The school aims to bring together internationally renowned researchers in the field, as well as students and postdoctoral researchers interested in working on applied topology topics. Four courses, problem-solving workshops, and a few talks will be offered. On Saturday, a social activity is also planned.
We are happy to announce that we will offer accommodation, in shared rooms, during the school program—arriving on Nov. 6 and leaving on Nov. 13—as well as meals during the days the school has activities. Hotel reservations will be made directly by the organizing committee. However, please note that no financial support for transportation will be provided.
Due to limited capacity, participation in this event will be confirmed following a review of registration applications, with preference given to those already working in the field of applied topology. We encourage you to submit your application as soon as possible to increase your chances of securing a spot. You will receive a confirmation email if your application is approved.
Thank you for your overwhelming response! Registration is now closed.
Courses
Discrete Morse Theory (DMT)
Jesús González CINVESTAV
Jesús González earned his Bachelor's degree from the National Polytechnic Institute (ESFM-IPN) and went on to complete his Master's and Ph.D. at the University of Rochester. Since June 1994, he has been a leading researcher in the Department of Mathematics at CINVESTAV, where his work encompasses topology, geometry, algebra, and homotopy theory, with practical applications in robotics and big data analysis.
Throughout his career, Jesús has supervised 32 theses at various levels—11 doctoral, 14 master's, and seven Bachelor's. He is currently mentoring a doctoral and a master's student. His research contributions include over 50 original articles published in prestigious international journals, and he has presented at more than 120 conferences and congresses worldwide. With invitations as a plenary speaker on ten occasions, Jesús's expertise and contributions are widely recognized in his field.
Bio
In 1997, Robin Forman introduced a combinatorial adaptation of classical Morse theory used in the study of smooth manifolds. Like its differentiable predecessor, discrete Morse theory (DMT) provides efficient access to topological aspects of smooth manifolds through their triangulations. Given its nature, the use of DMT has not been limited to theoretical contexts; these techniques have found fertile and particularly fruitful ground in applications relevant to our modern technological life, such as data analysis and robotics. In this course we will review the foundations of DMT and its use in various computational problems of algebraic topology and its applications.
Abstract
Introduction to Persistent Homology (IPH)
Jesús Rodríguez Viorato CIMAT
Jesús Rodríguez Viorato is a topologist whose research focuses on knot theory, exploring areas such as the universality of knots, the Kervaire conjecture, contact manifolds, and cable knots. He is also keen on applying topological methods to other disciplines, with a particular interest in computational linguistics.
Over the years, Jesús has held several distinguished academic positions, including a Visiting Assistant Professorship at the University of Iowa in 2017 and a postdoctoral fellowship at the Center for Research in Mathematics (CIMAT) in 2015. He earned his Ph.D. in Mathematics from the National Autonomous University of Mexico (UNAM) in 2014.
Bio
This mini-course provides a brief introduction to the persistent homology of point clouds. We will focus on the most common structures encountered in applications, discussing their definitions and presenting the foundational results that support the theory. The course is divided into three parts. In the first part, we will cover simplicial homology by defining simplicial complexes and their homology, along with methods for computation. In the second part, we will explore persistent homology by introducing constructions of complexes associated with point clouds, such as the Čech complex, and explain how to generate persistent modules from them, emphasizing the importance of the Nerve Theorem. In the final part, we will introduce the Stability Theorem by defining the bottleneck distance and showing how persistent homology remains stable under small perturbations in the point cloud. The only prerequisite for this course is basic knowledge of linear algebra with mod two coefficients.
Abstract
Introduction to Topological Data Analysis (TDA)
Paweł Dłotko Dioscuri Centre in TDA
Paweł Dłotko is the leader of the Dioscuri Centre in Topological Data Analysis. Previously he was working in Swansea University (UK), Inria Saclay, University of Pennsylvania and Jagiellonian University (from which he graduated in 2012).
Bio
In this lecture, we will embark on a comprehensive journey through the landscape of Topological Data Analysis (TDA), a powerful suite of techniques for quantifying and visualizing the shape of data, particularly in high-dimensional settings. TDA has seen broad applications across mathematics, physics, biology, medicine, and even social sciences, offering a rich interplay between theoretical depth and practical utility.
We will begin by introducing the classical framework of persistent homology, a fundamental concept in TDA, which allows us to detect and analyze multi-scale topological features in data. Building upon this, we will explore its extensions, including Euler characteristic curves and profiles, that provide more refined ways to summarize the shape and structure of data. Moving forward, we will highlight the use of these topological tools in diverse fields of mathematics, particularly focusing on applications in statistical hypothesis testing. This will demonstrate the robustness and versatility of TDA in analyzing complex datasets.
To complement the theoretical discussions, we will delve into topological visualization techniques, showcasing how ideas from algebraic topology can help us grasp the layout and structure of high-dimensional point clouds. This will be followed by an introduction to Forman’s discrete Morse theory, which offers a discrete counterpart to classical Morse theory and can be a useful tool in the analysis of topological spaces derived from data.
Throughout the lecture, we will present both mathematical insights and algorithmic implementations. We will also provide software demonstrations, showing how these methodologies can be applied in real-world scenarios, with a strong emphasis on case studies and practical use cases.
Abstract
Applications of Topology to the Theory of Neural Networks (TNN)
Kathryn Lindsey Boston College
Kathryn Lindsey is an Associate Professor of Mathematics at Boston College, specializing in the geometric foundations of deep learning theory.
Her research uses geometry, topology, and dynamics to prove theorems about the functionality and efficiency of deep neural networks in supervised learning. In particular, her research focus is on the class of ReLU feedforward neural networks, which coincides with the class of finitely piecewise linear functions. She completed a Ph.D. in mathematics at Cornell University and a postdoc at the University of Chicago.
Bio
Recent applications of neural networks have been spectacularly successful (chatgpt, image recognition, etc.) and are rapidly transforming many aspects of modern society. However, many foundational questions about how and why they work (or don't) remain mysterious. Developing a rigorous mathematical theory of deep learning (deep learning refers to neural networks) is an active and rapidly advancing area of research.
This course will discuss applications of topology to the emerging mathematical theory of neural networks. No prior knowledge of neural networks is required; the first lecture will introduce the relevant theory. Examples of topological concepts that we will discuss include transversality, genericity of hyperplane arrangements, Morse theory, polyhedral complexes, and topological structures of fibers of quotient maps. Along the way, I will point out a variety of concrete, open, mathematical questions.
Abstract
Introduction to Applied Topology through Sage (Sage)
Gregory Alexander DePaul UC Davis
Greg DePaul is in his final year of his doctorate at UC Davis, studying intersections of applied topology and statistics.
Bio
Often a barrier fledgling applied topologists face is becoming acquainted with the variety of tools used in performing tasks in computational geometry and topology. This course aims to rectify this by introducing in an interactive and welcoming manner how you can use Sage to solve problems in applied topology. This will be done through a series of jupyter notebooks that will be published in advance to allow students ample time to prepare for this mini-course.
Lectures will include an (1) introduction to Topology in the Python / Sage environment, (2) Building Simplicial Complexes, (3) TDA with Vietoris Rips, and (4) Constructing More Complex Filtrations. Much of the tooling is based on the GUDHI (Geometry Understanding in Higher Dimensions) Library.
Abstract
Talks
Talk 1: Applications of Tame Geometry
Carlos Alfonso Ruíz Guido Colegio de Matemáticas Bourbaki
Abstract
Talk 2: Ribbon graphs and the DNA reporter strand problem
Vinicio Antonio Gómez Gutiérrez UNAM
Abstract
Talk 3: Studying convective flows with TDA
Juan Ahtziri González Lemus UMSNH
Abstract
Talk 4: Beyond Pairwise Interactions
Rolando Kindelan Nuñez Universidad de Chile
Abstract
Talk 5: Applied knot theory
José Ángel Frías García UNAM
Abstract
Timetable
🚌 The bus departs everyday at:
- 8:00 from Hampton Inn Mérida
- 8:10 from Siglo 21 Hotel
| Time | Thursday, Nov 7 |
|---|---|
| 09:00-10:00 | DMT |
| 10:00-10:30 | Coffee break |
| 10:30-11:30 | IPH |
| 11:30-11:45 | Break |
| 11:45-13:00 | DMT 2 |
| 13:00-14:15 | Lunch Time |
| 14:15-15:15 | IPH 2 |
| 15:15-15:30 | Break |
| 15:30-16:30 | DMT 3 |
| 16:30-17:30 | IPH 3 |
| Time | Friday, Nov 8 |
|---|---|
| 09:00-10:00 | TDA 1 |
| 10:00-10:30 | Coffee break |
| 10:30-11:15 | Talk 1 |
| 11:15-11:30 | Break |
| 11:30-13:00 | TDA 2 |
| 13:00-14:15 | Lunch Time |
| 14:15-15:30 | DMT 4 |
| 15:30-16:30 | Problem Session |
| 16:30-17:30 | TDA 3 |
| Time | Saturday, Nov 9 |
|---|---|
| 09:00-10:00 | TDA 4 |
| 10:00-10:15 | Coffee break |
| 10:15-11:15 | TDA 5 |
| 11:15-12:00 | Talk 2 |
| 12:00-13:00 | TDA 6 |
| 13:00-18:00 | Outdoor Networking Session |
| 18:00-19:00 | Back to Merida |
| Time | Monday, Nov 11 |
|---|---|
| 09:00-10:00 | TNN 1 |
| 10:00-10:30 | Coffee break |
| 10:30-11:30 | Sage |
| 11:30-11:45 | Break |
| 11:45-13:00 | Academic development Q&A |
| 13:00-14:15 | Lunch Time |
| 14:15-15:45 | TNN 2 |
| 15:45-16:45 | Problem Session |
| 16:45-17:30 | Talk 3 |
| Time | Tuesday, Nov 12 |
|---|---|
| 09:00-10:00 | TNN 3 |
| 10:00-10:30 | Coffee break |
| 10:30-11:30 | Sage 2 |
| 11:30-11:45 | Break |
| 11:45-13:00 | TNN 4 |
| 13:00-14:15 | Lunch Time |
| 14:15-15:30 | Sage 3 |
| 15:30-15:45 | Break |
| 15:45-16:30 | Talk 4 |
| 16:30-16:45 | Break |
| 16:45-17:30 | Talk 5 |
Fees
$300 MXN - For local attendees
(no accommodation)
$600 MXN - For non-local attendees
(with accommodation)
Payments will be requested within one week after the acceptance notice.
Mérida, Yucatán
México is amongst the most popular holiday destinations in the world, and Yucatán is arguably one of its most exciting states to visit, with some of México’s most popular tourist attractions and destinations: from Mérida to Chichen Itza, from Cancún to Playa Del Carmen, Tulum, Cozumel and many other. Mérida is the capital city of Yucatán, about a four-hour drive from Cancún International Airport and a forty-minute drive from Progreso and the nearby coastline beaches. All participants will be hosted in the north of the city of Mérida, a few meters from the Siglo XXI Convention Center and less than 1 km from the Mundo Maya Museum, the location is surrounded by squares, restaurants, business areas, among others; the exit to Puerto Progreso is a couple of minutes away, and the Manuel Crescencio Rejón International Airport is 12 km away.
This event is funded by the CONAHCYT research grant CBF2023-2024-4059: “Interacciones topológico-computacionales” and the generous aid provided by our sponsors below.
Organizing Committee
José Carlos
Gómez Larrañaga
CIMAT Mérida
José Luis
León Medina
CIMAT Mérida
Bernardo
Villarreal
CIMAT Mérida