Seminar GEOTOP-A
First presential meeting in Merida, Mexico
Scientific Programme
You can download the Book of Abstracts and programme directly from HERE.
AR = Audiovisual Room “Dr. Eduardo Urzaiz”, AUD1 = Auditorium 1 “Manuel Cepeda Peraza”, AUD2 = Auditorium 2 “Salón de Consejo”
Keynote speakers
DNA
1) Sophie Jackson - University of Cambridge
The What, Where, How and Why of Topological Knots in Proteins
For decades it was thought that topological knots would never be formed by the polypeptide chain of any protein, knotting being incompatible with folding mechanisms. However, we now know that many proteins fold and form three-dimensional structures in which the chain crosses itself and threads through loop(s) to form knots. Proteins with very deep knots, i.e., a large part of the chain has passed through a knotting loop to form the knot have been identified, and four different classes of knots have been found embedded in protein strucutres: 3-1, 4-1, 5-2, and 6-1 knots. In addition, recently it has been established that a single polypeptide chain can contain more than one knot - several examples of tandem trefoil knotted proteins have been characterised. With the advent of the machine-learning based protein structure algorithm AlphaFold, several new classes of knotted protein have been predicted although their knotted structures have not yet been verified experimentally. Over twenty years, numerous experimental and computational studies on knotted proteins have investigated how such structures might form, in addition, to the properties of the knotted structure and whether they differ significantly or not from unknotted proteins. In this talk, I will review the field and explain 1) what knots are found in proteins and where they are within the folded structures, 2) the mechanisms by which knotted may fold, i., how the knots get there, and 3) why proteins may have evolved to form knotted structures. The talk will provide background on twenty years of research as well as discussing some state-of-the-art studies on designing proteins with novel knotted folds, as well as watching knotted proteins unfold and translocate through narrow pores.
2) Cristian Micheletti - International School for Advanced Studies (SISSA)
Dynamics and mechanics of knotted DNA and RNAs: insights from molecular dynamics simulations
I will report on a series of studies where we used molecular dynamics simulations and various models to study how the properties of DNA and RNAs are affected by the presence of knots and other forms of structural entanglement[1]. I will first consider model DNA plasmids that are both knotted and supercoiled, and discuss how the simultaneous presence of knots and supercoiling creates long-lived multi-strand interlockings that might may be relevant for the simplifying action of topoisomerases. I next consider how entangled nucleic acids behave when driven through narrow pores[2-4], a setting that models translocation through the lumen of enzymes, and discuss the biological implication for a certain class of viral RNAs[4]. [1] L. Coronel, A. Suma and C. Micheletti, "Dynamics of supercoiled DNA with complex knots", Nucleic Acids Res. (2018) 46 , 7533 [2] A. Suma, V. Carnevale and C. Micheletti, Nonequilibrium thermodynamics of DNA nanopore unzipping, Phys. Rev. Lett., (2023), 130 048101 [3] A. Suma, A. Rosa and C. Micheletti, Pore translocation of knotted polymer chains: how friction depends on knot complexity, ACS Macro Letters, (2015), 4 , 1420-1424 [4] A. Suma, L. Coronel, G. Bussi and C. Micheletti, "Directional translocation resistance of Zika xrRNA” Nature Communications (2020), 11 , art no. 3749
PHYS
3) Mitchell Berger - University of Exeter
Continuous topological measures: helicity, winding, and higher order winding
Many measures of topological complexity are discrete: for example the linking number between two closed curves is an integer. However, some topological invariants can be continuous. The winding number of two curves extending between parallel planes, with fixed end points provides a simple example. We will discuss how winding numbers work in more complicated geometries such as spheres, cubes, and closed surfaces in general. On the way, we will need Gauss-Bonnet. Also we will touch on higher order winding related to the Borromean rings.
4) Louis H Kauffman - University of Illinois at Chicago
Reconnection Numbers of Knotted Vortices
This talk will discuss a model in combinatorial topology for vortex reconnection. We discuss the reconnection number R(K) of a knotted vortex with knot type K. The reconnection number is the least number of reconnections needed to obtain an unknotted curve from the given knot type K. This least number applies as a lower bound to the number of reconnections that can occur in a physical realization in terms of actual vortices and will therefore be of use experimentally. We show how, in many cases (e.g torus knots and links and positive knots and links) it is possible to derive formulas for the reconnection number. The technique on the topological side uses Khovanov Homology and the Rasmussen invariant.
CTRS
5) Yuliy Baryshnikov - University of Illinois at Urbana-Champaign
On Spaces of Coverings
Consider a relation between two topological spaces. A finite collection is a covering if for any , one has for one of the points in . (For example, if is a metric space, and is the relation of being at the distance , then is a covering if the union of -balls around 's cover .) The topology of the space of coverings is important, if unexplored, topic in several applied disciplines, from multi-agent systems to sociology. In this talk we discuss some examples where the homotopy type of these spaces can be explicitly computed.
6) Pablo Soberón - City University of New York
New results on envy-free distributions
An envy-free distribution problem refers to a problem in which a resource has to be distributed among participants so that each receives their preferred part. Since the existence of envy-free distribution of one-dimensional resources was proved by Stromquist and Woodall in the 1980's, several extensions and generalizations have been proven. In this talk, we discuss two variations. In the first one, we have more participants than pieces to allocate, and we are interested in the weakest conditions on the preferences of the participants to find an envy-free distribution for a subset of them. In the second, we discuss high-dimensional generalizations of this problem, and seek to find common generalizations of the Woodall-Stromquist theorem and the ham sandwich theorem. The methods used to prove the existence of envy-free partitions involve studying the topological degree of some associated maps and extensions of Birkhoff's theorem on doubly stochastic matrices.
7) Ismar Volić - Wellesley College
Simplicial complexes and political structures
Simplicial complexes and their topology are a natural tool for modeling interactions in a system and revealing its deeper underlying structures. We will discuss how simplicial complexes can be used to study political systems in which coalitions are represented by simplices. Some basic topological constructions can then easily be translated into political situations such as merging of parties, introduction of mediators, or delegation of power. The topological point of view also supplies a refined point of view on game-theoretic notions like the Banzhaf and Shapley-Shubik power indices of agents in a political system. We will also present a generalization to hypergraphs which captures an even richer collection of political dynamics concepts. Time permitting, a recasting of some classical results from social choice theory in topological and category-theoretic terms will also be mentioned.
DAMLAI
8) Pawel Dlotko - Dioscuri Centre in Topological Data Analysis, Mathematical Institute, Polish Academy of Sciences
Data, their shape, and beyond
In contemporary science we are exposed to vast amounts of data. Understanding them is often helpful, sometimes essential, to make considerable progress in the field. Mathematics, and mathematical statistics, offer a wealth of tools allowing for better understanding of data. Most tools concentrate on the quantitative characterization of data, rather than understanding their layout, or shape. To fill in the gap, in my Dioscuri Centre in Topological Data Analysis, we are developing new techniques to quantify the shape of data and provide visualizations which, in the next step, deliver new knowledge. Our methods apply for a large variety of inputs, including high dimensional samples, time series, images, correlation patterns and more. In this talk, I will give a brief and intuitive overview of our methods with a hope that you may find them beneficial in your research. A showcase of the current usages of our methodology will provide both an important motivation for, and driving force to, our research.
9) Radmila Sazdanovic - NC State University
The shape of relations: knots and other stories
Topological Data Analysis provides tools for discovering relevant features of data by analyzing the shape of a point cloud. In this context we develop tools for visualizing maps between high dimensional spaces with the goal of discovering relations between data sets with expected correlations. Additionally we are adapting TDA tools to analyzing infinite data sets where representative sampling is impossible or impractical and using them in synergy with ML techniques. Most of the examples will focus on analyzing relations between knot invariants with additional examples in game theory and cancer genomics.
TCLS
10) Alexander Dranishnikov - University of Florida
On some variations of TC and the LS-category
I'll define a new version of Topological Complexity (TC) of a space denoted as dTC which, I think, fits better for motion planning for some autonomous systems. As the classical TC the new dTC is homotopy invariant. Also, dTC has a corresponding analog of the LS-category, denoted d-cat. In the talk I'll present some computations and estimates for both dTC(X) and d-cat(X) as well as a comparison with the classical TC(X) and cat(X).
TDA
11) Yusu Wang - UC San Diego
Graph learning models: theoretical understanding, limitations and enhancements
Graph data is ubiquitous in many application domains. The rapid advancements in machine learning also lead to many new graph learning frameworks, such as message passing (graph) neural networks (MPNNs), graph transformers and higher order variants. In this talk, I will describe some of our recent journey in attempting to provide better (theoretical) understanding of these graph learning models (e.g, their representation power and limitations in capturing long range interactions in graphs), the pros and cons of different models, and ways to further enhance them in practice. This talk is based on multiple pieces of work with various collaborators, whom I will mention in the talk.
Oral contributions
DNA
12) Arron Bale - Durham University
Using writhe to provide limits on entanglement for protein backbones and identify shared helical domains
There is much interest in developing quantitative methods to compare different protein structures or identify common sub-structures across protein families. We present methods for studying and comparing protein structures based on the writhe of a smoothed representation of their amino-acid backbone. The smoothed backbone is a minimal representation that preserves the essential entanglement of secondary structures, highlighting the link between global entanglement and secondary structure. This method is able to identify similar protein entanglement for structures which may be seen as distinct via more commonly used distance-based methods. Secondly, by studying the distribution of entanglement across a representative sample of proteins, we show that there exists a length independent upper limit on the entanglement of protein backbones with respect to their secondary structure. Using fundamental properties of the writhe, we show that this upper limit corresponds to helical structure. We show that these helical structures are present across various length scales in proteins, acting as maximal entanglement domains. The presence of helices is also used to highlight structural similarities in many distinct CATH domains, with implications in the field of structure based protein design. With this, we develop a writhe comparison metric that can identify topologically similar subsections of proteins.
13) James Bozeman - American University of Malta
Identification of Potential Z-DNA Forming Subsequences in Genomes
In this talk we report on results from a Short Term Scientific Mission (STSM) done under the auspices of the EU-COST Action grant EUTOPIA, CA17139, as well as on more recent work in this area. The STSM was performed at the Universidad de Campinas, Brazil with Dr. Veronica Andrea Gonzalez Lopez and Dr. Jesus Enrique Garcia, who have used a local metric from a distance measure between samples coming from discrete Markovian processes to decide if 2 independent samples are governed by the same stochastic law. They applied this technique to the stochastic profile of strains of the Zika virus utilizing the 4 bases of DNA, finding the probability of one base following another in the genomic sequence. Subsequent to this we applied the methodology to SARS Covid-19. It is well known what sequences of bases in DNA are likely to be in the left-handed form, i.e. the Z-DNA conformation. In particular, d(GC)n > d(CA)n > d(CGGG)n > d(AT)n. These Z-DNA forming sequences (ZFS) have been found in the full DNA sequences of SARS Covid-19, rodent parvoviruses, salmonella and some carcinogens. During the STSM we examined the stochastic profiles of such sequences in the fasta format to determine the probability of these occurring. Novelly, we found such sequences in the Epstein-Barr virus, which to the best of our knowledge had not been checked previously for ZFS, and calculated the probabilities of those subsequences occurring. Other new work included applying the partition Markov model to one cancer-causing chemical with ZFS and one without and comparing the two, which can lead to a classification of them. Note that there are anti-Z-DNA antibodies associated with these cancer-causing chemicals. Finally we will present our current work, including implications for the 3-d conformation of the molecule, applying the idea of microsatellites to the repeated sequences known to be in left-handed form, and using the Levenshtein distance for measuring the difference between two sequences.
14) Yair Augusto Gutierrez Fosado - University of Edinburgh
Topological Elasticity and Programmable degradation of Physical Gels with Limited Valence
Networks made by colloids and polymers are fundamental components of virtually any material around us, and even a substance as familiar as water exhibits a complex network structure [1]. Our understanding of the relationship between the microscopic topology of these networks and their macroscopic properties is limited [2]. Most theories rely on chemical composition, fraction of branching points, mesh size and fractal dimension to predict the mechanical properties of soft colloidal and polymeric gels [3]. However, it is increasingly clear that an accurate description of a material’s behaviour also requires a quantitative understanding of topological motifs created by the microscopic building blocks [4,5]. Classic theories fail most dramatically in the case of physical networks with limited valence, where the connectivity of the building blocks is constrained and are thus expected to display unconventional network structures. Here, we investigate the elasticity of DNA nanostar hydrogels - a model system for networks with limited valence - by coupling rheology measurements, confocal imaging and molecular dynamics simulations. We discover that these networks display a large degree of interpenetration and that loops within the network are topologically linked, forming a percolating network-within-network structure. Strikingly, we discover that the onset of topological links between shortest loops fully determines the elasticity of these physical gels. Our findings highlight the emergence of "topological elasticity" as a previously overlooked mechanism in generic network-forming liquids and physical gels. Therefore, we expect this mechanism will inspire the design of new materials with topologically-controllable behaviours. Finally, I will also report on the design of active DNA hydrogels with time-varying programmable degradation by restriction enzymes. [1] A. Neophytou et al, Nature Physics, 18, 1248-1253 (2022). [2] M. Zhong, el al, Science 353, 1264 (2016). [3] M. Bantawa, Nature Physics(2023). [4] M. Kapnistos, et al, Nature materials 7, 997 (2008). [5] J. Smrek, el at, Science Advances 7, 1 (2021).
15) Nataša Jonoska - University of South Florida
Mathematical model for DNA building blocks in crystallographic nanostructures
Bottom-up self-assembly of DNA nanostructures have been proposed for variety of biotech uses such as templates for new materials, substrates for targeted drug delivery, as well as long term information storage. The rationally-designed 3D DNA motif, the tensegrity triangle, is the first molecule used to provide DNA crystallographic assemblies. The sequence design possibilities of these building blocks give ever-increasing geometric complexities to form vast arrays of three-dimensional structures. We show a mathematical model based in topological and geometric constraints that explain the formation of certain types of tensegrity triangle structures as well as their chiral topology as left- or right-handed. The mathematical model is also experimentally verified.
17) Alexander Klotz - California State University, Long Beach
Kinetoplast DNA as a model material
Kinetoplasts DNA is a network consisting of thousands of topologically linked circular DNA molecules, which form part of a gene-editing network in the mitochondria of trypanosome parasites. When isolated from the cell, kinetoplasts from Crithidia fasciculata take the form of micron-scale elastic membranes with positive Gaussian curvature. In this talk I will overview the recent work from myself and collaborators exploring the kinetoplast as a model system for two-dimensional polymer physics, filling a similar role that viral genomic DNA has served in the study of linear polymer physics. I will also discuss some more recent work from my group attempting to clarify the nature of the kinetoplast's network topology using graph theory simulations and single-molecule experiments.
18) Kenneth Millett - University of California
Capturing Entanglement in Molecular Systems
Topological entanglement is a structural phenomena in biological systems having impacts on material and functional characteristics. It can be captured through the measurement of linking and of knotting traditionally found in knot theory but extended to includes measures for open strands rather that restricted to closed loops. Following a brief review of some 30 years research effort to quantify entanglement we will report on some recent work with Eleni Panagiotou and on the entanglement of colonies of California blackworms.
19) Joseph Starr - University of Iowa
The Tanglenomicon: Tabulation of two string tangles
"The Most Important Missing Infrastructure Project in Knot Theory" -Dr. Dror Bar-NatanThere are a number of great knot and link tables available to researchers; be that mathematicians, biologists, physicists, and many other domains. However, with only knot and link tables we are in the position of a chemist with a table of fatty acids but no periodic table. Our group at University of Iowa are striving to build that periodic table of knot "elements", the two string tangles. With the goal of increasing availability of tangle data for researchers everywhere.
20) De Witt Sumners - Florida State University
Tangles in DNA and Fluid Vortices
Knots and links in DNA can interfere with vital cellular life processes, but they can also provide experimental insight into the mechanism of enzymes that facilitate these processes. This talk will discuss the topology of DNA site-specific recombinase, an enzyme that can switch genes on and off. When this enzyme binds to two sites on DNA, this forms a 2-string protein-DNA tangle, and the tangle model uses topological change in tangle structure to compute enzyme binding and mechanism [1]. Very similar to DNA recombination is vortex reconnection in fluids, and the topology of anti-parallel vortex reconnection will be discussed [2]. [1] C. Ernst and D. W. Sumners. A calculus for rational tangles: applications to DNA recombination, Math. Proc. Camb. Phil. Soc. 108(1990), 489-515. [2] C. E. Laing, Renzo L. Ricca, D.W. Sumners, Conservation of writhe helicity under anti-parallel reconnection, Nature Scientific Reports 5 : 9224 | DOI: 10.1038/srep09224 (2015).
21) Lynn Zechiedrich - Baylor College of Medicine
So you think you know the structure of DNA? I thought I did too!
By regulating access to the primary code, supercoiling and looping can be thought of as secondary DNA codes. Toward “cracking” these secondary codes, we discovered that supercoiling and loop length-dependent site-specific base pair disruptions (Fogg et al. (2021) Nat Commun 12:5683) facilitate very sharp bending to allow DNA to adopt novel DNA conformations (Irobalieva, Fogg et al. (2015) Nat Commun 6:8440). Molecular dynamics simulations revealed that one flipped base, with enough negative supercoiling, expands into a series of adjacent flipped bases to form denaturation bubbles (Randall et al. (2009) Nucleic Acids Res 37:5568); these simulations were verified and we further discovered that site-specific base pair disruptions at one site cause site-specific base pair disruptions at distant sites (Fogg et al. (2021) Nat Commun 12:5683), a remarkable “action at a distance” with major biological consequences. In this talk, I will discuss new findings demonstrating that cations should also be considered a secondary code because they dramatically affect the interplay of supercoiling and looping-dependent site-specific base-pair disruptions and DNA shape to regulate DNA activity. Funded by NIH R35 GM141793.
PHYS
23) Yasuhide Fukumoto - Institute of Mathematics for Industry, Kyushu University
Isovortical perturbations and wave energy of compressible neutral and conducting fluids based on Frieman-Rotenberg equation
According to Arnold's theorem, a steady state of the ideal incompressible Euler flow is characterized as an extremum of the kinetic energy, with respect to isovortical perturbations that preserve all the Casimirs or topological invariants. This statement carries over to the compressible non-isentropic flows and the compressible magnetohydrodynamics as well. For the latter, isovortical perturbations are extended to isomagnetovortical perturbations. We shall show how this structure is manifestly embodied by the Howard-Gupta (HG) equation for the dynamics of perturbations of a neutral fluid and the Frieman-Rotenburg (FR) equation for that of an electrically conducting fluid. Self-adjointness of the force operators in the HG an FR equations facilitates to derive the energy formula for waves on steady states. Examples are given of the energy formula for compressible baroclinic flows.
24) Miguel Ángel García Ariza - Instituto de Investigaciones en Matemáticas Aplicadas y Sistemas, UNAM
Newton's Law of Cooling as a Gradient Flow and Asymmetric Relaxation
I present Newton's Law of Cooling as a gradient flow on a statistical manifold. This allows for a geometric characterization of asymmetric relaxation to equilibrium. In particular, I show that ideal gases warm up faster than they cool down, provided that they are initially equidistant to a given thermal bath.
25) Max Lipton - Massachusetts Institute of Technology
Variations of the Möbius Knot Energy
Physical knot theory is the study of how analytically defined quantities for explicitly parametrized simple closed curves relate to classical topological knot invariants. In this talk, I will discuss the M\"obius knot energy, and some of the landmark results of Freedman-He-Wang and Agol-Marques-Neves. I will then turn to some of my recent work on Morse-theoretic methods with the electric potential induced by a charged knot, including various visualizations of critical points and equipotential surfaces. Finally, I will talk about open problems and future directions.
26) Xin Liu and Renzo Ricca - Beijing University of Technology / University of Milano-Bicocca
Geometric picture for algebraic invariants of knots/links with applications in physics and biology
We propose a geometric picture for polynomial invariants of knots/links: using orthogonal polynomial basis (such as Legendre, Hermitian, etc.) to span an algebraic space, such that a knot polynomial obtains a set of coordinates, which implies a knot/link could be represented by a point in the algebraic space [3]. The method is useful in physical and biological practice. Recent laboratory and numerical experiments in classical/quantum fluids [1,2] and recombinant DNA plasmids [4,5] show that a knot complex system may experience an cascade evolution from a high-topological complexity state to a low-complexity state. In terms of the above picture we provide a geometric interpretation for the observations: an evolving procedure is regarded as geodesic route connecting the knot/link point and the origin of the algebraic space, with monotonically decreasing complexity degree, while the intermediate stages of the procedure (connected by topologically non-conservative transitions) correspond to a series of midpoints on the geodesic pathway. This is joint work with Renzo Ricca (U Milano-Biccoca, Italy) and Xinfei Li (Guangxi U Science & Technology, China). [1] Kleckner D., Irvine W.T.M. 2013 Creation and dynamics of knotted vortices, Nature Physics 9, 253. [2] Kleckner D., Kauffman L.H., Irvine W.T.M. 2016 How superfluid vortex knots untie, Nature Physics 12, 650–655. [3] Liu X., Ricca R.L. & Li X.-F. 2020 Minimal unlinking pathways as geodesics in knot polynomial space. Communications Physics 3, 136. [4] Scheeler M.W. et al. 2014 Helicity conservation by flow across scales in reconnecting vortex links and knots. PNAS USA 111,15350. [5] Stolz R., et al. 2017 Pathways of DNA unlinking: A story of stepwise simplification. Scientific Reports 7, 12420.
27) Jason Parsley - Wake Forest University
Rotation Classes of Petal Knots
A petal diagram of a knot consists of a central point and n unnested loops; a petal diagram exists for every knot type. Petal diagrams may be indexed by cyclic permutations, which provides an easily accessible model of random knotting, important for many applications. In this talk, we describe multiple isotopies on the set of petal diagrams via their effects on cyclic permutations. We give updates on how our understanding of these permutations leads to conclusions about petal knots. This is joint work with Eric Rawdon and Yunxin Yao.
28) Oscar Ramírez - Instituto de Ciencias Nucleares (ICN), UNAM
Symmetries in Cartan's Formalism and their connection with Principal Fiber Bundles
In the context of gauge and gravity theories written in terms of differential forms, an algorithm to read off its local symmetries can be constructed. Usually, the relevant symmetries are gauge invariance and diffeomorphisms, for which a commutator of infinitesimal generators can be defined acting on the dynamical fields. In this short talk, I will discuss this algorithm and how the algebra generated by the commutator is related with the principal fiber bundle associated with the corresponding physical theory.
29) Renzo Ricca - University of Milano-Bicocca
Quantum Vortex Knots And Links Under Zero Helicity Condition
Here we show that quantum vortex defects evolve under conservation of the zero helicity condition [1,2]. As a result knots and links present must interchange linking, writhe and twist during evolution to conserve their zero total linkage, with new interesting physical effects. [1] Belloni, A. & Ricca, R.L.. (2023) On the zero helicity condition for quantum vortex defects. J. Fluid Mech. 963, R2. [2] Sumners, De W.L., Cruz-White, I.I. & Ricca, R.L. (2021) Zero helicity of Seifert framed defects. J. Phys. A: Math. Theor. 54, 295203.
30) Takashi Sakajo - Kyoto University
Topological Flow Data Analysis for Blood Flows Inside a Heart
Complex vortex patterns of blood flow in the heart play an important role in an efficient blood flow supply from the heart to the organs. Recent progress in medical imaging and computer technology such as echocardiography and cardiac MRI has recently yielded blood flow visualization tools. On the other hand, however, there are still few mathematical theories to clearly define the vortex flow structures such as size and location, or change over time in the main chamber in the heart. Although the function of the vortex blood flow inside the left ventricle is highly unstable and complex, we propose a new mathematical theory to extract topological features of the flows in the heart in terms of discrete graphs, called partially cyclically ordered rooted tree (COT) representations, thereby identifying well-organized vortex flow structures as topological vortex structures and characterizing healthy blood flows as well as inefficient flow patterns in the diseased heart. Developing an image processing software based on mathematical theory, we have conducted the topological classification of 2D blood flow patterns obtained by the visualization tools. This realizes a new image processing characterizing healthy blood flow patterns as well as inefficient patterns in diseased hearts.
31) Nicola Sansonetto - University of Verona
Dynamics of a large number of charged vortices in the plane
We consider a system N interacting charged point vortices in the plane, with N 'large'.We investigate the dynamics in the relative equilibria when the charges have different power and different signs. Then, in the case the particles have the same sign, weinvestigate the dynamics for the limits of the number of particles that goes to infinity.
32) Koya Shimokawa - Ochanomizu University
Crossing number and signature of links
In this talk, we will discuss the relationship between the crossing number and the signature of a link. We show that the crossing number bounds the absolute value of the signature, and a link can be determined when the absolute value of the signature is close to its crossing number. As an application, we discuss the characterization of links obtained from the torus link by an anti-parallel reconnection and the classification of such reconnections. This is a joint work with Kai Ishihara and Kei Okada.
33) Pablo Vázquez-Montejo - CONAHCYT - Universidad Autónoma de Yucatán
Equatorial deformation of a spherical fluid membrane
Many biophysical processes involve the interaction of biopolymers with membranes, for instance the cytokinesis and the endocytosis involve the formation of a neck and its subsequent scission due to the constriction by a filament or a protein complex. As an approach to understand the physical properties of such celullar processes, a theoretical analysis of the deformation of a spherical fluid vesicle by a circular rigid ring is presented. The configurations of the vesicle are determined by minimizing the bending energy, quadratic in the curvature of the membrane. Deformed vesicles with spontaneous curvature preserving the axial symmetry and with their area or volume fixed are examined. The axial symmetry provides a first integral of the Euler-Lagrange equation, which is solved with boundary conditions reflecting the constraint imposed by the ring. The force exerted by the ring gives rise to a discontinuity in the derivative of the membrane curvature across the ring. As the equatorial radius is constricted the behavior of the vesicle is similar either for fixed area or volume, it first adopts a prolate shape, followed by a transition to a non-convex gourd shape, which at the maximum constriction morphs into two spheres connected by a narrow neck. The geometry of the neck and the constrictive force are examined in detail. As the equatorial radius is stretched, the vesicle adopts an increasingly oblate shape, but remains convex. If the area is fixed the configuration of maximum stretching consists of a pair of discs joined along their boundary, whereas if the volume is fixed the vesicle adopts a discocyte shape.
CTRS
35) Marco Tulio Angulo - Institute of Mathematics, UNAM
Assembly archetypes in ecological communities
An instrumental discovery in comparative and developmental biology is the existence of assembly archetypes that synthesize the vast diversity of organisms' body plans---from legs and wings to human arms---into simple, interpretable, and general design principles. In this talk, I describe how we combine a novel mathematical formalism based on Category Theory with experimental data to show that similar ''assembly archetypes'' exist at the larger organization scale of ecological communities when assembling a species pool across diverse environmental contexts, particularly when species interactions are highly structured. We applied our formalism to clinical data, discovering two assembly archetypes differentiating healthy and unhealthy human gut microbiota. The concept of assembly archetypes and the methods to synthesize them can pave the way to discovering the general assembly principles of the ecological communities we observe in nature. This is joint work with Hugo Flores (UNAM), Omar Antolin (UNAM), and Serguei Saavedra (MIT)
37) Armando Castañeda - Instituto de Matemáticas, UNAM
Combinatorial topology techniques for concurrent data structures
Since early 90s, a number of combinatorial topology techniques have been successfully applied to distributed computing, which has derived in a rich and solid theory of distributed computing. However, the theory has mostly be applied to study distributed problems that can be defined through the task formalism. Roughly speaking, a task is the equivalent of a function in a distributed setting. Very little is known about topology techniques applied to distributed problems that cannot be specified as tasks. A prominent example are concurrent data structures, which are of great interests in the design of real-world concurrent and distributed systems. In this talk we will see an attempt to extend the theory to encompass this type of problems.
39) Matthias Fuegger - CNRS, LMF, ENS Paris-Saclay, Université Paris-Saclay
Algorithm synthesis in dynamic systems
Systems where processes communicate via messages in communication-closed rounds are a versatile model covering classical static networks with faults to dynamically changing networks. Design of distributed algorithms for such networks is difficult. In the talk we report on recent and ongoing work in collaboration with Bérard, Bollig, Bouyer, and Sznajder on automated algorithm synthesis in dynamic systems: in particular the case of n=2 processes with immediate acknowledge of messages.
40) Emmanuel Godard - Université Aix-Marseille
A Simple Computability Characterization for Colorless Tasks in General Submodels
There are many distributed models that can be represented by simplicial subdivisions. We consider here models where the subdivision is in addition also mesh-shrinking. The most used model with these properties is the Iterated Immediate Snapshot model, which is known to be equivalent to the standard wait-free model. In this setting, we consider general sub-models M of such models, that is arbitrary subsets of executions, and arbitrary colorless tasks. Based upon the geometrization mapping geo introduced in [4] to investigate set-agreement, we give a simple necessary and sufficient condition to have a colorless task (I, O, ∆) solvable under M in the form of the existence of a continuous function f : geo(I × M) −→ |O| carried by ∆. This necessary and sufficient condition for colorless tasks was already known for full models like the Iterated Immediate Snapshot model [10, Th. 4.3.1], so our result is an extension of the characterization to any submodels. Note also that the presentation is simpler than other general characterizations, since the associated topology is simpler to understand and manipulate that the general abstract ones. As an example of its simplicity, we can now derive the characterization of the computability of set-agreement on submodels from [4] by a direct application of the No-Retraction theorem of standard topology textbook. Joint work with Yannis Coutouly (Université Aix-Marseille)
42) Eric Goubault - Ecole Polytechnique
Epistemic Logic via Combinatorial Topology
Abstract simplicial complexes are a generalization of the notion of graph, which contain not only vertices and edges, but also cells of higher dimension such as triangles or tetrahedra. In algebraic topology, they are used to describe topological spaces in a combinatorial way. In this talk, I will present a new semantics for epistemic logic where we replace the usual Kripke frame by a (decorated) simplicial complex. By doing so, we uncover some higher-dimensional topological information which is hidden in Kripke models. Evaluating the satisfiability of an epistemic formula in a simplicial complex model amounts to inspecting various possible paths in the underlying topological space. This has particularly deep applications to distributed computing, and if time permits, we will also mention further generalizations, with pre-simplicial sets and hypergraphs, and also temporal extensions to epistemic logics. This talk is intended to be of a (short) tutorial nature, and is part of joint work with Jeremy Ledent, Sergio Rajsbaum and Roman Kniazev.
43) Vinicio Antonio Gómez-Gutiérrez - Universidad Nacional Autónoma de México
On some open problems about the CTLNs
The Combinatorial Threshold Linear Networks (CTLNs) are a family of ordinary differential equations inspired in neural networks. There are many open problems about them, I found very interesting the problems of graph rules posed by Carina Curto and Katherine Morrison in January 2023 (their article is online on ArXiV). In this talk, I explain some work in progress about some of these problems.
44) Sophia Knight - University of Minnesota-Duluth
Using domain theory to model knowledge, belief, and information in multi-agent systems
The focus of this work is developing a domain theoretical model of knowledge, belief, and information in multi-agent systems. I begin by reviewing the connections between domain theory and topology, and the use of domain theory to represent information in constraint systems that underlie traditional process calculi. Then I describe multi-agent domains and their representations of knowledge, belief, and information. I discuss dynamic process calculi based on these domains, and their semantics. Finally I explore the expansion of these systems to infinite groups of agents. In a strictly logical context, portraying the common and distributed knowledge of such infinite agent groups presents challenges. However, when situated within the domain-theoretic constraint system framework, these knowledge operators find a natural and elegant representation.
45) Isaac Bernardo Lara Nuñez - Centro de Estudios Económicos, El Colegio de México
A Generalization of Arrow's Impossibility Theorem through Combinatorial Topology
We present a generalization of Arrow’s impossibility theorem. Instead of assuming the unrestricted domain, we propose a domain restriction called the class of polarization and diversity over triples. The domains in this class are defined by requiring profiles in which society is strongly, but not completely, polarized over how to rank triples of alternatives, as well as some profiles that violate the valuerestriction condition. To prove this result, we use the combinatorial topology framework started by Rajsbaum and Raventós-Pujol in 2022, which in turn is based on the algebraic topology approach started by Baryshnikov in 1993. While Rajsbaum and Raventós-Pujol employed this approach to study Arrow’s impossibility theorem and domain restrictions for the case of two voters and three alternatives, we extend it for the general case of any finite number of alternatives and voters. Although allowing for arbitrary (finite) alternatives and voters results in simplicial complexes of high dimension, our findings illustrate that complexes can be effectively analyzed by examining their 2-skeleton, even within the context of domain restrictions at the level of the 2-skeleton.
46) Thomas Nowak - ENS Paris-Saclay
Topological Characterization of Task Solvability in General Models of Computation
The famous asynchronous computability theorem (ACT) relates the existence of an asynchronous wait-free shared memory protocol for solving a task with the existence of a simplicial map from a subdivision of the simplicial complex representing the inputs to the simplicial complex representing the allowable outputs. The original theorem relies on a correspondence between protocols and simplicial maps in finite models of computation that induce a compact topology. This correspondence, however, is far from obvious for computational models that induce a non-compact topology, and indeed previous attempts to extend the ACT have failed. We show first that in every non-compact model, protocols solving tasks correspond to simplicial maps that need to be continuous. This correspondence is then used to prove that the approach used in ACT that equates protocols and simplicial complexes actually works for every compact model, and to show a generalized ACT, which applies also to non-compact computation models. Our study combines combinatorial and point-set topological aspects of the executions admitted by the computational model.
48) Joseph Root - University of Chicago
The Topology of Incentives in Social Choice
Social choice rules are used by voters who can choose to act strategically. For this reason, rules which are ''strategy-proof" are simple to use and are especially valuable in practice. However, in many settings, no nontrivial strategy-proof rules exist. This dilemma is explored through the lens of topology.
89) Armando Castañeda - Instituto de Matemáticas, UNAM
Asynchronous Robot Gathering: A Brief Tutorial on the Combinatorial Topology Approach to Distributed Computing
Since early 90s, a number of combinatorial topology techniques have been successfully applied to distributed computing, which has derived in a rich and solid theory of distributed computing. However, the theory has mostly been applied to study distributed problems that can be defined through the task formalism. In this talk we will see a brief tutorial of the approach through the asynchronous gathering problem, where autonomous robots are required on the same spot in a space that is modelled as a graph.
DAMLAI
49) Manuel Cuerno - Universidad Autónoma de Madrid
Topological Data Analysis (TDA) on ATM
Airports and Air Traffic Management Systems are complex sociotechnical structures that are highly interdependent. This particular feature makes them difficult to analyze and understand. Globally the air transport operations at airports and air traffic management systems integrate the interaction of over 32,000 in-service aircraft operated by more than 1,300 commercial airlines. Although flight trajectory data sets offer a big potential to grasp the features and behaviour of such intricate system, they can be complex and high-dimensional. Sparse data sets are affected by inconsistencies, errors, and high levels of variability. Flight trajectory data sets are difficult to analyse due to several reasons, from the huge interconnection of all its factor to the continuous changes this type of data suffers. All these factors make it extremely difficult to extract insights from the data to derive operational patterns and detect operational anomalies. To overcome all this difficulties we have implemented the usage of Topological Data Analysis (TDA) for the analysis of airport patterns and anomalies out of spatiotemporal flight trajectories. TDA is a powerful analytical technique that can help to overcome some of the limitations of existing research works in the field of analysing high-dimensional flight trajectory data for the identification of common traffic patterns in airports. Overall, TDA offers a powerful tool for analysing complex and high-dimensional aviation data sets. By identifying topological features and patterns, TDA can reveal hidden relationships and help airlines and airports make better decisions about flight scheduling, maintenance, and safety. TDA can help airport operators and stakeholders better understand complex data and identify patterns and insights that can be used to improve airport operations and the passenger experience. In this work, we propose to use TDA to analyze flight trajectory data and identify patterns in the movement of aircraft, and determine the relationships between different variables involved in the spatial and temporal flight trajectory and delays to identify common patterns and anomalies in airport operation and congestion, and help to recognize underlying causes of delays and develop more effective strategies for reducing them. We aim to demonstrate the efficacy of TDA through an analysis of real-world data, specifically the Spanish network of airports during the Summer Season of 2018, as classified by AENA (a Spanish public company incorporated as a public limited company that manages general interest airports in Spain). This talk is based on joint work by Manuel M Cuerno, Luis Guijarro, Rosa María Arnaldo Valdés and Víctor Fernando Gómez Comendador (https://arxiv.org/abs/2304.08906) and, hopefully, we will be able to present some new discoveries.
50) Mauricio Che Moguel - Durham University
Generalised metric spaces of persistence diagrams
In this talk, I will introduce a construction of metric spaces inspired by the metric structures defined on the space of persistence diagrams, widely known as the p-Wasserstein distance and Bottleneck distance. These metric structures are rooted in the field of optimal transport theory. The outcome of our construction is a family of functors that take metric pairs as inputs and yield pointed metric spaces as outputs. I will present a continuity result concerning this construction, specifically in relation to the natural notions of Gromov-Hausdorff convergence within the classes of metric pairs and pointed metric spaces. Additionally, I will discuss generalisations of well-known properties of the geometry of the space of persistence diagrams endowed with these metrics, such as completeness, separability, geodesicity, curvature bounds, and the existence of barycenters (Frechet means) for certain types of probability measures. This is based on joint projects with Fernando Galaz-Garcia, Luis Guijarro, Ingrid Membrillo-Solis, and Motiejus Valiunas.
51) Juan Carlos Diaz Patiño - Instituto de Neurobiología de la UNAM
Variability of topological features on brain functional networks in precision resting-state fMRI.
Nowadays, much scientific literature discusses Topological Data Analysis (TDA) applications in Neuroscience. Nevertheless, a fundamental question in the field is, how different are fMRI in one individual over a short time? Are they similar? What are the changes between individuals? This talk presents the approach used to study resting-state functional Magnetic Resonance Images (fMRI) with TDA methods using the Vietoris-Rips filtration over a weighted network and looking for statistical differences between their Betti Curves and also a vectorization method using the Minimum Spanning Tree.
53) José Frías - UNAM
Embedded Graph Polynomials and the study of 3-manifolds
Heegaard splittings are essential in the study of topology and geometry of 3-manifolds. Given a Heegaard splitting of a 3-manifold M, the boundaries of two systems of decomposing disks for the two handlebodies determine a graph embedded in the Heegaard surface, whose vertices are the intersection points between the two families of curves. There are many polynomials that can be computed for graphs embedded in surfaces, such as the Tutte, Penrose or the ribbon graph polynomials. We aim to use techniques in machine learnig and data analysis to uncover relations between graph polynomials computed for graphs induced by Heegaard splittings and topological properties of the corresponding 3-manifolds, as in the case of knot polynomials studied by R. Sazdanovic, P. Dlotko and collaborators.
54) Marissa Masden - ICERM/Brown University
Combinatorial and PL Morse Methods for Accessing Level Set Topology of ReLU Neural Networks
ReLU neural networks are a class of highly-parametrized functions commonly used in machine learning for classification and regression. These functions divide their input space into a canonical polyhedral complex (Grigsby and Lindsey, 2022), whose face poset structure can be seemingly arbitrarily complicated. In order to better understand these functions' topological expressivity, that is, the ability of a given neural network architecture to express the correct sublevel set topology for the task on which it is trained, we discuss how combinatorial techniques can be used to understand the face poset of the canonical polyhedral complex, giving access to explicit Betti number computations as well as PL Morse methods for obtaining empirical measurements of the topology of level and sublevel sets of these functions. In addition, we examine the behavior of ReLU neural network functions when trained on certain toy classification tasks, and discuss how their decision boundary (level set) topology during training experimentally depends on architecture.
55) Ingrid Membrillo Solis - University of Westminster / University of Southampton
Data-driven applications of geometry and topology in complex systems dynamics
A complex system is formed by entities that, through their interactions and dependencies, give rise to a unified whole with properties and behaviour distinct from those of its constituent parts. Examples include the human brain, living cells, organisms, soft matter materials, the earth's global climate, ecosystems and the economy. The dependencies and relations among the entities that form the systems produce high dimensional non-linear dynamics that make the modelling, classification and prediction of the dynamics of the complex systems a major challenge. In this talk, we will present two data-driven frameworks, based on persistent homology and discrete vector fields, for the study of complex dynamical systems. We will show how these topological and geometric frameworks can be efficiently used for tracking the time-evolution of the systems, ultimately leading to dimensionality reduction, phase space representation, mode decomposition and global attractor characterisation of complex systems dynamics.
56) Astrid Arena Olave Herrera - Michigan State University (United States)
Revealing brain network dynamics during the emotional state of suspense using topological data analysis
Suspense is an affective state ubiquitous in human life, from art to quotidian events. However, little is known about the behavior of large-scale brain networks during suspenseful experiences. To address this question, we examined the continuous brain responses of participants watching a suspenseful movie, along with reported levels of suspense from an independent set of viewers. We employ sliding window analysis and Pearson correlation to measure functional connectivity states over time. Then, we use Mapper, a topological data analysis tool, to obtain a graphical representation that captures the dynamical transitions of the brain across states and we anchored the topological characteristics of the combinatorial object with the measured suspense. Our analysis revealed changes in functional connectivity within and between the Salience, Fronto-Parietal, and Default networks associated with suspense. In particular, the functional connectivity between the Salience and Fronto-Parietal networks increased with the level of suspense. In contrast, the connections of both networks with the Default network decreased. Together, our findings reveal specific dynamical changes in functional connectivity at the network level associated with variation in suspense, and suggest topological data analysis as a potentially powerful tool for studying dynamic brain networks.
57) Jesús Rodríguez Viorato - CONAHCyT - CIMAT
Topological Analysis of ArXiv Math Research Papers
We created a point cloud of the ArXiv math papers using Latent Semantic Analysis and computed its persistent homology. We interpret the holes detected by persistent homology as a hidden connection between different articles. We believe this could be used to suggest research readings. We made different computations trying to find evidence of the above interpretation. For instance, we define a homological-based index that measures the size of the hole that a research paper fills. We compare this index with the number of citations. We will present a comprehensible synthesis of our computations. This ongoing work is in collaboration with Adrián Pastor López, Fernando Sánchez Vega, and Miguel Angel Ruiz.
58) Hannah Santa Cruz Baur - Penn State University
Hodge Laplacians on Sequences
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 introduce a Hodge-theoretic approach to spectral theory and dimensionality reduction for probability distributions on sequences and simplicial complexes. We will demonstrate that this Hodge theory has desirable properties with respect to natural null-models, where the underlying vertices are independent. In particular, we will see that for the case of independent vertices in simplicial complexes, the appropriate Laplacians are multiples of the identity and thus have no meaningful Fourier modes. For the null model of independent vertices in sequences, we will see that the appropriate Hodge Laplacian has an integer spectrum with high multiplicities, and describe its eigenspaces.
59) Pablo Suárez-Serrato - Instituto de Matemáticas/UNAM
Computation, learning, and undecidability in Hamiltonian dynamics
Similarly to the growth of Applied Topology, the uses and applications of Geometry are now expanding into scientific, computational, and engineering domains. First, we'll review the recent history of this burgeoning Applied Geometry area. I'll mention a couple of collaborations, developing and implementing algorithms inspired by the marked length spectrum that classify complex networks (with Eliassi-Rad and Torres) and analyzing digital images using a variant of curve-shortening flow (with Velazquez Richards). Then, I'll present joint work with Evangelista and Ruiz Pantaleón on computational Poisson geometry and its applications to learning symbolic expressions of Hamiltonian systems. We developed and released two Python packages that perform symbolic and numerical computation of objects in Poisson geometry. We then used them to train neural networks (hybrids with CNN and LSTM components) that learn symbolic expressions of Hamiltonian vector fields. Finally, I'll briefly mention the theoretical limitations of computationally analyzing Hamiltonian dynamics. I recently constructed an example of a Hamiltonian flow on the 4-sphere that is Turing complete. Therefore the most general cases of Hamiltonian learning problems are undecidable.
TCLS
61) Andrés Angel - Universidad de los Andes
Groupoid LS category and equivariant LS-cat, the finite and the continuous case.
By studying group actions from the perspective of groupoids we are led to a natural notion of groupoid LS category based on a path groupoid construction. We prove that when the group is finite this groupoid LS category is precisely the equivariant LS-category. For non finite compact Lie groups we are actually led to a notion of homotopy between G-spaces that is more general than G-homotopy and a more general notion of G LS-cat. This talk is based on work in progress with Hellen Colman and the articles: Ángel, A., & Colman, H. (2023). Free and based path groupoids. Algebraic & Geometric Topology, 23(5), 1959-2008. Ángel, A., & Colman, H. (2021). G-category versus orbifold category. Topological Methods in Nonlinear Analysis, 1-19.
62) Jose Calcines - Universidad de La Laguna
Proper Topological Complexity
Many classical homotopy invariants, such as Lusternik-Schnirelmann category, fail to capture the behavior of a space at infinity. In 1992, R. Ayala, E. Domínguez, A. Márquez, and A. Quintero successfully introduced a proper homotopy invariant of the Lusternik-Schnirelmann type. These invariants were subsequently deeply explored in several follow-up papers. The objective of our work is to introduce a version of topological complexity within the proper context, following a similar approach and spirit to that of proper Lusternik-Schnirelmann category. However, the category of spaces and proper maps lacks many crucial properties. For instance, it does not support products, and there is no concept of a proper function space denoted as since the cylinder does not have a right adjoint. Furthermore, there is no concept of a proper fibration, and notably, the bi-evaluation map does not make sense within the proper setting. These limitations hinder our ability to define a proper notion of topological complexity. Our solution arises from considering the category of exterior spaces. An exterior space is essentially a topological space equipped with a specific filter of open subsets, referred to as an externology (which can be likened to a neighborhood system at infinity). The category of exterior spaces exhibits superior properties compared to the category of spaces and proper maps. Specifically, it is known to be a complete and cocomplete category, as well as a model category in the sense of Quillen. Moreover, it encompasses the category of spaces and proper maps as a full subcategory. Once we establish the concept of proper topological complexity within the framework of exterior spaces, we will delve into some properties and provide examples of computations.
63) Daniel Cohen - Louisiana State University
Supersolvable toric arrangements
A toric arrangement is a finite collection of codimension one subtori in a complex torus. If the intersection pattern of these subtori satisfy the combinatorial condition of supersolvabiilty, the complement of the toric arrangement sits atop a tower of fiber bundles. We investigate the topological complexity of the complement in this context.
Joint work with Christin Bibby and Emanuele Delucchi.
64) Dan Guralnik - University of Florida
Reduced Nerves of Good Covers for Path Planning
Symbolic methods for path planning such as temporal logic-based planning for robotic systems currently do not account for topological information about the system's configuration space . A natural way to incorporate such information would be to reason over the nerve of a good cover refining the witness sets of the atomic propositions used for task specification.%The complexes , however, are, in a sense, too large to provide a faithful representation of system capabilities: paths in may exist which do not lift to paths in , potentially confounding the execution of symbolic plans computed over the nerve . In this talk we will introduce a canonical construction of a sub-complex ---the reduced nerve of ---having the property that paths in correspond to paths in , while remains homotopy-equivalent to whenever is a good cover. Joint work with Yu Wang.
65) Teresa Hoekstra-Mendoza - CIMAT-Guanajuato
On the higher topological complexities of anchored configuration spaces of graphs
Configuration spaces constitute a well-studied class of topological spaces. In usual configuration spaces, two or more particles are not allowed to occupy the same space, since we want to avoid collisions. In anchored configuration spaces collisions are allowed, but we require that a certain pre-determined discrete set of points is always occupied by at least one particle.In this talk I am going to present some bounds for the higher topological complexities of the anchored configuration space of the circle with two anchors. I shall also talk about the anchored configuration space of banana string graphs which are graphs obtained by gluing together a certain number of banana graphs.
66) José Luis León Medina - CIMAT-Mérida
The topological complexity of non-k-equal spaces
The non-k-equal spaces are a type of generalization of configuration spaces where we allow less than k collisions or equalities in the coordinates of n-tuples. In this talk, I will describe the techniques for finding the Lusternik-Schnirelmann category and topological complexity for non-k-spaces over n-tuples of real numbers. Also, some conjectures about extending these results for the case of k parabolic arrangements will be presented.
67) Stephan Mescher - Martin Luther University of Halle-Wittenberg
New lower bounds on sectional categories of subgroup inclusions
The contents of this talk are joint work with Arturo Espinosa Baro. The sectional category of a subgroup inclusion is a generalization of the TC of an aspherical space. Both sequential TCs of aspherical spaces and parametrized TC of epimorphisms can be expressed as special cases of this construction as well. In the talk, we will present a generalization of joint work with M. Farber, namely the construction of a spectral sequence from whose behavior we can derive cohomological lower bounds for sectional categories of subgroup inclusions which improve the standard lower bounds for sectional categories. As applications, we will present consequences for various flavours of TCs of aspherical spaces.
68) Michael Farber - Queen Mary University of London
Sequential Parametrized Motion Planning And Its Complexity
The approach of parametrized motion planning was introduced recently by D. C. Cohen, M. Farber, and S. Weinberger. In this presentation, we introduce sequential parametrized motion planning. A sequential parametrized motion planning algorithm produces a motion of the system which is required to visit a prescribed sequence of states, in a certain order, at specified moments of time. The sequential parametrized algorithms are universal as the external conditions are not fixed in advance but constitute part of the algorithm’s input. In this talk, we presentan upper bound and a lower bound for the sequential parametrized topological complexity. Further, we obtain the sequential parametrized topological complexity of the Fadell - Neuwirth fibration. In the language of robotics, sections of the Fadell - Neuwirth fibration are algorithms for moving multiple robots avoiding collisions with other robots and obstacles in the Euclidean space.(This is a joint work with Amit Kumar Paul)
69) Petar Pavesic - University of Ljubljana
Diameters of homotopy sets
Enrique Macias-Virgos and David Mosquera-Lois have recently defined a distance function on homotopy sets . They proved that many concepts from Lusternik-Schnirelmann category and topological complexity can be recovered as a distance between suitably chosen maps. In fact, one can even avoid reference to specific maps and show that and . In our talk we will discuss general properties of and present some interesting examples.
71) Lucile Vandembroucq - Centre of Mathematics, University of Minho
On the Lusternik-Schnirelmann of $\mathbf{Gr}_2(\mathbb{R}^n)$
For some specific integers and , we construct a non-trivial map from the Grassmannian of -dimensional planes of , , to the projective space . By using some of these maps as well as a theorem due to A. Dranishnikov, we obtain new estimates of the Lusternik-Schnirelmann category of for small values of . This is a joint work with R. Brasil and A.C. Ferreira.
TDA
72) Alberto Alcalá Alvarez - UNAM
Topological Music Analysis (TMA)
In the present work we describe a framework for applying different topological data analysis (TDA) techniques to data extracted from a music fragment given as a score in traditional Western notation. We first consider different sets of points in Euclidean spaces of different dimensions that correspond to musical events in the score, and obtain their persistent homology features. Then we introduce two families of simplicial complexes that can be used to model chords and chord sequences, and leverage homology to compute their salient features. Finally, we show the results of applying the described methods to the analysis and stylistic comparison of fragments from music pieces in different styles.
73) Javier Arsuaga - University of California, Davis
Topological data analysis identifies breast cancer genes in molecular subtypes
Copy number changes play an important role in the development of cancer and are commonly associated with changes in gene expression. Persistence curves, such as Betti curves, have been used to detect copy number changes; however, it is known these curves are unstable with respect to small perturbations in the data. We briefly discuss the stability of lifespan and Betti curves by providing bounds on the distance between persistence curves of Vietoris–Rips filtrations built on data and slightly perturbed data in terms of the bottleneck distance. Next, we perform simulations to compare the predictive ability of Betti curves, lifespan curves (conditionally stable) and stable persistent landscapes to detect copy number aberrations. We use these methods to identify significant chromosome regions associated with the four major molecular subtypes of breast cancer: Luminal A, Luminal B, Basal and HER2 positive. Identified segments are then used as predictor variables to build machine learning models which classify patients as one of the four subtypes. We find that no single persistence curve outperforms the others and instead suggest a complementary approach using a suite of persistence curves. In this study, we identified new cytobands associated with three of the subtypes: 1q21.1-q25.2, 2p23.2-p16.3, 23q26.2-q28 with the Basal subtype, 8p22-p11.1 with Luminal B and 2q12.1-q21.1 and 5p14.3-p12 with Luminal A. Further analysis of the regions in 2q and 5p revealed genes associated with breast cancer survival and these include FGF10, RICTOR, and Drosha.
74) Greg DePaul - University of California, Davis
A novel filtered complex from multidimensional Gaussian balls
Vietoris Rips is a widely used filtration, despite being a rigid topology as well as susceptible to outliers. Instead, we propose to utilize density estimation and classical linear fitting methods to build a novel filtered complex from multidimensional Gaussian balls that are meant to approximate a tangent space. Because this construction leverages density estimation, it is often more robust against outliers than traditional TDA methods. One application of this technique can be used for resolving singularities present in point cloud data in the hopes of recognizing intrinsic dimensionality of a dataset.
75) Shaday Guerrero Flores - Centro de Ciencias Matematicas UNAM Campus Morelia
Exploring the Horizontal Gene Transfer through TDA
The study of horizontal gene transfer is fundamental, as it is one of the main routes by which bacteria acquire antibiotic resistance. ESKAPE bacteria (Enterococcus faecium, Staphylococcus aureus, Klebsiella pneumoniae, Acinetobacter baumannii, Pseudomonas aeruginosa, and Enterobacter spp.) are particularly concerning as they are responsible for the majority of hospital infections and have high resistance to available antibiotics. Topological Data Analysis (TDA) has emerged as a powerful tool for understanding complex structures in large datasets. In this paper, we employ techniques such as persistence homology from TDA to investigate the horizontal gene transfer (HGT) among ESKAPE bacteria, a group of pathogens that represent a significant threat due to their resistance to multiple antibiotics. Our research particularly focused on the horizontal transfer of genes associated with antimicrobial resistance.
76) Davide Gurnari - Dioscuri Centre in Topological Data Analysis, Mathematical Institute, Polish Academy of Sciences
Harmonic Persistent Homology for disentangling multiway interaction in data
We discuss Harmonic Persistent Homology and how it can be used to associate concrete subspaces of cycles to each homology class, the harmonic representatives. We provide recent insights relating the notion of essential simplices to harmonic representatives weights. Finally, we show a computational pipeline that leverages harmonic cycles to obtain a ranking of highly expressed genes in different breast cancer subtypes. This is joint work with Aldo Guzmán-Sáenz, Filippo Utro, Saugata Basu and Laxmi Parida.
78) Sara Kalisnik Hintz - ETH
Magnitude, Alpha Magnitude and Applications
Magnitude is an isometric invariant for metric spaces that was introduced by Leinster around 2010, and is currently the object of intense research, since it has been shown to encode many known invariants of metric spaces. In recent work, Govc and Hepworth introduced persistent magnitude, a numerical invariant of a filtered simplicial complex associated to a metric space. Inspired by Govc and Hepworth’s definition, we introduced alpha magnitude. Alpha magnitude presents computational advantages over both magnitude as well as Rips magnitude, and is thus an easily computable new measure for the estimation of fractal dimensions of real-world data sets. I will also briefly talk about work in progress, a clustering algorithm based on the alpha magnitude. This is joint work with Miguel O'Malley and Nina Otter.
80) José Martín Mijangos Tovar - IIMAS UNAM
Divergence measures over the set of persistence diagrams
Given a filtration of simplicial complexes one can apply persistent homology and summarize the result in barcodes or persistence diagrams. Then, in order to extract statistical information from these barcodes, sometimes one computes statistical indicators over the length of its bars. An issue with this approach is that infinite bars must be deleted or cut to finite ones; however, so far there is no systematic way to perform this procedure. With the aim of accomplishing this by minimizing certain functions, and motivated by ideas of information geometry, we have proposed divergence measures over the set of persistence diagrams that generalize the standard Wasserstein and bottleneck distance. In this talk I will introduce you to these divergence measures as well as their properties.
81) Juan Antonio Pichardo - El Colegio de la Frontera Norte
The structure of non-trivial cycles in street networks
The structure of road and street networks is fundamental to understanding various urban phenomena. Remarkably, the centrality measures of nodes are correlated with the spatial distribution of economic activities, congestion levels, and structural changes in cities, among others. Most central nodes generate cycles associated with cities' evolution (transition) and congestion levels. Nevertheless, these cycles are identified intuitively because vertices in the same neighborhood can have very different centralities. In this direction, we propose the use of persistent homology over the subset of central nodes to identify potential cycles. These cycles are compared with the intuitive cycles and the cycles generated by central edges. We present experimental results for several cases of street networks.
82) Alexander Smith - University of Minnesota
Topological and Geometrical Data Analysis in Industrial Process Monitoring and Anomaly Detection
Datasets can be viewed as mathematical objects (e.g., point clouds, matrices, graphs, images, fields/functions) that have shape. This shape can describe the space that data populates (e.g., data that lies on a manifold) or can be used to understand the complex structures contained within data (e.g., the multi-scale organization of self-assembled materials). Data shape can be exploited to improve the effectiveness of data analysis methods or provide connections between complex materials and their physical and chemical properties. However, quantifying shape is difficult to do with common methods based on statistics, signal processing, or with the use of machine learning. Topology and geometry are fields of mathematics that provide tools for the characterization and quantification of the shape of data directly. In this talk I will discuss how data taken from industrial processes, such as time series and images, can be represented as a shape and how that shape can be analyzed through topological and geometrical methods such as the Euler characteristic (EC) and Riemannian manifold geometry. I will provide a brief overview of these methods and illustrate how exploiting the topology and geometry of data can provide improvements in data-centric tasks such as dimensionality reduction, anomaly detection, and statistical process control in the context of textile production, chemical process systems, and granular material manufacturing.
83) Francesca Tombari - Max Planck Institute for Mathematics in the Sciences, Leipzig
Invariants and structures for tame parametrised chain complexes
A persistence module can be seen as a functor from a poset to the category of finite-dimensional vector spaces. Tame persistence modules, in particular, have been extensively studied in the past years as the result of a synthesis of the homological information of geometric objects when filtered according to certain functions. The problem of defining suitable invariants for tame persistence modules still attracts many researchers in the field. Our work about tame functors indexed by generic posets differs from others since we consider as landing category the one of chain complexes. One may observe that it is not possible to characterise indecomposables in this category, as for generic persistence modules. However, it is possible to characterise a class of those, called cofibrant, when the indexing poset has dimension one, once we define a model structure on the category of tame parametrised chain complexes. In addition, we show a technique to construct indecomposables via a glueing technique. This talk is based on joint work with Wojciech Chachólski, Barbara Giunti and Claudia Landi.
84) Jonathan Emmanuel Treviño Marroquín - CIMAT, Guanajuato
An introduction to Semi-coarse Spaces
The category of semi-coarse spaces is one of several topological constructs that have been used to establish foundations of Topological Data Analysis and develop discrete homotopy; other examples of these categories are Čech closure spaces and Choquet Spaces. In this talk we introduce these spaces, which allows us to study finite graphs and compact metric spaces at ‘medium’ scales similar to the way coarse geometry allows the study of spaces at 'large' scales. Our main target in this exposition is to develop the beginnings of homotopy theory for semi-coarse spaces. This is joint work with Antonio Rieser.
85) Ziga Virk - University of Ljubljana and institute IMFM
Mapping spaces of persistence diagram into the Hilbert space with controlled distortion
Stability is one of the most important properties of persistent homology. Similar inputs yield similar persistence diagrams. In this context, the space of persistence diagrams is typically equipped with the bottleneck metric. In order to apply statistical tools or further data analytic techniques to collections of persistence diagrams, we thus need to use a map from the space of persistence diagrams into a Euclidean or Hilbert space. In the past decade dozens of such maps have been proposed, including persistence landscape and persistence images. These maps are typically stable (Lipschitz). However, none of them has explicit lower bounds on distortion and hence they provide no control on the loss of information. In this talk we will present Lipschitz maps from certain spaces of persistence diagrams into Hilbert and Euclidean spaces with explicit distortion functions. The maps are fairly geometric, consisting essentially of bottleneck distances to specific landmark diagrams, and are thus easily implementable. The idea for the construction comes from the quantification of certain classical constructions in dimension theory.
86) Iris Yoon - Wesleyan University
Topological tracing of encoded circular coordinates between neural populations
Recent developments in in vivo neuroimaging in animal models have made possible the study of information coding in large populations of neurons and even how that coding evolves in different neural systems. Topological methods, in particular, are effective at detecting periodic, quasi-periodic, or circular features in neural systems. Once we detect the presence of circular structures, we face the problem of assigning semantics: what do the circular structures in a neural population encode? Are they reflections of an underlying physiological activity, or are they driven by an external stimulus? If so, which specific features of the stimulus are encoded by the neurons? To address this problem, we introduced the method of analogous bars (Yoon, Ghrist, Giusti 2023). Given two related systems, say a stimulus system and a neural population, or two related neural populations, we utilize the dissimilarity between the two systems and Dowker complexes to find shared features between the two systems. We then leverage this information to identify related features between the two systems. In this talk, I will briefly explain the mathematics underlying the analogous bars method. I will then present applications of the method in studying neural population coding and propagation on simulated and experimental datasets. This work is joint work with Gregory Henselman-Petrusek, Lori Ziegelmeier, Robert Ghrist, Spencer Smith, Yiyi Yu, and Chad Giusti.
87) Lori Ziegelmeier - Macalester College
Topological Data Analysis of Knowledge Networks
Knowledge networks can organize complex data by constructing graphs where nodes are concepts or ideas and edges represent connections of significance. Understanding the structure of these knowledge networks to uncover how science progresses over time is of interest to researchers studying the “Science of Science.” In this project, we are interested in understanding cycles or holes within a network, which can be thought of as gaps in knowledge. We use topological data analysis, and in particular, persistent homology filtered through time where the nodes represent scientific concepts and edges between two nodes are added at the time when they appear together in an abstract of a scientific paper. We study properties of these knowledge gaps in multiple dimensions such as when they form, when they no longer remain, and the concepts and papers that make up the cycles. We observe that papers involved in the knowledge gaps are cited more frequently than papers that are not.
Posters
88) Miguel Angel Ruiz Ortiz - CIMAT
A persistent-homology-based turbulence index & some applications of TDA on financial markets
Topological Data Analysis (TDA) is a modern approach to Data Analysis focusing on the topological features of data; it has been widely studied in recent years and used extensively in Biology, Physics, and many other areas. However, financial markets have been studied slightly through TDA. Here we present a quick review of some recent applications of TDA on financial markets, including applications in the early detection of turbulence periods in financial markets and how TDA can help to get new insights while investing. Also, we propose a new turbulence index based on persistent homology -- the fundamental tool for TDA -- that seems to capture critical transitions in financial data; we tested our index with different financial time series (S&P500, Russel 2000, S&P/BMV IPC and Nikkei 225) and crash events (Black Monday crash, dot-com crash, 2007-08 crash and COVID-19 crash).