundant in the sense that the dimension of their joint space is strictly bigger than the dimension of their work space. The additional degrees of freedom allow motion paths that avoid critical configurations of joints, and we will discuss how difficult it is to construct predictable singularity-free motion plans that perform a given set of tasks. This is joint work with Edward Haug and Adrian Peidro.

Singularity-free motion planning for redundant parallel manipulators

The concept of a persistence module was introduced in the context of topological data analysis. In its original incarnation a persistence module is defined to be a functor from the poset of nonnegative real numbers with theory natural order to the category of vector spaces and homomorphisms. These are referred to as single parameter persistence modules and are a fundamental and useful concept in topological data analysis when the source data depends on a single parameter. The concept naturally lends itself to generalisation, and one may consider persistence modules as functors from an arbitrary poset (or more generally an arbitrary small category) to some abelian target category. In other words, a persistence module is simply a representation of the source category in the target abelian category. As such much research was dedicated to studying persistence modules in this context. Unsurprisingly, it turns out that when the source category is more general than a linear order, then its representation type is generally wild. In particular, keeping in mind that persistence module theory is supposed to be applicable, computability of general persistence modules is very limited. In this talk I will describe the background and motivation for persistence module theory and introduce a new set of ideas for local analysis of persistence module by methods borrowed from spectral graph theory and multivariable calculus.

Differential Calculus for Modules over Posets

Graph Neural Networks (GNNs) are proven to be a powerful machinery for learning  complex dependencies in multivariate spatio-temporal processes. However, most existing GNNs have inherently static architectures, and as a result, do not explicitly account for time dependencies of the encoded knowledge and are limited in their ability to simultaneously infer latent time-conditioned relations among entities. We postulate that such hidden time-conditioned properties may be captured by the tools of multipersistence, i.e., an emerging machinery in topological data analysis which allows us to quantify dynamics of the data shape along multiple geometric dimensions. We propose to summarize inherent time-conditioned topological properties of the data as time-aware multipersistence Euler-Poincaré surface and prove its stability. We then construct a supragraph convolution module which simultaneously accounts for the extracted intra- and inter-dependencies in the data.  We illustrate the utility of the proposed approach in application to forecasting highway traffic flow, blockchain Ethereum token prices, and COVID-19 hospitalizations.

Coupling Time-Aware Multipersistence Knowledge Representation with Graph Convolutional Networks for Time Series Forecasting

A hyperplane arrangement in $\mathbb{R}^d$ divides space into two sets via a chessboard coloring. Given a set of measures, we can attempt to split each into two equal parts using the chessboard coloring of a hyperplane arrangement. Special cases of this problem include the classic ham sandwich theorem by Banach or the necklace splitting theorem by Hobby and Rice. We present a new common generalization of many mass partition results of this kind. Surprisingly, the proof methods are not topological, breaking a long tradition in the area. During this talk, we will describe the results that can be generalized this way and the reach of this new non-topological approach. Joint work with Alfredo Hubard.

Bisecting masses with hyperplane arrangements

I'll present the so called Localized Induction Approximation that describes the dynamics of a vortex filament according to the Binormal Curvature Flow (BF). I'll give a result about the desingularization of the Biot-Savart integral proved with Marco A. Fontelos within the framework of Navier-Stokes equations. Some particular examples regarding BF obtained with Valeria Banica will be also considered. These examples allow to connect BF with the so called Riemann non-differentiable function and the Frisch-Parisi approach to turbulence.

The binormal flow and the evolution of viscous vortex filaments

This talk is focused on new techniques for the topological data analysis (TDA) of labelled point cloud data. Well-established filtrations in TDA for a point cloud data include the Čech, Vietoris-Rips, and alpha filtrations. Bauer and Edelsbrunner [BE16] demonstrated that the Čech filtration can be simplicially collapsed onto the alpha filtration, showing that they are homotopy equivalent. Recent techniques in data collection across fields like cancer biology, geospatial analysis and ecology have produced chromatic (labeled) data that express interactions among points of different colors. In such cases, it is crucial to understand not only the overall spatial structure of the data but also the spatial relationships among subsets defined by their labels. The chromatic alpha filtration [Mon+24] is a generalization of the alpha filtration that captures these relationships, making it particularly useful for multi-species data in TDA. In this talk we introduce the chromatic Delaunay-Čech and chromatic Delaunay-Rips filtrations as computationally efficient alternatives to the chromatic alpha filtration. We use generalized discrete Morse theory to demonstrate that the Čech, chromatic Delaunay-Čech, and chromatic alpha filtrations are interconnected through simplicial collapses, extending Bauer and Edelsbrunner's results from non-chromatic to chromatic contexts. Our findings offer theoretical support for the application of chromatic Delaunay-Čech and chromatic Delaunay-Rips filtrations, and we illustrate their computational advantages through numerical experiments. This is joint work with A. Natarajan, T. Chaplin and A. Brown from University of Oxford (arXiv:2405.19303). [BE16] Ulrich Bauer and Herbert Edelsbrunner. “The Morse theory of Čech and Delaunay complexes”. In: Transactions of the American Mathematical Society 369.5 (Dec. 27, 2016), pp. 3741-3762. DOI:10.1090/tran/6991. [Mon+24] Sebastiano Cultrera di Montesano et al. Chromatic Alpha Complexes. Feb. 7, 2024. arXiv: 2212.03128

Morse Theory for Chromatic Delaunay Triangulations

The Vietoris-Rips filtration, the standard filtration on metric data in topological data analysis, is notoriously sensitive to outliers and can be insensitive to variations in density. A natural solution is to consider 2-parameter persistence, treating density and spatial scale as separate parameters. In this talk, I will present results on the stability, robustness, and computability of 2-parameter persistence. A main focus will be Sheehy's subdivision-Rips bifiltration, the only density-sensitive bifiltration on metric data known to satisfy a strong robustness property. This filtration is too large to compute directly, but we will see that it can be approximated by much smaller objects. Our results reveal an apparent tension between robustness and computability for 2-parameter persistence, which in spite of substantial progress, is not yet fully understood. The talk will be based on three papers, the first with Andrew Blumberg and the others with KenMcCabe: <a href="https://link.springer.com/article/10.1007/s10208-022-09576-6" target="_blank">https://link.springer.com/article/10.1007/s10208-022-09576-6</a> <a href="https://arxiv.org/abs/2406.07679" target="_blank">https://arxiv.org/abs/2406.07679</a> <a href="https://arxiv.org/abs/2408.16716" target="_blank">https://arxiv.org/abs/2408.16716</a>

Robustness and Computability of 2-Parameter Persistent Homology

Hodge Laplacians have been previously proposed as a natural tool for understanding higher-order interactions in networks and directed graphs. In this talk, we will cover a Hodge-theoretic approach to spectral theory and dimensionality reduction for probability distributions on sequences and simplicial complexes.  We will introduce a feature space based on the Laplacian eigenvectors associated  to a set of sequences, and will see these eigenvectors capture the underlying geometry of our data. Furthermore, we will show this Hodge theory has desirable properties with respect to natural null-models, where the underlying vertices are independent. Specifically, we will see the appropriate Hodge Laplacian has an integer spectrum with high multiplicities, and describe its eigenspaces. Finally, we will cover a simple proof showing the underlying cell complex of sequences has trivial reduced homology.

Hodge Laplacians on Sequences

Methods of Topological Data Analysis are now an important part of modern data-driven scientific discovery. This talk will provide an overview of recent results, theoretical and experimental, that arise from the interactions between topology and physics. We will discuss topological characteristics that can be used to track the time evolution of physical system and to detect its phase transitions. We will then discuss a topological quantification of disorder where it can be used to detect “sufficiently ordered systems” which, although irregular, still display interesting physical characteristics. We will end with a discussion of the uses of topology in the design of physical systems. 

Topological insights into physical phenomena

The Mobius energy of a knot is a useful analytic tool which can yield information about classical knot invariants. Freedman, He, and Wang proved the existence of curves with a given prime knot type which minimizes the Mobius energy, and they also proved the minimizers are $C^{1,1}$. Shortly after, He proved the minimizers are analytic using nonlocal techniques involving pseudodifferential calculus. I will discuss these methods and how they may apply to unresolved problems regarding the Mobius energy.

Pseudodifferential Methods and the Mobius Knot Energy

In this talk, we provide a summary of the analysis of a large sample of random equilateral polygons in spherical confinement. The analysis illustrates the dependence of the knot spectrum and of geometric properties of the polygons on the lengths of the polygons as well as the radius of confinement. The geometric properties are sometimes also influenced by the knotting complexity. Since our polygons are rooted at the center of the confinement sphere, the presentation also addresses the question of what might happen for a confinement sphere with a radius less than 1. The generation process for the spherical polygons is rigorous, however, the analysis are only based on numerical results.

Random polygons in spherical confinement

We review a well-known method to compute the braid index. Using this method, we can give a compute the braid index of all alternating Montesinos knots and links and all non alternating pretzel knots and links. The method uses just information that can be read of a minimal diagram of the knot or link.

On the braid index of knots and links

The topological complexity and its higher versions are homotopy invariants which were introduced by M. Farber and Y. Rudyak in order to give a topological measure of the complexity of the motion planning problem. We will discuss some properties of these invariants for closed manifolds with abelian fundamental group. In particular, we will give sufficient conditions for the (higher) topological complexity of such a manifold to be non-maximal. This is based on joint works with N. Cadavid, D. Cohen, J. González and S. Hughes.

On the (higher) topological complexity of manifolds with abelian fundamental group

There is an increased necessity for mathematical study of self-assembly of various phenomena ranging from nano-scale structures, material design, crystals and nano devices. We present a range of topological questions associated with DNA self-assembly and three dimensional structures. The questions vary from topological graph theory related to DNA strand routing of a three-dimensional mesh, to questions in knot theory related to structural embeddings in 3D.

Topological models for studying DNA self-assembly

In seminal papers in the 1960s Vladimir Arnold introduced the idea that the motion of an ideal fluid can be considered as a geodesic in the space of volume-preserving maps from the fluid domain to itself. This viewpoint places fluid dynamics, on any Riemannian manifold, in an abstract setting which also incorporates Lie group structure. Although this theory is profound and beautiful, at first sight it has little bearing for the everyday applications of fluid dynamics. However it turns out that the process of Lagrangian averaging (namely averaging over fluid parcels in an ensemble of fluid flows, contrasted with Eulerian averaging at a fixed point), is best understood using the ideas of pull-backs and Lie derivatives on a general manifold, even though one ultimately applies these notions in ordinary three-dimensional space. This talk will be very much aimed at fluid dynamicists rather than professional geometers, and will outline Arnold’s ideas, and applications to the Generalised Lagrangian Mean Theory put forward by David Andrews and Michael McIntyre, and subsequent related theories, particularly of Andrew Soward and Paul Roberts. This is joint work with Jacques Vanneste (University of Edinburgh).

 The geometry of Lagrangian averaging in ideal fluid dynamics

One of the most fundamental issues in evolutionary biology is how unicellular life transitioned to multicellular life. How — and why — was it that the simplest single-celled organisms that emerged from the primordial soup evolved into organisms with many cells and cell types dividing up life’s processes? This talk will describe recent experimental and theoretical advances in understanding the architecture of organisms that serve as models of this evolutionary transition. I will discuss the shape-shifting properties of certain choanoflagellates (the closest living relatives of animals), the recent discovery of common probability distributions of cellular neighborhood volumes in yeast and alga, as well as embryonic 'inversion' and the spontaneous curling of the extracellular matrix of green algae. These studies together shed light on the fundamental question, "How do cells produce structures external to themselves in an accurate and robust manner?"

 The Geometry of Multicellular Life

How can mathematics help us understand the brain? In recent years, a field called topological data analysis (TDA) has offered powerful new ways to study complex systems (from protein shapes to brain networks) by capturing their underlying structure. In neuroscience, these tools help us uncover the hidden patterns that shape how brain cells connect and communicate. The Topological Morphology Descriptor (TMD, [1]), turns the branching shapes of neurons into mathematical “barcodes” that summarize their structure. This approach allows us to classify, cluster and compare neurons across different types and species. In this talk, I will present recent results in the topological representation of brain cells, focusing on neurons. I will then demonstrate how algebraic topology provides insights into the relationships between single neurons and networks, allowing us to bridge different computational scales. A central question in neuroscience concerns the organizational principles that distinguish the human brain from other species. Our findings [2] suggest that human neurons are strikingly more complex than those in other animals. In particular, human pyramidal cells, the most abundant cell type in the cortex, form denser, more interconnected networks. This greater dendritic complexity, a unique characteristic of human neurons, may underlie the enhanced computational power and cognitive flexibility of the human cortex. [1] Kanari et al. 2018. A Topological representation of branching neuronal morphologies [2] Kanari et al. 2025. Of mice and men: Dendritic architecture differentiates human from mouse neuronal networks

From Neurons to Networks: Exploring the Brain Through Algebraic Topology

We present recent advances in the topological analysis of protein structures, combining mathematical invariants, persistent homology, and machine learning. We explore how topological analysis captures subtle geometric features of folded chains, enabling efficient recognition of knots, links, θ-curves, and lasso motifs inside proteins. Finally, we show how integrating these tools, ranging from combinatorial invariants to recurrent neural architectures, reveals new insights into the organization, dynamics, and function of entangled proteins, illustrating the deep interplay between topology and modern structural biology.

Topological analysis of knotted proteins

We implement a new version of the BFACF algorithm combined with the Wang-Landau method to sample lattice polymers with a theta shape. The initial goal is to understand how the three "arms" scale in length, and if this resembles the scaling of a large knotted polygon in dilute solution. Other shapes like "tadpoles" are also studied. These branching structures can potentially be used to model R-loops and other complex polymer molecules. This is work in progress, in collaboration with Aleks Owczarek and James Gleeson.

 Lattice models of theta-shaped polymers and other branching structures.

Classical manifold learning relies on estimation of the tangent bundle of a manifold from a finite sample, i.e., the first derivative.  A natural next question to consider is second derivative information --- estimation and application of curvature from finite point clouds.  This talk will survey the landscape on estimating various kinds of curvature for point clouds and on applications of discrete curvature measures in geometric data analysis.  The work discussed includes joint work with Hickok, Saidi, and Rieck.

Curvature in geometric data analysis

One of the main scientific fields where TDA has been very successfully applied has been in Neuroscience. In this talk, we will provide a brief introduction to functional magnetic resonance imaging, including its various types of studies. Then we will present several results obtained using TDA, from both the Mapper algorithm and Persistent Homology perspectives.

Applications of TDA to the study of the human brain connectome.

This talk has two parts. The first main part will discuss recent breakthroughs concerning an inherently interdisciplinary project between mathematicians, physicists, chemists, and computer scientists that attempts to produce structures in three-dimensional Euclidean space from graph embeddings on triply-periodic minimal surfaces. Exploring the different graphs embeddings naturally leads to a new application, relevant for materials science, structure formation, and knot theory, of the mapping class group (MCG) of a surface, a prominent object that has received considerable attention in pure mathematics. We explain how to apply the MCG to the construction of candidates for new crystalline structures from graph embeddings on triply-periodic minimal surfaces, making use of the intrinsically hyperbolic nature of the surfaces for promising three-periodic structures. We then give an overview of new results on MCGs that facilitates an enumeration of isotopy classes of graph embeddings with a given group of symmetries and conclude with a catalogue of three-dimensional structures that have resulted from the approach. In the second part of the talk, we discuss ongoing work on analyzing the resulting entangled structures using methods from topological data analysis. We explain how the hyperbolic context has inspired novel results concerning approximations of persistent homology computations of natural filtrations for point sets of bounded doubling dimension.

Three-dimensional entangled graphs from mapping class groups and approximate persistence computations

 In this talk, we review properties and applications of quantum link invariants constructed from infinite-dimensional representations of quantum groups at generic values of $q$. When these invariants were introduced by the speaker and Ciprian Manolescu approximately five years ago, they highlighted a new and surprising role of Spin$^c$ structures that was rather mysterious at the time and was not expected in complex Chern-Simons theory. Since then, the role of Spin$^c$ structures was understood thanks to many works --- including Akhmechet-Johnson-Krushkal, Moore-Tarasca, Harichurn-Nemethi-Svoboda, among others --- and plays an important part in connections to Heegaard Floer homology and categorification of quantum $U_q (sl_2)$ invariants of 3-manifolds at generic $q$. This opens a new path from quantum topology to the study of exotic smooth structures on 4-manifolds.

A two-variable series for knot complements: recent developments and applications

Understanding the folding of RNA sequences into three-dimensional structures is a fundamental challenge in molecular biology. A key aspect amenable to mathematical analysis is characterizing the branching of an RNA secondary structure, which is a critical molecular characteristic yet too often difficult to predict correctly. Using combinatorial models (i.e. plane trees/noncrossing perfect matchings) and methods (e.g. convex polytopes and their normal fans), we give results that characterize different types of branching landscapes. Not only does this yield insights into RNA structure formation, but also suggests interesting new directions for further mathematical analysis.

On barrier height and other problems in RNA branching landscapes

Complex systems like the brain, climate, and next-generation artificial intelligence rely on higher-order interactions that extend beyond simple pairwise relationships. These many-body interactions are captured by higher-order networks [1]. By integrating algebraic topology with non-linear dynamics, theoretical physics and machine learning, this talk reveals the critical role of topology in shaping the dynamics of such systems [2]. The research highlights how topological signals, dynamical variables defined on nodes, edges, triangles, and other higher-order structures, drive phenomena such as topological synchronization, pattern formation, and triadic percolation. The surprising result that emerges from this research is that topological operators including the Topological Dirac operator, offer a common language for treating complexity, AI algorithms, and quantum physics. These findings not only advance the understanding of the underlying mechanisms in neuroscience and climate science but also pave the way for transformative machine learning algorithms inspired by theoretical physics. We conclude the seminar discussing the role of geometry in shaping dynamics. In particular we will provide an overview of the emergent field of Statistical Mechanics of Geometry, a promising new information theory framework for unifying quantum gravity captured by the Gravity from Entropy [3] approach, complex systems and the theory of computation. [1] Bianconi, G., 2021. Higher-order networks: An introduction to Simplicial Complexes. Cambridge University Press. [2] Millán, A.P., Sun, H., Giambagli, L., Muolo, R., Carletti, T., Torres, J.J., Radicchi, F., Kurths, J. and Bianconi, G., 2025. Topology shapes dynamics of higher-order networks. Nature Physics, pp.1-9. [3] Bianconi, G., 2025. Gravity from entropy. Physical Review D, 111(6), p.066001.

Topology shapes dynamics of higher-order networks and more

This talk discusses knotted vortices and how they tend to degenerate to unknots and unlinks by reconnection processes. We define a reconnection number of knots and links and show how aspects of four dimensional topology allow the computation of the reconnection number. This allows us to see that in some cases of experiments with vortices in water and with computer models of vortex knots, the processes find the least number of reconnections to undo the knot. In other cases the physical process is less efficient than the best mathematical reconnection. Our main result is an exact determination of the reconnection number R(K) for K any positive link. In particular, we show the Theorem: If K is an unsplittable knot or link with a positive crossing diagram with c(K) crossings and s(K) Seifert circles, then the reconnection number of K is exactly twice the genus of the Seifert spanning surface of K when K is a knot, and for both knots and links we have the formula R(K) = c(K) - s(K) + 1. The talk will give many examples and remind about work on vortices of Kelvin, Aleeksenko, Kleckner and Irvine, Ricca and others. We will also connect with recent work with Hall, Tai and Smalyukh. The talk will be self-contained but will use results from Khovanov Homology and the Rasmussen invariant whose consequences will be stated clearly. There are many questions about vortex reconnection, and we hope that this talk will stimulate even more questions.

Reconnection Numbers of Knotted Vortices

Advancements in sequencing technologies have produced a wealth of genomic data. Through recent methods enabled by the development of artificial intelligence, these data can be converted, for example, into phylogenies that reflect evolutionary history, or into molecular structures that reveal biological functions and properties. While analyzing the resulting structural data can advance biological research by providing novel approaches and insights, a major challenge is the lack of efficient and accurate structure encodings. In this talk, we introduce an encoding technique from topological combinatorics that represents tree structures by a series of graph polynomials. We show that these tree-distinguishing polynomials enable efficient, accurate and interpretable structure analysis using modern data analytics tools. We demonstrate their applications in comparing phylogenies and extracting evolutionary information, as well as predicting and understanding the formation of pathogenic nucleic acid structures.

Tree polynomial encoding of structural data in life sciences

Physics-informed neural networks (PINNs) provide a mesh-free way of searching for solutions of differential equations by turning their residuals, constraints, or variational principles into an optimisation problem. While standard applications focus on Euclidean domains, many fundamental problems in geometric analysis involve nonlinear PDEs posed on manifolds having non-trivial topology and/or geometry. In this talk, after an introduction to PINNs, I will present a PINN-based approach to the problem of constructing minimal surfaces in hyperbolic space with prescribed asymptotic boundary, motivated by a conjecture of Joel Fine. This approach, developed in collaboration with Marco Usula, produced substantial nontrivial evidence corroborating the conjecture, by showing that the number of double points on the numerical surfaces matches theoretical predictions (see arXiv:2605.26234). Along the way, I will emphasise all aspects of our method that can be generalised into useful guidelines for the use of PINNs in geometric analysis and differential geometry. Numerical results like the ones mentioned above represent valuable insights into deep geometric conjectures and highlight the role of PINNs in the broader and rapidly evolving field of AI for mathematical discovery.

Geometry-Aware Neural Networks for Geometric Analysis

A central challenge in artificial intelligence (AI)-driven molecular science lies in efficiently representing molecular data and developing learning architectures that capture intrinsic structure-function relationships. In this work, we introduce advanced mathematics-based molecular representations and learning frameworks. Molecular structures and interactions are encoded using high-order topological and algebraic representations, including Rips complexes, Alpha complexes, Neighborhood complexes, Dowker complexes, Hom-complexes, Tor-algebras, Rhomboid tilings, Sheaves, etc. Building on these foundations, we design physics-informed geometric and topological deep learning models that systematically integrate high-order, multiscale, and periodic information of molecular systems. These models have been successfully applied to diverse molecular datasets across chemistry, biology, and materials science, demonstrating their versatility and effectiveness in uncovering complex structural-functional relationships.

Mathematical AI for Molecular Sciences: from topological data analysis to topological deep learning

This talk is an introduction to Khovanov Homology, a knot and link invariant that arises from the Kauffman Bracket model of the Jones polynomial by categorification: The states in the bracket summation are upleveled to become objects in a category and resmoothngs of states become morphisms in the category. Khovanov defined a natural chain complex structure associated with this category, and the homology of this chain complex becomes the new invariant. The graded Euler characteristic of the Khovanov homology yields the bracket polynomial or the Jones polynomial, depending on grading choices. We will further discuss how the Khovanov homology allows the definition of the Rasmussen invariant that can estimate the minimal genus of a knot or link in the four ball. This talk will discuss Khovanov homology and Rasmussen invariant for both classical and virtual knot theory and we will prepare the ground for a following talk where the Rasmussen invariant will be used to determine the minimal number of reconnections needed to unknot certain knotted vortices.

Introduction to Khovanov Homology

In this talk, we provide a new construction for studying parameterized persistence, called a canopy. We do this by making a simple but major modification in the persistence bundle representation information: namely, rather than tracking a point in the persistence diagram, we instead track some choice of pairs of simplices that created said point. This viewpoint is a combinatorial version of tracking the chain complex information rather than just the output of persistence. We show how to construct the canopies from any filtered filtration function, proving, using the algebraic structure of filtered chain complexes, that different choices of pairs result in homeomorphic structures. Finally, we showcase the power of our approach by using canopies to define vines even in the presence of points with multiplicity; to discuss monodromy; and to obtain some immediate results linking non-trivial monodromy in the persistent homology transform with the existence of non-Hausdorff points in the canopy.

Canopies: A Generalization of Vines and Vineyards for Parameterized Persistence

The textures and phases of liquid crystals are replete with geometric motifs, and the geometric approach to elasticity underpins a large portion of nonlinear theories. Despite this, the basic characterisation of topology comes from the homotopy theory without particular attention to geometric features. I will describe our recent work developing geometric approaches to liquid crystal topology, describing cholesteric point defects and topological chirality, and the geometric features of bend distortions, illustrated by applications to the twist-bend nematic phase.

Geometric Topology of Liquid Crystal Textures.

I will introduce in this talk a variant of topological complexity, that can be applied to help classify directed spaces, and has applications to the study of dynamical systems in the large, such as differential inclusions. Directed topological complexity looks for specific partitions {F1,…,Fn,…} of the set of reachable states of some directed space X, such that there is a continuous section from each of the Fi to the space of directed paths which is right inverse to the end points map. As in the classical case, this sheds an interesting light on a number of directed invariants, and we discuss in particular dicontractibility. We will also show some examples of calculations on directed graphs, non-positively curved cubical complexes, and directed spheres. This ongoing work with Aurélien Sagnier and Michael Farber.

Directed topological complexity.

Knotted proteins are believed to be functionally advantageous and to provide extra stability to protein chains [1]. Twenty years of their investigation suggests that they fold via a slipknot conformation, across the native twisted loop. During the talk I will show that diversity of identified folds of knotted proteins and their locations in cells is still growing and surprising us. Moreover recently we went one step further and found that proteins can be even more entangled than knots – they also form lassos and links, which consist of several components. Based on the search through the entire Protein Data Bank we identified several sequentially nonhomologous chains that form a Hopf link, a Solomon link [2], and various types of lassos. I will show that topological properties of these proteins are related to their function and stability. Finally I will explain how the presence of links affects folding pathways of proteins and present new reaction coordinate to study entangled proteins. During my talk I will also present knotted TrmD which is the leading antimicrobial drug target owing to its essen­tiality for bacterial growth, its broad conservation across bacterial species, and its substantial differences from the human and archaeal counterpart Trm5. I will also present our achievements in designing selective inhibitor for TrmD, based on combining theoretical and experimental methods [3]. All entangled proteins – 7% of proteins deposited in the PDB – are collected in databases: KnotProt, LassoProt and LinkProt. [1] JI Sulkowska, E Rawdon, KC Millett, J Onuchic, A Stasiak, PNAS (2012) 109: E1715-23 [2] P Dabrowski-Tumanski, JI Sulkowska, PNAS (2017) 114, 3415–3420 [3] T Christian, et al., Nature S&MB (2016) 23, 941-948

Mysteries of entanglement – proteins, life and physics. Biological role of knotting.

In this paper we study on-line local routing algorithms for plane geometric networks. Our algorithms take advantage of the geometric properties of planar to route messages using only local information available at the nodes of a network. We pay special attention to on-line local routing algorithms which guarantee that a message reaches its destination. A message consists of data that have to be sent to a destination node, i.e. the message itself plus a finite amount of space that keeps only a constant amount of data to aid a message to reach its destination, e.g. the address of the starting and destination nodes, a constant number of nodes visited, etc. Local means that at each site we have at our disposal only local information regarding a node and its neighbors, i.e. no global knowledge of the network is available at any time, other that the network is planar and connected. We then develop location aided local routing algorithms for wireless communication networks, in particularly cellular telephone networks.

Local solutions for global problems in wireless networks

There are now more than 1700 protein chains that are known to contain some type of topological knot in their polypeptide chains in the protein structure databank. Although this number is small relative to the total number of protein structures solved, it is remarkably high given the fact that for decades it was thought impossible for a protein chain to fold and thread in such a way as to create a knotted structure. There are four different types of knotted protein structures that contain 31, 41, 52 and 61 knots and over the past 15 years there has been an increasing number of experimental and computational studies on these systems. The folding pathways of knotted proteins have been studied in some detail, however, the focus of this talk is to address the fundamental question “Why are there knots in proteins?” It is known that once formed, knotted protein structures tend to be conserved by nature. This, in addition to the fact that, at least for some deeply knotted proteins, their folding rates are slow compared with many unknotted proteins, has led to the hypothesis that there are some properties of knotted proteins that are different from unknotted ones, and that this had resulted in some evolutionary advantage over faster folding unknotted structures. In this talk, I will review the evidence for and against this theory. In particular, how a knot within a protein chain may affect the thermodynamic, kinetic, mechanical and cellular (resistance to degradation) stability of the protein will be discussed.

Why are there knots in proteins?

In this talk, the basics of discrete Morse theory will first be shortly summarized. We will then describe two generalizations which both have applications to mathematical models associated with motion. Parametric discrete Morse theory can be applied to tracking the motion of features in a sequence of images or scenes, and fiberwise discrete Morse theory to robot motion planning.

Discrete Morse theory in action.

I will discuss how to determine vanishing polynomials on a fixed finite sample of points and what to do with that information. For example, from the equations defining a variety one can learn the degree and the dimension of the variety. One can also construct ellipsoid complexes which, based on the experiments, strengthen the topological signal for persistent homology. All the algorithms needed are made available in a Julia package.

Learning Algebraic Varieties from Samples.

Quantum computation can be regarded as a methodical study of the structure preparation, evolution, and measurement of quantum systems in a computational framework. In a quantum system, a physical state corresponds to a unit length vector in a complex space (Hilbert space or finite dimensional complex space depending on the problem). A physical process corresponds to the application of a unitary transformation to a state. A measurement of a state projects that state to one of a number of orthogonal possibilities represented by an orthonormal basis for the complex space. The probability of a measurement is the absolute square of its coefficient in the state vector. A quantum computer consists in a unitary transformation and the capability to prepare states, apply the unitary transformation and measure the results. The statistics of repeated measurement gives the information for the computation. Thus the key to quantum computing is the design of unitary transformations that can accomplish a particular task. There are a number of designs that are very interesting such as Shor’s method of factoring integers on a quantum computer. The design of actual quantum computers is very difficult because it is hard to control measurement and keep the system from decohering in its interaction with the environment. It is thought that topological physical situations, such as the behaviour of anyons in the quantum Hall effect or the behaviour of Majorana Fermions in nano wires, may hold the key to the desired fault tolerant computation. In both of these instances one can formulate the unitary transformations in terms of representations of the Artin Braid group that are related to the physics of the situation. It is thought that such topological phases of physical systems will be resistant to perturbation and will make scalable quantum computing possible. It is the purpose of this talk to describe braiding related to both the quantum Hall effect (the Fibonacci model) and unitary braiding related to Majorana Fermions (braiding from Clifford algebras). The talk will be self-contained and as pictorial as possible.

Unitary Braiding and Topological Quantum Computing.

Hydrogen bonds stabilize the crystal structures of most proteins. In the aqueous environment, they lie at the edge of stability and hence are easily formed and broken, especially those with high free energy. Using a Boltzmann-like formalism, empirical sampling of their geometry leads to free energy estimates, thus allowing a priori predictions of protein regions primed for structural rearrangement. One among diverse applications involves viral glycoproteins and capsids, namely those proteins employed by viruses to fuse with or penetrate the host cell. Here the methods are used to identify sites of interest for vaccine/drug/testing development, in particular for the SARS CoV-2 virus that causes COVID-19.

Backbone Free Energy Estimator, a New Geometric Tool in Structural Biology.

Genomic technologies have revolutionized the field of genetics. In particular, the identification of genetic elements has greatly benefited from statistical genetics tools that, through association studies, test thousands of genetic elements for a given phenotype. In this talk I will present a methodology that combines topological data analysis with statistical genetics to identify genetic signatures (in the form of copy number aberrations) in breast cancer. The proposed topological encoding of these chromosome aberrations allows to: (1) map chromosome aberrations across the cancer genome using the zero homology group (2) build predictive models using machine learning techniques (3) identify some co-amplifications of the genome using the first homology group (4) identify cancer driving genes when incorporating a fiber bundle-like structure to the data.

Computational homology in statistical genetics: applications to cancer genomic data.

Entanglement in polymeric systems has been central theme since, at least, the work of Sir Sam Edwards, Pierre-Gilles de Gennes, and Jacques des Cloizeaux in the 60’s. Beginning with a somewhat historical discussion tracing these early developments, I will sketch some key elements that have catalyzed my own polymer physics research before focusing polymer gels and the seminal theory of Eleni Panagiotou on polymer entanglement in periodic boundary system models of polymer gels. I will then trace how her dissertation lead to a fruitful collaboration with me and others some whose results will be described next. These concern, time permitting, ideal systems, polyethylene gels, one-dimensional systems, the determination of entanglement length, linking matrices associated to gels, and the relationship between entanglement and viscoelasticity.

Entanglement of Polymers

I explain what is an energy conserving, Hamiltonian time crystal. I show how it arises and how it moves, in terms concrete examples. But according to no-go theorems Hamiltonian time crystals are not supposed to exist, thus I start and show a way around these no-go theorems: I explain why a timecrystalline dynamics needs to take place on a pre-symplectic rather than a symplectic phase space. For this the Hamiltonian needs to support conserved charges, and the occurrence of a time crystal becomes relates to the wider concept of spontaneously broken symmetries. Except that in the case of a time crystal, the symmetry breaking takes place in a dynamical context. I then analyze in detail a general family of time crystalline Hamiltonians It describes the dynamics of string-like objects, such as a molecular chain. I show how the topology of the chain, its knottiness, has a strong effect to its timecrystalline character.

Hamiltonian time crystals - why do they exist and how do they move

In this this lecture, I will provide an informal introduction to a range of topics in fluid dynamics that have a topological character. These topics include flows with boundary singularities, Lagrangian chaos, frozen-in fields, magnetohydrodynamic analogies, fast and slow dynamo mechanisms, magnetic relaxation, minimum energy states, knotted flux tubes, vortex reconnection and the finite-time singularity problem. I will conclude with a number of open questions concerning these topics.

Some Topological Aspects of Fluid Dynamics.

Topological properties of DNA molecules play important role in biology. The ability of cells to control DNA topology is essential for their survival, and topology also controls the geometrical properties of the molecule which influence its function. Advances in the theory of DNA elasticity clarified our understanding of this role and other phenomena associated with biological function and storage of DNA. Classical elastic rod models have been applied to DNA supercoiling, knotting, and cyclization, while base-pair level models of DNA made it possible to incorporate the effects of nucleotide sequence and the negative charge of DNA in new mesoscale models of complex DNA-protein assemblies, which yielded insights into the role of DNA deformability in gene regulation. More recently, simulations of the dynamics of elastic rods submerged in a viscous incompressible fluid has been conducted in order to gain understanding of the behavior of topologically constrained DNA molecules. This lecture will give an overview of recent research on the above topics.

On the interplay between DNA topology, elasticity, and its biological role.

The following is known as the geometric hypothesis of Banach: let V be an m-dimensional Banach space and suppose all n-dimensional subspaces of V are isometric, n > 1. In 1932, Banach conjectured that under this hypothesis V is a Hilbert space. Gromov proved in 1967 that the conjecture is true for n=even and Dvoretzky derived the same conclusion under the hypothesis m=infinity and V a real Banach space, V. Millman proved the analogous when V is a complex Banach space . We prove this conjecture for V a real Banach space of dimension n=4k+1, with the possible exemption of 133. The ingredients of the proof are classical homotopy theory, irreducible representations of the orthogonal group and convex geometry (ellipsoids). At the end of the lecture, will discuss the case in which V is a complex Banach space of dimension n=4k+1.

On the Geometric Hypothesis of Banach.

Signalling pathways can be modelled as a biochemical reaction network. When the kinetics follow mass-action kinetics, the resulting mathematical model is a polynomial dynamical system. I will overview approaches to analyse these models with steady-state data using computational algebraic geometry and statistics. Then I will present how to analyse such models with time-course data using differential algebra and geometry for model identifiability. Finally, I will present how topological data analysis can provide additional information to distinguish these models and experimental data from wild-type and mutant molecules. These case studies showcase how computational geometry, topology and dynamics can provide new insights in the biological systems, specifically how changes at the molecular scale (e.g. MEK WT and mutants) result in phenotypic changes (e.g. fruit fly mutations).

Previous talks

October 16, 2026

Louis H Kauffman

Reconnection Numbers of Knotted Vortices

October 2, 2026

Pengyu Liu

Tree polynomial encoding of structural data in life sciences

September 18, 2026

Tancredi Schettini Gherardini

Geometry-Aware Neural Networks for Geometric Analysis

September 4, 2026

Kelin Xia

Mathematical AI for Molecular Sciences: from topological data analysis to topological deep learning

August 28, 2026

Louis H Kauffman

Introduction to Khovanov Homology

August 21, 2026

Barbara Giunti

Canopies: A Generalization of Vines and Vineyards for Parameterized Persistence

Previous talks

October 16, 2026

Louis H KauffmanReconnection Numbers of Knotted Vortices

October 2, 2026

Pengyu LiuTree polynomial encoding of structural data in life sciences