### Title and abstracts

All times are shown in the current Mexico City time zone, CDT (UTC -5).
The complete program follows the mini-course descriptions below.

### Mini-Course Descriptions

1) Thursday, September 23rd, 9:00 am
Monday, September 27th, 9:00 am
Wednesday, September 29th, 9:00 am

Ngoc Tran, University of Texas at Austin
Title: Recursive random partitions: probability, geometry, and machine learning.
Abstract: Recursive random partitions in dimension one are staples of Bayesian statistics, with various applications to topics modeling and clustering. They also have a nice theory linking combinatorial stochastic processes, fragmentation and coalescence, with applications to population genetics. Over the last five years, recursive random partitions in higher dimensions have yielded a new class of random forests that are efficient, easy to compute. They are also the first class of random forest that achieve the minimax rate in regression tasks. Moreover, there are new, powerful ways to think about recursive random partitions in higher dimensions using stochastic geometry, which promises a lot of future interactions between this field and machine learning.

This mini-course gives a flavor of the techniques and open problems in this field.

2) Thursday, September 23rd, 10:15 am
Monday, September 27th, 12:10 pm
Wednesday, September 29th, 12:10 pm

Benjamin Schweinhart, George Mason University
Title: Percolation and Topology
Abstract: Various models of percolation are fundamental in statistical mechanics; classically, they study the emergence of a giant component in random structures. From early in the mathematical study of percolation, geometry and topology have been at the heart of the subject. Indeed, Frisch and Hammersley wrote in 1963 that, Nearly all extant percolation theory deals with regular interconnecting structures, for lack of knowledge of how to define randomly irregular structures. Adventurous readers may care to rectify this deficiency by pioneering branches of mathematics that might be called stochastic geometry or statistical topology.''

This mini-course will overview the interaction of percolation theory and stochastic topology. In my first talk, I will provide a brief introduction to classical percolation theory. Next, I will cover previous work on topological events in higher dimensional percolation including M. Aizenman et al.'s area/perimeter law on the probability that a curve is bounded by a surface area in bond percolation in $Z^3$, and O. Bobrowski and P. Skraba's definition of homological percolation on the torus. The final talk will describe joint work with collaborators P. Duncan and M. Kahle establishing sharp phase transitions for homological percolation in two percolation models on the torus.

3) Thursday, September 23rd, 12:10 pm
Monday, September 27th, 10:15 am
Wednesday, September 29th, 10:15 am

Tomasz Kaczynski, University of Sherbrooke
Title: Computational Homology, Dynamics, and Data
Abstract: Algebraic topology has been primarily applied to two independent research areas: One in dynamical systems, via the Conley-Morse theory for providing rough classifications of dynamical behaviours, and the other one in Imaging Science, merging with the new field of Topological Data Analysis, via persistent homology. It can be beneficial to think about both applications simultaneously, as the dynamics of flows is a background for Morse theory, while the Morse theory is a background for persistent homology.

The plan of the mini-course is as follows:

Lecture 1. Overview of classical applications. Homology of cubical and simplicial complexes.

Lecture 2. Basic terminology and concepts of the classical flow theory in continuous and discrete time. A historical overview of the quest for designing combinatorial analogues of continuous vector fields, aimed at understanding and classifying their dynamics. Experimenting with Morse connection graphs in Imaging.

Lecture 3. Basics of persistent homology. Examples of applications: point-cloud data and shape similarity. Author's contributions to multi-parameter persistence.

### Wednesday, September 22nd

8:45 am Welcome and Opening Remarks, Victor Rivero, Director General of CIMAT

9:00 am Yohai Reany, Technion - Israeli Institute of Technology
Title: Cycle Registration in Persistent Homology with Applications in Topological Bootstrap

Abstract: In this talk we will present a novel approach for comparing the persistent homology representations of two spaces (filtrations). Commonly used comparison methods are based on numerical summaries such as persistence diagrams and persistence landscapes, along with suitable metrics (e.g., Wasserstein). These summaries have several advantages, especially for computational purposes, but they are merely a marginal of the actual topological information that persistent homology can provide. We propose a novel approach for comparing between two topological representations directly in the data space. We do so by defining a correspondence relation between individual persistent cycles of two different spaces and devising a method for computing this correspondence. Our matching of cycles is based on both the persistence intervals and the spatial placement of each feature. We demonstrate our new framework in the context of topological inference, where we use statistical bootstrap methods in order to differentiate between real features and noise in point cloud data.

9:40 am Pablo Camara, University of Pennsylvania Medical School
Title: Applications of combinatorial Laplacians in data analysis: examples in genomics

Abstract: A prevailing paradigm for data analysis involves comparing groups of samples to statistically infer features that discriminate them. However, many modern applications in data analysis do not fit well into this paradigm because samples cannot be naturally arranged into groups. In such instances, graph techniques can be used to rank features according to their degree of consistency with an underlying metric structure without the need to cluster samples. Here, we extend graph methods for feature selection to abstract simplicial complexes and present a general framework for clustering-independent analysis. Combinatorial Laplacian scores take into account the topology spanned by the data and reduce to the ordinary Laplacian score when restricted to graphs. We show the utility of this framework with several applications to the analysis of gene expression and multi-modal genomic data. Our results provide a unifying perspective on topological data analysis and manifold learning approaches to the analysis of large-scale datasets.

10:20am Coffee Break

11:00 am Miguel O'Malley, Wesleyan University
Title: (Persistent) Magnitude and Data

Abstract: Magnitude is an isometric invariant for metric spaces that has been introduced by Leinster in 2010. In recent work, Hepworth and Govc introduced persistent magnitude, a numerical invariant of a filtered simplicial complex associated to a metric space. In ongoing work, we study how magnitude and persistent magnitude can be applied to the analysis of point cloud data: we investigate their stability properties, and their discriminative power in distinguishing between different data sets exhibiting self-similar properties. We further introduce alpha magnitude, a new invariant inspired by Hepworth and Govc’s notion of persistent magnitude, and establish some of its key properties.

11:40 am Eliza O'Reilly, Caltech
Title: Stochastic Geometry for Machine Learning

Abstract: The Mondrian process in machine learning is a recursive partition of space with random axis-aligned cuts used to build random forests and Laplace kernel approximations, and it can be viewed as a special case of the stable under iterated (STIT) tessellation in stochastic geometry. We utilize this viewpoint to resolve open questions on the generalization of cut directions in the Mondrian process. Our results generalize the use of random partitions for kernel approximation and show minimax rates of convergence for random forest estimators built from STIT tessellations. This is based on joint work with Ngoc Tran.

12:20 pm Paul Duncan, The University of Ohio
Title: Duality in Percolation

Abstract: Though percolation theory is most often studied by probabilists, topological ideas have played an important role in its development. Topologically dual percolation systems can often be coupled to produce useful relationships in their properties. We will give an overview of the use of this technique, from the famous Harris-Kesten theorem to recent work on percolation with higher dimensional cells. We will also touch on some work in progress involving contracting loops in a random cubical complex and its relationship to entanglement percolation, in which vertices may be connected by topological links as well as paths.

### Thursday, September 23rd

9:00 am Ngoc Tran, University of Texas at Austin
Mini-Course 1: Recursive random partitions: probability, geometry, and machine learning
Lecture 1

10:15 am Benjamin Schweinhart, George Mason University
Mini-Course 2: Percolation and Topology
Lecture 1

11:30 Coffee Break

12:10 pm Tomasz Kaczynski, University of Sherbrooke
Mini-Course 3: Computational Homology, Dynamics, and Data
Lecture 1

### Friday, September 24th

9:00 am Marzieh Eidi, Max Planck Institute Leipzig
Title: Qualitative Shape Analysis with the help of Dynamical Systems

Abstract: Topology concerns with those parameters which are preserved under continuous deformation of a space. The number of k-dimensional holes is one main such parameter and computing them has been a challenging question for many years. In this seminar, I will talk about how to recover homology groups of both smooth and combinatorial settings, in a unifying perspective, based on dynamical systems operating on the object. Specifically, I will focus on some results on CW complexes which is a generalization of Forman's theory from discrete gradient vector fields to the general combinatorial vector fields when we are allowed to have non-trivial cycles. (This talk is based on a work under the supervision of professor Jost, https://arxiv.org/abs/2105.02567)

9:40 am Kevin Knudson, University of Floriday
Title: : Discrete Stratified Morse Theory
Abstract: In this talk I will discuss an extension of Forman's discrete Morse theory to the setting of stratified spaces and demonstrate its utility via various examples.

10:20 am Poster Session

11:00 am Rosemberg Toala
Title: Extremal {p,q}-animals

Abstract: An animal is a planar shape formed by attaching congruent regular polygons, known as tiles, along their edges. In 1976, Harary and Harborth gave upper and lower bounds for the number of edges an animal with n tiles has. Their work focused on the three types of euclidean animals: triangular, square, and hexagonal. In this talk, I will present similar extremality results for animals living on hyperbolic tessellations. I will focus on minimizing the edge perimeter of a hyperbolic animal with n tiles. This is achieved by studying the graph parameters of spiral animals.

11:40 am Alexander Smith, University of Wisconsin
Title: Topological Data Analysis: Applications in Molecular Simulation

Abstract: Statistics, machine learning, and signal processing are dominant paradigms used to analyze data; unfortunately, such techniques provide limited capabilities to capture topological features of data. This limitation often requires models that are over-parameterized and that are difficult to interpret. Topological Data Analysis is an area of mathematics that integrates topology, geometry, and statistics and introduces powerful descriptors to characterize data topology, such as the Euler characteristic (EC). The EC does not require strict statistical assumptions for data (e.g., stationarity, isotropy), generalizes to high dimensions, and provides dimensionality reduction capabilities.
In this talk, I discuss the basic concepts of the EC, and demonstrate its application to data from molecular dynamics (MD) simulations. I will show how the EC can be used to quantify both 2-dimensional and 3-dimensional MD data taken from simulations of self-assembled monolayers and catalysis of biomass-derived molecules. The EC can be used as a pre-processing step for these datasets, which simplifies the models needed to perform classification and regression tasks (i.e., complex CNNs simplified to linear regression models). I will also explore the connections between the characterized topology of these systems and their statistics via random field theory which aids us in the physical interpretation of these complex simulations.

12:20 pm Industry Panel Discussion

Panelists:
Natalia Garcia, RealLife and Université Libre de Bruxelles
Aldo Guzmán-Sáenz, IBM Research
Leo Betthauser, Microsoft Research
Moderator: Erika Roldán

### Monday, September 27th

9:00 am Ngoc Tran, University of Texas at Austin
Mini-Course 1: Recursive random partitions: probability, geometry, and machine learning
Lecture 2

10:15 am Tomasz Kaczynski, University of Sherbrooke
Mini-Course 3: Computational Homology, Dynamics, and Data
Lecture 2

11:30 am Poster Session

12:10 pm Benjamin Schweinhart, George Mason University
Mini-Course 2: Percolation and Topology
Lecture 2

### Tuesday, September 28th

9:00 am Omer Bobrowski, Technion - Israel Institute of Technology
Title: Homological Connectivity in Random Čech Complexes
Abstract: A well-known phenomenon in random graphs is the phase-transition for connectivity, proved first by Erdős Rényi in 1959. In this talk we will discuss a high-dimensional analogue of this phenomenon which we refer to as "homological connectivity". Briefly, homological connectivity is the point where the homology of a simplicial filtration stops changing. The model we study is the Čech complex generated over a spatial Poisson point process.
We will show that there is a sequence of sharp phase transitions (for different degrees of homology) and also explore the behavior of the complex inside each critical window.

9:40 am D. Yoeshwaran, Indian Statistical Institute, Bangalore
Title: Poisson process approximation for critical points of random distance function.
Abstract: We shall consider extremal critical points (along with scaled critical radius) of a random distance function and show a Poisson process approximation result for the same. This is obtained as a consequence of a general Poisson process approximation result for stabilizing functionals of Poisson processes that arise in stochastic geometry. The bounds are derived for the Kantorovich-Rubinstein distance between a point process and an appropriate Poisson point process. This is a joint work with Omer Bobrowski (Technion) and Matthias Schulte (Hamburg Institute of Technology).

10:20am Coffee Break

11:00 am Takashi Owada, Purdue University
Title: Convergence of persistence diagram in the sparse regime

Abstract: The objective of this paper is to examine the asymptotic behavior of persistence diagrams associated with \v{C}ech filtration. A persistence diagram is a graphical descriptor of a topological and algebraic structure of geometric objects. We consider \v{C}ech filtration over a scaled random sample $r_n^{-1}\mathcal X_n = \{ r_n^{-1}X_1,\dots, r_n^{-1}X_n \}$, such that $r_n\to 0$ as $n\to\infty$. We treat persistence diagrams as a point process and establish their limit theorems in the sparse regime: $nr_n^d\to0$, $n\to\infty$. In this setting, we show that the asymptotics of the $k$th persistence diagram depends on the limit value of the sequence $n^{k+2}r_n^{d(k+1)}$. If $n^{k+2}r_n^{d(k+1)} \to \infty$, the scaled persistence diagram converges to a deterministic Radon measure almost surely in the vague metric. If $r_n$ decays faster so that $n^{k+2}r_n^{d(k+1)} \to c\in (0,\infty)$, the persistence diagram weakly converges to a limiting point process without normalization. Finally, if $n^{k+2}r_n^{d(k+1)} \to 0$, the sequence of probability distributions of a persistence diagram should be normalized, and the resulting convergence will be treated in terms of the $\mathcal M_0$-topology.

11:40 Hugo Rincon, Technische Universität Wien
Title: A combinatorial topology perspective on distributed computing

Abstract: Distributed computing is a rapidly growing field of study largely motivated by it's many practical applications. Technologies such as internet, cloud computing, blockchain, among others shape the way we live today.
In this talk, we present a brief introduction to the topological framework for analyzing distributed system models. We will show how to model the input, output and communication model of a distributed systems using simplicial complexes. In particular we will talk about wait-free shared memory computability, which is interestingly characterized as a fully topological feature.

### Wednesday, September 29th

9:00am Ngoc Tran, University of Texas at Austin
Mini-Course 1: Recursive random partitions: probability, geometry, and machine learning
Lecture 3

10:15pm Tomasz Kaczynski, University of Sherbrooke
Mini-Course 3: Computational Homology, Dynamics, and Data
Lecture 3

11:30 Coffee Break

12:10am Benjamin Schweinhart, George Mason University
Mini-Course 2: Percolation and Topology
Lecture 3

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